Micropolar Fluid Flow Between Two Inclined Parallel ...

Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition IMECE2017 November 3-9, 2017, Tampa, Florida, USA

IMECE2017-72528

Micropolar Fluid Flow between Two Inclined Parallel Plates Abbas Hazbavi

Sajad Sharhani

Department of Mechanical Engineering Ahvaz Branch Islamic Azad University Ahvaz, Iran Email: [email protected]

Department of Mechanical Engineering Ahvaz Branch Islamic Azad University Ahvaz, Iran Email: [email protected]

ABSTRACT

behavior by Eringen [1-2]. The microscopic effects arising from the microrotations of fluid elements and local structure are considered in this model, it also explains the flow characteristics of geomorphological sediments, colloidal suspensions, polymeric additives, lubricants, liquid crystals, hematological suspensions, etc. In micropolar fluid model, the microrotation vector is used to simulate the rotation of fluid particles; this independent kinematic vector makes it unique from other non-Newtonian fluids. The mathematical aspects of micropolar fluid flow theory introduced by Łukaszewicz [3]. Guram and Anwar [4] numerically investigated the micropolar fluid flow between a rotating and a stationary disk. Rashidi et al. [5] to study the steady, incompressible and laminar micropolar fluids flow reduced governing equations to nonlinear ordinary differential equations by using similarity transformation.

In this study, the hydrodynamic characteristics are investigated for magneto-micropolar fluid flow through an inclined channel of parallel plates with constant pressure gradient. The lower plate is maintained at constant temperature and upper plate at a constant heat flux. The governing equations which are continuity, momentum and energy are are solved numerically by Explicit Runge-Kutta. The effect of characteristic parameters is discussed on velocity and microrotation in different diagrams. The nonlinear parameter affected the velocity microrotation diagrams. An increase in the value of Hartmann number slows down the movement of the fluid in the channel. The application of the magnetic field induces resistive force acting in the opposite direction of the flow, thus causing its deceleration. Also the effect of pressure gradient is investigated on velocity and microrotation in different diagrams.

In addition, the effects of a magnetic field on a micropolar fluid have great importance in engineering applications. To date, many studies have targeted the issue of micropolar fluid and the research is continuing. A comprehensive glossary of magnetomicropolar fluid is collected by M. Duran et al [6] and Ahmadi et al [7]. Pranesh and Kiran [8] studied the rayleighbenard magneto convection in a micropolar fluid with maxwellcattaneo law using the Galerkin technique. Md. Ziaul Haque et al [9] studied numerically the behavior of micropolar fluid on steady MHD free convection and mass transfer through a porous medium with constant heat and mass fluxes has been studied. A. Tetbirt et al [10] analyzed convective heat transfer of two different types of immiscible fluids in a vertical channel in the existence of a magnetic field. M. Ramzan et al [11]

Keywords: Magneto-Micropolar, Hartmann Number

INTRODUCTION In several industrial applications such as chemical engineering, material processing engineering, aerospace engineering, biomechanics, slurry technologies, etc., the Newtonian fluid is not sufficient to characterize flow properties. Such properties must be described in non-Newtonian fluid models, but unfortunately many non-Newtonian fluid models cannot exhibit the behavior of fluids in industrial processes. One of mathematical model was proposed to explain micropolar fluid

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studied the micropolar fluid flow through a permeable stretching sheet in existence of thermal radiation, joule heating, partial slip and magneto hydrodynamic (MHD) with convective boundary conditions. His results showed that micro-rotation and velocity profiles decrease with the increase of slip parameter value. However, increase in temperature distribution is seen with gradual mounting values of thermal radiation parameter. G. Swapna et al [12] investigated the heat and mass transfer of a mixed convection magneto micropolar fluid flow over a non-permeable linearly stretching cylinder inserted in a porous medium in the presence of thermal radiation and first order chemical reaction with convective boundary condition. K. Batool M. Ashraf [13] numerically analyzed the heat and mass transfer characteristics towards a heated shrinking sheet immersed in an electrically conducting incompressible micropolar fluid in the case of a transverse magnetic field existence. The Hall and ionic effects of an incompressible micropolar fluid flow through a pipe of circular cross-section is studied by D. Srinivasacharya and Mekonnen Shiferaw [14]. Hazbavi [15] analyzed the second law of magnetorheological rotational flow in Taylor–Couette geometry with viscous dissipation. His results showed that the total entropy generation number decreases as the Hartmann number and fluid elasticity increase and it increases with increasing Brinkman number.

