Radiative Heat Transfer Flow of Micropolar Fluid with Variable Heat Flux in a Porous Medium

Abstract. A two-dimensional steady convective flow of a micropolar flui d past a vertical porous flat plate in the presence of radiation with v ariable heat flux has been analyzed numerically. Using Darcy-Forchheimer model the c orresponding momentum, microrotation and energy equations have been solved numeri cally. The local similarity solutions for the flow, microrotation and heat transfer char acteristics are illustrated graphically for various material parameters. The effects o f he pertinent parameters on the local skin friction coefficient, plate couple stress and the heat transfer are also calculated. It was shown that large Darcy parameter leads to decrease the v locity while it increases the angular velocity as well as temperature of the micropola r fluids. The rate of heat transfer in weakly concentrated micropolar fluids is higher than strongly concentrated micropolar fluids.


Introduction
A micropolar fluid is the fluid with internal structures in which coupling between the spin of each particle and the macroscopic velocity field is taken into account.It is a hydrodynamical framework suitable for granular systems which consist of particles with macroscopic size.The dynamics of micropolar fluids, originated from the theory of Eringen [1] has been a popular area of research.This theory may be applied to explain the flow of colloidal suspensions (Hadimoto and Tokioka [2]), liquid crystals (Lockwood et al. [3]), polymeric fluids, human and animal blood (Ariman et al. [4]) and many other situations.
Ahmadi [5] presented solutions for the flow of a micropolar fluid past a semi-infinite plate taking into account microinertia effects.Soundalgekar and Takhar [6] studied the flow and heat transfer past a continuously moving plate in a micropolar fluid.Gorla [7] studied mixed convection in a micropolar fluid from a vertical surface with uniform heat flux.Rees and Pop [8] studied free convection boundary layer flow of micropolar fluids from a vertical flat plate while Mohammadein and Gorla [9] studied the same flow bounded by stretching sheet with prescribed wall heat flux, viscous dissipation and internal heat generation.Aissa and Modammadein [10] studied joule heating effects on a micropolar fluid past a stretching sheet with variable electric conductivity.Hassanien et al. [11] studied unsteady magnetohydrodynamic micropolar fluid flow and heat transfer over a vertical porous plate through a porous medium in the presence of thermal and mass diffusion with a constant heat source.Rahman and Sattar [12] studied magnetohydrodynamic convective flow of a micropolar fluid past a vertical porous plate in the presence of heat generation/absorption.
Markin and Mahmood [13] obtained similarity solutions for the mixed convection flow over a vertical plate for the case of constant heat flux condition at the wall.Perdikis and Raptis [14] studied heat transfer of a micropolar fluid in the presence of radiation.Later Raptis [15] studied the same fluid flow past a continuously moving plate in the presence of radiation.Chen and Char [16] have studied the suction and injection on a linearly moving plate subject to uniform wall temperature and heat flux.El-Arabawy [17] studied the effect of suction/injection on a micropolar fluid past a continuously moving plate in the presence of radiation.Ogulu [18] studied the oscillating plate-temperature flow of a polar fluid past a vertical porous plate in the presence of couple stresses and radiation.Recently, Rahman and Sattar [19] studied transient convective heat transfer flow of a micropolar fluid past a continuously moving vertical porous plate with time dependent suction in the presence of radiation.
In the present work we investigate the thermal radiation interaction of the boundary layer flow of micropolar fluid past a heated vertical porous plate embedded in a porous medium with variable suction as well as heat flux at the plate.The local similarity solutions are then obtained numerically for various parameters entering into the problem using shooting method and discussed the results from the physical point of view.

