Fluid Flow and Heat Transfer in Rotating Porous Media

SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY THERMAL ENGINEERING AND APPLIED SCIENCE

Peter Vadasz

Fluid Flow and Heat Transfer in Rotating Porous Media 123

SpringerBriefs in Applied Sciences and Technology Thermal Engineering and Applied Science

Series editor Francis A. Kulacki, Minneapolis, MN, USA

More information about this series at http://www.springer.com/series/10305

Peter Vadasz

Fluid Flow and Heat Transfer in Rotating Porous Media

123

Peter Vadasz Department of Mechanical Engineering Northern Arizona University Flagstaff, AZ USA

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2193-2530 ISSN 2193-2549 (electronic) SpringerBriefs in Thermal Engineering and Applied Science ISBN 978-3-319-20055-2 ISBN 978-3-319-20056-9 (eBook) DOI 10.1007/978-3-319-20056-9 Library of Congress Control Number: 2015941493 Springer Cham Heidelberg New York Dordrecht London © The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)

Preface

There is a large volume of literature dealing with convection in porous media, including an excellent book by Nield and Bejan (2013) that covers all relevant topics. However, although the topic of heat convection in rotating porous media is covered there, it is impossible to dedicate much detail in a book covering such a large number of topics. In addition, the topic of isothermal flows in rotating porous media is outside the scope of that book. A deeper reflection into the effects of rotation in porous media is therefore desirable and this is the main purpose of this book. It introduces the reader into the realm of modeling theory in porous media and presents classifications of flow and heat transfer in rotating porous media and their corresponding applications. Then different configurations of theoretical solutions and some experimental results are presented sequentially. The book is anticipated to be of interest to scientific researchers, university professors, and graduate students as well as practicing engineers.

v

Contents

1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Porous Media Definitions and the Continuum Approach . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 4 5

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Modeling of Flow and Heat Transfer in Porous Media. . . . 2.1 Modeling of Flow in Porous Media . . . . . . . . . . . . . . . 2.1.1 Darcy Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Darcy Equation in a Rotating Frame of Reference and Other Extended Forms . . . . . . . . . . . . . . . . 2.2 Modeling of Heat Transfer in Porous Media . . . . . . . . . 2.2.1 Finite Heat Transfer Between the Phases . . . . . . . 2.2.2 Thermal Equilibrium Between the Phases . . . . . . 2.2.3 Equation of State . . . . . . . . . . . . . . . . . . . . . . 2.3 Natural Convection and Buoyancy in Porous Media . . . . 2.3.1 Homogeneous Porous Media . . . . . . . . . . . . . . 2.3.2 Heterogeneous Porous Media . . . . . . . . . . . . . . 2.4 Non-Darcy Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Classification of Rotating Flows in Porous Media . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Isothermal Flow in Heterogeneous Porous Media. . . 3.1 General Background . . . . . . . . . . . . . . . . . . . . 3.2 Taylor-Proudman Columns and Geostrophic Flow in Rotating Porous Media . . . . . . . . . . . . . . . . . 3.3 Flow Through Heterogeneous Porous Media in a Rotating Square Channel . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Natural Convection Due to Thermal Buoyancy of Centrifugal Body Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Temperature Gradients Perpendicular to the Centrifugal Body Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Temperature Gradients Collinear with the Centrifugal Body Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Effects on Natural Convection . . . . . 5.1 Coriolis Effect on Natural Convection Due Buoyancy of Centrifugal Body Forces . . . 5.2 Coriolis Effect on Natural Convection Due Buoyancy of Gravity Forces . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Other Effects of Rotation on Flow and Natural Convection in Porous Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Natural Convection Due to Thermal Buoyancy of Combined Centrifugal and Gravity Forces . . . . . . . . . . . . . . . . . . . . . 6.2 Onset of Convection Due to Thermohaline (Binary Mixture) Buoyancy of Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . 6.3 Finite Heat Transfer Between the Phases and Temperature Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Anisotropic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Applications to Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Applications to Solidification of Binary Alloys . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

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Abstract

The effect of rotation is shown to have a significant impact on the flow in porous media. In isothermal systems this effect is limited to the effect of the Coriolis acceleration on the flow. It is shown that similarly as in non-porous domains, Taylor-Proudman columns and geostrophic flows might also exist in porous media. The effect of rotation on the flow in a heterogeneous porous channel is presented, showing the existence of a secondary flow in a plane perpendicular to the axial imposed flow. In non-isothermal systems the effect of rotation is expected in natural convection. Then the rotation may affect the flow through two distinct mechanisms, namely thermal buoyancy caused by centrifugal forces and the Coriolis force (or a combination of both). Since natural convection may be driven also by the gravity force and the orientation of the buoyancy force with respect to the imposed thermal gradient has a distinctive impact on the resulting convection, a significant number of combinations of different cases arise in the investigation of the rotation effects in non-isothermal systems. Results pertaining to some of these cases are presented. However, the lack of extensive experimental confirmation is particularly noticed. Although some experimental results were reported, it is recommended that the effort of getting experimental confirmation of theoretical results be expanded. This recommendation is valid for the investigation of isothermal systems as well.

ix

Nomenclature

Da êx ; êy ; êz ên êg êx Ek; EkD Fr g h H H ke K L Mf p pr Pe Pr; Pre Q_ r Rag ; Rax Re T TC ; TH Ta

Darcy number, defined by Eq. (2.15) Unit vectors in the x, y and z direction, respectively Unit vector normal to the boundary, positive outwards Unit vector in the direction of the acceleration of gravity Unit vector in the direction of the angular velocity of rotation Porous media Ekman number, defined by Eq. (2.13) Porous media Froude number, defined by Eq. (2.8) Acceleration of gravity Macro-level integral heat transfer coefficient for the heat conduction at the fluid-solid interface The front aspect ratio of the porous layer, equals H =L The height of the layer Effective thermal conductivity Permeability, dimensional The length of the porous layer Heat capacity ratio, defined by Eq. (2.24) Pressure, dimensionless Dimensionless reduced pressure, generalized to include the constant components of the centrifugal and gravity terms Peclet number, defined by Eq. (2.27) Prandtl number and effective Prandtl number, defined by Eqs. (2.28) and (2.39), respectively Internal heat generation, dimensionless Coordinate in the radial direction Gravity and centrifugal Rayleigh numbers, defined by Eqs. (2.53) and (2.54), respectively Reynolds number, defined by Eq. (2.15) Temperature, dimensionless Coldest and hottest walls temperatures, respectively, dimensional Porous media Taylor number (reciprocal of Ekman number), defined by Eq. (5.9) xi

xii

Nomenclature

u, v, w V Va W W X x0 x y z

Horizontal x and y, and vertical components of the filtration velocity, respectively Filtration velocity, dimensionless Vadasz number introduced by Straughan (2001), defined by Eq. (2.40) The width of the layer The top aspect ratio of the porous layer, equals W =L Position vector, equals xêx þ yêy þ zêz The dimensionless offset distance from the rotation axis Horizontal length coordinate Horizontal width coordinate Vertical coordinate

Greek Symbols ae b bT dij g / l q x mo w

Effective thermal diffusivity Thermal expansion coefficient Dimensionless thermal expansion coefficient, equals b DTc Kronecker delta function The reciprocal of the offset distance from the rotation axis, equals 1=x0 ¼ Rax =Rax0 Porosity Fluid’s dynamic viscosity Density, dimensionless Angular velocity of the rotating porous domain Fluid’s kinematic viscosity Stream function

Subscripts ∗ c cr s f

Dimensional values Characteristic values Critical values Solid phase Fluid phase

Chapter 1

Introduction

Abstract The introduction chapter introduces the topic, its theoretical significance, as well as its applications in engineering and geophysics. Then a conceptual introduction of the porous media, its definitions and the continuum approach adopted in this book follows.

Keywords Packed bed agitated vessels Food processing Chemical processing Centrifugal filtration Rotating machinery Porous media continuum approach

1.1 General Background The study of flow in rotating porous media is motivated by its theoretical significance and practical applications in geophysics and engineering. Flows in porous geological formations subject to earth rotation, the flow of magma in the earth mantle close to the earth crust (Fowler 1990) represent examples of geophysical applications. Among the applications of rotating flow in porous media to engineering disciplines, one can find the food process industry, chemical process industry, centrifugal filtration processes, and rotating machinery. More specifically, packed bed mechanically agitated vessels are used in the food processing and chemical engineering industries in batch processes. The packed bed consists of solid particles or fibers of material, which form the solid matrix while fluid flows through the pores. As the solid matrix rotates, due to the mechanical agitation, a rotating frame of reference is a necessity when investigating these flows. The role of the flow of fluid through these beds can vary from drying processes to extraction of soluble components from the solid particles. The molasses in centrifugal crystal separation processes in the sugar milling industry and the extraction of sodium alginate from kelp are just two examples of such processes. Another important application of rotating flows in porous media is in the design of a multi-pore distributor in a gas-solid fluidized bed. A multi-pore distributor is a device, which is constructed from foraminous materials, wire compacts, filter cloth, © The Author(s) 2016 P. Vadasz, Fluid Flow and Heat Transfer in Rotating Porous Media, SpringerBriefs in Thermal Engineering and Applied Science, DOI 10.1007/978-3-319-20056-9_1

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1

Introduction

compressed fibers, sintered metal, or similar (Whitehead 1985). Research results (Davidson and Harrison 1963) showed that the porous distributor allowed a more even expansion of the bed than the other distributors and its design affected the behavior of the bed over most of its height. An even distribution of the gas is necessary to avoid instability in the fluidized bed, which can break down proper fluidization. A commonly used solution to avoid maldistribution of gas and bed instability is cyclic interchange fluidization (CIF) (Kvasha 1985), where the distributor is rotating at constant angular velocities which vary between 20 and 2500 rpm, depending on the size of the bed (the higher its diameter, the lower the angular velocity). Some examples of applications of the cyclic interchange fluidization are the highly exothermic synthesis of alkylchlorsilanes polymer filling of composites, treatment of finely dispersed solids, drying of paste-like polymers, permanganate of potash and iodine (Kvasha 1985). Therefore, evaluating the flow field through a porous rotating distributor becomes a design necessity. Modeling of flow and heat transfer in porous media is also applied for the design of heat pipes using porous wicks and includes effects of boiling in unsaturated porous media, surface tension driven flow with heat transfer and condensation in unsaturated porous media. Plumb (1991) presented a comprehensive review of the heat transfer in unsaturated porous media flow with particular applications to the heat pipe technology. Again, when the heat pipe is used for cooling devices, which are subject to rotation the corresponding centrifugal, and Coriolis effects become relevant as well. The macro-level porous media approach is gaining an increased level of interest in solving practical fluid flow problems, which are too difficult to solve by using a traditional micro-level approach. As such Direct Chill (DC) casting models apply the Darcy law to predict the heat transfer, fluid flow and ultimately the thermal stresses in the solidified metal. Such a model was applied by Katgerman (1994) to analyze the heat transfer phenomena during continuous casting of aluminum alloys. When centrifugal casting processes are being considered, rotation effects become relevant to the problem. The porous medium approach is also used in processing of composite materials. Güçeri (1994) states that “most of the studies of resin transfer molding (RTM) processes and structural reaction injection molding (SRIM) treat the flow domain as an anisotropic porous medium and preform a Darcy flow analysis utilizing a continuum model.” The electrocatalysis of the oxygen reduction reaction (ORR) in alkaline media on ultra-thin porous coating rotating electrodes is another application of transport phenomena in rotating porous media and its application was discussed by Lima and Ticianelli (2004) in their study that used experimental methods. Additional applications of the porous medium approach are discussed by Nield and Bejan (2013), and Bejan (2013) in comprehensive reviews of the fundamentals of heat convection in porous media. Bejan (2013) mentions among the applications of heat transfer in porous media the process of cooling of winding structures in high-power density electric machines. When this applies to the rotor of an electric machine, say generator (or motor), rotation effects become relevant as well. Mohanty (1994) presented a study of natural and mixed convection in rod arrays motivated by safety

1.1 General Background

3

related thermal-hydraulic modeling of nuclear reactors with particular attention to the rod-bundle geometry. The author concluded that “bundle average experimental friction factor values in forced convection are better explained through a porous medium model” and “the porous medium parameters so derived also yield quantitative corroboration of the flow through vertical bundles induced solely by buoyancy.” The porous media approach was also successfully applied to simulate complex transport phenomena in mass and heat exchangers (Robertson and Jacobs 1994) and in cooling of electronic equipment (Vadasz 1991). Additional applications of the porous media approach are the flow of liquids in biological tissues like the human brain, the cardiovascular flow of blood in the human heart or other physiological processes, pebble bed nuclear reactors, and cooling of turbine blades in the high pressure and hot portion of a turbo-expander. Regarding the last application, such a cooling process enables the expander inlet gas temperature to increase beyond the allowed metal temperature, bringing a significant contribution to the cost-effectiveness of the expander. The cooling process occurs by injecting air through channels in the internal part of the blade. As long as the geometry of the channels is not too complicated the traditional heat transfer approach can be applied to evaluate the cooling performances. However, for complicated channel geometry the porous medium approach will prove again the most effective way of simulating the phenomenon. With the emerging utilization of the porous medium approach in non-traditional fields, including some applications in which the solid matrix is subjected to rotation (like physiological processes in human body subject to rotating trajectories, cooling of electronic equipment in a rotating radar, cooling of turbo-machinery blades, or cooling of rotors of electric machines) a thorough understanding of the flow in a rotating porous medium becomes essential. Its results can then be used in the more established industrial applications like food processing, chemical engineering or centrifugal processes, as well as to the less traditional applications of the porous medium approach. Reviews of the fundamentals of flow and heat transfer in rotating porous media were presented by Vadasz (1994, 1997, 1998, 2000, 2002a, b). No reported results were found on isothermal flow in rotating porous media prior to 1994. Pioneering research results on natural convection in rotating porous media were reported by Rudraiah et al (1986), Patil and Vaydianathan (1983), Jou and Liaw (1987a, b), and Palm and Tyvand (1984). Nield (1991) while presenting a comprehensive review of the stability of convective flows in porous media found that the effect of rotation on convection in a porous medium attracted limited interest. The main reason behind the limited interest for this type of flow is first the fact that isothermal flow in homogeneous porous media following Darcy law is irrotational (Bear 1991) hence the effect of rotation on this flow is insignificant. However, for a heterogeneous medium with spatially dependent permeability or for natural convectionin a non-isothermal homogeneous porous medium the flow is not irrotational anymore, hence the effects of rotation become significant. In some applications these effects can be small e.g., when the porous media Ekman number is high. Nevertheless, the effect of rotation is of interest even then, as it may generate secondary flows in planes perpendicular to the main flow direction. Even

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Introduction

when these secondary flows are weak, it is essential to understand their source, as they might be detectable in experiments. To back up this claim, it is sufficient to look at the corresponding rotating flows in pure fluids (non-porous domains). There, the Coriolis effect and secondary motion in planes perpendicular to the main flow direction are controlled by the Ekman number. Experiments (Hart 1971, Johnston et al 1972, and Lezius and Johnston 1976) showed that this secondary motion is detectable even for very low or very high Ekman numbers, although the details of this motion may vary considerably according to pertaining conditions. It is therefore expected to obtain secondary motion when a solid phase is present (a porous matrix) in a similar geometric configuration, although its details cannot be a priori predicted based on physical intuition only. This creates a strong motivation to investigate the effect of rotation on the fluid flow in isothermal heterogeneous porous media. For high angular velocity of rotation or extremely high permeability, conditions pertaining to some engineering applications, the Ekman number can become of unit order of magnitude or lower, and then the effect of rotation becomes even more significant. The same motivation applies for investigating the effect of rotation on natural convection in porous media. A significant effort leading to a large volume of publications on this topic was evident during the past 20 years, producing a good fundamental understanding of the related phenomena. The methodology adopted in this book consists of a presentation of the dimensionless equations governing the flow and transport phenomena in a rotating frame of reference, followed by a classification of problems, i.e. isothermal rotating flows in heterogeneous porous media, and natural convection in rotating homogeneous porous systems. Then each class of problems is analyzed and solved. Conclusions regarding the effect of rotation are then drawn and discussed.

1.2 Porous Media Definitions and the Continuum Approach A porous medium can be defined as a two-phase system, one phase being solid and the second phase being fluid, while both phases are interconnected. The fluid phase is present in the pores (voids) and therefore the first definition related to the porous medium is the porosity /, a dimensionless quantity defined by the ratio of the volume occupied by the effective voids to the total volume of the porous matrix. In addition, the permeability of a porous medium, K , represents the ability of the porous medium to allow fluid flow through its pores. The more permeable a porous medium is, the easier it is for the fluid to flow through it. The permeability has units of area, ½m2 in SI, but the more common unit used is the darcy, i.e. 1 darcy ¼ 9:869233 1013 m2 (or the millidarcy). There is a wide range of physical systems that comply with the definition of a porous medium. Soil, sand, fractured rock, fabric, pebble bed containers and reactors, human tissues like brain, lungs, hair, skin are only a few examples. In principle, in order to evaluate the flow in a porous

1.2 Porous Media Definitions and the Continuum Approach

5

medium at the pore-level (micro-level) one needs to know the governing equations, which we do, i.e. they are the Navier-Stokes equations, and the boundary conditions on the solid-fluid interface, which we do, i.e. these are the impermeability and non-slip conditions on any solid-fluid interface, i.e. the velocity vector is zero there. However, the only unknown and extremely difficult to formulate mathematically is the shape function of this solid-fluid interface on which we need to set the velocity to be equal to zero. An alternative approach is therefore needed and it was developed in the sense of defining average values of variables over a Representative Elementary Volume (REV), which is large enough for statistical averages to be meaningful but small enough relative to the size of the porous domain in order to permit the assumption of a continuum, i.e. the variables can change their values continuously in the porous domain, while the space coordinates represent the coordinates of the REV centroid, which also vary continuously within the porous domain. As such instead of the pore-level velocity vector one obtains a macro-level filtration velocity, an average of the former over an REV. Similarly all other dependent variable are also values averaged over an REV. The different models associated with this macro-level continuum approach of a porous medium will be introduced in the next section. They consist of averaging the corresponding micro-level governing equations subject to sensible assumptions and approximations. Some of the models, such as the Darcy model, have been originally derived directly from experimental data.

