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Eur. Phys. J. Plus (2014) 129: 41
DOI 10.1140/epjp/i2014-14041-3
Thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating Machireddy Gnaneswara Reddy
Eur. Phys. J. Plus (2014) 129: 41 DOI 10.1140/epjp/i2014-14041-3
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating Machireddy Gnaneswara Reddya Department of Mathematics, Acharya Nagarjuna University Ongole Campus, Ongole, A.P.-523 001, India Received: 4 October 2013 / Revised: 11 November 2013 c Societ` Published online: 13 March 2014 – a Italiana di Fisica / Springer-Verlag 2014 Abstract. An analytical study for the problem of mixed convection with thermal radiation and firstorder chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable surface embedded in a porous medium has been presented. Slip boundary condition is applied at the porous interface. The heat equation includes the terms involving the viscous dissipation, radiative heat flux, Ohmic dissipation, the internal absorption and absorption of radiation, whereas the mass transfer equation includes the effects of chemically reactive species of first order. The dimensionless governing equations for this investigation are formulated and the non-linear coupled differential equations are solved analytically using the perturbation technique. Comparisons with previously published work on special cases of the problem are performed and results are found to be in excellent agreement. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of magnetic field, thermal radiation, chemical reaction and Ohmic dissipation. It is observed that the effect of magnetic field, heat source and thermal radiation is to decrease the velocity, temperature profiles in the boundary layer. The effect of increasing the values of rarefaction parameter is to increase the velocity in the momentum boundary layer. Further, it is found that increasing the value of the chemical reaction decreases the concentration of species in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient, local Nusselt number and local Sherwood number at the surface are discussed.
1 Introduction Mixed-convection flows with simultaneous heat and mass transfer under the influence of a magnetic field and chemical reaction arise in many transport processes both naturally and in many branches of science and engineering applications. They play an important role in many industries, viz. in the chemical industry, power and cooling industry for drying, chemical vapour deposition on surfaces, cooling of nuclear reactors and magnetohydrodynamic (MHD) power generators. Such processes occur when the effects of buoyancy forces in forced convection or the effects of forced flow in free convection become significant. Many transport processes exist in nature and in industrial applications, in which the simultaneous heat and mass transfer occurs as a result of combined buoyancy effects of the diffusion of chemical species. A few fields of interest in which combined heat and mass transfer play an important role in the design of chemical processes in equipment, formation and dispersion of fog, distribution of temperature and moisture over agriculture fields. A comprehensive description of the theoretical work for both laminar and turbulent mixedconvection boundary layer flows has been given in a review paper by Chen and Armaly [1] and in the book by Pop and Ingham [2]. Magnetohydrodynamics plays an important role in agriculture, engineering and petroleum industries. The problem of mixed convection under the influence of a magnetic field has attracted numerous researchers in view of its applications in geophysics and astrophysics. Soundalgekar et al. [3] investigated the problem of free convection effects on the Stokes problem for a vertical plate with a transverse applied magnetic field, whereas Elbasheshy [4] studied the MHD heat and mass transfer problem along a vertical plate under the combined buoyancy effects of thermal and species diffusion. Gnaneswara Reddy and Bhaskar Reddy [5] analyzed the mass transfer and heat generation effects on MHD free convection flow past an inclined vertical surface in a porous medium. a
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It is well known that the boundary condition for a viscous fluid at a solid wall obeys the no-slip condition, i.e., the fluid velocity matches the velocity of the solid boundary. However, in many practical applications, the particle adjacent to a permeable surface no longer takes the velocity of the surface but the particle at the surface has a finite tangential velocity which slips along the surface. The flow regime is called a slip-flow regime, and this effect cannot be neglected. Sharma and Chaudhury [6] studied the effect of variable suction on transient free convective viscous incompressible flows past a vertical plate in a slip-flow regime. Hayat et al. [7] investigated the flow of an elasticoviscous fluid past an infinite wall with time-dependent suction. In the above-mentioned studies the radiation effect is ignored. When technological processes take place at high temperatures, the thermal radiation heat transfer becomes very important and its effects cannot be neglected (Siegel and Howell [8], Modest [9]). Recent developments in hypersonic flights, missile re-entry rocket combustion chambers, gas-cooled nuclear reactors and power plants for inter-planetary flights have focused the attention of researchers on thermal radiation as a mode of energy transfer, and emphasize the need for the inclusion of radiative transfer in these processes. The interaction of radiation with mixed-convection flows past a vertical plate was investigated by Hossain and Takhar [10]. Aboeldahab [11] studied the radiation effect in heat transfer in an electrically conducting fluid at stretching surface. Abo-Eldahab and Elgendy [12] presented the radiation effect on convective heat transfer in an electrically conducting fluid at a stretching surface with variable viscosity and uniform free stream. Seddeek [13] examined the effect of radiation and variable viscosity on unsteady force convection flows in the presence of an align magnetic