Mathematical formulation
Let us consider a steady two-dimensional flow of a viscous, incompressible micropolar fluid of temperature T ∞ past a heated vertical porous flat plate immersed in a porous medium and there is a suction velocity v 0 (x) at the plate.Consider a variable surface heat flux such that the temperature at the surface of the plate is proportional to x m (x measures the distance from the leading edge along the surface of the plate and m is a constant).The flow is assumed to be in the x-direction, which is taken along the plate in the upward direction and y-axis is normal to it.The flow configuration and the coordinate system are shown in the Fig. 1.
The fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation.The radiative heat flux in the x-direction is considered negligible in comparison to the y-direction.
Within the framework of the above-noted assumptions, we assume that the Boussinesq and boundary layer approximations hold and using the Darcy-Forchheimer model, the governing equations relevant to the problem in the presence of radiation are given by: where u, v are the velocity components along x, y co-ordinates respectively, ν a = µ+S ρ is the apparent kinematic viscosity, µ is the coefficient of dynamic viscosity, S is the microrotation coupling coefficient (also known as the coefficient of gyro-viscosity or as the vortex viscosity), ρ is the mass density of the fluid, σ is the microrotation component normal to the xy-plane, b is the empirical constant, K ′ is the Darcy permeability, ν s = (µ + S 2 )j (see Rees and Bassom [20]) is the microrotation viscosity or spin-gradient viscosity, j is the micro-inertia density, T is the temperature of the fluid in the boundary layer, T ∞ is the temperature of the fluid outside the boundary layer, U ∞ is the velocity of the fluid far away from the plate, c p is the specific heat of the fluid at constant pressure, k is the thermal conductivity, q r is the radiative heat flux, g 0 is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion.
By using Rosseland approximation (see Raptis [15]), q r takes the form where σ 1 , the Stefan-Boltzmann constant and k 1 is the mean absorption coefficient.We assume that the temperature differences within the flow are sufficiently small such that T 4 may be expressed as a linear function of temperature.This is accomplished by expanding T 4 in a Taylor's series about T ∞ and neglecting higher-order terms, thus By using ( 5) and ( 6) equation ( 4) gives The corresponding boundary conditions for the above problem are given by, Positive and negative values for v 0 indicate blowing and suction respectively, while v 0 = 0 corresponds to an impermeable plate.When microrotation parameter n = 0 we obtain σ = 0 which represents no-spin condition i.e. the microelements in a concentrated particle flow-close to the wall are not able to rotate as stated by Jena and Mathur [21].
The case n = 0.5 represents vanishing of the anti-symmetric part of the stress tensor and represents weak concentration.For this case Ahmadi [5] suggested that in a fine particle suspension the particle spin is equal to the fluid velocity at the wall.The case corresponding to n = 1 be representative of turbulent boundary layer flows (see Peddison and McNitt [22]).In equation ( 8), the last condition represents non-uniform heat flux at the surface of the plate.

Transformations of the model
In order to obtain local similarity solution of the problem we adopted the following nondimensional variables which have been used by many authors in the literature (see for example [12,15,17]): where ψ is the stream function.Since u = ∂ψ ∂y and v = − ∂ψ ∂x we have from equation ( 9) that Here f is non-dimensional stream function and prime denotes differentiation with respect to η.
The thermal boundary conditions depend on the type of heating process being considered, that is the prescribed heat flux.Define the temperature distribution as follows where D is a constant and L is some characteristic length.Now substituting equations ( 9)-( 11) into equations ( 2), ( 3) and ( 7) we obtain the following non-dimensional ordinary differential equations which are locally similar as some of the parameters depend on the local coordinate x, where K = S ρνa is the coupling parameter, γ = ( is the local buoyancy parameter, Da is the Darcy parameter, Da = K ′ U∞ νax is the modified Darcy number, Fs = bx K ′ is the modified Forchheimer number, G 1 = Sx ρjU∞ is the vortex viscosity parameter, G 2 = νs ρjνa is the spin gradient viscosity parameter, P n is the Prandtl number.
The transformed thermal boundary conditions may be written as But as θ ′ (0) must be equal to −1, this implies (as could be concluded from equation ( 15)) that m = 2 and b 0 should satisfy the relation b 0 = D L 2 .Then the corresponding boundary conditions (8) become, where νaU∞ is the suction velocity at the plate for V 0 > 0.

Skin friction coefficient, plate couple stress and Nusselt number
The quantities of chief physical interest are the skin friction coefficient, plate couple stress and the Nusselt number (rate of heat transfer).The equation defining the wall shear stress is The local skin friction coefficient is defined as Thus from equation (18) we see that the local values of the skin friction coefficient C f is proportional to f ′′ (0).The equation defining the plate couple stress is The dimensionless couple stress is defined by Thus the local couple stress in the boundary layer is proportional to g ′ (0).We may define a non-dimensional coefficient of heat transfer, which is known as Nusselt number as follows: where h(x) = qw(x) Tw−T∞ and q w (x) = −κ( ∂T ∂y ) y=0 − 4σ1 3κ1 ( ∂T 4 ∂y ) y=0 is the quantity of heat transferred through unit area.Now the rate of heat transfer, in terms of the dimensionless Nusselt number, given by Thus from equation (22) we see that the local Nusselt number Nu x is reciprocal to θ(0).Hence the numerical values proportional to C f , M x and Nu x are calculated from equations (18), (20) and (22) are shown in Figs.10-12.