References Bear J (1991) Dynamics of fluids in porous media. Elsevier, New York, pp 131–132 (reprinted by Dover 1991) Bejan A (2013) Convection heat transfer, 4th edn. Wiley, Hoboken Davidson JF, Harrison D (1963) Fluidised particles. Cambridge University Press, New York, p 7 Fowler AC (1990) A compaction model for melt transport in the earth asthenosphere Part I: The base model. In: Raya MP (ed) Magma transport and storage. Wiley, Chichester, pp 3–14 Güçeri SI (1994) Fluid flow problems in processing composites materials. In: Proceedings of the 10th international heat transfer conference, vol 1, Brighton, UK, pp 419–432 Hart JE (1971) Instability and secondary motion in a rotating channel flow. J Fluid Mech 45:341–351 Johnston JP, Haleen RM, Lezius DK (1972) Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J Fluid Mech 56:533–557 Jou JJ, Liaw JS (1987a) Transient thermal convection in a rotating porous medium confined between two rigid boundaries. Int Commun Heat Mass Transfer 14:147–153 Jou JJ, Liaw JS (1987b) Thermal convection in a porous medium subject to transient heating and rotation. Int J Heat Mass Transfer 30:208–211 Kvasha VB (1985) Multiple-spouted gas-fluidized beds and cyclic fluidization: operation and stability. In: Davidson R Clift JF, Harrison D (ed) Fluidization, 2nd edn. Academic Press, London, pp 675–701 Lezius DK, Johnston JP (1976) Roll-cell instabilities in a rotating laminar and turbulent channel flow. J Fluid Mech 77:153–175

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Introduction

Lima FHB, Ticianelli EA (2004) Oxygen electrocatalysis on ultra-thin porous coating rotating ring/disk platinum and platinum–cobalt electrodes in alkaline media. Electrochim Acta 49:4091–4099 Mohanty AK (1994) Natural and mixed convection in rod arrays. In: Proceedings of the 10th international heat transfer conference, GK-7, vol 1, Brighton, UK, pp 269–280 Nield DA (1991) The stability of convective flows in porous media. In: Kakaç S, Kilkis B, Kulacki FA, Arniç F (eds) Convective heat and mass transfer in porous media. Kluwer Academic Publ., Dordrecht, pp 79–122 Nield DA, Bejan A (2013) Convection in porous media, 4th edn. Springer, New York Palm E, Tyvand A (1984) Thermal convection in a rotating porous layer. J Appl Math Phys (ZAMP) 35:122–123 Patil PR, Vaidyanathan G (1983) On setting up of convection currents in a rotating porous medium under the influence of variable viscosity. Int J Eng Sci 21:123–130 Robertson ME, Jacobs HR (1994) An experimental study of mass transfer in packed beds as an analogy to convective heat transfer. In: Proceedings of the 9th international heat transfer conference, vol 1, pp 419–432, Brighton, UK Rudraiah N, Shivakumara IS, Friedrich R (1986) The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed porous medium. Int J Heat Mass Transf 29:1301–1317 Vadasz P (1991) On the evaluation of heat transfer and fluid flow by using the porous media approach with application to cooling of electronic equipment. Proceedings of the 5th Israeli conference on packaging of electronic equipment, Herzlia, Israel, pp D4.1–D4.6 Vadasz P (1994) Fundamentals of flow and heat transfer in rotating porous media. In: Heat transfer, PA, vol 5. Taylor and Francis, Bristol, pp 405–410 Vadasz P (1997) Flow in rotating porous media. In: du Plessis P, Rahman M (eds) Fluid transport in porous media from the series advances in fluid mechanics, vol 13, pp 161–214, Computational Mechanics Publications, Southampton Vadasz P (1998) Free convection in rotating porous media. In: Ingham DB, Pop I (eds) Transport phenomena in porous media. Elsevier Science, Oxford, pp 285–312 Vadasz P (2000) Fluid flow and thermal convection in rotating porous media. In: Vadfai K (ed) Handbook of porous media. Marcel Dekker, New York, pp 395–439 Vadasz P (2002a) Heat transfer and fluid flow in rotating porous media. In: Hassanizadeh SM, Schotting RJ, Gray WG, Pinder GF (eds) Computational methods in water resources, vol 1, Development in water science, vol 47, pp 469–476. Elsevier, Amsterdam Vadasz P (2002b) Thermal convection in rotating porous media. In: Trends in heat, mass and momentum transfer, vol 8, pp 25–58, Research Trends, Trivandrum, Kerala, India Whitehead AB (1985) Distributor characteristics and bed properties. In: Davidson R Clift JF, Harrison D (eds) Fluidization, 2nd edn. Academic Press, London, pp 173–199

Chapter 2

Modeling of Flow and Heat Transfer in Porous Media

Abstract Modeling of flow and heat transfer in rotating porous media is introduced in this chapter. The governing equations are presented and transformed into dimensionless form in order, among others, to identifying conditions for neglecting terms in these equations. The classification of different types of rotating flows in porous media concludes the chapter. Keywords Porous media modeling forms

Rotating porous media

Dimensionless

2.1 Modeling of Flow in Porous Media A common equation to all models of flow in porous media is the continuity equation representing the mass balance of the fluid. Regardless of what kind of dynamic model is being used the continuity equation is to be used in conjunction with the latter. The continuity equation for a non-deformable porous medium (the porosity is constant, / ¼ const:) is presented in the form /

@ q þ V r q þ q r V ¼ 0 @ t

ð2:1Þ

where / is the porosity of the porous medium defined as the ratio between the volume of effective pores to the total volume of the porous matrix (voids + solid). The definition of effective pores is applicable to pores that are not completely isolated from fluid flow, i.e. interconnected pores. In Eq. (2.1) q is the density of the fluid, V ¼ u êx þ v êy þ w êz is the filtration velocity vector and u ; v ; w are its components in the x ; y and z direction, respectively, in a Cartesian coordinate system.

© The Author(s) 2016 P. Vadasz, Fluid Flow and Heat Transfer in Rotating Porous Media, SpringerBriefs in Thermal Engineering and Applied Science, DOI 10.1007/978-3-319-20056-9_2

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2 Modeling of Flow and Heat Transfer in Porous Media

When the flow is definitely incompressible the density is identically constant, i.e. q ¼ qo ¼ const: and Eq. (2.1) above becomes r V ¼ 0

ð2:2Þ

Even when the flow is only approximately incompressible Eq. (2.2) can be used as a very accurate approximation. An example of such an application that is widely used in this book refers to the Boussinesq approximation (Boussinesq 1903) that is acceptable in reference to buoyancy driven flows, and indicates that the density can be regarded as constant in all terms of the governing equations except the buoyancy terms in the momentum equation. Therefore, subject to Boussinesq approximation (Boussinesq 1903) Eq. (2.2) will be used in this book to represent a good approximation of the continuity Eq. (2.1). When relaxation of the latter approximation is applied in particular circumstances we shall revert to the accurate form of the continuity Eq. (2.1). Dimensionless form of the continuity equation The two forms of the continuity equation presented above can be presented in a dimensionless form by using a reference value of density say qo , a characteristic velocity uc , a characteristic length lc and a characteristic time that can be defined in terms of the latter two in the form tc ¼ lc =uc Mf (The coefficient Mf represents the ratio between the effective heat capacity of the fluid and effective heat capacity of the porous medium, a coefficient that will be defined later. The choice of including this coefficient in the characteristic time is done for later convenience in notation). By definition, any characteristic or reference values are constant. We distinguish between the two only for physical reasons. A reference value typically applies to material properties, which act as system parameters in the equations and may or may not change, while a characteristic value applies to the dependent and independent variables that naturally always change. The specific choice of qo , uc and lc will be made in each instance when we deal with a specific problem. For now all we need to remember is that they are constant values. Then we may introduce the dimensionless position vector, the dimensionless filtration velocity, the dimensionless density and the dimensionless r operator (in Cartesian coordinates) as X¼

X x y z ¼ êx þ êy þ êz ¼ x êx þ y êy þ z êz ; lc lc lc lc

V q ; q ¼ ; uc qo @ @ @ @ @ @ êx þ êy þ êz êx þ êy þ êz ¼ r ¼ lc r ¼ lc @ x @ y @ z @x @y @z V¼

ð2:3aÞ ð2:3bÞ ð2:3cÞ

2.1 Modeling of Flow in Porous Media

9

leading to the dimensionless form of Eq. (2.1) /

@q þ V rq þ q r V ¼ 0 @t

ð2:4Þ

and the dimensionless form of Eq. (2.2) rV ¼0

ð2:5Þ

2.1.1 Darcy Model The most fundamental model of flow in porous media is the Darcy model. It represents the momentum balance for the fluid and can be presented in the form V ¼

K r p q gêg l

ð2:6Þ

where K is the permeability of the porous matrix (carrying units of m2 or darcy, 1 13 2 m ), l is the dynamic viscosity of the fluid (cardarcy unit = 9:869233 10 rying units of Pa s ¼ N s m2 ), g is the acceleration due to gravity (carrying units of m/s2), q is the density of the fluid carrying units of kg/m3), and êg is a unit vector in the gravity acceleration direction. The permeability of the porous medium is a property of the matrix and is independent of the fluid flowing through it. It can be regarded as an effective cross sectional area perpendicular to the direction of the flow and hence represents a measure of how permeable the matrix is. The Darcy equation accounts for the lumped effect of the friction forces between the fluid and the solid matrix on the filtration velocity, in addition to the gravitational forces. The forces between the fluid particles within a Representative Elementary Volume are being assumed negligible and therefore neglected. The latter are of the order of 2 magnitude of Darcy number Da ¼ K lc , i.e. OðDaÞ, which for typical porous media has very small values. Dimensionless form of the Darcy equation The dimensionless form of the Darcy equation, assuming constant values of permeability and dynamic viscosity, can be obtained by introducing a characteristic pressure difference Dpc and by using the same characteristic and reference values as used for the continuity equation in the previous section. Again, except for the fact that the characteristic pressure difference that we introduce, Dpc , is constant we do not need to specify anything else about Dpc . This will be done later when dealing with specific problems. Therefore, the dimensionless form of the Darcy equation is

10


V ¼ Np rp þ Frq êg

ð2:7Þ

where two dimensionless groups, a pressure number Np and a porous media Froude number Fr, emerged in the form Np ¼

K Dpc g K ; Fr ¼ l uc lc m uc

ð2:8Þ

where m ¼ l =qo is the kinematic viscosity of the fluid.

2.1.2 Darcy Equation in a Rotating Frame of Reference and Other Extended Forms Darcy equation can be extended to include additional body forces, such as centrifugal and Coriolis effects in the case of a rotating porous matrix, or an additional term that accounts for time dynamics of the evolving values of the filtration velocity, i.e. a time derivative term. For the former, rotation effects introduce the centrifugal and Coriolis terms as follows V ¼

K 2q r p q gêg þ q x ðx X Þ þ x V l /

ð2:9Þ

where x is the angular velocity of rotation (carrying units of s1 ), / is porosity, and X is the position vector measured from the axis of rotation. The third term in the brackets represents the centrifugal force while the fourth term represents the Coriolis acceleration. Another extension to the Darcy equation is applicable when fast transients or high frequency effects are of interest. Then, the time resolution obtained by assuming a very fast reaction of Darcy flow to changes and therefore the quasi-steady approximation which is inherent in the formulation of the Darcy law is not sufficient and a time derivative of the filtration velocity needs to be incorporated leading to q @ V l þ V ¼ r p þ q gêg / @ t K

ð2:10Þ

Combining (2.9) with (2.10) yields the Darcy equation in a rotating frame of reference extended to include fast transients and high frequency effects. q @ V l 2q þ V ¼ r p þ q gêg q x ðx X Þ x V / @ t K / ð2:11Þ

2.1 Modeling of Flow in Porous Media

11

Dimensionless forms of the extensions to Darcy equation Assuming a constant angular velocity of rotation, x ¼ const:, we can transform Eq. (2.9) into the following dimensionless form V ¼ Np rp þ Fr q êg Cn q êx ðêx X Þ

q êx V EkD

ð2:12Þ

where two additional dimensionless groups emerged, a centrifugal number Cn and a porous media Ekman number EkD defined in the form Cn ¼

K x2 lc /m ; EkD ¼ m uc 2x K

ð2:13Þ

and where m ¼ l =qo is the kinematic viscosity of the fluid, and Eq. (2.11) is transformed into the dimensionless form DaReMf @ V q êx V þ V ¼ Np rp þ Fr q êg Cn q êx ðêx XÞ q @t EkD / ð2:14Þ where the additional dimensionless groups that emerged are the Darcy number Da, and the Reynolds number Re Da ¼

K uc l c ; Re ¼ l2c m

ð2:15Þ

and where the definition of Mf will follow later. Note that despite the fact that it is the porous media filtration velocity that emerged in the definition of the Reynolds number, the corresponding length scale is a macro-level length scale, not the poresize. However the Reynolds number appears in Eq. (2.14) in a product combination with the Darcy number, bringing therefore the pore-scale effects into account too.

2.2 Modeling of Heat Transfer in Porous Media Similarly as in fluid flow, the equation governing the heat transfer in porous media is the energy equation averaged over the phases of the porous medium. Fourier law is assumed for the conduction heat fluxes in each phase.

12


2.2.1 Finite Heat Transfer Between the Phases Assuming different temperatures associated with the different phases, i.e. Ts for the solid phase temperature and Tf for the fluid phase temperature one obtains the following equations of energy balance ð1 /Þqs cs /qf cp;f

@ Ts ¼ ð1 /Þ~ks r2 Ts þ h Tf Ts @ t

@ Tf þ /qp;f cp V r Tf ¼ /~kf r2 Tf h Tf Ts @ t

ð2:16Þ ð2:17Þ

where qs , cs , and ~ks are the density, specific heat, and thermal conductivity of the solid phase material, and qf , cp;f , and ~kf are the density, the constant pressure specific heat, and the thermal conductivity of the fluid phase material. The following notation is useful cs ¼ ð1 uÞqs cs ; cf ¼ u qf cp;f

ð2:18Þ

representing the solid phase and fluid phase effective heat capacities, respectively, and ks ¼ ð1 uÞ~ks ; kf ¼ u ~kf

ð2:19Þ

representing the effective thermal conductivities of the solid and fluid phases, respectively. The term h Tf Ts represents the rate of heat generation per unit volume in the solid phase within the REV due to the heat transferred from the fluid over the fluid-solid interface. The coefficient h [ 0, carrying units of W m−3 K−1, is a macro-level integral heat transfer coefficient for the heat conduction at the fluidsolid interface (averaged over the REV) that is assumed independent of the phases’ temperatures and independent of time. Note that this coefficient is conceptually distinct from the convection heat transfer coefficient and is anticipated to depend on the thermal conductivities of both phases as well as on the surface area to volume ratio (specific area) of the medium. Subject to the notation (2.18), (2.19), (2.16) and (2.17) become @ Ts ¼ ks r2 Ts þ h Tf Ts @ t

ð2:20Þ

@ Tf þ cf V r Tf ¼ kf r2 Tf h Tf Ts @ t

ð2:21Þ

cs cf

2.2 Modeling of Heat Transfer in Porous Media

13

2.2.2 Thermal Equilibrium Between the Phases When the temperature difference between the solid and fluid phases is very small, conditions applicable to most circumstances when for example the thermal conductivity of the solid phase is not extremely small, then a sensible approximation would be to set the temperatures of the phases equal to each other, i.e. T ¼ Ts Tf . These conditions are identified as Local Thermal Equilibrium (LTE or Lotheq), while the former case of distinct phase temperatures is identified as Local Thermal Non Equilibrium (LTNE) or Lack of Local Thermal Equilibrium (LaLotheq). Then the two Eqs. (2.20) and (2.21) can be combined by adding them to yield ce

@ T þ cf V r T ¼ ke r2 T @ t

ð2:22Þ

where ce ¼ cs þ cf , ke ¼ ks þ kf are the effective heat capacity and effective thermal conductivity of the porous medium, and the interface heat transfer term disappeared as the amount of heat gained by the solid as it transferred from the fluid is equal to the amount of heat lost by the fluid as it transferred to the solid via the fluid-solid interface at each point in space. Dividing Eq. (2.22) by ce yields @ T þ Mf V r T ¼ ~ae r2 T @ t

ð2:23Þ

where ~ae ¼

cf ke ; Mf ¼ ce ce

ð2:24Þ

are an effective thermal diffusivity of the porous medium, and the heat capacity ratio, i.e. the ratio between the effective heat capacity of the fluid phase and the effective heat capacity of the porous medium, respectively. Dimensionless forms of the energy equation for local thermal equilibrium The derivation of the dimensionless form of the energy Eq. (2.23) follows the definition of the dimensionless temperature T¼

ðT To Þ DTc

ð2:25Þ

where To is a reference value of temperature and DTc is a characteristic temperature difference to be explicitly defined for every specific problem. Introducing the definition of the adjusted effective thermal diffusivity ae ¼ ~ae Mf ¼ ke cf the dimensionless energy equation becomes

14


@T 1 þ V rT ¼ r2 T @t Pe

ð2:26Þ

where Peclet number emerged as an additional dimensionless group, defined in the form Pe ¼

uc lc ¼ Pr Re ae

ð2:27Þ

where the Prandtl number Pr is defined by Pr ¼

m ae

ð2:28Þ

2.2.3 Equation of State A relationship between the density, temperature, and pressure (and solute concentration when the fluid is a solution of soluble substances, e.g. salt in water, alcohol in water, etc.) is needed in order to complete the model formulation. If the fluid is compressible such as a gas, in certain circumstances one may use as an approximation the ideal gas equation q ¼

1 p Rg T

ð2:29Þ

where Rg is the specific gas constant (i.e. the universal gas constant divided by the molecular mass of the gas). For liquids and also for gases (even when Eq. (2.29) applies) one may use a linear approximation relating the density to temperature and pressure, when temperature and pressure differences are not excessively large. Then q ¼ qo 1 bT ðT To Þ þ bp ðp po Þ

ð2:30Þ

where qo is a reference value of density obtained when the temperature is To and the pressure is po . The coefficients bT and bp are the thermal expansion coefficient (carrying units of K1 ) and the pressure compression coefficient (carrying units of Pa1 ), respectively, defined in the form bT ¼ bp ¼

1 @ q 1 @ q q @ T qo @ T

ð2:31Þ

1 @ q 1 @ q q @ p qo @ p

ð2:32Þ

2.2 Modeling of Heat Transfer in Porous Media

15

It can be observed that by using the ideal gas Eq. (2.29) with (2.31) and (2.32) it leads to the following thermal expansion and pressure compression coefficients: bT ¼ 1=T and bp ¼ 1=p , respectively. Dimensionless forms of equation of state The dimensionless form of the linear approximation of the equation of state can be obtained by using the definition of the dimensionless temperature from Eq. (2.25) and the dimensionless pressure in the form p ¼ ðp po Þ=Dpc . Then Eq. (2.30) becomes q ¼ 1 b T T þ bp p

ð2:33Þ

where q ¼ q =qo is the dimensionless density, and bT ¼ bT DTc , bp ¼ bp Dpc are the dimensionless thermal expansion and pressure compression coefficients, respectively. Typically, for most fluid flows bp bT , especially for incompressible flows, i.e. flows of liquids. Therefore a common approximation of the dimensionless equation of state would be q ¼ ½ 1 bT T

ð2:34Þ

When the fluid phase is a solution of a soluble substance in a liquid, such as salt in water, then the density would depend also on the concentration S. Then an extension of the equation of state (2.34) would be required in the form q ¼ ½1 bT T þ bS S

ð2:35Þ

where S ¼ ðS So Þ=DSc , and bS ¼ bS DSc is the dimensionless compression coefficient due to solute concentration, and S is the solute concentration. In such cases an additional transport equation for mass (solute) transfer accounting for the molecular diffusion and the convection of the solute within the fluid phase would be required too.