field. Aydin and Kaya [14] stuided the effect of radiation on MHD mixed convection flow about a permeable vertical plate. Gnaneswara Reddy [15] analyzed Thermal radiation and viscous dissipation effects on MHD marangoni convection flow over a permeable flat surface with heat generation/absorption. Very recently, Gnaneswara Reddy [16] investigated the influence of thermal radiation, viscous dissipation and hall current on the MHD convection flow over a stretched vertical flat plate. The combined effects of heat and mass transfer with chemical reaction are of great importance to engineers and scientists because of its almost universal occurrence in many branches of science and engineering, and hence received a considerable amount of attention in recent years. The study of the chemical reaction with heat transfer in a porous medium has important engineering applications, e.g., tubular reactors, oxidation of solid materials and synthesis of ceramic materials. There are two types of reactions, such as i) homogeneous reaction and ii) heterogeneous reaction. A homogeneous reaction occurs uniformly throughout a given phase, whereas the heterogeneous reaction takes place in a restricted region or within the boundary of a phase. The effects of a chemical reaction depends on whether the reaction is heterogeneous or homogeneous. A chemical reaction is said to be of first order, if the rate of the reaction is directly proportional to the concentration itself. In many industrial processes involving flow and mass transfer over a moving surface, the diffusing species can be generated or absorbed due to some kind of chemical reaction with the ambient fluid which can greatly affect the flow and hence the properties and quality of the final product. These processes take place in numerous industrial applications, such as the polymer production and the manufacturing of ceramics or glassware. Thus we are particularly interested in the cases in which diffusion of the species and the chemical reaction occur at roughly the same speed in analyzing the mass transfer phenomenon. Das et al. [17] have studied the effect of a homogeneous first-order chemical reaction on the flow past an impulsively started infinite vertical plate. Kandasamy et al. [18] investigated the effects of chemical reaction, heat source and thermal stratification on heat and mass transfer in MHD flow over a vertical stretching surface. The problem of combined heat and mass transfer of an electrically conducting fluid in MHD natural convection adjacent to a vertical surface is analyzed by Chen [19] by taking into account the effects of Ohmic heating and viscous dissipation but neglecting the chemical reaction of the species. Ghaly and Seddeek [20] have investigated the effect of chemical reaction, heat and mass transfer on a laminar flow along a semi-infinite horizontal plate with temperature-dependent viscosity. Pal and Talukdar [21] presented a perturbation analysis of an unsteady MHD mixed convective heat and mass transfer in a boundary layer flow with thermal radiation and chemical reaction. Recently, Gnaneswara Reddy [22] studied chemically reactive species and radiation effects on a MHD convective flow past a moving vertical cylinder. In all the above investigations, the effect of Ohmic heating are not considered in the problem of coupled heat and mass transfer in the presence of a magnetic field. However, it is more realistic to include the Ohmic effect in order to explore the impact of the magnetic field on the thermal transport in the boundary layer. The effect of Ohmic heating on the MHD free convective heat transfer has been examined by Hossain [23] for a Newtonian fluid. Abo-Eldahab and Abd El-Aziz [24] studied the effect of Ohmic heating on mixed convection boundary layer flow of a micropolar fluid from a rotating cone by considering power law variation in surface temperature. Chaudhary et al. [25] have analyzed the effect of radiation on heat transfer in MHD mixed-convection flow with simultaneous thermal and mass diffusion from an infinite vertical plate with viscous dissipation and Ohmic heating. Abd El-Aziz [26] studied the effect of Ohmic heating on combined heat and mass transfer of viscous incompressible fluid having temperature-dependent viscosity as well as thermal conductivity in a MHD three-dimensional flow over a stretching surface. Recently, Pal and Mondal [27] analyzed the effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink and Ohmic dissipation. The objective of the present study is to analyze the effects of buoyancy force and first-order chemical reaction in a two-dimensional MHD flow, heat and mass transfer of a viscous incompressible fluid past a permeable vertical plate
Eur. Phys. J. Plus (2014) 129: 41
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Fig. 1. Physical model and coordinate system.
embedded in a porous medium in the presence of viscous dissipation, Ohmic dissipation, heat absorption, absorption of radiation and thermal radiation. We have also considered the slip velocity with temperature and concentration boundary conditions at the plate surface in the present paper. The classical model introduced by Cogley et al. [28] is used for the radiation effect as it has the merit of simplicity and enables us to introduce a linear term in temperature in the analysis for optically thin media. The effects of various physical parameters on the velocity, temperature and concentration profiles as well as on local skin-friction coefficient and local Nusselt number are discussed. The validation of the analysis has been performed by comparing the present results with those available in the open literature [25, 29] and a very good agreement has been established. Possible new emerging engineering areas of the type of the problem considered in the present paper can be found in many industries such as in powder industry and in generating electric power, in which electrical energy is extracted directly from moving electrically conducting fluid. Also, the results would be useful in many practical areas related to the diffusive operations which involve the molecular diffusion of species with chemical reactions.