Numerical solution
The set of nonlinear ordinary differential equations ( 12)-( 14) with boundary conditions (16) have been solved by using sixth order Runge-Kutta method along with Nachtsheim-Swigert [23] shooting iteration technique with V 0 , γ, K, λ, Fs, G 1 , G 2 , P n and n as prescribed parameters.A step size of ∆η = 0.001 was selected to be satisfactory for a convergence criterion of 10 −6 in all cases.The value of η ∞ was found to each iteration loop by the statement η ∞ = η ∞ + ∆η.The maximum value of η ∞ , to each group of parameters V 0 , γ, K, λ, Fs, G 1 , G 2 , P n and n is determined when the value of the unknown boundary conditions at η = 0 not change to successful loop with error less than 10 −6 .
In order to verify the effects of the step size ∆η we ran the code for our model with three different step sizes as ∆η = 0.01, ∆η = 0.001, and ∆η = 0.005 and in each case we found excellent agreement among them.Fig. 2(a)-(c), respectively, shows the velocity, microrotation and temperature profiles for different step sizes.To assess the accuracy of our code, we reproduced the values of f ′′ (0), g ′ (0) and −θ ′ (0) which are proportional respectively to the local skin friction coefficient, plate couple stress and the rate of heat transfer coefficient considering the model of El-Arabawy [17].Tables 1-3 show the comparison of the data produced by our code and that of El-Arabawy.In fact the results show a close agreement, hence an encouragement for the use of the present code for our model.

Results and discussion
The present work generalized the problem of heating effects on a boundary layer of a micropolar fluid over the porous plate with variable heat flux in the presence of radiation.For the purpose of discussing the results, the numerical calculations are presented in the form of non-dimensional velocity, microrotation and temperature profiles.In the calculations the values of the local buoyancy parameter γ, coupling parameter K, suction parameter V 0 , radiative Prandtl number P n, Darcy parameter λ, Forchheimer number Fs, vortex viscosity parameter G 1 , and spin-gradient viscosity parameter G 2 are chosen arbitrarily to carryout the parameter survey.The case γ ≫ 1 corresponds to free convection, γ = 1 corresponds to mixed convection and γ ≪ 1 corresponds to forced convection.The existence of the free convection current is due to the difference in temperature of the plate T w and the temperature in the uniform flow, viz.T w − T ∞ .Hence the difference may be positive, zero or negative depending upon T w > T ∞ .Then the values of γ will assume positive, zero or negative values.Physically γ < 0 corresponds to as externally heated plate as the free convection currents are carried towards the plate and γ > 0 corresponds to an externally cooled plate while γ = 0 corresponds to the absence of free convection currents.Since we are considering cooling problem, only positive values of γ are chosen.Fig. 3(a) shows the velocity profiles for different values of suction parameter V 0 for a cooling plate.It can be seen that for cooling of the plate the velocity profiles decrease monotonically with the increase of suction parameter indicating the usual fact that suction stabilizes the boundary layer growth.The free convection effect is also apparent in this figure.For a fixed suction velocity V 0 , velocity is found to increase and reaches a maximum value in a region close to the plate, then gradually decreases to one.Fig. 3(b) shows the microrotation (angular velocity) profiles for different values of suction parameter.The angular velocity g remains negative and decreases with the increase of V 0 close to the plate where viscosity is dominant.After a short distance from the plate (η ≈ 0.2) these profiles overlap and then increase with the increase of V 0 .Within the boundary layer region these profiles increase from − 1 2 f ′′ (0) to zero as η increases from zero to infinity.Fig. 3(c) indicates the temperature profiles showing the effect of suction parameter V 0 .It can be seen that temperature decreases with the increase of suction.Decelerated fluid particles close to the heated wall absorb more heat from the plate as a consequence the temperature of the fluid within the boundary layer increases.But when these decelerated fluid particles are sucked through the porous plate there is decease to the temperature profile.
Fig. 4(a) shows the velocity profiles for different values of coupling parameter K. From this figure we see that for cooling plate velocity increases with the increase of the coupling parameter K. Fig. 4(b) shows the effect of coupling parameter in the microrotation profiles.From this figure we observe that microrotation decreases monotonically as the increase of K. Fig. 4(c) shows the decreasing effect of coupling parameter on the temperature profiles.
The effect of modified Forchheimer number Fs on the velocity field is shown in Fig. 5(a).From this figure we see that velocity decreases with the increase of Fs.Fig. 5(b) shows the effect of in the microrotation profiles.From this figure we observe that Fs has increasing effect on the microrotation profiles.Forchheimer number has also increasing effect on the temperature profiles which is shown in the Fig. 5(c).
The effect of the local Darcy parameter λ on the velocity field is shown in the Fig. 6(a).From this figure we observe that velocity profiles decrease with the increase of λ.Darcy number is the measurement of the porosity of the medium, as the porosity of the medium increases, the value of Da increases and hence λ decreases.For large porosity of the medium fluid gets more space to flow, consequently fluid velocity increases.The case λ = 0 corresponds to a pure fluid, rather than a porous medium.Fig. 6(b) shows that microrotation increases with the increase of Darcy parameter λ.The effect of λ on the temperature profiles is shown in Fig. 6(c).From here we see that temperature profiles increase with the increase of Darcy parameter.Fig. 7(a) shows the effect of vortex viscosity parameter G 1 on the velocity profiles for cooling plate.From here we see minor decreasing effect of G 1 on the velocity profiles.Fig. 7(b) shows the effect of G 1 on the microrotation profiles.From this figure we see that microrotation increases very rapidly with the increase of the vortex viscosity parameter G 1 .It is also understood that as the vortex viscosity increases the rotation of the micropolar constituents gets induced in most part of the boundary layer except very close to the wall where kinematic viscosity dominates the flow.From Fig. 7(c) we found no effect of G 1 on temperature profiles.
In Fig. 8(a)-(c), respectively, we have varied the spin-gradient viscosity parameter G 2 keeping all other parameters value fixed.From Fig. 8(a) we see that velocity profiles increase slowly with the increase of spin-gradient viscosity parameter G 2 .The effect of G 2 is significant on the microrotation profiles.From Fig. 8(b) we see that microrotation decreases very rapidly with the increase of the spin-gradient viscosity parameter.From Fig. 8(c) we see that the values of G 2 introduce no effect on the temperature profiles.
The above-mentioned calculations have been done for a weakly concentrated micropolar fluid i.e. for the microrotation parameter n = 0.5.Here we have varied the microrotation parameter n, keeping all other parameters value fixed.Fig. 9(a) reveals that as n increases the concentration of the fluid decreases hence fluid velocity increases.Fig. 9(b) shows the microrotation profiles for different values of n.Microrotation decreases significantly with the increase of n.We observe that g increases from −nf ′′ (0) to zero as η increases from zero to infinity.From Fig. 9(c) we observe that temperature decreases as we go from strong concentration fluid to weak concentration fluid.
In Fig. 10(a)-(c), respectively, we have presented the physical parameters skin friction coefficients, plate couple stress and the rate of heat transfer for different values of the local buoyancy parameter γ and suction parameter V 0 .From these figures we see that skin friction C f , plate couple stress M x and Nusselt number Nu x increases rapidly with the increase of γ as well as with V 0 .
Fig. 11(a)-(c), respectively, show the skin friction coefficients, plate couple stress and the rate of heat transfer for different values of coupling parameter K and Darcy parameter λ.Here we found skin friction, plate couple stress and Nusselt number increase with the increase of K and decrease with the increase of λ.
The effect of microrotation parameter n and vortex viscosity parameter G 1 on the skin friction, plate couple stress and the rate of heat transfer are shown in Fig. 12(a)-(c), respectively.In these figures we found that skin friction, plate couple stress and Nusselt number increases with the increase of n for a fixed value of G 1 .From these figures it is also clear that C f , M x and Nu x increases with the increase of G 1 except for the case n = 0.For n = 0, C f and M x decreases while Nu x oscillates with the increase of G 1 .