2.3 Natural Convection and Buoyancy in Porous Media 2.3.1 Homogeneous Porous Media Natural convection is the effect of flow and convection heat transfer due to the existence of density gradients in a body force field (such as gravity or centrifugal force field). As density depends on temperature as demonstrated in the derivation of the equation of state, temperature gradients may create natural convective flows when a body force field is present. What characterizes natural convection is the lack of a known value of characteristic filtration velocity that can be applied upfront in a problem. No characteristic velocity can be specified because the latter is dictated by

16


the temperature gradients and their resulting buoyancy rather than being known upfront. Therefore a sensible choice of uc would be uc ¼ ae =lc . With this choice of uc the Froude number Fr, the pressure number Np , and the centrifugal dimensionless group Cn in Eqs. (2.12) and (2.14) become Fr ¼

g K l c K Dpc x 2 l2 K ; Np ¼ ; Cn ¼ c m ae l ae m ae

ð2:36Þ

Without loss of generality for the same reason as for the filtration velocity one can chose the characteristic pressure difference to be such that the pressure number Np becomes unity, i.e. Dpc ¼ l ae =K leading to Np ¼ 1. Also the Reynolds number in Eq. (2.14) renders into the reciprocal Prandtl number Re ¼

ae 1 ¼ Pr m

ð2:37Þ

and the Peclet number in Eqs. (2.27) and (2.26) equals one by definition Pe ¼

uc lc ae lc ¼ ¼1 ae lc ae

ð2:38Þ

One may define the effective Prandtl number in terms of the effective thermal diffusivity ~ ae (see Eq. (2.24)) Pre ¼

m Pr ¼ a~e Mf

ð2:39Þ

Then the coefficient to the time derivative term in Eq. (2.14) becomes DaReMf DaMf Da 1 ¼ ¼ ¼ /Pre Va / /Pr

ð2:40Þ

a new dimensionless group that Straughan (2001) named the Vadasz number (Va), or the Vadasz coefficient named by Straughan (2008) (see also Sheu 2006 and Govender 2010). An extremely useful approximation that is usually applied to natural convection problems is the Boussinesq approximation (Boussinesq 1903). Boussinesq approximation states that the density can be considered constant, i.e. q ¼ qo ¼ const: (and consequently its dimensionless value is q ¼ 1) everywhere except in the terms associated with body forces, where the variations of density are to be accounted for. According to this approximation the equation of state (2.34) will be substituted instead of q in the gravity and centrifugal terms in Eqs. (2.12) and (2.14), but q ¼ 1 will be substituted everywhere else. Introducing these results from Eqs. (2.36), (2.37), (2.38), (2.39), (2.40) with Np ¼ 1 and applying the Boussinesq

2.3 Natural Convection and Buoyancy in Porous Media

17

approximation (Boussinesq 1903) one obtains the following equations for natural convection to replace Eqs. (2.5), (2.14), and (2.26) rV ¼0 V ¼ rp þ Fr q êg Cn q êx ðêx X Þ

ð2:41Þ 1 êx V EkD

1 @V 1 êx V þ V ¼ rp þ Fr q êg Cn q êx ðêx X Þ Va @ t EkD @T þ V rT ¼ r2 T @t

ð2:42Þ ð2:43Þ ð2:44Þ

It becomes appealing in the application of the porous media models presented in the previous sections for problems of natural convection where buoyancy effects are to be investigated to introduce some identities that are extremely useful in what follows. These identities are êg ¼ r êg X

ð2:45Þ

1 êx ðêx X Þ ¼ r ðêx XÞ ðêx X Þ 2

ð2:46Þ

êx ðêx X Þ ¼ ðêx XÞ êx X

ð2:47Þ

and their proof is provided in the Appendix. Substituting these identities into Eqs. (2.42) and (2.43) and presenting the body force terms by using the artificial representation q ¼ q 1 þ 1 that will prove useful shortly, produces V ¼ rp þ Fr ðq 1Þ r êg X þ Fr r êg X

1 1 êx V Cn ðq 1Þ ½ðêx XÞ êx X þ Cn r ðêx X Þ ðêx X Þ 2 EkD

ð2:48Þ 1 @V þ V ¼ rp þ Fr ðq 1Þ r êg X þ Fr r êg X Va @ t

1 1 êx V Cn ðq 1Þ ½ðêx XÞ êx X þ Cn r ðêx X Þ ðêx XÞ 2 EkD

ð2:49Þ Introducing ðq 1Þ ¼ bT T from (2.34) and grouping all gradient terms under a common gradient operator yields

18


Cn V ¼ r p Fr êg X ðêx X Þ ðêx X Þ þ Fr bT T êg Cn bT T ½ðêx X Þ êx X 2 1 êx V EkD

ð2:50Þ

Cn 1 @V þ V ¼ r p Fr êg X ðêx X Þ ðêx X Þ þ Fr bT T êg Va @ t 2 1 êx V Cn bT T ½ðêx X Þ êx X EkD ð2:51Þ The term under the common gradient operator is a reduced pressure pr , defined as Cn pr ¼ p Fr êg X ðêx X Þ ðêx X Þ 2

ð2:52Þ

The product of bT by Fr and Cn yields two new dimensionless groups in the form of the gravity related Rayleigh number and the centrifugal Rayleigh number, respectively in the form Rag ¼ Fr bT ¼

bT DTc g K lc m ae

ð2:53Þ

Rax ¼ Cn bT ¼

bT DTc x2 l2c K m ae

ð2:54Þ

Equations (2.52), (2.53) and (2.54) transform Eqs. (2.50) and (2.51) into the following form V ¼ rpr þ Rag T êg Rax T ½ðêx X Þ êx X

1 êx V EkD

ð2:55Þ

1 @V 1 êx V ð2:56Þ þ V ¼ rpr þ Rag T êg Rax T ½ðêx XÞ êx X Va @ t EkD The particular case when êg ¼ êz and êx ¼ êz will be selected to analyze later. Subject to this orientation of the gravity and angular velocity of rotation Eqs. (2.55) and (2.56) take the form 1 êz V V ¼ rpr þ Rag T êz Rax T x êx þ y êy EkD

ð2:57Þ


1 @V 1 êz V þ V ¼ rpr þ Rag T êz Rax T x êx þ y êy Va @ t EkD

19

ð2:58Þ

The vector r ¼ ðx êx þ y êy Þ in Eqs. (2.57) and (2.58) represents the perpendicular radius vector from the axis of rotation to any point in the flow domain. Three dimensionless groups emerged in Eq. (2.57) when fast transients or high frequencies are not of interest. These dimensionless groups control the significance of the different phenomena. Therefore, the value of Ekman number ðEkD Þ controls the significance of the Coriolis effect and the ratio between the gravity related Rayleigh number Rag and the centrifugal Rayleigh number ðRax Þ controls the significance of gravity with respect forces as far as natural convection is concerned. to centrifugal This ratio is Rag Rax ¼ g x2 lc . When fast transients or high frequencies are of interest Eq. (2.58) is to be considered. In such a case one additional dimensionless group emerged, the Va number representing the ratio between two characteristic frequencies, i.e. the fluid flow frequency xm ¼ /m =K and the thermal diffusion frequency xa ¼ ~ae l2c , i.e. Va ¼ xm =xa ¼ /m l2c K ~ae ¼ /Pre =Da, or alternativelythe ratio between two time scales, i.e. the thermal diffusion time scale sa ¼ l2c ~ ae , and the fluid flow time scale sm ¼ K =/m , i.e. Va ¼ sa =sm ¼ /m l2c K ~ae ¼ /Pre =Da. In addition to such cases Eq. (2.58) should be used also when the Prandtl number is of the order of magnitude of Darcy number, i.e. Pre ¼ OðDaÞ i.e. a very small number (as Da\\1 in most porous media). Such small values of the Prandtl number are typical for liquid metals. In such cases too the time derivative term in Eq. (2.58) should be retained. Equations (2.41), (2.55) or (2.56), and (2.44) include three distinct mechanisms of coupling. Two of these couplings are linear. The first is the coupling between the pressure terms and the filtration velocity components, i.e., a coupling between Eq. (2.41) and the three components of Eq. (2.55) or (2.56). The second linear coupling is due to the Coriolis acceleration acting on the filtration velocity components in the plane perpendicular to the direction of the angular velocity. The third coupling is non-linear as it involves the gravity or centrifugal buoyancy terms in Eq. (2.55) or (2.56) and the energy Eq. (2.44). As V in Eq. (2.55) or (2.56) is dependent on temperature ðT Þ due to the buoyancy terms, it follows that the convective term V rT in the energy Eq. (2.44) causes the non-linearity. Therefore the coupling between the temperature, T, and the filtration velocity, V, is the only source of non-linearity in the Darcy’s formulation of flow and heat transfer in rotating porous media Eq. (2.55), or its extension Eq. (2.56).

2.3.2 Heterogeneous Porous Media In heterogeneous porous media the permeability and thermal conductivity can be dependent on the space variables, i.e. K ðx ; y ; z Þ and possibly ke ðx ; y ; z Þ.

20


In this book only cases when the permeability is allowed to vary in space are being considered. Then, a reference value of permeability Ko is being used in all previous definitions and the dimensionless permeability function is then defined in the form K ðx; y; zÞ ¼ K ðx ; y ; z Þ=Ko . Subsequently, following the same derivations one obtains the following set of equations for the heterogeneous porous media rV ¼0

1 êx V V ¼ K rpr þ Rag T êg Rax T ½ðêx X Þ êx X EkD

K @V 1 êx V þ V ¼ K rpr þ Rag T êg Rax T ½ðêx X Þ êx X Va @ t EkD

@T þ V rT ¼ r2 T @t

ð2:59Þ ð2:60Þ ð2:61Þ ð2:62Þ

Decoupling the equations is difficult without losing generality although the linear couplings can be resolved. Resolving the Coriolis related coupling is particularly useful. In doing so a Cartesian coordinate system is used and without loss of generality one can choose êx ¼ êz . A further choice is made, which reduces the generality of the problem, namely that the gravity acceleration is collinear with the z axis and directed downwards, i.e., êg ¼ êz . Some generality is lost, as problems where the direction of rotation and gravity are not collinear will not be represented by the resulting equations. Nevertheless, it is of interest to demonstrate this particular choice of system as it represents a significant number of practical cases. Subject to these choices (êx ¼ êz and êg ¼ êz ) Eq. (2.60) can be expressed by extending Eq. (2.57) to heterogeneous media in the following form (with the r dropped from pr and the D dropped from EkD for convenience) 1 þ K Ek1êz V ¼ K rp þ Rax T xêx þ yêy Rag Têz

ð2:63Þ

An important observation can be made by presenting Eq. (2.63) explicitly in terms of the three scalar components

@p þ Rax Tx @x

@p 1 þ Rax Ty K Ek u þ v ¼ K @y

@p þ Rag T w ¼ K @z u K Ek 1 v ¼ K

ð2:64Þ ð2:65Þ ð2:66Þ


21

where u; v, and w are the corresponding x; y, and z components of the filtration velocity vector V. It is now observed that the first two Eqs. (2.64) and (2.65) for the horizontal components of V are “Coriolis coupled”, while the third one is not. Since this type of coupling is linear it is possible to decouple them to obtain V H ¼ E ½rH p þ Rax T XH

ð2:67Þ

H

where V H ¼ uêx þ vêy is the horizontal filtration velocity, XH ¼ xêx þ yêy is the horizontal position vector, rH @=@xêx þ @=@yêy is the horizontal gradient operator and E is the following tensor operator H

E ¼ H

K 1 ½1 þ Ek 2 K 2 Ek 1 K

Ek 1 K 1

ð2:68Þ

The definition of the tensor operator E is extended to include the z component of V, thus leading to

H

V ¼ E ½rp þ B T

ð2:69Þ

where B ¼ Rax xêx þ Rax yêy Ragêz is the buoyancy coefficient vector and the tensor operator E is defined in the form

2 1 K 4 Ek 1 K E¼ ½1 þ Ek 2 K 2 0

Ek 1 K 1 0

3 0 5 0 ð1 þ Ek 2 K 2 Þ

ð2:70Þ

It is interesting to observe from Eqs. (2.69) and (2.70) that the Coriolis effect due to rotation is equivalent to a particular form of an anisotropic porous medium. The anisotropy is represented by the tensor operator E . It should be pointed out that the

analogy between the Coriolis effect and anisotropy was identified by Palm and Tyvand (1984) for the problem of gravity driven thermal convection in a rotating porous layer at marginal stability. It has been shown here and by Vadasz (1997, 2000) that this analogy is more general and applies also for centrifugally driven convection and for isothermal flows in rotating heterogeneous porous media (Rax ¼ Rag ¼ 0! B ¼ 0 in Eq. (2.69)). Subsequently Auriault et al. (2000, 2002) have obtained the same conclusion by using a method of multiple scale expansions.

22


2.4 Non-Darcy Models Although for a significantly high number of practical instances Darcy’s model in a rotating frame of reference is sufficient for representing the effects of rotation, nonDarcy models have been used as well. Their relevance and limitations are subject to professional discourse (e.g. Nield 1991, 1995, and Vafai and Kim 1990). Nevertheless there are circumstances where there is sufficient consensus that Darcy model falls short in representing accurately the fluid flow in porous media. In order to present the different relevant models, a dimensionless form of a general equation will be used based on a formulation by Hsu and Cheng (1990) as presented also by Nield (1991), but extended to include the centrifugal and Coriolis terms as dictated by the rotating frame of reference. Since the objective is to establish the validity conditions for different models, the basic dimensionless form of the mass balance (continuity equation), and energy equations are presented too. Therefore the mass, momentum and energy balance equations for a heterogeneous porous medium are presented in the form @q þ r ðqV Þ ¼ 0 ð2:71Þ @t

Mf ReD @ V ReD Da 2 F ReD qK þ 2 ðV rÞV þ V ¼ K rpr þ r V 1=2 jV jVþ / / @t Da /

ReD ReD 1 êx V ðq 1Þr êg X ðq 1Þêx ðêx X Þ q EkD Frg Frx /Mf

ð2:72Þ @T 1 þ V rT ¼ r2 T þ Q_ @t Pe

ð2:73Þ

where pr is the dimensionless reduced pressure generalized to include the constant components of the centrifugal and gravity terms as presented in Eq. (2.52), F is the dimensionless Forchheimer coefficient, and Q_ is a heat source term included for the eventuality of dealing with heat transfer problems where heat is generated within the porous medium, e.g. an electric current passing through an electrically conducting solid phase and producing Ohm’s heating. In Eqs. (2.71), (2.72) and (2.73) the following dimensionless groups or their combination emerged as the Reynolds number Re, the Darcy number Da, the Darcy-Reynolds number ReD , the Peclet number Pe, the Prandtl number Pr, the gravity related Froude number Frg , the Rossby number Ro, the Ekman number Ek, the centrifugal Froude number Frx , and the porous media Ekman number EkD defined in the form

2.4 Non-Darcy Models

uc l c ; m m Pr ¼ ae

Re ¼

Frg ¼

u2c ; g l c

Da ¼

23

Ko ; l2c

Ro ¼

ReD ¼ ReDa ¼

uc ; 2x lc

Ek ¼

uc Ko ; m l c

m Ro ; ¼ 2 2x lc Re

Pe ¼

uc l c ¼ Pr Re ; ae ð2:74Þ

Frx ¼

uc x lc

2 ¼ 4Ro2 ð2:75Þ

EkD ¼

/Ek /Ro /Ro /m ¼ ¼ ¼ Da Re Da ReD 2x Ko

ReD ReDa Ko g ¼ ¼ ; Frg Frg m uc

ReD ReDa ReDa Ko x2 lc ¼ ¼ ¼ Frx 4Ro2 Frx m uc

ð2:76Þ ð2:77Þ

where x is the angular velocity of rotation, m is the kinematic viscosity, and Ko is a reference value of permeability used to scale the dimensional permeability function K ðx ; y ; z Þ of the heterogeneous porous domain. Characteristic values of filtration velocity uc , length lc , time tc ¼ lc =uc Mf , pressure difference l ae =K , were used to transform the filtration velocity, space variables, time, and pressure into their dimensionless form. The characteristic value of temperature difference is DTc to be explicitly defined in each particular problem, and the density is rendered dimensionless in dividing it by the reference density value qo . As a result, the dimensionless groups presented in Eqs. (2.74)–(2.77) and/or their combination appear as coefficients of the different terms in the dimensionless equations. The physical interpretation of these dimensionless groups is now being discussed in order to perform an order of magnitude analysis and derive the validity conditions for assessment of the relative significance of each one of the terms in these equations. The Reynolds number Re defined in Eq. (2.74) represents the ratio between the inertial effects and viscous forces (here the viscous forces considered are those representing the shear stress between the fluid and solid phases). It does not appear in this form in the equations as Re is a Reynolds number related to the macroscopic length scale lc . In the equations the effect of Reynolds number is felt pffiffiffiffiffiffi via a combination of micro and macro scales via the microscopic length scale Ko in addition to the macroscopic one introduced by the Darcy number. The Darcy number Da defined in Eq. (2.74) is the square of the ratio between the microscopic pffiffiffiffiffiffi length scale Ko (because Ko is proportional to the square of the pore size D ) and the macroscopic length scale lc . The gravity related Froude number Frg expresses the ratio between inertial effects and the gravity body force. The Rossby number Ro defined in Eq. (2.75) is the ratio between convective inertial acceleration to the Coriolis force, while Ekman number Ek is a measure of how viscous forces compare to the Coriolis force. The relative significance of the centrifugal body force is defined in Eq. (2.75) by the centrifugal Froude number Frx , representing the ratio between convective inertial forces and the centrifugal body force.