2 Formulation of the problem Let us consider the steady two-dimensional laminar boundary layer flow of a viscous incompressible electrically conducting and heat absorbing fluid past a semi-infinite vertical permeable plate embedded in a uniform porous medium, which is subject to slip boundary condition, thermal and concentration buoyancy effects. The x∗ -axis is along the plate and y ∗ is perpendicular to the plate. The wall is maintained at a constant temperature Tw and concentration Cw higher than the ambient temperature T∞ and concentration C∞ , respectively. Also, it is assumed that there exists a homogeneous chemical reaction of first order with rate constant R between the diffusing species and the fluid. The concentration of the diffusing species is very small in comparison to other chemical species, the concentration of species far from the wall, C∞ , is infinitesimally small and hence the Soret and Dufour effects are neglected. The chemical reactions are taking place in the flow and all thermo-physical properties are assumed to be constant except density in the buoyancy terms of the linear momentum equation which is approximated according to the Boussinesq approximation. It is also assumed that viscous and electrical dissipation are negligible. A uniform transverse magnetic field of magnitude B0 is applied in the presence of radiation and concentration buoyancy effects in the direction of the y ∗ -axis. The transversely applied magnetic field and magnetic Reynolds number are assumed to be very small so that the induced magnetic field and the Hall effect are negligible. It is assumed that the porous medium is homogeneous and present everywhere in local thermodynamic equilibrium. The rest of the properties of the fluid and the porous medium are assumed to be constant. A schematic representation of the physical model and coordinates system is depicted in fig. 1. The governing equations for this investigation are based on the balances of mass, linear momentum, energy and concentration species. Taking into consideration these assumptions, the equations that describe the physical situation can be written in Cartesian frame of references, as follows: – Continuity equation: dv ∗ = 0, (1) dy ∗ i.e. v ∗ = −v0 (constant), dp∗ = 0 ⇒ p∗ is independent of y ∗ . dy ∗
(2) (3)
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Eur. Phys. J. Plus (2014) 129: 41
– Momentum equation: ρv ∗
du∗ d2 u∗ µ = µ ∗2 + gβ(T ∗ − T∞ ) + gβ ∗ (C ∗ − C∞ ) − σB02 u∗ − ∗ u∗ . ∗ dy K dy
(4)
– Energy equation: ρcp v ∗
dT ∗ d2 T ∗ +µ = α dy ∗ dy ∗2
du∗ dy ∗
2
−
∂qr∗ + σB02 u∗2 − Q0 (T ∗ − T∞ ) + Q∗1 (C ∗ − C∞ ). ∂y ∗
(5)
– Mass diffusion equation:
d2 C ∗ dC ∗ = D − R∗ (C ∗ − C∞ ), (6) dy ∗ dy ∗2 where x∗ , y ∗ are the dimensional distances along and perpendicular to the plate, respectively. g is the gravitational acceleration, T ∗ is the dimensional temperature of the fluid near the plate, T∞ is the free stream dimensional temperature, K ∗ is the permeability of the porous medium, C ∗ is the dimensional concentration, C∞ is the free stream dimensional concentration. β and β ∗ are the thermal and concentration expansion coefficients, respectively. p∗ is the pressure, cp is the specific heat of constant pressure, µ is viscosity of the fluid, qr∗ is the radiative heat flux, ρ is the density, σ is the magnetic permeability of the fluid, v0 is the constant suction velocity, v is the kinematic viscosity, D is the molecular diffusivity, Q0 is the dimensional heat absorption coefficient, Q∗1 is the coefficient of proportionality of the absorption of the radiation and R is the chemical reaction parameter. u∗ and v ∗ are the components of dimensional velocities along the x∗ and y ∗ directions, respectively, α is the fluid thermal diffusivity. The second and third terms on the RHS of the momentum equation (4) denote the thermal and concentration buoyancy effects, respectively. Also the second and fourth terms on the RHS of the energy equation (5) represent the viscous dissipation and Ohmic dissipation, respectively. The third and fifth terms on the RHS of eq. (5) denote the inclusion of the effect of thermal radiation and heat absorption effects, respectively. The radiative heat flux is given by Cogley et al. [28] as v∗
∂qr∗ = 4(T ∗ − T∞ )I ′ , ∂y ∗
(7)
∞ bλ where I ′ = 0 Kλw ∂e ∂T ∗ dλ, Kλw is the absorption coefficient at the wall and ebλ is Planck’s function. Under these assumptions, the appropriate boundary conditions for velocity involving slip flow, temperature and concentration fields are defined as √ K ∗ du∗ u∗ = u∗slip = , T ∗ = Tw , C ∗ = Cw at y = 0, (8) α1 dy ∗ u∗ → 0, T ∗ → T∞ , C ∗ → C∞ as y → ∞, (9) where Cw and Tw are the wall dimensional concentration and temperature, respectively. K ∗ is the permeability of the porous medium and α1 is the constant. Introducing the following non-dimensional quantities: v0 y ∗ , ν K ∗ v02 K= , v2 y=
θ=
T ∗ − T∞ Tw − T∞
u∗ ρgβ ∗ ν 2 (Cw − C∞ ) ρgβν 2 (Tw − T∞ ) σB02 v 2 , Gm = , M2 = , , Gr = 3 3 v0 v0 µ v0 µ v02 µ 4νI ′ v02 µcp Q0 ν Q∗ ν(Cw − C∞ ) , F = , E= , Pr = , φ= , Q1 = 12 2 2 α ρcp v0 ρcp v0 v0 (Tw − T∞ ) cp (Tw − T∞ ) (10) ∗ ∗ R ν C − C∞ ν , C= , Sc = , γ= 2 . Cw − C∞ D v0 u=
Using (7) and (10) in eqs. (4)–(6), we get the following non-dimensional equations: d2 u du 1 2 + Grθ + Gmφ − M + + u = 0, dy 2 dy K 2 du dθ d2 θ + Pr E − Pr (F + φ) θ + Pr EM 2 u2 + Q1 C = 0, + Pr dy 2 dy dy d2 C dC − ScγC = 0, + Sc dy 2 dy
(11) (12) (13)
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where Gr is the Grashof number, Gm is the solutal Grashof number, Pr is the Prandtl number, M is the magnetic field parameter, F is the radiation parameter, Sc is the Schmidt number, E is the Eckert number, φ is the heat source parameter, γ is the chemical reaction parameter, K is the permeability parameter and Q1 is the absorption of radiation parameter. The corresponding boundary condition (8) and (9) in dimensionless form are du , dy θ → 0,
θ = 1,
u = uslip = h u → 0, where h =
√
2 K v0 α1 ν
C=1
C→0
as
at y = 0,
(14) (15)
y → ∞,
is the rarefaction parameter.