Conclusions
In this paper, we have investigated numerically the radiative heat transfer flow of micropolar fluid past a vertical permeable flat plat embedded in a porous medium varying surface heat flux.Using usual similarity transformations the governing equations have been transformed into non-linear ordinary differential equations and were solved for similar solutions by using Nachtsheim-Swigert shooting iteration technique.Effects of the various parameters such as the local buoyancy parameter γ, coupling parameter K, suction parameter V 0 , local Darcy parameter λ, local Forchheimer number Fs, local vortex viscosity parameter G 1 , and spin-gradient viscosity parameter G 2 on the flow, microrotation, and temperature profiles are examined.The following conclusions can be drawn as a result of the numerical computations: 1. Skin friction coefficient increases with the increase of suction parameter, buoyancy parameter, coupling parameter, non-zero microrotation parameter and spin gradient viscosity parameter except for no spin condition, n = 0.This coefficient decreases with the increase of the Darcy parameter.
2. Plate couple stress increases with the increase of suction parameter, buoyancy parameter, coupling parameter, and non-zero microrotation parameter.
3. The rate of heat transfer increases with the increase of the suction parameter, buoyancy parameter, coupling parameter and non-zero microrotation parameter.
4. Large Darcy parameter leads to the decrease of the velocity profiles and increase of the microrotation as well as temperature profiles.
5. Radiation has significant decreasing effect on the velocity and temperature field while it has increasing effect on the angular velocity.