24


The dimensionless coefficients appearing in Eqs. (2.71), (2.72) and (2.73) are merely combinations of these dimensionless groups. As such ReD defined in Eq. (2.74) is the Reynolds number related to a microscopic pore size length scale, times the between the microscopic and macroscopic length scales, i.e. pratio ffiffiffiffiffiffi pffiffiffiffiffiffi Ko lc . Equation (2.76) defines the porous media Ekman ReD ¼ uc Ko m pffiffiffiffiffiffi number related to the microscopic pore size length scale Ko . As the microscopic pffiffiffiffiffiffi length scale Ko is by definition much smaller than the macroscopic length scale lc it is evident that Da\\1 for applications of transport phenomena in porous media. Therefore, even for reasonable high values of the macroscopic Reynolds number Re, the value of the microscopic Reynolds number is still much smaller than 1, i.e. ReD \\1 (see Eq. (2.74)). This is certainly true for Re numbers of unit order of magnitude. This situation breaks down only for values of Re as large as Re ¼ OðDa1 Þ, or for values of the microscopic Reynolds number in the order of pffiffiffiffiffiffi magnitude of uc Ko m ¼ O Da1=2 . Therefore, as long as these conditions are satisfied ReD \\1 and the terms including ReD as their only coefficient can be neglected when compared to the coefficient of V, which is 1.Yet, the terms including ReD in their coefficient as a combination of dimensionless groups can not be neglected as for example ReD Frg is usually of a unit order of magnitude despite the fact that ReD \\1. This is possible when Frg \\1 causing the combined group ReD Frg ¼ Oð1Þ. From Eq. (2.72) one can observe that the Brinkman term ðDa=/Þr2 V is small and insignificant when Da\\1. As this is always the case, this term is significant only in thin layers adjacent to a rigid boundary (or to a porous medium adjacent to a pure fluid interface), i.e. for length scales in the order of magnitude O Da1=2 - dimensionless, or O Ko1=2 - dimensional. This obvious conclusion is consistent with Nield (1983, 1991). Some studies discuss the case of sparsely packed porous media when the porosity can get closer to / 1. Then, substantially higher permeabilities are anticipated and one may argue that then Da ¼ Oð1Þ. This scenario is not realistic as it violates the very basic assumption made when adopting the continuum approach for a porous medium, i.e. that the size of the REV is substantially smaller than the macroscopic length scale for the statistical averages to be meaningful. When the porosity gets closer to / 1 in order to account for sparsely packed porous media the latter condition is violated and all derived porous media models are then void. New and different models are then needed to be developed, tested and validated experimentally in order replace the ones based on the porous media approach. Of particular interest is the question related to when the inertial terms in Eq. (2.72) are significant and which one of the different inertial terms is more significant than the others. Although the centrifugal and Coriolis terms can be regarded as inertial terms they are not being considered as such for the purpose of the present analysis. There are therefore three inertial terms, i.e. the convective inertial term ðV rÞV, the Forchheimer term jV jV, and the local time derivative term @ V=@ t. It is clear that a necessary condition for all these terms to be neglected is that ReD \\1. When ReD ¼ Oð1Þ these terms are no insignificant anymore.

2.4 Non-Darcy Models

25

Nevertheless, even for ReD ¼ Oð1Þ, when comparing the convective inertial term to the Forchheimer term one obtains the ratio between these terms as the ratio between their coefficients. Assuming F ¼ Oð1Þ and that / ¼ Oð1Þ this ratio becomes ½ðV rÞV term=½jV jV term ¼ O Da1=2 \\1. Therefore, since Da\\1 for the continuum approach to be valid, the convective inertial term is much smaller than the Forchheimer term, irrespective of the value of ReD . If ReD \\1 both terms are insignificantly small compared to the Darcy term V (third term on the left hand side of Eq. (2.72)). If ReD ¼ Oð1Þ then the convective inertial term can be neglected when compared to the Forchheimer term because Da1=2 \\1. This conclusion is in agreement with Nield (1991) although the arguments used are different. When steady state solutions are sought, the time derivative terms can be dropped out from Eqs. (2.71), (2.72) and (2.73). However, for time dependent solutions a comparison between the time derivative terms in Eqs. (2.71), (2.72) and (2.73) is needed. To perform such a comparison the equation of state relating the density to pressure and/ or temperature is required. For incompressible fluids or compressible fluids flowing at low Mach numbers and/or subjected to moderate temperature variations, a linear relationship between density, pressure and temperature as presented in Eq. (2.33) is an adequate approximation. Since in most circumstances bp \\bT the pressure related term in Eq. (2.33) can be neglected, leading to the final form of the equation of state as Eq. (2.34). Furthermore, by using the Boussinesq approximation (Boussinesq 1903) the density may be assumed constant q ¼ 1 ðq ¼ qo Þ everywhere except when it appears in a body force term in the momentum Eq. (2.72). This can be easily verified to be true for moderate temperature variations, i.e. for bT \\1, by introducing Eq. (2.34) into (2.71), and (2.72) and performing an order of magnitude analysis. Therefore, assuming the Boussinesq approximation, bT \\1 and Da\\1 Eqs. (2.71), and (2.72) take the form rV ¼0

ð2:78Þ

K

ReD bT Mf ReD @ V F ReD ReD bT þ V ¼ K rpr 1=2 jV jV T r êg X þ T êx ðêx X Þ / @t Frg Frx Da

1 êx V EkD

ð2:79Þ Therefore the time derivatives appear in the momentum Eq. (2.79) and the energy Eq. (2.73) involving two time scales, their ratio being the ratio of their coefficients in Eqs. (2.79) and (2.73) evaluated in the following form (assuming K ¼ Oð1Þ, / ¼ Oð1Þ, and Mf ¼ Oð1Þ) ðmomentum time scaleÞ=ðenergy time scaleÞ ¼ ReD . This result shows that the flow transients decay much faster, O Re1 D , than the energy transients and can therefore be neglected as long as ReD \\1. However, when ReD ¼ Oð1Þ both time derivatives should be retained. Of particular interest is the set of equations that apply to gravity or centrifugally driven natural convection in homogeneous porous media. For natural convection the characteristic filtration

26


velocity is uc ¼ ae =lc leading to Eqs. (2.37), (2.38), and (2.40), i.e. Re ¼ 1=Pr, Pe ¼ 1, and Mf ReD / ¼ Mf Da Re / ¼ Da=/Pre ¼ 1=Va. In addition, using this choice of characteristic filtration velocity the following dimensionless groups in Eq. (2.79) change form ReD ¼ Da=Pr, the gravity related Rayleigh number ReD bT Frg ¼ Rag ¼ b DTc g Ko lc =m ae , the centrifugal Rayleigh number ReD bT =Frx ¼ Rax ¼ b DTc x2 l2c Ko m ae , and ReD Da1=2 ¼ Da1=2 Pr. For homogenous media K ¼ 1 and Eqs. (2.79) and (2.73) take the form

1 @V F Da1=2 1 êx V þ V ¼ rpr þ jV jV þ Rag T r êg X Rax T êx ðêx X Þ þ Va @ t EkD Pr

ð2:80Þ @T þ V rT ¼ r2 T þ Q_ @t

ð2:81Þ

From these equations it is evident that for very small values of the Prandtl number, i.e. Pre ¼ OðDaÞ such that Va ¼ Oð1Þ, which is typical only for liquid metals, the time derivative term in Eq. (2.80) should be retained. This will occur in applications linked to solidification of binary alloys where the mushy layer at the interface between the liquid metal and the solidified material is considered a porous medium. However, even when Da\\Pre such that Va [ [ 1 there might be instances when the time derivative in Eq. (2.80) cannot be neglected. In cases when instability sets in as oscillatory and the frequency of oscillations is very high, or when chaotic solutions are anticipated high frequency modes will make the time derivative term in Eq. (2.80) significant. This exhaustive discussion of non-Darcy models became necessary as some studies, particularly on gravity-driven convection, use non-Darcy models, which are not justified and in some cases reach conclusions, which can be misleading. Nield (1983) pointed out some examples of such occurrences.

2.5 Classification of Rotating Flows in Porous Media Rotating flows in porous media can be dealt with by classifying them first into three major categories (a) Isothermal flows in heterogeneous porous media subject to rotation. (b) Convective flows in non-isothermal homogeneous porous media subject to rotation. (c) Convective flows in non-isothermal heterogeneous porous media subject to rotation. This book is concerned with the first two categories. The third category is not covered, simply because there are no reported research results on natural convection

2.5 Classification of Rotating Flows in Porous Media

27

in rotating heterogeneous porous media. Regarding category (a) above, it is evident from Eq. (2.60) subject to isothermal conditions, i.e. Rag ¼ Rax ¼ 0, that heterogeneity of the medium is an important condition for the effects of rotation to be significant. This can be observed, for example, from the basic Darcy’s law, which under homogeneous conditions (i.e. K ¼ const: ¼ 1) yields irrotational types of flows, i.e. the vorticity r V ¼ 0. The heterogeneity of the medium, K KðXÞ, introduces a non-vanishing vorticity, i.e. r V 6¼ 0. Non-isothermal flows, as a result of natural convection, allow also a non-vanishing vorticity. Convective flows in rotating porous media are further classified into three categories and each one of them can be separated into three cases. In order to justify this classification we proceed first by defining the phenomenon of natural convection. Natural convection is the phenomenon of fluid flow driven by density variations in a fluid subject to body forces. Therefore, there are two necessary conditions to be met in order to obtain natural convective flow; (i) the existence of density variations within the fluid and (ii) the fluid must be subjected to body forces. As density is in general a function of pressure, temperature and solute concentration (in the case of a binary mixture), i.e., q qðp; T; SÞ and its variation with respect to pressure is much smaller than that with respect to temperature or concentration (i.e., bp \\bT , leading to Eq. (2.34) q ¼ 1 bT T for example). Hence the convection can be driven either by temperature variations or by variations in solute concentration or by both. In the two latter cases Eq. (2.34) should be extended to include S in the form of Eq. (2.35) q ¼ 1 bT T þ bS S, the solute transport equation should be included in the model and Eq. (2.60) should be extended as well to account for the solute effect on density. Gravity, centrifugal forces, electromagnetic forces (in the case of liquid metals subject to an electric field) are only examples of body forces which represent the second necessary condition for natural convection to occur. Some of these body forces are constant, such as gravity for example, while others can vary linearly with the perpendicular distance from the axis of rotation like the centrifugal force. The lack of body forces, for example under micro-gravity conditions in the outer space, prevents occurrence of convection. The two conditions mentioned above are indeed necessary for natural convection to occur; however they are not sufficient. The relative orientation of the density gradient with respect to the body force is an important factor for providing the sufficient conditions for convection to occur. This is shown graphically in Fig. 2.1 for the particular case of thermal convection, where B represents the body force and rq ¼ bT rT is the direction of the density gradient. Given this basic introduction on the causes for the setup of convection the following classification of convective flows is introduced: (i) Convection due to thermal buoyancy. (ii) Convection due to thermo-solutal buoyancy. A third category related to convection due to solutal buoyancy alone could have been introduced but it would not bring any significant contribution to (i) above as

28


(b)

(a)

T

B

Unconditional Convection

(c) T

B

Conditional Convection

T

B

No Convection

Fig. 2.1 The effect of the relative orientation of the temperature gradient with respect to the body force on the setup of convection

the results obtained for thermal buoyancy alone can be easily converted to the third case by analogy. Three separate cases for each one of the above categories can be considered, depending on the driving body force, i.e., convection driven by gravity, convection driven by the centrifugal force and convection driven by both gravity and centrifugal force. In each of these cases the effect of Coriolis acceleration on the natural convective flow is another rotation effect of interest in the present book.

References Auriault JL, Geindreau C, Royer P (2000) Filtration law in rotating porous media. CR Acad Sci Paris 328(Serie II b):779–784 Auriault JL, Geindreau C, Royer P (2002) Coriolis effects on filtration law in rotating porous media. Transp Porous Media 48:315–330 Boussinesq J (1903) Theorie Analitique de la Chaleur. Gutheir-Villars, Paris, vol 2, p 172 Govender S (2010) Vadasz number influence on vibration in a rotating porous layer placed far away from the axis of rotation. J Heat Transf 132:112601/1-112601/4 Hsu CT, Cheng P (1990) Thermal dispersion in a porous medium. Int J Heat Mass Transf 33:700– 706 Nield DA (1983) The boundary correction for the Rayleigh-Darcy problem: limitations of the Brinkman equation. J. Fluid Mech 128:37–46 Nield DA (1991) The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int J Heat Fluid Flow 12(3):269–272 Nield DA (1995) Discussion on “Analysis of heat transfer regulation and modification employing intermittently emplaced porous cavities. J Heat Transf 117:554–555 Palm E, Tyvand A (1984) Thermal convection in a rotating porous layer. J Appl Math Phys (ZAMP) 35:122–123 Sheu L-J (2008) An autonomous system for chaotic convection in a porous medium using a thermal non-equilibrium model. Chaos Solitons Fractals 30:672–689. Springer, ISBN: 0387765417 Straughan B (2001) A sharp nonlinear stability threshold in rotating porous convection. Proc R Soc Lond A 457:87–93 Straughan B (2008) Stability and wave motion in porous media. (Applied Mathematical Sciences Series). Springer, ISBN: 0387765417

References

29

Vadasz P (1997) Flow in rotating porous media. In: du Plessis P, Rahman M (eds) Fluid transport in porous media from the series Advances in fluid mechanics, vol 13, pp 161–214. Computational Mechanics Publications, Southampton Vadasz P (2000) Fluid flow and thermal convection in rotating porous media. In: Vadfai Kambiz (ed) Handbook of porous media. Marcel Dekker, New York, pp 395–439 Vafai K, Kim SJ (1990) Analysis of surface enhancement by a porous substrate. J Heat Transf 112:700–706

Chapter 3

Isothermal Flow in Heterogeneous Porous Media

ABSTRACT Isothermal flow in roating porous media is shown to apply for conditions when the permeability of the porous medium is not constant, hence representing heterogeneous porous media. The demonstration of the possible existence of Taylor-Proudman columns in rotating porous media follows. Examples of secondary flows in a cross section perpendicular to the main flow direction are presented for a heterogeneous porous medium in a rotating square channel. Keywords Isothermal flow Proudman columns

Heterogeneous rotating porous media

Taylor-

3.1 General Background Considering Darcy’s regime under isothermal conditions (Rag ¼ Rax ¼ 0) yields the following equation from (2.63)

1 þ KEk 1êz V ¼ Krp

ð3:1Þ

V ¼ E rp

ð3:2Þ

or

where B ¼ 0 in Eq. (2.69) and E is the tensor operator defined by Eq. (2.70).

Further decoupling between the pressure and the filtration velocity is possible by applying the divergence operator (r) on Eq. (3.2) and making use of Eq. (2.5), leading to r2 p þ Ek 2 K 2

@2p @K @p þ c2 rH K rH p þ Ek 1 c3 JH ðK; pÞ ¼ 0 ð3:3Þ þ c1 2 @z @z @z


31

32

3 Isothermal Flow in Heterogeneous Porous Media

where c1 ; c2 and c3 are the coefficients c3 ¼

2 2 c3 ½1 Ek 2 K 2 ; c ; c ¼ ¼ 1 2 c3 K 2 ½1 þ Ek 2 K 2

ð3:4Þ

and JH ðK; pÞ is the horizontal Jacobian defined as JH ðK; pÞ ¼

@K @p @K @p @x @y @y @x

ð3:5Þ

Let us consider now a few cases of interest to demonstrate the application of the theory to some particular examples. When the heterogeneity can be represented by a vertical stratification K KðzÞ and therefore it is independent of the horizontal coordinates x and y, the horizontal Jacobian and the horizontal permeability gradient vanish, i.e., JH ¼ 0 and rH K ¼ 0, which yields r2 p þ Ek 2 K 2

@2p @K @p ¼0 þ c1 2 @z @z @z

ð3:6Þ

For a homogeneous medium K ¼ 1, and the equation for the pressure takes the form r2 p þ Ek 2 @ 2 p @z2 ¼ 0. Compressing the vertical coordinate by using the 1=2 produces for this case a simplified Laplace transformation f ¼ z ð1 þ Ek 2 Þ 2 equation in the following form @ p @ x2 þ @ 2 p @ y2 þ @ 2 p @ f2 ¼ 0. One can notice that except for the compression of the vertical coordinate the rotation does not have any significant effect in homogeneous media.