3 Method of solution It is difficult to obtain the closed-form solution of a set of coupled ordinary differential equations (11)–(13). However, these equations can be solved analytically after reducing them to a set of ordinary differential equations in dimensionless form. Thus we can represent the velocity u, temperature θ and concentration C in terms of power of the Eckert number E as in the flow of an incompressible fluid Eckert number is always less than unity, since the flow, due to the Joules dissipation, is superimposed on the main flow. Hence, we can assume u(y) = u0 (y) + Eu1 (y) + O(E 2 ), θ(y) = θ0 (y) + Eθ1 (y) + O(E 2 ), C(y) = C0 (y) + EC1 (y) + O(E 2 ).
(16) (17) (18)
Substituting (16)–(18) in eqs. (11)–(13) and equating the coefficient of zero-th powers of E (i.e. O(E 0 )), we get the following set of equations: u′′0 + u′0 + Grθ0 + GmC0 − pu0 = 0,
θ0′′ + Pr θ0′ − Pr(F + C0′′ + ScC0′ − ScγC0
(19)
φ)θ0 + Pr Q1 C0 = 0,
(20)
= 0.
(21)
Next, equating the coefficients of first order of E (i.e. O(E 1 )), we obtain u′′1 + u′1 + Grθ1 + GmC1 − pu1 = 0,
θ1′′ + Pr θ1′ − Pr(F + C1′′ + ScC1′ − ScγC1 where p = (M 2 +
1 K)
φ)θ1 +
Pr u20
(22) Pr M 2 u20
+
+ Pr Q1 C1 = 0,
(23)
= 0,
(24)
and the corresponding boundary conditions are
u0 = hu′0 , u0 → 0,
u1 = hu′1 , u1 → 0,
θ0 = 1, θ0 → 0,
θ1 = 0, θ1 → 0,
C0 = 1, C0 → 0,
C1 = 0 C1 → 0
at as
y = 0, y → ∞.
(25) (26)
Solving eqs. (19)–(24) with the help of boundary conditions (25) and (26), we get u0 = A7 e−A4 y + A5 e−A2 y + A6 e−A1 y , −A2 y
θ0 = (1 − A3 )e −A1 y
C0 = e
−A1 y
+ A3 e
(28)
,
(29)
,
−A4 y
u1 = A27 e
−A2 y
+ A15 e
−(A2 +A4 )y
+ A21 e
−A2 y
θ1 = A14 e C1 = 0,
(27)
+ A16 e
−2A2 y
+ A17 e
−2A4 y
+ A18 e
−(A1 +A2 )y
+ A19 e
−(A1 +A4 )y
+ A20 e
(30)
,
−2A1 y
+ A8 e
−2A1 y
−2A2 y
+ A9 e
where A’s are given in appendix A.
−2A4 y
+ A10 e
−(A1 +A2 )y
+ A11 e
−(A1 +A4 )y
+ A12 e
−(A2 +A4 )y
+ A13 e
,
(31) (32)
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0.45 Gr = Gr = Gr = Gr = Gr =
0.4 0.35 0.3
1.0 2.0 3.0 4.0 5.0
u
0.25 0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
y Fig. 2. Effect of Gr on velocity profiles against spanwise coordinate y.
The skin-friction coefficient, the Nusselt number and the Sherwood number are important physical parameters for this type of boundary layer flow which are defined and determined as follows: τw ∂u , Cfx = 2 = ρv0 ∂y y=0 Cfx = − [A4 A7 + A2 A5 + A1 A6 + E (A4 A27 + A2 A15 + 2A1 A16 + 2A2 A17 + 2A4 A18 +A19 (A1 + A2 ) + A20 (A1 + A4 ) + A21 (A2 + A4 ))] , ∂T ∗ ∂y ∗ y∗ =0 N ux ∂θ N ux = x , ⇒ =− Tw − T∞ Rex ∂y
(33)
y=0
N ux = [A2 (1−A3 )+A1 A3 + E (A2 A14 +2A1 A8 +2A2 A9 +2A4 A10 +A11 (A1 +A2 )+A12 (A1 +A4 )+A13 (A2 +A4 ))] , Rex (34) ∂C ∗ ∂y ∗ y∗ =0 Shx ∂C , Shx = x ⇒ = Cw − C∞ Rex ∂y y=0
Shx = −A1, Rex
where Rex =
v0 x ν
(35)
is the local Reynolds number.
4 Results and discussions The problem of the mixed convective boundary layer flow in a porous medium with Ohmic heating in the presence of magnetic field, absorption of radiation and thermal radiation using the classical model for the radiative heat flux has been considered. The solutions for velocity field and temperature field are obtained using the perturbation technique. Final results are computed for a variety of physical parameters which are presented by means of graphs. In order to get physical insight into the problem, the effects of various parameters encountered in the equations of the problem are analyzed on velocity, temperature and concentration fields with the help of figures. These results show the influence of the various physical parameters such as the thermal Grashof number Gr, solutal Grashof number Gm, magnetic field parameter M , porous permeability parameter K, Schmidt number Sc, heat absorption parameter φ, absorption of radiation parameter Q1 , chemical reaction parameter γ, thermal radiation parameter F and rarefaction parameter h on the velocity, temperature and the concentration profiles can be analyzed from figs. 2–18 and on skinfriction coefficient, local Nusselt number and local Sherwood number in tables 1–4. We can extract interesting insights
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0.7 Gm Gm Gm Gm Gm
0.6 0.5
= = = = =
1.0 2.0 3.0 4.0 5.0
u
0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
6
7
8
y Fig. 3. Effect of Gm on velocity profiles against spanwise coordinate y.