DV  P      (    )    V Dt  (  2    )(.V )   f   B 2U 0



D    V  2       Dt (     )(. )   l v v DE   p.V      q   B 2U 2 0 Dt where:



( (3) ( (4)

   (.V )2  2 : D  4 ( 1   V  V )2   (. )2   v 2 :     : ( )T v

( (5)

here, D/Dt is the material time derivative, Φ is the viscous dissipation, D denotes the deformation tensor with D=1/2(Vk,l+Vl,k); E the specific internal energy, q the heat flux, ρ is the density of fluid, V is the velocity vector, ω is the microrotation vector, p is the thermodynamic pressure, f is the body force vector, j is the micro- inertia density, l is the body couple vector, μ, λ and κ are the material constants (viscosity coefficients) and αv, βv, γ are the spin gradient coefficients. The stress tensor τkl and the couple stress tensor Mkl in the governing equations are given by:

Though most of the work has been done for micropolar fluid flow by considering magnetic field has not yet been addressed in the literature. Therefore, the objective of the present study is to investigate the micropolar fluid flow, between two parallel porous plates in present of magnetic field. The governing equations are numerically solved and the behavior of flow characteristics with pertinent flow is discussed.

 kl  ( p  Vr ,r ) kl   (Vk ,l  Vl ,k )   (Vl ,k  M kl  vr ,r kl  vk ,l  l ,k

PROBLEM FORMULATION

 ٍm )

klm

where δkl and ϵklm are the metric tensor and the covariant ε symbol, respectively. The following inequalities, derived from the Clausius–Duhem inequality must be satisfied by the material constants:

Consider the flow of an incompressible Micropolar Fluid between two infinite inclined parallel porous plates at y=+h and y=-h, respectively. Let the main flow be generated by a constant pressure gradient. Let the porous walls be such that a uniform vertical cross flow is generated (v=v0=constant). Positive v0 corresponds to injection at the lower plate and suction at the upper plate. In this geometry the main Reynolds number is Re=ρUh/μ, and the cross flow Reynolds number is Re=ρv0h/μ, where ℎ is the channel width, the density and a zero-shear-rate viscosity, are presented by ρ and μ respectively. In this study the top plate is subjected to a constant uniform heat flux q (Isoflux) while the bottom plate is kept at a constant temperature T1 (isothermal). In this case, the micropolar fluid flow described above evolves according to the governing equations which are MHD. The following is a common version of these equations [1, 2]:     ( V )  0 t

( (2)

3  2    0,

2    0

3 v   v    0,

  v .

 0

Eqs. (1)–(5) represent conservations of mass, momentum, angular momentum and energy, respectively. We remark that for    v  v    0 and vanishing l and f, microrotation ω becomes zero, and Eq. (2) reduces to the classical NavierStokes equations. Also we note that for κ = 0, the velocity V and the micro rotation ω are not coupled and the microrotations do not affect the global motion. Under these hypotheses and Boussinesq approximations the governing equations of micropolar fluid are: (

(1)

du 0 dx

2

(6)



  

d 2u du d p   v0    g *T T  T1  sin       B 2u  0 0 x dy 2 dy dy

problem is stiff. A convenient way of detecting stiffness is to directly estimate the dominant eigenvalue of the Jacobian of the problem (see [16] and [17]). In the present study, analysis and calculations are conducted for E1 micropolar fluid with k=5470×10-7, 𝛼0=0.9 [18] (corresponding to N=0.93352).

(7)



d 2 d du   jv0  2   0 2 dy dy dy

(8)

kf

d 2T dT du du d   c p v0  (    )( )2  2 ( 2   )   ( )2   B 2u 2  0 0 dy 2 dy dy dy dy

(9)

RESULTS AND DISCUSSION

Here u is the velocity vector in the x direction, Γ is the microrotation component, T is the temperature, g* is the acceleration due to gravity, Kf is the coefficient of thermal conductivity; βT is the coefficient of thermal expansion. The boundary conditions are as follow:

u  0,   0,

dT q  , at y  h dy k f

u  0,   0, T  T1 ,

For validation of the numerical method used in this study, we have compared our results with analytical solution of Ariman and Cakmak [19] in the absence of R, α with the fixed values of N=0.1, m=1 and A=-1. The quantitative comparison is shown in Table 1 and it is found to be in excellent agreement.

(10)

Table 1 Comparison analysis for the velocity calculated by the present method and that of analytical solution [19] for R = 0 and a = 0. η velocity Analytical Numerical -1 0 0 -0.8 0.3466 0.3406 -0.6 0.6168 0.6063 -0.4 0.8100 0.7964 -0.2 0.9261 0.9106 0 0.9648 0.9487 0.2 0.9261 0.9106 0.4 0.8100 0.7964 0.6 0.6168 0.6063 0.8 0.3466 0.3406 1 0 0

at y  h

By introducing the dimensionless parameters bellow:



T  T1 y u h , f ( )  , g ( )  ,  ( )  , qh h U0 U0 kf

(11)

The mathematical problem defined in Eqs. (6)–(9) are then transformed into a set of ordinary differential equations and their associated boundary conditions:

1 N Gr sin(  )  Haf '  P f ''  Rf '  g'  1 N 1 N Re 2  N '' 1 N ' g  a R( ) g  2g  f '  0 2 m N Br N (2  N ) '2  ''  R Pr  '  [ f '2  2 N ( g 2  gf ' )  g ] 1 N m2  BrHaf 2  0

(15) (16)

The velocity profiles for E1 micropolar fluid with N=0.93352 [18] are depicted as a functions of normal distance between two plates, for various micropolar viscosity, Hartmann numbers, cross flow Reynolds numbers, Prandtl numbers and Brinkman numbers are shown in Figs. 1, 2, 3, 4 and 5 respectively. Figs. 1-5 show the decreasing nature of velocity with increasing micropolar viscosity, Hartmann number, cross flow Reynolds number, Prandtl number and increase with the increase of Brinkman number. In the limit as N→0, Equations (15) and (16) reduce to the corresponding equations for a Newtonian viscous fluid. Hence, it can be observed that the velocity in the case of Newtonian viscous fluid is more than that of micropolar fluid. The velocity profile exhibits a curvature point within the space between two plates. As Hartmann number increases with increasing applied magnetic field, the velocity profiles declines monotonically due to resistive force. The movement of the fluid in the space between two plates is slowed down with an increase in the value of Hartmann number. The resistive force acting in the opposite direction of the flow with applied of the magnetic field, thus causing fluid flow deceleration.

(17)

Where prime denotes ordinary differentiation with respect to η, U0 is the characteristic velocity, N=κ/κ+μ is the coupling number, Pr=μcp/kf is the Prandtl number, Ha   B02 h2  is the Hartman

number, Gr   2 g *T gh4  2 k f

is the Grashof

number, m  h  (2   )  (   ) is the micropolar parameter, Br  U 02 hq is the Brinkman, P  h2 U 0 dp dx is the constant pressure gradient a   h2 is the micro2

2

inertial parameter.

f  0, g  0,    1, at   1 f  0, g  0,   01 , at   -1

(18)

The governing equations are solved numerically by Explicit Runge-Kutta with the Stiffness Test, which can be used to identify whether the method applied with the specified Accuracy Goal and Precision Goal tolerances to a given

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Fig. 1. Effect of micropolar viscosity on the velocity profile for micropolar fluid with (R=1, Re=1 and Pr= 0.2).

Fig. 4. Effect of Prandtl number on the velocity profile for micropolar fluid with (R=1, Re=1 and Gr=1).

Fig. 2. Effect of Hartmann number on the velocity profile for micropolar fluid with (R=1, Re=1 and Pr=0.2).

Fig. 5. Effect of Brinkman number on the velocity profile for micropolar fluid with (R=1, Re=1 and Pr=0.2). The microrotation profiles for mentioned micropolar fluid with N=0.93352 are showed as a functions of normal distance between two plates, for various micropolar viscosity parameter, Hartmann numbers, cross flow Reynolds numbers, Prandtl numbers and Brinkman numbers are shown in Figs. 6, 7, 8, 9 and 10 respectively. Figs. 6-10 show the profiles are increased near the lower boundary condition and decreased near the upper boundary condition with increasing micropolar viscosity parameter, Hartmann number, cross flow Reynolds number and Prandtl number except Brinkman number.

Fig. 3. Effect of cross flow Reynolds numbers on the velocity profile for micropolar fluid with (Re=1, Gr=1 and Pr=0.2).

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Fig. 9. Effect of Prandtl number on the microrotation profile for micropolar fluid with (R=1, Re=1 and Gr=1).

Fig. 6. Effect of micropolar viscosity on the microrotation profile for micropolar fluid with (R=1, Re=1 and Pr=0.2).

Fig. 10. Effect of Brinkman number on the microrotation profile for micropolar fluid with ((R=1, Re=1 and Pr=0.2).

Fig. 7. Effect of Hartmann number on the microrotation profile for micropolar fluid with (R=1, Re=1 and Pr=0.2).

CONCLUSION In this study, hydrodynamics of Micropolar Fluid between two infinite inclined parallel porous plates were investigated. In this study the top plate is subjected to a constant uniform heat flux q (isoflux) while the bottom plate is kept at a constant temperature (isothermal). This study was focused on the hydrodynamic characteristics and its dependency on various dimensionless parameters such as the Micropolar viscosity (N), Hartmann number (Ha), cross flow Reynolds number (R), the Prandtl number (Pr) and the Brinkman number (Br). As the Hartmann number increases with increasing applied magnetic field, the steady-state velocity declines monotonically due to resistive force. An increase in the value of Hartmann number slows down the movement of the fluid in the channel. The application of the magnetic field induces resistive force acting in the opposite direction of the flow, thus causing its deceleration. Moreover, the study shows that the presence of magnetic field tends to slowdown the fluid motion.

Fig. 8. Effect of cross flow Reynolds on the microrotation profile for micropolar fluid with (Re=1, Gr=1 and Pr=0.2).

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