3.2 Taylor-Proudman Columns and Geostrophic Flow in Rotating Porous Media To present the topic of Taylor-Proudman columns we consider Eq. (3.1) multiplied by ½Ek=K for êx ¼ êz and rescale the pressure in the form pR ¼ Ek p to yield

Ek þ êz V ¼ rpR K

ð3:7Þ

Given typical values of viscosity, porosity and permeability one can evaluate the range of variation of Ekman number in some engineering applications. There, the angular velocity may vary from 10 rpm to 10,000 rpm leading to Ekman numbers in the range from Ek ¼ 1 to Ek ¼ 103 . The latter value is very small, pertaining to the conditions considered in this section. Therefore, in the limit of Ek ! 0, say Ek ¼ 0, Eq. (3.7) takes the simplified form

3.2 Taylor-Proudman Columns and Geostrophic Flow in Rotating Porous Media

êz V ¼ rpR

33

ð3:8Þ

and the effect of permeability variations disappears. Taking the “curl” of Eq. (3.8) leads to r ðêz V Þ ¼ 0

ð3:9Þ

Evaluating the “curl” operator on the cross product of the left-hand side of Eq. (3.9) gives ðêz rÞV ¼ 0

ð3:10Þ

Equation (3.10) is identical to the Taylor-Proudman condition for pure fluids (nonporous domains); it thus represents the proof of the Taylor-Proudman theorem in porous media and can be presented in the following simplified form @V ¼0 @z

ð3:11Þ

The conclusion expressed by Eq. (3.11) is that V ¼ V ðx; yÞ, i.e. it cannot be a function of z, where z is the direction of the angular velocity vector. This means that all filtration velocity components can vary only in the plane perpendicular to the angular velocity vector. The consequence of this result can be demonstrated by considering a particular example that was presented by Vadasz (1994) (see Greenspan (1980) for the corresponding example in pure fluids). Figure 3.1 shows a closed cylindrical container filled with a fluid saturated porous medium. The topography of the bottom surface of the container is slightly changed by fixing securely a small solid object. The container rotates with a fixed angular velocity x . Any forced horizontal flow in the container is expected to adjust to its bottom topography. However, since Eq. (3.11) applies for each component of V it applies in particular to w, i.e. @ w=@ z ¼ 0. But the impermeability conditions at the top and bottom solid boundaries require V ên ¼ 0 at z ¼ hðr; hÞ and at z ¼ 1, where hðr; hÞ represents the bottom topography in polar coordinates ðr; hÞ. The combination of these boundary conditions with the requirement that, @ w=@ z ¼ 0 yields w ¼ 0 anywhere in the field. Hence, a flow over the object as described qualitatively in Fig. 3.2a becomes impossible as it introduces a vertical component of velocity. Therefore, the resulting flow may adjust around the object as presented qualitatively in Fig. 3.2b. However, since this flow pattern is also independent of z, it extends over the whole height of the container resulting in a fluid column above the object, which rotates as a solid body. This will demonstrate a Taylor-Proudman column in porous media, as presented qualitatively in Fig. 3.1. It should be pointed out that the column in porous media should be expected in the macroscopic sense, i.e., the diameter of the column (and naturally the diameter of the obstacle) must

34


z *

r Fig. 3.1 A closed cylindrical container filled with a fluid saturated porous medium. A solid object fixed at the bottom and a qualitative description of a Taylor-Proudman column in porous media

z

(b)

y

r

(a)

x

Impossible type of flow

Fig. 3.2 a An impossible type of flow over the object. b The flow adjusts around the object (as seen from above) and extends at all heights creating a column above the object, which behaves like a solid body

have macroscopic dimensions which are much greater than the microscopic characteristic size of the solid phase, or any other equivalent pore scale size. Experimental confirmation of these theoretical results is necessary. However, because of the inherent difficulty of visualizing the flow, i.e., the Taylor-Proudman column through a granular porous matrix, an indirect method was adopted. The rationale behind the proposed method was to place a porous layer in-between two pure fluid layers inside a cylinder (see Fig. 3.3a). A disturbance in the form of a small fixed obstacle was created on the bottom wall in the fluid layer located below the porous layer causing the appearance of a Taylor-Proudman column in this layer. As the fundamental property of a TaylorProudman column is that it does not allow for variations of velocity in the vertical direction, the proposed experimental setup should allow us to detect TaylorProudman columns in the top undisturbed fluid layer located above the porous layer, if the column exists in the porous layer as a result of the disturbance in the bottom fluid layer. This means that a small object located at the bottom of the lower

3.2 Taylor-Proudman Columns and Geostrophic Flow in Rotating Porous Media

35

Fig. 3.3 a A closed cylindrical container divided into a porous layer and two fluid layers above and below the porous layer. A solid object is fixed at the bottom surface of the container. b Qualitative description of the anticipated Taylor-Proudman column as observed in the preliminary runs of the experiment

fluid layer should create a Taylor-Proudman column, which extends upwards through the porous layer into the upper fluid layer. It is because of this expectation that the top fluid layer will “feel” through the buffer porous layer the small object located at the bottom of the lower fluid layer, that some colleagues of the author named the effect resulting from the experiment “The Princess and the Pea”. Of course another possibility exists which is consistent with Eq. (3.11). This could for example imply that the filtration velocity is zero in a rotating porous layer, meaning that the whole fluid in a rotating porous layer rotates as a solid body instantaneously (almost). The experimental apparatus consists of a record player turntable adjusted to allow variable angular velocity in the designed range and to provide a better dynamic balance when the cylindrical container is placed securely on the rotating plate. Preliminary results confirm the appearance of Taylor-Proudman columns in the top fluid layer as shown qualitatively in Figs. 3.3b and 3.4. A further significant consequence of Eq. (3.11) is represented by a geostrophic type of flow. Taking the z-component of Eq. (3.11) yields @ w=@ z ¼ 0, and the continuity Eq. (2.5) becomes two dimensional @u @v þ ¼0 @x @y

ð3:12Þ

Therefore the flow at high rotation rates has a tendency towards two-dimensionality and a stream function, w, can be introduced for the flow in the x y plane u¼

@w @w ; v¼ @y @x

ð3:13Þ

36


Fig. 3.4 A Taylor-Proudman column as observed in the top fluid layer in the preliminary runs of the experiment

which satisfies identically the continuity Eq. (3.12). Substituting u and v with their stream function representation given by Eq. (3.13) into Eq. (3.7) yields @ w @ pR ¼ @x @x

ð3:14Þ

@ w @ pR ¼ @y @y

ð3:15Þ

As both the pressure and the stream function can be related to an arbitrary reference value, the conclusion from Eqs. (3.14) and (3.15) is that the stream function and the pressure are the same in the limit of high rotation rates (Ek ! 0). This type of geostrophic flow means that isobars represent streamlines at the leading order, for Ek ! 0.

3.3 Flow Through Heterogeneous Porous Media in a Rotating Square Channel In the previous section the case of high rotation rates (i.e.,Ek 1) was considered. Let us consider now the case when small rotation rates are of interest, i.e., Ek 1. This is particularly useful in many practical applications. The problem of an axial flow through a long rotating square channel filled with fluid saturated porous material is considered. The axial flow is imposed through an axial pressure gradient while the channel rotates about an axis perpendicular to the horizontal walls (Fig. 3.5). With a homogeneous porous medium the permeability is constant throughout the flow domain resulting in a uniform distribution of the filtration velocity and the effect of rotation does not affect the flow. However, for heterogeneous porous

3.3 Flow Through Heterogeneous Porous Media in a Rotating Square Channel

37

z *

x H* y H* Fig. 3.5 A heterogeneous fluid saturated porous medium in a rotating square channel subject to axial flow in the x direction imposed through an axial pressure gradient in the x direction

media, the permeability is spatially dependent, thus leading to secondary circulation in a plane perpendicular to the imposed fluid motion. The particular case where the permeability varies only along the vertical coordinate is considered. This assumption is compatible with the assumption of developed flow in pure fluids (non-porous domains). As Ek 1, an expansion of the dependent variables is used in the form V ¼ V 0 þ Ek 1 V 1 þ O Ek 2

ð3:16Þ

p ¼ p0 þ Ek 1 p1 þ O Ek 2

ð3:17Þ

The substitution of these expansions in Eqs. (3.1) and (2.5) yields a hierarchy of partial differential equations at different orders. At the leading order a basic flow solution is found in the form u0 ¼ K ðzÞ; v0 ¼ w0 ¼ 0

ð3:18Þ

To the first order a two-dimensional continuity equation for the y and z components of V 1 in the y z plane allows for the introduction of a stream function defined as v1 ¼ @ w1 =@ z, w1 ¼ @ w1 =@ y. Hence, the equation governing the flow in the y z plane takes the form @ 2 w1 @ 2 w1 dðln K Þ @ w1 dK ¼ K þ dz @ z dz @ y2 @ z2

ð3:19Þ

which has to be solved subject to the impermeability boundary conditions at the walls w1 ¼ 0 at y ¼ 0; 1 and at z ¼ 0; 1. An analytical solution to Eq. (3.19) subject to the rigid walls boundary conditions was presented by Vadasz (1993) for a particular monotonic form of the permeability function, i.e. K ¼ ecz . The solution is a single vortex rotating counterclockwise in the y z plane as presented in Fig. 3.6a. The forcing term ðKdK=dzÞ

38


(a)

(b)

(c)

z

z

z

y

y

y

Fig. 3.6 The flow field in the y–z plane of a rotating heterogeneous porous channel; (a) results of analytical solution for a permeability increasing upwards, (b) qualitative results for a permeability decreasing upwards, (c) qualitative results for a permeability decreasing in the bottom half and increasing in the top half of the domain

in Eq. (3.19) is responsible for the single vortex and the resulting flow direction. A monotonic function K ðzÞ keeps the sign of the forcing term unchanged for all values of z. Therefore, the single vortex turns counter-clockwise when K increases in the z-direction and clockwise when K decreases, as presented qualitatively in Fig. 3.5b. When K ðzÞ is not monotonic, e.g., when it decreases in the first half of the domain (z 2 ½0; 0:5) and increases in the second half (z 2 ½0:5; 1), the sign of the forcing term in Eq. (3.19) changes accordingly leading to two vortices as presented qualitatively in Fig. 3.5c. This secondary circulation in a plane perpendicular to the main flow direction is a result of the Coriolis effect on the main forced flow. Extending the problem to a case which applies to all values of Ekman number requires the introduction of a scaling for V and K in the form V ¼ V=Ek and K ¼ K=Ek. Then, subject to the same assumptions of developed flow, implying also K K ðzÞ, the following equation is obtained from Eq. (2.5) 2 i @ 2 w h 2 i d ln K @ w @2w h dK ¼ K þ 1þ K þ K 1 @ y2 @ z2 dz dz @ z @w u¼K 1þ @z

ð3:20Þ ð3:21Þ

where the stream function was redefined in terms of v and w in the form v ¼ @w @z, w ¼ @w @y. Monotonic, non-monotonic antisymmetric, as well as non-monotonic but symmetric variations of permeability were investigated by Vadasz and Havstad (1999) by using a finite-differences numerical method (Havstad and Vadasz 1999) to solve the three-dimensional fluid flow problem formulated by Eqs. (3.20) and (3.21). The authors established the accuracy of the method by using a co-ordinate transform which allows mesh refinement to be tailored to the boundary layer position and thickness for a wide range of Ekman

3.3 Flow Through Heterogeneous Porous Media in a Rotating Square Channel

39

number values. The results, corresponding to the three different classes of permeability functions for Ek ¼ 1, provide a confirmation of the analytical solutions regarding the nature of the secondary flows. In addition, the significant back-impact of this secondary flow on the basic axial flow via an axial flow deficiency was identified as well.

References Greenspan HP (1980) The theory of rotating fluids. Cambridge University Press, Cambridge, pp 5–18 Havstad MA, Vadasz P (1999) Numerical solution of the three dimensional fluid flow in a rotating heterogeneous porous channel. Int J Numer Methods Fluids 31(2):411–429 Vadasz P (1993) Fluid flow through heterogeneous porous media in a rotating square channel. Transp Porous Media 12:43–54 Vadasz P (1994) On Taylor-Proudman columns and geostrophic flow in rotating porous media. SAIMechE R&D J 10(3):53–57 Vadasz P, Havstad MA (1999) The effect of permeability variations on the flow in a rotating porous channel. ASME J Fluids Eng 121(3):568–573

Chapter 4

Natural Convection Due to Thermal Buoyancy of Centrifugal Body Forces

Abstract Natural convection in rotating porous media due to thermal buoyancy created by the centrifugal body force is presented. The distinction between the cases where the temperature gradients are aligned with the direction of the centrifugal body force or perpendicular to the latter are discussed separately. In the second case stability conditions have to be established, i.e. the convection does not occur unconditionally. A spectral system is then used to analyze the nonlinear effects leading to a system of ordinary differential equations for the amplitudes of convection. They predict a transition to chaotic solutions (weak turbulence) at certain values of the parameters. Keywords Porous media natural convection weak trubulence

Centrifugal buoyancy Chaos and

4.1 General Background Considering Darcy’s regime for a homogeneous porous medium (K ¼ 1) subject to a centrifugal body force and neglecting gravity (Rag Rax 1) Eqs. (2.59), (2.60) and (2.62) with K ¼ 1 and Rag ¼ 0 represent the mathematical model for this case. The objective in the first instance is to establish the convective flow under small rotation rates, then Ek 1, and as a first approximation the Coriolis effect can be neglected, i.e., Ek ! 1. Following these conditions the governing equations when heat generation is absent (i.e., Q_ ¼ 0) become (using identity (47) to revert back) rV ¼0

ð4:1Þ

V ¼ ½rp Rax T êx ðêx X Þ

ð4:2Þ

@T þ V rT ¼ r2 T @t

ð4:3Þ


41

4 Natural Convection Due to Thermal Buoyancy …

42

The three cases corresponding to the relative orientation of the temperature gradient with respect to the centrifugal body force as presented in Fig. 2.1 are considered. Case 1(a) in Fig. 2.1 corresponds to a temperature gradient, which is perpendicular to the direction of the centrifugal body force and leads to unconditional convection. The solution representing this convection pattern is the objective of the investigation. Cases 1(b) and 1(c) in Fig. 2.1 corresponding to temperature gradients collinear with the centrifugal body force represent a stability problem. The objective is then to establish the stability condition as well as the convection pattern when this stability condition is not satisfied.

4.2 Temperature Gradients Perpendicular to the Centrifugal Body Force An example of a case when the imposed temperature is perpendicular to the centrifugal body force is a rectangular porous domain rotating about the vertical axis, heated from above and cooled from below. For this case the centrifugal buoyancy term in Eq. (4.2) becomes Rax T x êx leading to V ¼ rp Rax T x êx

ð4:4Þ

An analytical two-dimensional solution to this problem (see Fig. 4.1) for a small aspect ratio of the domain was presented by Vadasz (1992). The solution to the non-linear set of partial differential equations was obtained through an asymptotic expansion of the dependent variables in terms of a small parameter representing the aspect ratio of the domain. The convection in the core region far from the sidewalls was the objective of the investigation. To first order accuracy, the heat transfer coefficient represented by the Nusselt number was evaluated in the form 2

* x*

z *

TH*

z=1 T =0 x

T =0 x

z=0

TC * x= 0

L*

H*

x x =L

Fig. 4.1 A rotating rectangular porous domain heated from above, cooled from below, and insulated on its sidewalls (courtesy Elsevier Science Ltd (Vadasz 1994a))

4.2 Temperature Gradients Perpendicular to the Centrifugal Body Force

2 Rax þO H Nu ¼ 1 þ 24

43

ð4:5Þ

where Nu is the Nusselt number and the length scale used in the definition of Rax , Eq. (2.54), was lc ¼ H . A different approach was used by Vadasz (1994a) to solve a similar problem without the restriction of a small aspect ratio. A direct extraction and substitution of the dependent variables was found to be useful for de-coupling the non-linear partial differential equations, resulting in a set of independent non-linear ordinary differential equations, which was solved analytically. To obtain an analytical solution to the non-linear convection problem we assume that the vertical component of the filtration velocity, w, and the temperature T are independent of x, i.e., @ w=@ x ¼ @ T=@ x ¼ 0 8 x 2 ð0; LÞ, being functions of z only. It is this assumption that will subsequently restrict the validity domain of the results to moderate values of Rax (practically Rax \5). Subject to the assumptions of two-dimensional flow v ¼ 0 and @ðÞ=@ y ¼ 0 and that w and T are independent of x the governing equations take the form @ u dw þ ¼0 @ x dz u¼

ð4:6Þ

@p Rax x T @x

ð4:7Þ

@p @z

ð4:8Þ

w¼

d2 T dT ¼0 w d z2 dz

ð4:9Þ

The method of solution consists of extracting T from Eq. (4.7) and expressing it explicitly in terms of u ; @ p=@ x and x. This expression of T is then introduced into Eq. (4.9) and the derivative @=@ x is applied to the result. Then, substituting the continuity Eq. (4.6) in the form @ u=@ x ¼ d w=d z and Eq. (4.8) into the results yields a non-linear ordinary differential equation for w in the form d3 w d2 w w ¼0 d z3 d z2

ð4:10Þ

An interesting observation regarding Eq. (4.10) is the fact that it is identical to the Blasius equation for boundary layer flows of pure fluids (non-porous domains) over a flat plate. To observe this, one simply has to substitute wðzÞ ¼ f ðzÞ=2 to obtain 2f 000 þ ff 00 ¼ 0, which is the Blasius equation. Unfortunately, no further analogy to the boundary layer flow in pure fluids exists, mainly because of the quite different boundary conditions and because the derivatives [dðÞ=d z] and the flow


44

(w) are in the same direction. The solutions for the temperature T and the horizontal component of the filtration velocity u, are related to the solution of the ordinary differential equation u0 u000 u002 þ uu02 ¼ 0

ð4:11Þ

where ðÞ0 stands for dðÞ=d z and u ¼ x uðzÞ T ðzÞ ¼

ð4:12Þ

1 ½P þ uðzÞ Rax

ð4:13Þ

where P is a constant defined by Z1 P ¼ Rax

TðzÞdz

ð4:14Þ

0

The relationship (4.14) is a result of imposing a condition of no net flow through R1 any vertical cross-section in the domain, stating that 0 udz ¼ 0. The following boundary conditions are required to the solution of (4.10) for w: w ¼ 0 at z ¼ 0 and z ¼ 1 representing the impermeability condition at the solid boundaries and T ¼ 0 at z ¼ 0 and T ¼ 1 at z ¼ 1. Since @ u=@ x ¼ u according to Eq. (4.12), then following the continuity Eq. (4.6) u ¼ dw=dz and the temperature boundary conditions can be converted into conditions in terms of w by using Eq. (4.13), leading to the following complete set of boundary conditions for w: dw ¼P dz

ð4:15Þ

dw ¼ P þ Rax dz

ð4:16Þ

z ¼ 0 : w ¼ 0 and z ¼ 1 : w ¼ 0 and

Equations (4.15) and (4.16) represent four boundary conditions, while only three are necessary to solve the third order Eq. (4.10). The reason for the fourth condition comes from the introduction of the constant P, whose value remains to be determined. Hence, the additional two boundary conditions are expressed in terms of the unknown constant P and the solution subject to these four conditions will determine the value of P as well. A method similar to Blasius’s method of solution was applied to solve Eq. (4.10). Therefore, wðzÞ was expressed as a finite power series and the objective of the solution was to determine the power series coefficients. Once the solution for wðzÞ and P was obtained, u and T were evaluated by using u ðzÞ ¼ dw=dz and Eqs. (4.12) and (4.13).

4.2 Temperature Gradients Perpendicular to the Centrifugal Body Force

45

Then, for presentation purposes a stream function w was introduced to plot the results (u ¼ @ w=@ z, w ¼ @ w=@ x). An example of the flow field represented by the streamlines is presented in Fig. 4.2 for Rax ¼ 4 and for an aspect ratio of 3 (excluding a narrow region next to the sidewall at x ¼ L). Outside this narrow region next to x ¼ L the streamlines remain open on the right hand side. They are expected to close in the end region. Nevertheless, the streamlines close on the left-hand side, throughout the domain. The reason for this is the centrifugal acceleration, which causes u to vary linearly with x, thus creating (due to the continuity equation) a non-vanishing vertical component of the filtration velocity w at all values of x. The local Nusselt number Nu, representing the local vertical heat flux was evaluated as well by using the definition Nu ¼ j@ T=@ zjz¼0 and using the solution for T. A comparison between the heat flux results obtained from this solution and the results obtained by Vadasz (1992) using an asymptotic method was presented graphically by Vadasz (1994a). The two results compare well as long as Rax is very small. However, for increasing values of Rax the deviation from the linear relationship pertaining to the first order asymptotic solution (Nu ¼ 1 þ Rax =24, according to Vadasz 1992) was evident.