0.7 M M M M M
0.6 0.5
= = = = =
0.0 1.0 2.0 3.0 5.0
u
0.4 0.3 0.2 0.1 0 0
1
2
3
4
5
6
7
8
y Fig. 4. Effect of M on velocity profiles against spanwise coordinate y.
regarding the influence of all the parameters that govern this problem. In the present study following default parameter values are adopted for computations: Gr = 2.0, Gm = 2.0, M = 2.0, K = 0.5, h = 0.2, Pr = 0.7, E = 0.01, F = 2.0, φ = 1.0, Q1 = 2.0, Sc = 0.6 and γ = 0.5. All graphs and tables therefore correspond to these values unless specifically indicated on the appropriate graph and table. Figures 2 and 3 illustrate the influence of the thermal and solutal buoyancy force parameters Gr and Gm, respectively. The thermal Grashof number signifies the relative effect of the thermal buoyancy (due to density differences) force to the viscous hydrodynamic force in the boundary layer flow. It is found, from fig. 2, that an increase in Gr leads to an increase in the value of the velocity field. In addition to this fact, it is further noted that the peak value of velocity increases rapidly near the surface as the value of Gr decreases. The solutal Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. A similar phenomenon is seen in fig. 3, which shows that the velocity increases with an increase in the value of the solutal Grashof number Gm. This is due to the fact that boundary layer thickness increases with an increase in the value of Gr or Gm.
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0.35 K K K K K
0.3 0.25
= = = = =
0.5 1.0 1.5 2.0 3.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
y Fig. 5. Effect of K on velocity profiles against spanwise coordinate y.
0.35 h= h= h= h= h=
0.3 0.25
0.0 0.1 0.2 0.3 0.5
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
y Fig. 6. Effect of h on velocity profiles against spanwise coordinate y.
The effect of the magnetic field on velocity profiles in the boundary layer is depicted in fig. 4. The curve with magnetic parameter M = 0 corresponds to a non-MHD flow and in other four curves the magnetic parameter is taken in increasing order. The magnetic parameter is found to decelerate the velocity of the flow field to an appreciable amount, due to the magnetic pull of the Lorentz force acting on the flow field. It is interesting to note that the effect of the magnetic field is to decrease the value of the velocity profile throughout the boundary layer. The effect of the magnetic field is more prominently seen at the point of peak value, i.e. the peak value drastically decreases with an increase in the value of the magnetic field, this is because the presence of a magnetic field in an electrically conducting fluid introduces a force called the Lorentz force, which acts against the flow if the magnetic field is applied in the normal direction, as in the present problem. This type of resisting force slows down the fluid velocity as shown in this figure. The effect the reaction permeability parameter K on the velocity profiles is shown in fig. 5. As depicted in this figure, the effect of increasing the value of porous permeability is to increase the value of the velocity component in the boundary layer, due to the fact that the drag is reduced by increasing the value of the porous permeability on
Eur. Phys. J. Plus (2014) 129: 41
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0.35 Pr = Pr = Pr = Pr = Pr =
0.3 0.25
0.7 1.0 2.0 5.0 7.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
y Fig. 7. Effect of Pr on velocity profiles against spanwise coordinate y.
1 Pr = 0.7 Pr = 1.0 Pr = 2.0 Pr = 5.0 Pr =7.0
0.8
θ
0.6
0.4
0.2
0 0
1
2
3
4
5
6
y Fig. 8. Effect of Pr on temperature profiles against spanwise coordinate y.
the fluid flow, which results in an increased velocity. It is clearly seen that as K increases the velocity profiles across the boundary layer increases, since the resistance offered by the porous medium decreases as the permeability of the porous medium increases. The variation of the velocity profile with rarefaction parameter is represented in fig. 6. The curve with rarefaction parameter h = 0 corresponds to no slip flow and in other four curves the rarefaction parameter (slip flow) is taken in increasing order. It is evident from this figure that the velocity distribution increases rapidly near the plate and then decreases exponentially far away from the plate till it attains the minimum values as y → ∞. The effect of increasing the values of rarefaction parameter is to increase the velocity in the momentum boundary layer with formation of sharp peak near the surface. Thus the effect of h is more prominently observed very close to the surface which ultimately vanished far away from the plate. Figures 7 and 8 illustrate the velocity and temperature profiles for different values of the Prandtl number Pr. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing velocity (fig. 7). From fig. 8, it is observed that
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Eur. Phys. J. Plus (2014) 129: 41
0.35 F F F F F
0.3 0.25
= = = = =
1.0 2.0 3.0 4.0 5.0
u
0.2 0.15 0.1 0.05 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
y Fig. 9. Effect of F on velocity profiles against spanwise coordinate y.
1 F F F F F
0.8
= = = = =
1.0 2.0 3.0 4.0 5.0
θ
0.6
0.4
0.2
0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
y Fig. 10. Effect of F on temperature profiles against spanwise coordinate y.
an increase in the Prandtl number results in a decrease of the thermal boundary layer thickness and in a general lower average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from the heated plate more rapidly than for higher values of Pr. Hence in the case of smaller Prandtl numbers as the boundary layer is thicker and the rate of heat transfer is reduced. The effect of the radiation parameter F on the velocity and temperature is shown in figs. 9 and 10, respectively. The radiation parameter F defines the relative contribution of conduction heat transfer to thermal radiation transfer. It is obvious that an increase in the radiation parameter results in decreasing velocity within the boundary layer. From these figures one can see that there is a decrease in the value of horizontal velocity with an increase in the radiation parameter F , which shows the fact that an increase in the radiation parameter decreases the velocity in the boundary layer due to the decrease in the boundary layer thickness. It is seen from fig. 10 that the increase of the radiation parameter F leads to a decrease of the boundary layer thickness and thereby a decrease in the value of the heat transfer in the presence of thermal and solutal buoyancy force.
Eur. Phys. J. Plus (2014) 129: 41
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0.35
φ φ φ φ φ
0.3 0.25
= = = = =
0.0 1.0 2.0 3.0 4.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
y Fig. 11. Effect of φ on velocity profiles against spanwise coordinate y.
1
φ φ φ φ φ
0.8
0.0 1.0 2.0 3.0 4.0
θ
0.6
= = = = =
0.4
0.2
0 0
1
2
3
4
5
6
y Fig. 12. Effect of φ on temperature profiles against spanwise coordinate y.
For different values of the heat absorption parameter φ, the velocity and temperature profiles are plotted in figs. 11 and 12. It is observed that the velocity profile, by increasing the value of the heat absorption parameter φ and the boundary layer thickness, decreases with the increase in the absorption parameter as shown in fig. 11. A similar effect is seen by increasing the value of the absorption parameter, as depicted in fig. 12. This is due to the fact that the thermal boundary layer absorbs energy, which causes the temperature to fall considerably with increasing the value of internal heat absorption parameter φ. The influence of absorption of the radiation parameter Q1 on the dimensionless velocity and temperature field is shown in figs. 13 and 14, respectively. This is clearly observed for the case of increasing the value of the absorption of the radiation parameter, due to the increase in the buoyancy force which accelerates the flow rate, which is shown in fig. 13. From fig. 14, it is seen that the effect of absorption of radiation parameter is to increase temperature in the boundary layer as the radiated heat is absorbed by the fluid which is responsible for the increase in the temperature of the fluid very close to the porous boundary layer and its effect diminished far away from the porous boundary.
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Eur. Phys. J. Plus (2014) 129: 41
0.35 Q1 = Q1 = Q1 = Q1 = Q1 =
0.3 0.25
0.0 1.0 2.0 3.0 4.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
y Fig. 13. Effect of Q1 on velocity profiles against spanwise coordinate y.
1 Q1 = Q1 = Q1 = Q1 = Q1 =
0.8
1.0 2.0 3.0 4.0 5.0
θ
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
y Fig. 14. Effect of Q1 on temperature profiles against spanwise coordinate y.
Figures 15 and 16 illustrate the velocity and concentration profiles for different values of the Schmidt number Sc. The Schmidt number embodies the ratio of the momentum to the mass diffusivity. The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. It is found, from fig. 14, that at very low values of the Schmidt number (e.g., Sc = 0.2), there is increase in the peak velocity near the plate (y ≈ 0.8), whereas for higher values of the Schmidt number (e.g., Sc = 1.0), the peak shifts closer to the plate (y ≈ 0.4). It is observed that as the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity. The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers. Figure 17 depicts the effect of the chemical reaction parameter γ on the velocity profiles for generative chemical reaction. This figure shows that the velocity decreases with increasing the rate of chemical reaction γ. Hence an increase in the chemical reaction rate parameter leads to a fall in the momentum boundary layer. The effect the reaction rate parameter γ on the species concentration profiles for generative chemical reaction is shown in fig. 18. It is noticed
Eur. Phys. J. Plus (2014) 129: 41
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0.35 Sc Sc Sc Sc Sc
0.3 0.25
= = = = =
0.2 0.6 0.78 0.94 1.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
y Fig. 15. Effect of Sc on velocity profiles against spanwise coordinate y.
1 Sc = 0.2 Sc = 0.6 0.8
Sc = 0.78 Sc = 0.94 Sc = 1.0
C
0.6
0.4
0.2
0 0
1
2
3
4
5
6
7
8
y Fig. 16. Effect of Sc on concentration profiles against spanwise coordinate y.
that there is a marked effect of increasing the value of the chemical reaction rate parameter γ on the concentration distribution in the boundary layer. It is clearly observed from this figure that the concentration of species value of 1.0 at vertical plate decreases till it attains the minimum value of zero at the end of the boundary layer and this trend is seen for all the values of the reaction rate parameter. Further, it is observed that increasing the value of the chemical reaction decreases the concentration of species in the boundary layer, and this is due to the fact that a destructive chemical reduces the solutal boundary layer thickness and increases the mass transfer. From these two figures it is observed that the peak value attains near the porous boundary surface. In order to verify the accuracy of the present results, we have considered the analytical solutions obtained by Chaudhary et al. [25] and Pal and Talukdar [29] and we have computed the numerical results for skin-friction coefficient, local Nusselt number and local Sherwood number. These computed results are tabulated in table 1. It is interesting to observe from this table that the present results (under some limiting conditions) are in very good agreement with the computed results obtained from the analytical solutions of Chaudhary et al. [25] and Pal and Talukdar [29], which clearly shows the correctness of the present analytical solutions and computed results.
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Eur. Phys. J. Plus (2014) 129: 41
0.35
γ γ γ γ γ
0.3 0.25
= = = = =
0.0 0.2 0.5 1.0 2.0
u
0.2 0.15 0.1 0.05 0 0
1
2
3
4
5
6
7
8
y Fig. 17. Effect of γ on velocity profiles against spanwise coordinate y.