4.3 Temperature Gradients Collinear with the Centrifugal Body Force The problem of stability of free convection in a rotating porous layer when the temperature gradient is collinear with the centrifugal body force was treated by Vadasz (1994b, 1996a) for a narrow layer adjacent to the axis of rotation (Vadasz 1994b) and distant from the axis of rotation (Vadasz 1996a), respectively. The problem formulation corresponding to the latter case is presented in Fig. 4.3. In order to include explicitly the dimensionless offset distance from the axis of rotation x0 , and to keep the coordinate system linked to the porous layer, Eq. (4.4) was presented in the form V ¼ rp ½Raxo þ Rax xT êx

ð4:17Þ

=0 = 0.518 = 1.036 =0 = 1.295 = 0.777 = 0.259 =0

Fig. 4.2 Graphical description of the resulting flow field; five streamlines equally spaced between their minimum value wmin ¼ 0 at the rigid boundaries and their maximum value wmax ¼ 1:554. The values in the figure correspond to every other streamline (courtesy Elsevier Science Ltd (Vadasz 1994a))


46

z'

z'

z

2 * x'*

*

z

z=H

2

x' *

T=1

T=0

*

y'

x' x0

x

x 0

y

x'

W

x

Fig. 4.3 A rotating fluid saturated porous layer distant from the axis of rotation and subject to different temperatures at the sidewalls (Vadasz 1996a)

Two centrifugal Rayleigh numbers appear in Eq. (4.17); the first one, Raxo ¼ bT DTc x2 x0 L Ko ae m , represents the contribution of the offset distance from the rotation axis to the centrifugal buoyancy, while the second, Rax ¼ bT DTc x2 L2 Ko ae m , represents the contribution of the horizontal location within the porous layer to the centrifugal buoyancy. The ratio between the two centrifugal Rayleigh numbers is g¼

Rax 1 ¼ Raxo x0

ð4:18Þ

and can be introduced as a parameter in the equations transforming Eq. (4.17) in the form V ¼ rp Raxo ½1 þ gxT êx

ð4:19Þ

From Eq. (4.19) it is observed that when the porous layer is far away from the axis of rotation then g 1 (x0 1) and the contribution of the term g x is not significant, while for a layer close enough to the rotation axis g 1 (x0 1) and the contribution of the first term becomes insignificant. In the first case the only controlling parameter is Raxo while in the latter case the only controlling parameter is Rax ¼ gRaxo . The flow boundary conditions are V ên ¼ 0 on the boundaries, where ên is a unit vector normal to the boundary. These conditions stipulate that all boundaries are rigid and therefore non-permeable to fluid flow. The thermal boundary conditions are: T ¼ 0 at x ¼ 0, T ¼ 1 at x ¼ 1 and rT ên ¼ 0 on all other walls representing the insulation condition on these walls.

4.3 Temperature Gradients Collinear with the Centrifugal Body Force

47

The governing equations accept a basic motionless conduction solution in the form ½V b ; Tb ; pb ¼ 0; x; Raxo ðx2 2 þ gx3 3Þ þ C

ð4:20Þ

The objective of the investigation was to establish the condition when the motionless solution (4.20) is not stable and consequently a resulting convection pattern appears. Therefore a linear stability analysis was employed, representing the solution as a sum of the basic solution (4.20) and small perturbations in the form ½V; T; p ¼ ½V b þ V 0 ; Tb þ T 0 ; pb þ p0

ð4:21Þ

where ðÞ0 stands for perturbed values. Solving the resulting linearized system for the perturbations by assuming a normal modes expansion in the yand z directions, and hðxÞ in the x direction, i.e., T 0 ¼ Aj hðxÞexp rt þ i jy y þ jz z , and using the Galerkin method to solve for hðxÞ one obtains at marginal stability, i.e., for r ¼ 0, a homogeneous set of linear algebraic equations. This homogeneous linear system accepts a non-zero solution only for particular values of Raxo such that its determinant vanishes. The solution of this system was evaluated up to order 7 for different values of g, representing the offset distance from the axis of rotation. However, useful information was obtained by considering the approximation at order 2. At this order the system reduces to two equations which lead to the characteristic values of Raxo in the form

R0;c

h i b ð1 þ aÞ2 þð4 þ aÞ2 ¼ 2a b2 c2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2 b2 ð1 þ aÞ2 þð4 þ aÞ2 4 b2 c2 ð1 þ aÞ2 ð4 þ aÞ2 2a b2 c2

ð4:22Þ

where the following notation was used Ro ¼

Raxo Rax j2 g 2 256g2 ; c ¼ ; R ¼ ; a ¼ ; b ¼ 1 þ 2 p2 p2 p2 81p4

ð4:23Þ

and j is the wavenumber such that j2 ¼ j2y þ j2z while the subscript c in Eq. (4.22) represents characteristic (neutral) values (values for which r ¼ 0). A singularity in the solution for Ro;c , corresponding to the existence of a single root for Ro;c , appears when b2 ¼ c2 . This singularity persists at higher orders as well. Resolving for the value of g when this singularity occurs shows that it corresponds to negative g values implying that the location of the rotation axis falls within the boundaries of the porous domain (or to the right side of the hot wall - a case of little interest due to its inherent unconditional stability). This particular case will be discussed later in


48

(a)

(b)

4

8

8

R o,cr

5

2

4 3

1

-1

0

log 10 η

1

2

Ra ω /π 2

6 Rcr

0 -2

6

R cr

R o,cr

3

Raω ,cr Raω 0,cr + =1 7.81π 2 4π 2

7

7

5

Unstable

4 3

2

2

1

1

0

0

Stable

0

0.5

1

1.5

2

2.5

Raωο /π

3

3.5

4

2

Fig. 4.4 a The variation of the critical values of the centrifugal Rayleigh numbers as a function of g; b The stability map on the Rax Raxo plane showing the division of the plane into stable and unstable zones (Vadasz 1996a)

this section. The critical values of the centrifugal Rayleigh number as obtained from the solution up to order 7 are presented graphically in Fig. 4.4a in terms of both Ro;cr and Rcr as a function of the offset parameter g. The results presented in Fig. 4.4 are particularly useful in order to indicate the stability criterion for all positive values of g. It can be observed from the figure that as the value of g becomes small, i.e. for a porous layer far away from the axis of rotation, the critical centrifugal Rayleigh number approaches a limit value of 4p2 . This corresponds to the critical Rayleigh number in a porous layer subject to gravity and heated from below. For high values of g it is appropriate to use the other centrifugal Rayleigh number R, instead of Ro , by introducing the relationship R ¼ gRo (see Eqs. (4.18) and (4.23)) in order to establish and present the stability criterion. It is observed from Fig. 4.4a that as the value of g becomes large, i.e. for a porous layer close to the axis of rotation, the critical centrifugal Rayleigh number approaches a limit value of 7:81p2 . This corresponds to the critical Rayleigh number for the problem of a rotating layer adjacent to the axis of rotation as presented by Vadasz (1994b). The stability map on the Rax Raxo plane is presented in Fig. 4.4b, showing that the plane is divided between the 2 stable and unstable zones by the straight line 2 Rax;cr 7:81p þ Raxo;cr 4p ¼ 1. The results for the convective flow field are presented graphically in Fig. 4.5 following Vadasz, (1996a), where it was concluded that the effect of the variation of the centrifugal acceleration within the porous layer is definitely felt when the box is close to the axis of rotation, corresponding to an eccentric shift of the convection cells towards the sidewall at x ¼ 1. However, when the layer is located far away from the axis of rotation (e.g. x0 ¼ 50) the convection cells are concentric and symmetric with respect to x ¼ 1=2 as expected for a porous layer subject to gravity and heated from below (here “below” means the location where x ¼ 1).


49

Fig. 4.5 The convective flow field at marginal stability for three different values of x0 ; 10 stream lines equally divided between wmin and wmax . At x0 ¼ 1010 : wmin ¼ 1:378; wmax ¼ 1:378, at x0 ¼ 0:02 : wmin ¼ 1:374; wmax ¼ 1:374 and at x0 ¼ 50 : wmin ¼ 1:319; wmax ¼ 1:319 (Vadasz 1996a)

Although the linear stability analysis is sufficient for obtaining the stability condition of the motionless solution and the corresponding eigenfunctions describing qualitatively the convective flow, it cannot provide information regarding the values of the convection amplitudes, nor regarding the average rate of heat transfer. To obtain this additional information, Vadasz and Olek (1998) analyzed and provided a solution to the non-linear equations by using Adomian’s decomposition method to solve a system of ordinary differential equations for the evolution of the amplitudes. This system of equations was obtained by using the first three relevant Galerkin modes for the stream function and the temperature in the form w ¼ 2h

pz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cðR 1Þ X ðtÞ sinðpxÞ sin H

ð4:24Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pz ðR 1Þ 2cðR 1Þ Z ðtÞ sinð2pxÞ ð4:25Þ Y ðtÞ sinðpxÞ cos þ T ¼xþ H pR pR where c ¼ H 2 ðH 2 þ 1Þ, h ¼ ðH 2 þ 1Þ H, H being the layer’s aspect ratio, 2 2 R ¼ n p h , n ¼ Raxo þ Rax =2, and X; Y; Z the possibly time dependent ampitudes of convection. In this model it was considered of interest including the time derivative term in Darcy’s equation in the form ð1=VaÞ @ V=@ t, where Va ¼ / Pre =Da, and Da; Pre are the Darcy and the effective Prandtl numbers, 2

50


respectively, defined as Da ¼ Ko L2 and Pre ¼ m =~ae (see Eq. (2.58) with Rag ¼ 0 and Ek ! 1). The reason for including the time derivative term in the Darcy equation was the fact that one anticipates oscillatory and possibly chaotic solutions that some might have very high frequencies. Then, the following equations were obtained for the time evolution of the amplitudes X; Y; Z X_ ¼ a ðY X Þ

ð4:26Þ

Y_ ¼ R X Y ðR 1Þ X Z

ð4:27Þ

Z_ ¼ 4c ðX Y Z Þ

ð4:27Þ

where a ¼ cVa p2 , and R is a rescaled Rayleigh number introduced above following Eq. (4.25). The results obtained are presented in Fig. 4.6 in the form of projection of trajectories data points onto the Y X and Z X planes. Different transitions as the value of R varies are presented and they relate to the convective fixed point which is a stable simple node in Fig. 4.6a, a stable spiral in Fig. 4.6b, c, and loses stability via an inverse Hopf bifurcation in Fig. 4.6d, where the trajectory describes a limit cycle, moving towards a chaotic solution presented in Fig. 4.6e, f. A further transition from chaos to a periodic solution was obtained at a value of R slightly above 100, which persists over a wide range of R values. This periodic solution is presented in Fig. 4.6g, h for R ¼ 250. Previously in this section (see Eq. (4.22)) a singularity in the solution was identified and associated with negative values of the offset distance from the axis of rotation. It is this resulting singularity and its consequences, which were investigated by Vadasz (1996b) and are the objective of the following presentation. As this occurs at negative values of the offset distance from the axis of rotation it implies that the location of the rotation axis falls within the boundaries of the porous domain, as presented in Fig. 4.7. This particular axis location causes positive values of the centrifugal acceleration on the right side of the rotation axis and negative values on its left side. The rotation axis location implies that the value of x0 is not positive. It is therefore convenient to explicitly introduce this fact in the problem formulation specifying explicitly that x0 ¼ jx0 j. As a result Eq. (4.19) can be expressed in the form V ¼ rp Rax ½x jx0 jT êx

ð4:28Þ

The solution for this case is similar to the previous case leading to the same characteristic equation for Rc at order 2 as obtained previously in Eq. (4.22) for Ro;c , with the only difference appearing in the different definition of b and c as follows

1 256 j x0 j ; c2 ¼ b¼ 2 81p4

ð4:29Þ


(a)

51

(b) Y versus X

Y versus X

R= 1.1

1.1

R= 10

1.15 1.1

1.08

1.05

1.06

Y

Y

1 0.95

1.04

0.9

1.02 0.85

1

1

1.02

1.04

1.06

1.08

0.8 0.85

1.1

0.9

0.95

1

X

1.05

(d)

(c) Y versus X

Y

R= 23

1.15

1.15

versus

R=24.32

X

1.3 1.2

1.05

1.1

Y

1.1

Y

1.1

X

1

1 0.9

0.95

0.8

0.9 0.85 0.92 0.94 0.96 0.98

1

0.7 0.8 0.85 0.9 0.95

1.02 1.04 1.06 1.08

1

1.05 1.1

1.15 1.2

X

X

(e)

(f) Y versus X

Z versus X

R= 26

4

R= 26

2

3 1.5

2

0

1

Z

Y

1

-1 0.5

-2 -3 -4 -2.1 -1.6 -1.1 -0.6 -0.1 0.4

0 0.9

1.4

1.9

-2

-1.5

-1

-0.5

X

0

0.5

1

1.5

2

X

(g)

(h) Y versus X

Z versus X

R= 250

R= 250

1.4 4

1.3 1.2

2

Z

Y

1.1 0

1 -2

0.9 0.8

-4 -2

-1.5

-1

-0.5

0

X

0.5

1

1.5

2

0.7 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Fig. 4.6 Different transitions in natural convection in a rotating porous layer (courtesy Elsevier Science Ltd (Vadasz and Olek 1998))


52

z z' z'

z 2 * x'*

* 2 x' * *

2 *

x' *

2 *

z=H

x'

T=0

T=1

*

y

x x x0

x'

y' 0

W

x

x'

Fig. 4.7 A rotating porous layer having the rotation axis within its boundaries and subject to different temperatures at the sidewalls (courtesy Elsevier Science Ltd (Vadasz 1996b))

The singularity is obtained when b2 ¼ c2 , corresponding to b ¼ c or b ¼ c. Since b is uniquely related to the offset distance jx0 j and c ¼ 16 9p2 is a constant, one can relate the singularity to specific values of jx0 j. At order 2 this corresponds to jx0 j ¼ 0:3199 and jx0 j ¼ 0:680. It was shown by Vadasz (1996b) that the second value jx0 j ¼ 0:680 is the only one, which has physical consequences. This value corresponds to a transition beyond which, i.e., for jx0 j 0:68, no positive roots of Rc exist. It therefore implies an unconditional stability of the basic motionless solution for all values of R if jx0 j 0:68. The transitional value of jx0 j was investigated at higher orders showing jx0 j 0:765 at order 3 and the value increases with increasing the order. The indications are that as the order increases the transition value of jx0 j tends towards the limit value of 1. The results for the critical values of the centrifugal Rayleigh number expressed in terms of Rcr vs. jx0 j are presented graphically by Vadasz (1996b), who concluded that increasing the value of jx0 j has a stabilizing effect. The results for the convective flow field as obtained by Vadasz (1996b) are presented in Figs. 4.8, 4.9 and 4.10 for different values of jx0 j. Keeping in mind that to the right of the rotation axis the centrifugal acceleration has a destabilizing effect while to its left a stabilizing effect is expected; the results presented in Fig. 4.8b, c reaffirm this expectation showing an eccentric shift of the convection cells towards the right side of the rotation axis. When the rotation axis is moved further towards the hot wall, say at jx0 j ¼ 0:6 as presented in Fig. 4.9a, weak convection cells appear even to the left of the rotation axis. This weak convection becomes stronger as jx0 j increases, as observed in Fig. 4.9b for jx0 j ¼ 0:7 and formation of boundary layers associated with the primary convection


(a)

(c)

(b)

z'

z'

53

z'

*

*

*

x'

x'

x'

| x0 | = 0

| x0 | = 0.3

| x0 | = 0.5

Fig. 4.8 The convective flow field at marginal stability for three different values of jx0 j; 10 streamlines equally divided between wmin and wmax (courtesy Elsevier Science Ltd (Vadasz 1996b))

(a)

(b)

z'

z' *

*

x'

| x0 | = 0.6

x'

| x0 | = 0.7

Fig. 4.9 The convective flow field at marginal stability for two different values of jx0 j; 10 streamlines equally divided between wmin and wmax (courtesy Elsevier Science Ltd (Vadasz 1996b))


54

(a) z

z'

(b)

*

z'

z

*

x' x' | x 0 | = 0.9 | x 0 | = 0.8

Fig. 4.10 The convective flow field at marginal stability for two different values of jx0 j; 10 streamlines equally divided between wmin and wmax (courtesy Elsevier Science Ltd (Vadasz 1996b))

*

T=1

T=0

T=1

T=1

T=0

| x 0 | = 0.5

z' *

x'

x' | x0| = 0

(d)

z'

*

*

T=0

(c)

z'

T=0

(b)

T=1

(a) z'

x'

x' | x0 | = 0.6

| x0 | = 0.7

Fig. 4.11 The convective temperature field at marginal stability for four different values of jx0 j; 10 isotherms equally divided between Tmin ¼ 0 and Tmax ¼ 1 (courtesy Elsevier Science Ltd (Vadasz 1996b))

cells is observed to the right of the rotation axis. These boundary layers become more significant for jx0 j ¼ 0:8 as represented by sharp streamlines gradients in Fig. 4.10a. When jx0 j ¼ 0:9 Fig. 4.10b shows that the boundary layers of the primary convection are well established and the whole domain is filled with weaker secondary, tertiary and further convection cells. The results for the isotherms corresponding to values of jx0 j ¼ 0 ; 0:5; 0:6 and 0:7 are presented in Fig. 4.11 where the effect of moving the axis of rotation within the porous layer, on the temperature is evident.