1
γ γ γ γ γ
0.8
0.0 0.2 0.5 1.0 2.0
C
0.6
= = = = =
0.4
0.2
0 0
1
2
3
4
5
6
7
8
y Fig. 18. Effect of γ on concentration profiles against spanwise coordinate y. Table 1. Comparison of the present results with those of Chaudhary et al. [25] and Pal and Talukdar [29] with different values of M for Cfx N ux /Rex and Shx /Rex . Chaudhary et al. [25] and Pal and Talukdar [29]
Present results (K = ∞, φ = 0, Q1 = 0, γ = 0, h = 0)
M
Cfx
N ux /Rex
Shx /Rex
Cfx
N ux /Rex
Shx /Rex
0.0
2.5542
1.5743
−0.6000
2.5542
1.5743
−0.6000
1.0
1.5620
1.5800
−0.6000
1.5620
1.5800
−0.6000
2.0
1.1218
1.5817
−0.6000
1.1218
1.5817
−0.6000
3.0
0.8752
1.5825
−0.6000
0.8752
1.5825
−0.6000
4.0
0.7177
1.5829
−0.6000
0.7177
1.5829
−0.6000
Eur. Phys. J. Plus (2014) 129: 41
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Table 2. Numerical values of the skin-friction coefficient, the local Nusselt number and the local Sherwood number. Gr
Gm
M
K
h
Cfx
N ux /Rex
Shx /Rex
2.0
2.0
2.0
0.5
0.2
0.7765
1.6280
−0.9245
4.0
2.0
2.0
0.5
0.2
1.1259
1.6266
−0.9245
2.0
4.0
2.0
0.5
0.2
1.2042
1.6259
−0.9245
2.0
2.0
4.0
0.5
0.2
0.4030
1.7690
−0.9245
2.0
2.0
2.0
1.0
0.2
0.8591
1.5859
−0.9245
2.0
2.0
2.0
0.5
0.4
0.5648
1.6280
−0.9245
Table 3. Numerical values of the skin-friction coefficient, the local Nusselt number and the local Sherwood number. Pr
F
φ
Q1
Cfx
N ux /Rex
Shx /Rex
0.7
2.0
1.0
2.0
0.7765
1.6280
−0.9245
1.0
2.0
1.0
2.0
0.7632
1.8468
−0.9245
0.7
3.0
1.0
2.0
0.7522
1.9457
−0.9245
0.7
2.0
2.0
2.0
0.7630
1.7965
−0.9245
0.7
2.0
1.0
4.0
0.8001
1.4165
−0.9245
Table 4. Numerical values of the skin-friction coefficient, the local Nusselt number and the local Sherwood number. Sc
γ
Cfx
N ux /Rex
Shx /Rex
0.6
0.5
0.7765
1.6280
−0.9245
0.78
0.5
0.7432
1.6688
−1.1263
0.6
1.0
0.7425
1.6697
−1.1307
1.0
1.0
0.6766
1.7775
−1.6180
Tables 2–4 show the effect of thermal Grashof number Gr, solutal Grashof number Gm, magnetic field parameter M , permeability parameter K, rarefaction parameter, Prandtl number Pr, radiation parameter F , heat source parameter φ, absorption of radiation parameter Q1 , Schmidt number Sc and chemical reaction rate parameter γ on skin-friction coefficient Cfx , Nusselt number N ux and Sherwood number Shx . From table 2, it can be seen that the effect of Gr or Gm is to increase the skin-friction coefficient Cfx , whereas an opposite effect is seen on the values of the skin-friction coefficient by increasing the values of M , K and h. It can be clearly observed that the values of the skin-friction coefficient Cfx decrease with the increase in Pr, F and φ and the increase in the values of Q1 . Further, it is analyzed that the values of the Nusselt number increase with an increase in the values of the parameters Pr, F and φ, whereas an opposite effect can be seen on the values of the Nusselt number by decreasing the values of Q1 (see table 3). Further it is found, from table 4, that the values of skin-friction coefficient Cfx decrease with the increase in Sc or γ, whereas an opposite effect can be seen on the values of the Nusselt number by decreasing the values of γ. It is also analyzed that the Sherwood number decreases with an increase in the values of the parameters Sc or γ.
5 Conclusions In this study, a numerical analysis is presented to investigate the influence of thermal radiation and magnetic field on the steady heat and mass transfer mixed convective boundary layer slip flow by taking into account homogeneous chemical reaction of first order, absorption of radiation in the presence of Ohmic dissipation and viscous dissipation. A perturbation technique has been introduced, which transforms the momentum energy and mass transfer equations into ordinary differential equations. These equations are then solved analytically and computed results are presented to illustrate the details of the flow, heat and mass transfer characteristics and their dependency on material parameters. From the computed results the following conclusions are drawn: 1) The effect of the magnetic field is to decelerate the velocity of the flow field to an appreciable amount throughout the boundary layer, whereas the velocity distribution across the boundary layer increases with increasing the values of the radiation parameter, since the thermal radiation enhances the convective flow.
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Eur. Phys. J. Plus (2014) 129: 41
2) It is found that the velocity, as well as the temperature, decreases with an increase in heat thermal radiation parameter and source parameter, whereas a reverse trend is seen by increasing the values of the thermal radiation parameter. 3) The velocity decreases with the increase in the values of chemical reaction parameter, Prandtl number and Schmidt number, whereas a reverse trend is seen with increasing the permeability parameter. 4) The effect of increasing the values of the rarefaction parameter is to increase the velocity in the momentum boundary layer with the formation of a sharp peak near the surface. 5) The concentration distribution decreases at all points of the flow field with increasing in the chemical reaction parameter and the Schmidt number. 6) It can also be concluded that the presence of chemical reaction and thermal radiation decreases the skin-friction coefficient, whereas the presence of porous medium and heat source increases the skin-friction coefficient. The author is very thankful to the editor and reviewers for their encouraging comments and constructive suggestions to improve the presentation of this manuscript.
Appendix A. √
Sc2
Pr + Pr2 +4 Pr(F + φ)
+ 4Sc Pr Q1 , , A2 = , A3 = 2 2 2 A1 − Pr A1 − Pr(F + φ) √ −(GrA3 + Gm) −(A5 (1 + hA2 ) + A6 (1 + hA1 )) −Gr(1 − A3 ) 1 + 1 + 4p A4 = , A6 = , A7 = , A5 = 2 , 2 A2 − A2 − p A21 − A1 − p (1 + hA4 ) − Pr A25 (A22 + M 2 ) − Pr A27 (A24 + M 2 ) − Pr A26 (A21 + M 2 ) A8 = , A , A , = = 9 10 4A21 − 2 Pr A1 − Pr(F + φ) 4A22 − 2 Pr A2 − Pr(F + φ) 4A24 − 2 Pr A4 − Pr(F + φ) −2 Pr A6 A7 (A1 A4 + M 2 ) −2 Pr A5 A6 (A1 A2 + M 2 ) A11 = , A = , 12 (A1 + A2 )2 − Pr(A1 + A2 ) − Pr(F + φ) (A1 + A4 )2 − Pr(A1 + A4 ) − Pr(F + φ) −2 Pr A5 A7 (A2 A4 + M 2 ) , A14 = −(A8 + A9 + A10 + A11 + A12 + A13 ), A13 = (A2 + A4 )2 − Pr(A2 + A4 ) − Pr(F + φ) −GrA14 −GrA8 −GrA9 −GrA10 , A16 = , A17 = , A18 = , A15 = 2 A2 − A2 − p 4A21 − 2A1 − p 4A22 − 2A2 − p 4A24 − 2A4 − p −GrA11 −GrA12 −GrA13 A19 = , A20 = , A21 = , (A1 + A2 )2 − (A1 + A2 ) − p (A1 + A4 )2 − (A1 + A4 ) − p (A2 + A4 )2 − (A2 + A4 ) − p A22 = A19 (A1 + A2 ), A23 = A20 (A1 + A4 ), A24 = A21 (A2 + A4 ), A25 = 2(A1 A16 + A2 A17 + A4 A18 ), −(h(A2 A15 + A25 + A22 + A23 + A24 ) + A26 ) . A26 = A15 + A16 + A17 + A18 + A19 + A20 + A21 , A27 = (1 + hA4 ) A1 =
Sc +
References 1. T.S. Chen, B.S. Armaly, Mixed Convection in External Flow (Wiley, 1987). 2. I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media (Oxford Pergamon Press, 2001). 3. V.M. Soundalgekar et al., Nucl. Eng. Res. 53, 309 (1979). 4. E.M.A. Elbasheshy, Int. J. Eng. Sci. 34, 515 (1997). 5. M. Gnaneswara Reddy, N. Bhaskar Reddy, J. Appl. Fluid Mech. 4, 7 (2011). 6. P.K. Sharma, R.C. Chaudhury, Emirates J. Eng. Sci. 8, 33 (2003). 7. T. Hayat et al., Acta Mech. 153, 133 (2002). 8. R. Siegel, J.R. Howell, Thermal Radiation Transfer, 3rd edition (Hemisphere, 1992). 9. M.F. Modest, Radiative Heat Transfer, 2nd edition (Academic Press, 2003). 10. M.A. Hosssain, H.S. Takhar, Heat Mass Transfer 31, 243 (1996). 11. M. Aboeldahab Emad, J. Phys. D. Appl. Phys. 33, 3180 (2000). 12. E.M. Abo-Eldahab, M.S. Elgendy, Phys. Scr. 62, 321 (2000). 13. M.A. Seddeek, Int. J. Heat Mass. Transfer 45, 931 (2002). 14. O. Aydin, A. Kaya, Heat Mass Transfer 45, 239 (2008). 15. M. Gnaneswara Reddy, Brit. J. Eng. Technol. 1, 17 (2012).
Eur. Phys. J. Plus (2014) 129: 41 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
M. Gnaneswara Reddy, to be published in Ain Shams Eng. J. (2013). U.N. Das et al., Forsch. Ing. 60, 284 (1994). R. Kandasamy et al., Int. J. Heat Mass Transfer 48, 4557 (2005). C.H. Chen, Int. J. Eng. Sci 42, 699 (2004). A.Y. Ghaly, M.A. Seddeek, Chaos Solitons Fractals 19, 61 (2004). D. Pal, B. Talukdar, Commun. Nonlinear Sci. Numer. Simulat. 15, 1813 (2010). M. Gnaneswara Reddy, Ain Shams Eng. J. 4, 879 (2013). M.A. Hossain, Int. J. Heat Mass Transfer 35, 3485 (1992). E.M. Abo-Eldahab, M. Abd El-Aziz, Int. Commun. Heat Mass Transfer 31, 751 (2004). R.C. Chaudhary et al., Rom. J. Phys. 51, 715 (2006). M. Abd El-Aziz, Meccanica. 42, 375 (2007). D. Pal, H. Mondal, Commun. Nonlinear Sci. Numer. Simulat. 15, 1553 (2010). A.C. Cogley et al., AIAA J. 6, 551 (1968). D. Pal, B. Talukdar, Commun. Nonlinear Sci. Numer. Simulat. 15, 2878 (2010).
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