References

55

References Vadasz P (1992) Natural convection in rotating porous media induced by the centrifugal body force: the solution for small aspect ratio. ASME J Energy Res Technol 114:250–254 Vadasz P (1994a) Centrifugally generated free convection in a rotating porous box. Int J Heat Mass Transf 37(16):2399–2404 Vadasz P (1994b) Stability of free convection in a narrow porous layer subject to rotation. Int Commun Heat Mass Transf 21(6):881–890 Vadasz P (1996a) Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp Porous Media 23:153–173 Vadasz P (1996b) Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int J Heat Mass Transf 39(8):1639–1647 Vadasz P, Olek S (1998) Transitions and chaos for free convection in a rotating porous layer. Int J Heat Mass Transf 41(11):1417–1435

Chapter 5

Coriolis Effects on Natural Convection

Abstract The effect of the Coriolis acceleration on natural convection is presented in this chapter. When the imposed thermal gradient is perpendicular to the direction of the centrifugal body force unconditional natural convection occurs. Then the Coriolis effect is reflected in creation of secondary flows in a cross section perpendicular to the direction of the main convection. When the natural convection occurs due to gravitational buoyancy and the thermal gradient is parallel to the direction of the gravity body force natural convection occurs conditionally. Stability analysis is then required and presented for stationary or possibly oscillatory convection. Weak nonlinear solutions identify then the direction of the bifurcations for different values of the controlling parameters. Keywords Coriolis acceleration gravity Stability

5.1

Centrifugal buoyancy

Buoyancy due to

Coriolis Effect on Natural Convection Due to Thermal Buoyancy of Centrifugal Body Forces

In a previous section centrifugally driven natural convection was discussed under conditions of small rotation rates, i.e. Ek 1. Then, as a first approximation the Coriolis effect was neglected. It is the objective of the present section to show the effect of the Coriolis acceleration on natural convection even when this effect is small, i.e., Ek 1. A long rotating porous box is considered as presented in Fig. 5.1. The possibility of internal heat generation is included but the case without heat generation, i.e., when the box is heated from above and cooled from below is dealt with separately. This case was solved by Vadasz (1993) by using an asymptotic expansion in terms of two small parameters representing the reciprocal Ekman number in porous media and the aspect ratio of the domain. Equations (2.64), (2.65), and (2.66) were


57

58

5

Fig. 5.1 Three-dimensional convection in a long rotating porous box (courtesy: The ASME (Vadasz 1993))


*

z* 2

*

x *

HOT y * COLD

H*

L* H* x*

used with Rag ¼ 0 ; K ¼ 1 and neglecting the component of the centrifugal acceleration in the y direction (a small aspect ratio). Then an expansion of the form ½V; T; p ¼

1 X 1 X

H m Ekn ½V mn ; Tmn ; pmn

ð5:1Þ

m¼0 n¼0

is introduced in the equations, where H is the aspect ratio, i.e. H ¼ H =L . To leading order the zero powers of the aspect ratio H m and Ek n are used, which yields v00 ¼ w00 ¼ 0 ; T00 ¼ z ; u00 ¼

Rax x ½2z 1 2

ð5:2Þ

This solution holds for the core region of the box and is presented in Fig. 5.2c, f. To orders 1 in Ekn and 0 in H m the Coriolis effect is first detected in a plane perpendicular to the leading order natural convection flow leading to the following analytical solution for the stream function w01 in the y z plane w01 ¼

1 X 1 16Rax x X sin½ð2i 1Þpy sin½ð2j 1Þpz h i 4 2 2 p i¼1 j¼1 ð2i 1Þð2j 1Þ ð2i 1Þ þð2j 1Þ

ð5:3Þ

and the corresponding temperature solution is T01 ¼

1 X 1 16Rax x X cos½ð2i 1Þpy sin½ð2j 1Þpz h i2 p5 i¼1 j¼1 ð2j 1Þ ð2i 1Þ2 þð2j 1Þ2

ð5:4Þ

5.1 Coriolis Effect on Natural Convection …

59

(a) (d) z

(e) z

1

1

1

1

0.5

0.5

0.5

0.5

0.5

0

0

0

0

0

(b) z

(c) z

1

1

0.5

0

*

z

0

u00 / xRa

0

u00 / xRa

0

u00 / xRa

x

(f) z 1

T00

x

0

0

0

T00

T00

Fig. 5.2 The leading order convection for the core region; a filtration velocity, with heat generation, perfectly conducting top and bottom walls; b filtration velocity, with heat generation, perfectly conducting top wall, insulated bottom wall; c filtration velocity, no heat generation, heating from above; d temperature, as in (a); e temperature, as in (b); f temperature, as in (c)

From the solutions it was concluded that the Coriolis effect on natural convection is controlled by the combined dimensionless group r¼

Rax 2bT DTc x3 L H Ko2 ¼ Ek ae m2 /

ð5:5Þ

The flow and temperature fields in the plane y z, perpendicular to the leading order natural convection plane as evaluated through the analytical solution, are presented in Fig. 5.3c in the form of streamlines and isotherms. The single vortex in this plane is a consequence of the monotonic variation of the natural convection flow field in the core region, i.e., by the sign of @ u00 =@ z. By extending this argument to evaluate the natural convection flow and temperature field in a similar box to that in Fig. 5.1 but including internal heat generation and subject to different top and bottom boundary conditions, the following cases were considered by Vadasz (1995a, b): (i) a uniform rate of internal heat generation and perfectly conducting top and bottom walls, i.e., Q ¼ 1 ; z ¼ 0 :T ¼ 0 and z ¼ 1 : T ¼ 0. (ii) a uniform rate of internal heat generation, perfectly conducting top wall and adiabatic bottom wall, i.e., Q ¼ 1 ; z ¼ 0 : @ T=@ z ¼ 0 and z ¼ 1 : T ¼ 0. As a result a basic flow u00 , at the leading order, was evaluated in the form

60

5

(a)


(c)

(b) T= 0

z

z

z

y

T= 0

T= 1

T= 0

y

T z

=0

y

T= 0

Fig. 5.3 The flow and temperature field in the y z plane; a with heat generation, perfectly conducting top and bottom walls; b with heat generation, perfectly conducting top wall, insulated bottom wall; c no heat generation, heating from above

(a) for boundary conditions - set (i) Rax x 2 1 z zþ u00 ¼ 2 6 (b) for boundary conditions - set (ii) u00

Rax x 2 1 z ¼ 2 3

ð5:6Þ

ð5:7Þ

The graphical description of the leading order core solutions in terms of u00 =Rax x is presented in Fig. 5.2a, b and the solutions for T00 in Fig. 5.2d, e. Figure 5.2a, d correspond to the boundary conditions—set (i), while Fig. 5.2b, e correspond to the boundary conditions—set (ii). Because of the change of sign of the gradient @ u00 =@ z in case (i) (see Eq. (2.63) and Fig. 5.2a) a double vortex secondary flow as presented in Fig. 5.3a was obtained in the y z plane. Since the gradient @ u00 =@ z for case (ii) does not change sign a single vortex secondary flow is the result obtained in the y z plane for this case, as shown in Fig. 5.3b. A comparison of the flow direction of this vortex with the direction of the vortex resulting from the analytical solution for the case without heat generation while heating from above (see Fig. 5.3c), shows again the effect of the basic flow vertical gradient @ u00 =@ z on the secondary flow, i.e. a positive gradient is associated with an counter-clockwise flow and a negative gradient favors a clockwise flow. The resulting effect of these secondary flows on the temperature is presented by the dashed lines in Fig. 5.3, representing the isotherms.

5.2 Coriolis Effect on Natural Convection Due to Thermal Buoyancy …

5.2

61

Coriolis Effect on Natural Convection Due to Thermal Buoyancy of Gravity Forces

The problem of a rotating porous layer subject to gravity and heated from below (see Fig. 5.4) was originally investigated by Friedrich (1983) and by Patil and Vaidyanathan (1983). Both studies considered a non-Darcy model, which is probably subject to the limitations as shown by Nield (1991). Friedrich (1983) focused on the effect of Prandtl number on the convective flow resulting from a linear stability analysis as well as a non-linear numerical solution, while Patil and Vaidyanathan (1983) dealt with the influence of variable viscosity on the stability condition. The latter concluded that variable viscosity has a destabilizing effect while rotation has a stabilizing effect. Although the non-Darcy model considered included the time derivative in the momentum equation the possibility of convection setting-in as an oscillatory instability was not explicitly investigated by Patil and Vaidyanathan (1983). It should be pointed out that for a pure fluid (non-porous domain) convection sets in as oscillatory instability for a certain range of Prandtl number values (Chandrasekhar 1961). This possibility was explored by Friedrich (1983), which presents stability curves for both monotonic and oscillatory instability. Jou and Liaw (1987) investigated a similar problem of gravity driven thermal convection in a rotating porous layer subject to transient heating from below. By using a non-Darcy model they established the stability conditions for the marginal state without considering the possibility of oscillatory convection. An important analogy was discovered by Palm and Tyvand (1984) who showed, by using a Darcy model, that the onset of gravity driven convection in a rotating porous layer is equivalent to the case of an anisotropic porous medium. The critical Rayleigh number was found to be h i2 Rag;cr ¼ p2 ð1 þ TaÞ1=2 þ1

Fig. 5.4 A rotating fluid saturated porous layer heated from below (courtesy: Cambridge University Press (Vadasz 1998))

ð5:8Þ

62

5


where Ta is the Taylor number defined here as Ta ¼

2x Ko / m

2 ð5:9Þ

and the corresponding critical wave number is pð1 þ TaÞ1=4 . The porosity is missing in Palm and Tyvand (1984) definition of Ta. Nield (1999) has pointed out that these authors and others have omitted the porosity from the Coriolis term. This result, Eq. (5.8) (amended to include the correct definition of Ta), was confirmed by Vadasz (1998) for a Darcy model extended to include the time derivative term (see Eq. (2.56) with Rax ¼ 0), while performing linear stability as well as weak non-linear analyses of the problem to provide differences as well as similarities with the corresponding problem in pure fluids (non-porous domains). As such, Vadasz (1998) found that, in contrast to the problem in pure fluids, overstable convection in porous media at marginal stability is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr\1). Moreover, it was also established by Vadasz (1998) that in the porous media problem the critical wave number in the plane containing the streamlines for stationary convection is not identical to the critical wave number associated with convection without rotation, and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless, it was evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e. the higher the viscosity the less stable is the fluid. An example of stability curves for overstable convection is presented in Fig. 5.5 for

Ta = 5 20

R c(st)

18

= 0.

16

= 0.4

= 0.2 =0.6

2 (ov)

Rac /

Fig. 5.5 Stability curves for overstable gravity driven convection in a rotating porous layer heated from below (c ¼ Va p2 , R ¼ Rag p2 ) (courtesy: Cambridge University Press (Vadasz 1998))

=0.8

14 12 10 8 6

0

0.5

1

1.5

/

2

2.5

5.2 Coriolis Effect on Natural Convection Due to Thermal Buoyancy … Fig. 5.6 Stability map for gravity driven convection in a rotating porous layerheated from below (c ¼ Va p2 , R ¼ Rag p2 ) (courtesy: Cambridge University Press (Vadasz 1998))

63

35 2 i, cr =

30

(ov) R cr =

0 (st)

R cr

25

Stationary Convection

20 15

Overstable Convection

10 5 0

0

40

80

120

160

200

Ta

Ta ¼ 5, where j is the wave number. The upper bound of these stability curves is represented by a stability curve corresponding to stationary convection at the same particular value of the Taylor number, while the lower bound was found to be independent of the value of Taylor number and corresponds to the stability curve for overstable convection associated with Va ¼ 0. Two conditions have to be fulfilled for overstable convection to set in at marginal stability, i.e., (i) the value of Rayleigh number has to be higher than the critical Rayleigh number associated with overstable convection, and (ii) the critical Rayleigh number associated with overstable convection has to be smaller than the critical Rayleigh number associated with stationary convection. The stability map obtained by Vadasz (1998) is presented in Fig. 5.6, which shows that the Ta c (c¼ Va p2 ) plane is divided by a continuous curve (almost a straight line) into two zones, one for which convection sets in as stationary, and the other where overstable convection is preferred. The dotted curve represents the case when the necessary condition (i) above is fulfilled but condition (ii) is not. Weak non-linear stationary as well as oscillatory solutions were derived, identifying the domain of parameter values consistent with supercritical pitchfork (in the stationary case) and Hopf (in the oscillatory case) bifurcations. Unfortunately due to a typo affecting the sign of one of the nonlinear terms in the weak nonlinear analysis the direction of the bifurcations presented might be incorrect. The identification of the tricritical point corresponding to the transition from supercritical to subcritical bifurcations was presented on the c Ta parameter plane. The possibility of a codimension-2 bifurcation, which is anticipated at the intersection between the stationary and overstable solutions, although identified as being of significant interest for further study, was not investigated by Vadasz (1998).

64

5


References Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. Oxford University Press, Oxford Friedrich R (1983) The effect of Prandtl number on the cellular convection in a rotating fluid saturated porous medium. ZAMM 63:246–249 (in German) Jou JJ, Liaw JS (1987) Thermal convection in a porous medium subject to transient heating and rotation. Int J Heat Mass Transf 30:208–211 Nield DA (1991) The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int J Heat Fluid Flow 12(3):269–272 Nield DA (1999) Modeling the effect of a magnetic field or rotation on flow in a porous medium: momentum equation and anisotropic permeability analogy. Int J Heat Mass Transf 42:3715–3718 Palm E, Tyvand A (1984) Thermal convection in a rotating porous layer. J Appl Math Phys (ZAMP) 35:122–123 Patil PR, Vaidyanathan G (1983) On setting up of convection currents in a rotating porous medium under the influence of variable viscosity. Int J Eng Sci 21:123–130 Vadasz P (1993) Three-dimensional free convection in a long rotating porous box. J Heat Transf 115:639–644 Vadasz P (1995a) Three-dimensional free convection in a long rotating porous box. J Heat Transf 115:639 Vadasz P (1995b) Coriolis effect on free convection in a rotating porous box subject to uniform heat generation. Int J Heat Mass Transf 38(11):2011–2018 Vadasz P (1998) Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J Fluid Mech 376:351–375

Chapter 6

Other Effects of Rotation on Flow and Natural Convection in Porous Media

Abstract Combined gravity and centrifugal buoyancy due to thermal gradients are introduced in this chapter, followed by onset of convection in a binary mixture saturating the rotating porous layer. Effects of lack of local thermal equilibrium or local thermal non-equilibrium are then discussed. The case when the porous medium is anisotropic and the resulting effects on convection are then introduced. Applications of rotating porous media to nanofluids and solidification of binary alloys conclude this chapter.

Keywords Gravity and centrifugal buoyancy Thermohaline convection Lack of local equilibrium Local thermal Non-equilibrium Anisotropic porous media Nanofluids Solidification of binary alloys

6.1 Natural Convection Due to Thermal Buoyancy of Combined Centrifugal and Gravity Forces Chapter 4 dealt with the onset of natural convection due to thermal buoyancy caused by centrifugal body forces. Consequently the effect of gravity was neglected, i.e., Rag ¼ 0. For this assumption to be valid the following condition has to be satisfied: Rag Rax ¼ g x2 L 1. It is of interest to investigate the case when Rag Rax and both centrifugal and gravity forces are of the same order of magnitude. Vadasz and Govender (1998, 2001) present investigations of cases corresponding to the ones presented in Sect. 4.3, including the effect of gravity in a direction perpendicular to the centrifugal force. Figure 4.3 is still applicable to the present problem with the slight modification of drawing the gravity acceleration g in the negative z direction. The notation remains the same and Eq. (4.19) becomes V ¼ rp Raxo ½1 þ g xTêx þ Rag Têz


ð6:1Þ

65

6 Other Effects of Rotation on Flow and Natural Convection …

66

where g ¼ 1=x0 ¼ Rax =Raxo represents the reciprocal of the offset distance from the axis of rotation. The approach being the same as in Sect. 4.3, the solution is expressed as a sum of a basic solution and small perturbations as presented in Eq. (4.21). However, because of the presence of the gravity component in Eq. (6.1), a motionless conduction solution is not possible any more. Therefore, the basic solution far from the top and bottom walls is obtained in the form 1 wb ¼ Rag x ; Tb ¼ x; 2 1 1 1 pb ¼ Rag z Raxo x2 þ gx þ const: 2 2 3 ub ¼ vb ¼ 0;

ð6:2Þ

Substituting this basic solution into the governing equations and linearizing the result by neglecting terms that include products of perturbations, which are small, yields a set of partial differential equations for the perturbations. Assuming a normal modes expansion in the y and z directions in the form T 0 ¼ Aj hðxÞ exp rt þ i ðjy y þ jz zÞ

ð6:3Þ

where jy and jz are the wave numbers in y and z directions respectively, i.e., j2 ¼ j2y þ j2z , and using the Galerkin method, the following set of linear algebraic equations is obtained at marginal stability (i.e., for r ¼ 0) M nh i X

2 2 m2 p2 þ j2 j2 Raxo ð2 þ gÞ dml þ m¼1

"

#

16m l j2 Raxo 8 m ljz Rag 2 2 2 2 i p l þ m þ 2 j dmþl;2p1 am ¼ 0 2 2 2 2 p ðl m Þ p2 ðl2 m2 Þ ð6:4Þ

pffiffiffiffiffiffiffi for l ¼ 1; 2; 3; . . .; M and i ¼ 1. In Eq. (6.4) dml is the Kronecker delta function and the index p can take arbitrary integer values, since it stands only for setting the second index in the Kronecker delta function to be an odd integer. A particular case of interest is the configuration when the layer is placed far away from the axis of rotation, i.e. when the length of the layer L is much smaller than the offset distance from the rotation axis x0 . Therefore for x0 ¼ ðx0 =L Þ ! 1 or g ! 0 the contribution of the term gx in Eq. (6.1) is not significant. Substitution of this limit into Eq. (6.4) and solving the system at the second order, i.e. M ¼ 2, yields a quadratic equation for the characteristic values of Raxo . This equation has no real solutions for values of a ¼ j2z p2 beyond a transitional value . atr ¼ ð27p3 16 Rag Þ2 . This value was evaluated at higher orders too, showing that for M ¼ 10 the transitional value varies very little with Rag , beyond a certain Rag value around 50p. The critical values of Raxo were evaluated for different values of

6.1 Natural Convection Due to Thermal Buoyancy …

67

Rg ð¼ Rag pÞ and the corresponding two-dimensional convection solutions in terms of streamlines are presented graphically for the odd modes in Fig. 6.1a, showing the perturbation solutions in the x z plane as skewed convection cells when compared with the case without gravity. The corresponding convection solutions for the even modes are presented in Fig. 6.1b, where it is evident that the centrifugal effect is felt predominantly in the central region of the layer, while the downwards and upwards basic gravity driven convection persists along the left and right boundaries, respectively, although not in straight lines. Beyond the transition value of a, the basic gravity driven convective flow (Eq. 6.2) is unconditionally stable. These results were shown to have an analogy with the problem of gravity driven convection in a non-rotating, inclined porous layer (Govender and Vadasz 1995). Qualitative experimental confirmation of these results was presented by Vadasz and Heerah (1998) by using a thermosensitive liquid-crystal tracer in a rotating HeleShaw cell. When the layer is placed at an arbitrary finite distance from the axis of rotation no real solutions exist for the characteristic values of Raxo corresponding to any values of c other than c ¼ jz Rag ¼ 0. In the presence of gravity Rag 6¼ 0, and c ¼ 0 can be satisfied only if jz ¼ 0. Therefore the presence of gravity in this case has no other role but to exclude the vertical modes of convection. The critical centrifugal Rayleigh numbers and the corresponding critical wave numbers are the same as in the corresponding case without gravity as presented in Sect. 4.3. However, the eigenfunctions representing the convection pattern are different as they exclude the vertical modes replacing them with a corresponding horizontal mode in the y direction. Therefore, a cellular convection in the x y plane is superimposed to the basic convection in the x z plane.

6.2 Onset of Convection Due to Thermohaline (Binary Mixture) Buoyancy of Gravity Forces Limited research results are available on natural convection in a rotating porous medium due to thermohaline buoyancy caused by gravity forces. Chakrabarti and Gupta (1981) investigated a non-Darcy model, which includes the Brinkman term as well as a non-linear convective term in the momentum equation (in the form ðV rÞ V). Therefore the model’s validity is subject to the limitations pointed out by Nield (1991). Both linear and non-linear analyses were performed and overstability was particularly investigated. Overstability is affected in this case by both the presence of a salinity gradient and by the Coriolis effect. Apart from the thermal and solutal Rayleigh numbers and the Taylor number, two additional parameters affect the stability. These are the Prandtl number Pr ¼ m =ae , and the Darcy number Da ¼ Ko H2 , where H is the layer’s height. The authors found that, in the range of values of the parameters, which were considered, the linear stability results favor setting-in of convection through a mechanism of overstability. The results for non-linear steady convection show that the system becomes unstable to finite


68

z

(a)

z z z'

T=0

T=1

T=1

T=0

T=1

T=0

c

T=1

T=0

z

x'

x

x Rg=0

x

Rg=5

x

Rg =10

(b)

Rg =20

z z z'

z

x

x

x' Rg=5

T=1

T=1

T=0

T=0

T=1

T=0

c

Rg =10

x Rg =20

Fig. The convective flow field (streamlines) at marginal stability for different values of 6.1 Rg ¼ Rag p ; (a) the odd modes; (b) the even modes (Vadasz and Govender 1998)

6.2 Onset of Convection …

69

amplitude steady disturbances before it becomes unstable to disturbances of infinitesimal amplitude. Thus the porous layer may exhibit subcritical instability in the presence of rotation. These results are surprising at least in the sense of their absolute generality and the authors mention that further confirmation is needed in order to increase the degree of confidence in these results. A similar problem was investigated by Rudraiah et al. (1986) while focusing on the effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed porous medium. A non-Darcy model identical to the one used by Chakrabarti and Gupta (1981) was adopted by Rudraiah et al. (1986); however the authors spelled out explicitly that the model validity is limited to high porosity and high permeability which makes it closer to the behavior of a pure fluid system (nonporous domain). It is probably for this reason that the authors preferred to use the non-porous medium definitions for Rayleigh and Taylor numbers which differ by a factor of Da and Da2 , respectively, from the corresponding definitions for porous media. It is because of these definitions that the authors concluded that for small values of Da number the effect of rotation is negligible for values of Ta\106 . This means that rotation has a significant effect for large rotation rates, i.e., Ta [ 106 . If the porous media Taylor number had been used instead, i.e., the proper porous media scales, then one could have significant effects of rotation at porous media Taylor numbers as small as Ta [ 10. Hence, the results presented by Rudraiah et al. (1986) are useful provided Da ¼ Oð1Þ which is applicable for high permeability (or sparsely packed) porous layers. Marginal stability as well as overstability were investigated and the results show different possibilities of existence of neutral curves by both mechanisms, i.e., monotonic as well as oscillatory instability. In this regard the results appear more comprehensive in the study by Rudraiah et al. (1986) than in Chakrabarti and Gupta (1981). The finite amplitude analysis was performed by using a severely truncated representation of a Fourier series for the dependent variables. As a result a seventh-order Lorenz model of double diffusive convection in a porous medium in the presence of rotation was obtained. From the study of steady, finite amplitude analysis the authors found that subcritical instabilities are possible, depending on the parameter values. The effect of the parameters on the heat and mass transport was investigated as well, and results presenting this effect are discussed in Rudraiah et al. (1986).

6.3 Finite Heat Transfer Between the Phases and Temperature Modulation Lack of local thermal equilibrium (LaLotheq) or local thermal non-equilibrium (LTNE) implies distinct temperature values between the solid and fluid phases within the same REV. Malashetty et al. (2007) presented the linear stability and the onset of convection in a porous layer heated from below and subject to rotation, accounting for the Coriolis effect as in Vadasz (1998) but allowing for distinct

70


temperature values between the solid and fluid phases, i.e. lack of local thermal equilibrium (LaLotheq), or local thermal non-equilibrium (LTNE). The nonlinear part of the analysis was undertaken by using a truncated mode spectral system, such as the one used by Vadasz and Olek (1998) but adapted for the LaLotheq conditions. The effect of finite heat transfer between the phases leading to lack of local thermal equilibrium was investigated also by Govender and Vadasz (2007) while investigating also the effect of mechanical and thermal anisotropy on the stability of a rotating porous layer heated from below and subject to gravity. The topic of anisotropic effects is discussed in the next section. Bhadauria (2008) investigated the effect of temperature modulation on the onset of thermal instability in a horizontal fluid-saturated porous layer heated from below and subject to uniform rotation. An extended Darcy model, which includes the time derivative term, has been considered, and a time-dependent periodic temperature field was applied to modulate the surfaces’ temperature. A perturbation procedure based on small amplitude of the imposed temperature modulation was used to study the combined effect of rotation, permeability, and temperature modulation on the stability of the fluid saturated porous layer. The correction of the critical Rayleigh number was calculated as a function of amplitude and frequency of modulation, the porous media Taylor number, and the Vadasz number. It was found that both rotation and permeability suppress the onset of thermal instability. Furthermore, the author concluded that temperature modulation could either promote or retard the onset of convection.

6.4 Anisotropic Effects The effect of anisotropy on the stability of convection in a rotating porous layer subject to centrifugal body forces was investigated by Govender (2006). The Darcy model extended to include anisotropic effects and rotation was used to describe the momentum balance and a modified energy equation that included the effects of thermal anisotropy was used to account for the heat transfer. The linear stability theory was used to evaluate the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. It was found that the convection was stabilized when the thermal anisotropy ratio (which is a function of the thermal and mechanical anisotropy parameters) increased in magnitude. Malashetty and Swamy (2007), and Govender and Vadasz (2007) investigated the Coriolis effect on natural convection in a rotating anisotropic fluid-saturated porous layer heated from below and subject to gravity as the body force. Malashetty and Swamy (2007) assumed local thermal equilibrium while Govender and Vadasz (2007) dealt with lack of local thermal equilibrium (LaLotheq), or local thermal non-equilibrium (LTNE). Malashetty and Swamy (2007) used the linear stability theory as well as a nonlinear spectral method. The linear theory was based on the usual normal mode technique and the nonlinear theory on a truncated Galerkin analysis. The Darcy

6.4 Anisotropic Effects

71

model extended to include a time derivative and the Coriolis terms with an anisotropic permeability was used to describe the flow through the porous media. A modified energy equation including the thermal anisotropy was used. The effect of rotation, mechanical and thermal anisotropy parameters and the Prandtl number on the stationary and overstable convection was discussed. It was found that the effect of mechanical anisotropy is to prefer the onset of oscillatory convection instead of the stationary one. It was also found (just as in Vadasz 1998) that the existence of overstable motions in case of rotating porous media is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The steady finite amplitude analysis was performed using the truncated Galerkin modes to find the Nusselt number. The effect of various parameters on heat transfer was investigated. Govender and Vadasz (2007) analyzed the stability of a horizontal rotating fluid saturated porous layer exhibiting both thermal and mechanical anisotropy, subject to lack of local thermal equilibrium (LaLotheq), or local thermal non-equilibrium (LTNE). All of the results were presented as a function of the scaled inter-phase heat transfer coefficient. The results of the linear stability theory have revealed that increasing the conductivity ratio and the mechanical anisotropy has a destabilizing effect, whilst increasing the fluid and solid thermal conductivity ratios is stabilizing. In general it was found that rotation has a stabilizing effect in a porous layer exhibiting mechanical or thermal (or both mechanical and thermal) anisotropy.

6.5 Applications to Nanofluids An interesting recent application is related to nanofluids. A nanofluid is a suspension of nanoparticles or nanotubes in a liquid. When the liquid is saturating a porous matrix one deals with nanofluids in porous media. Rana and Agarwal (2015) investigated the natural convection in a rotating porous layer saturated by a nanofluid and a binary mixture. This implies that the nanoparticles are suspended in a binary mixture, e.g. in a water and salt solution. Therefore double-diffusive convection is anticipated. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. The neutral and critical Rayleigh numbers for stationary and oscillatory convection have been obtained in terms of various dimensionless parameters. The authors concluded that the principle of exchange of stabilities is applicable in the present problem, while more amount of heat is required in the nanofluid case for convection to set-in. Agarwal et al. (2011) considered the convection in a rotating anisotropic porous layer saturated by a nanofluid. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium the Darcy model has been used. Using linear stability analysis the expression for the critical Rayleigh number has been obtained in terms of various dimensionless parameters. Agarwal et al. (2011) indicate that bottom-heavy and top-heavy arrangements of nanoparticles tend to prefer oscillatory and stationary modes of convection, respectively.

72


6.6 Applications to Solidification of Binary Alloys During solidification of binary alloys the solidification front between the solid and the liquid phases is not a sharp front but rather a mushy layer combining liquid and solid phases each one being interconnected. It is not surprising therefore that the treatment of this mushy layer follows all the rules applicable to a porous medium. Natural convection due to thermal as well as concentration gradients occurs in the mushy layer resulting in possible creation of freckles that might affect the quality of the cast. When such a process occurs in a system that is subject to rotation, centrifugal buoyancy as well as Coriolis effects are relevant and essential to be included in any model of this process. Govender and Vadasz (2002b) investigated such a system via a weak nonlinear analysis for moderate Stefan numbers applicable to stationary convection in a rotating mushy layer. Consequently Govender and Vadasz (2002a) investigated a similar system via a weak nonlinear analysis for moderate Stefan numbers applicable to oscillatory convection in a rotating mushy layer. A near-eutectic approximation and large far-field temperature were employed in both papers in order to decouple the mushy layer from the overlying liquid melt. The parameter regimes in terms of Taylor number for example where the bifurcation is subcritical or supercritical were identified. In the case of oscillatory convection increasing the Taylor number lead to a supercritical bifurcation.

References Agarwal S, Bhadauria BS, Siddheshwar PG (2011) Thermal instability of a nanofluid saturating a rotating anisotropic porous medium. Spec Top Rev Porous Media Int J 2(1):53–64 Bhadauria BS (2008) Effect of temperature modulation on the onset of Darcy convection in a rotating porous medium. J Porous Media 11(4):361–375 Chakrabarti A, Gupta AS (1981) Nonlinear thermohaline convection in a rotating porous medium. Mech Res Commun 8(1):9–22 Govender S (2006) On the effect of anisotropy on the stability of convection in rotating porous media. Transport in Porous Media 64(4):413–422 Govender S, Vadasz P (1995) Centrifugal and gravity driven convection in rotating porous media—an analogy with the inclined porous layer. ASME-HTD 309:93–98 Govender S, Vadasz P (2002a) Weak non-linear analysis of moderate Stefan number oscillatory convection in rotating mushy layers. Transp Porous Media 48(3):353–372 Govender S, Vadasz P (2002b) Weak non-linear analysis of moderate Stefan number stationary convection in rotating mushy layers. Transp Porous Media 49(3):247–263 Govender S, Vadasz P (2007) The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp Porous Media 69(1):55–66 Malashetty MS, Swamy M, Kulkarni S (2007) Thermal convection in a rotating porous layer using a thermal nonequilibrium model. Phys Fluids 19(054102):1–16 Nield DA (1991) The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int J Heat Fluid Flow 12(3):269–272 Rana P, Agarwal S (2015) Convection in a binary nanofluid saturated rotating porous layer. J Nanofluids 4:1–7

References

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Rudraiah N, Shivakumara IS, Friedrich R (1986) The effect of rotation on linear and non-linear double-diffusive convection in a sparsely packed porous medium. Int J Heat Mass Transf 29:1301–1317 Vadasz P (1998) Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J Fluid Mech 376:351–375 Vadasz P, Govender S (1998) Two-dimensional convection induced by gravity and centrifugal forces in a rotating porous layer far away from the axis of rotation. Int J Rotating Mach 4 (2):73–90 Vadasz P, Govender S (2001) Stability and stationary convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int J Eng Sci 39 (6):715–732 Vadasz P, Heerah A (1998) Experimental confirmation and analytical results of centrifugallydriven free convection in rotating porous media. J Porous Media 1(3):261–272 Vadasz P, Olek S (1998) Transitions and chaos for free convection in a rotating porous layer. Int J Heat Mass Transf 41(11):1417–1435

Appendix

The objective of this appendix is to prove identities (2.45), (2.46) and (2.47). Let us therefore represent the unit vector êx in terms of the unit vectors linked to the Cartesian coordinate system êx ; êy and êz in the form êx ¼ axêx þ ayêy þ azêz

ðA:1Þ

where ax ; ay and az are the cosines of the angle between êx and the corresponding axis directions respectively. Then the following relationship between the a’s is established a2x þ a2y þ a2z ¼ 1

ðA:2Þ

Evaluating the cross product ðêx X Þ by using Eq. (A.1) and X ¼ x êx þ y êy þ z êz , and working out the inner product ðêx XÞ ðêx X Þ yields ðêx XÞ ðêx X Þ ¼ a2y þ a2z x2 þ a2x þ a2z y2 þ a2x þ a2y z2 2ay az yz 2ax az xz 2ax ay xy ðA:3Þ Then applying the gradient operator on Eq. (A.3) one obtains 1 r½ðêx X Þ ðêx XÞ ¼ xêx þ yêy þ zêz 2 ax x þ ay y þ az z axêx þ ayêy þ azêz

ðA:4Þ

It can be observed that the terms in the first brackets on the right hand side of Eq. (A.4) represent the vector X, the terms in the second brackets represent ðêx XÞ

© The Author(s) 2016 P. Vadasz, Fluid Flow and Heat Transfer in Rotating Porous Media, SpringerBriefs in Thermal Engineering and Applied Science, DOI 10.1007/978-3-319-20056-9

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Appendix

and the terms in the last brackets stand for the unit vector êx according to Eq. (A.1). Therefore Eq. (A.4) becomes 1 r½ðêx XÞ ðêx X Þ ¼ X ðêx X Þ êx 2

ðA:5Þ

It can be observed from Eq. (A.5) that the contribution of the term ðêx X Þ êx is to cancel out the component of X in the êx direction and to keep only the components of X, which are perpendicular to the axis of rotation. To evaluate now the triple cross product êx ðêx X Þ we use the following vector identity, which is valid for any three vectors A; B and C, i.e. : A ðB CÞ ¼ ðA CÞ B ðA BÞ C. Then choosing A ¼ B ¼ êx and C ¼ X yields êx ðêx X Þ ¼ ðêx XÞ êx X

ðA:6Þ

By comparing Eqs. (A.6) with (A.5) we obtain 1 êx ðêx XÞ ¼ r½ðêx X Þ ðêx XÞ 2

ðA:7Þ

Equations (A.6) and (A.7) represent the required proof of identities (46) and (47), respectively. For proving identity (45) let us represent the unit vector êg in terms of the unit vectors linked to the Cartesian coordinate system êx ; êy and êz in the form êg ¼ hxêx þ hyêy þ hzêz

ðA:8Þ

where hx ; hy and hz are the cosines of the angle between êg and the corresponding axis directions respectively. Then the following relationship between the h's is established h2x þ h2y þ h2z ¼ 1

ðA:9Þ

Evaluating the dot product êg X by using Eq. (A.8) and X ¼ x êx þ y êy þ z êz yields êg X ¼ hx x þ hy y þ hz z

ðA:10Þ

Taking the gradient of Eq. (A.10) produces r êg X ¼ hxêx þ hyêy þ hzêz |{z} ¼ êg ðA8Þ

Equation (A.11) represents the required proof of identity (2.45).

ðA:11Þ

Index

A Anisotropic effects, 70 Anisotropic porous media, 2, 21, 61, 65, 71 Application to nanofluids, 71 B Binary alloys, 26, 72 Buoyancy due to gravity, 42, 46, 67 C Centrifugal body force, 41, 42, 45, 57 Centrifugal buoyancy, 16, 19, 42, 46, 65, 72 Centrifugal filtration, 1 Chaos turbulence, 41 Chemical processing, 1 Continuum approach, 4, 5, 24 Coriolis acceleration, 10, 19, 28, 57 Coriolis effect, 57, 59, 61, 70 D Darcy equation, 10 Darcy model, 5, 9, 22, 61, 62, 69–71 Dimensionless forms, 11, 13, 15 E Equation of state, 14–16, 25 Extended forms, 10 F Finite heat transfer, 12, 69 Flow, 36, 65 Food processing, 1, 3 G General background, 1, 31, 41 Geostrophic flow, 32 Gravity, 16, 65 Gravity forces, 61, 65, 67

H Heat transfer, 2, 7 Heterogeneous porous media, 14, 19, 21, 27, 31, 36, 37 Homogeneous porous media, 3, 15, 25, 26 I Isothermal flow, 3, 21, 27, 31 L Lack of local equilibrium, 13, 69–71 Local thermal non-equilibrium, 69–71 M Modeling of flow, 2, 7 Modelling of heat transfer, 2, 7, 11 N Nanofluids, 71 Natural convection and buoyancy, 4, 15, 27, 36, 41, 57, 59, 61, 65, 70 Non-Darcy models, 22, 26 O Onset of convection, 67 Other effects of rotation, 65 P Packed bed agitated vessels, 1 Phase, 12, 13, 69 Porous media, 2, 4, 7, 11, 15, 26, 27, 36, 65 Porous media modeling, 17, 24 Porous media natural convection, 3, 4, 25, 67, 72 R Rotating flows, 26 Rotating frame, 10

© The Author(s) 2016 P. Vadasz, Fluid Flow and Heat Transfer in Rotating Porous Media, SpringerBriefs in Thermal Engineering and Applied Science, DOI 10.1007/978-3-319-20056-9

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78 Rotating machinery, 1 Rotating porous media, 1–3, 19, 27, 32, 36, 71 Rotating square channel, 36 S Solidification, 26, 72 Stability, 3, 21, 42, 45, 47–50, 52, 61–63, 67, 69–71 T Taylor-Proudman columns, 32, 34

Index Temperature gradients, 42, 45 Temperature modulation, 69 Thermal buoyancy, 41, 57, 61, 65 Thermal equilibrium, 13 Thermohaline buoyancy, 67 Thermohaline convection, 65, 67 W Weak turbulence, 41