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ISRN Thermodynamics
Volume 2012 (2012), Article ID 146263, 8 pages
http://dx.doi.org/10.5402/2012/146263
Research Article

Magnetohydrodynamics and Radiation Effects on Unsteady Convection Flow of Micropolar Fluid Past a Vertical Porous Plate with Variable Wall Heat Flux

Department of Mathematics, Acharya Nagarjuna University Ongole Campus, Ongole 523 001, AP, India

Received 28 March 2012; Accepted 19 April 2012

Academic Editors: J. K. Brennan, A. Ghoufi, and H. Hirao

Copyright © 2012 M. Gnaneswara Reddy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An analysis is presented for the problem of the unsteady two-dimensional laminar flow of a viscous incompressible micropolar fluid past a vertical porous plate in the presence of a transverse magnetic field and thermal radiation with variable heat flux. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which it absorbs the micropolar fluid with a suction velocity varying with time. The Rosseland approximation is used to describe radiative heat transfer in the limit of optically thick fluids. The effects of flow parameters and thermophysical properties on the flow temperature fields across the boundary layer are investigated. The method of solution can be applied for small perturbation approximation. Numerical results of velocity profiles of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient and the couple stress coefficient at the wall are prepared with various values of the fluid properties.

1. Introduction

The classical Navier-Stokes theory does not describe adequately the flow properties of polymeric fluids, colloidal suspensions, and fluids containing certain additives. Eringen [1] proposed the theory of micropolar fluids which shows microrotation effects as well as microinertia. The theory of thermomicropolar fluids was developed by Eringen [2] by extending his theory of micropolar fluid. Physically, they represent fluids consisting of randomly oriented particles suspended in a viscous medium. The flow characteristics of the boundary layer of micropolar fluid over a semi-infinite plate in different situations have been studied by many authors in (Ahmadi [3], Peddieson and McNitt [4], Takhar and Soundagekhar [5], and Gorla [6]). The study of the flow and heat transfer for a micropolar fluid past porous plate embedded in a porous medium has attracted the interest of many investigators in view of its applications in many engineering problems such as oil exploration, chemical catalytic reactors, thermal insulation, and geothermal energy extractions (Raptis [7]). Sharma and Gupta [8] considered thermal convection in micropolar fluids in porous medium. Kim [9] presented an analysis of an unsteady convection flow of a micropolar fluid past a vertical porous plate embedded in a porous medium.

The study of magnetohydrodynamics (MHD) plays an important role in agriculture, engineering, and petroleum industries. The problem of free convection under the influence of a magnetic field has attracted the interest of many researchers in view of its applications in geophysics and astrophysics. In addition, this type of flow has attracted the interest of many investigators in view of its applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, and geothermal energy extractions. Kim [10] studied unsteady MHD convection flow of polar fluids past a semi-infinite vertical-moving porous plate in a porous medium. Soundalgekar [11] obtained approximate solutions for the two-dimensional flow of an incompressible, viscous fluid past an infinite porous vertical plate with constant suction velocity normal to the plate, the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents. Raptis [12] studied mathematically the case of unsteady two-dimensional natural convective heat transfer of an incompressible, electrically conducting viscous fluid via a highly porous medium bound by an infinite vertical porous plate. El-Hakiem et al. [13] studied the effect of viscous and Joule heating on MHD-free convection flow with variable plate temperature in a micropolar fluid in the presence of uniform transverse magnetic field using the Kellerbox implicit scheme. Rahman and Sattar [14] analyzed magnetohydrodynamic convective flow of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption.

In the context of space technology and in the processes involving high temperatures, the effects of radiation are of vital importance. Recent developments in hypersonic flights, missile reentry, rocket combustion chambers, power plants for interplanetary flight, and gas-cooled nuclear reactors, have focused attention on thermal radiation as a mode of energy transfer and emphasized the need for improved understanding of radiative transfer in these processes. Raptis [15] studied the flow of a micropolar fluid past a continuous moving plate by the presence of radiation. Abo-Eldahab and Ghonaim [16] analyzed the radiation effects on heat transfer of a micropolar fluid through a porous medium. Kim and Fedorov [17] considered the case of mixed convection flow of a micropolar fluid past a semi-infinite moving vertical porous plate with varying suction velocity normal to the plate in the presence of radiation.

The purpose of this work is to study radiation effects on unsteady Magnetohydrodynamics (MHD) mixed convection flow of micropolar fluid with variable heat flux has received a little attention. We also consider the free stream to consist of a mean velocity over which is superimposed an exponentially varying with time. The effects of various governing parameters on the velocity, microrotation and temperature, skin-friction coefficient, and the wall couple stress coefficient are shown in figures and table and discussed in detail.

2. Mathematical Analysis

An unsteady two-dimensional mixed convection flow of a laminar incompressible micropolar fluid past a vertical porous plate in the presence of a transverse magnetic field and thermal radiation with variable heat flux is considered. The 𝑥-axis is taken along the vertical porous plate in an upward direction and 𝑦-axis is taken normal to the plate. The acceleration of gravity 𝑔 is in a direction opposite to 𝑥-coordinate. The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible. Viscous and Darcy resistance terms are taken into account as the constant permeability porous medium. It is also assumed here that the size of holes in the porous plate is significantly larger than a characteristic microscopic length scale of the micropolar fluid to simplify formulation of the boundary conditions. Due to the assumption of infinite plate, the flow variables are functions of 𝑦 and the time 𝑡 only. It is also assumed that the free stream to consist of a mean velocity and temperature over which are superimposed an exponentially varying with time. Under the usual Boussenesq’s approximation, the equation of continuity, linear momentum, microrotation, and energy can be written as

continuity equation: 𝜕𝑣𝜕𝑦=0,(1)

linear momentum equation: 𝜕𝑢𝜕𝑡+𝑣𝜕𝑢𝜕𝑦1=𝜌𝜕𝑝𝜕𝑥+𝜈+𝜈𝑟𝜕2𝑢𝜕𝑦2+𝑔𝛽𝑓𝑇𝑇𝜈𝐾𝑢𝜎𝐵20𝜌𝑢+2𝜈𝑟𝜕𝜔𝜕𝑦,(2)

angular momentum equation: 𝜌𝑗𝜕𝜔𝜕𝑡+𝑣𝜕𝜔𝜕𝑦𝜕=𝛾2𝜔𝜕𝑦2,(3)

energy equation: 𝜕𝑇𝜕𝑡+𝑣𝜕𝑇𝜕𝑦𝜕=𝛼2𝑇𝜕𝑦21𝑘𝜕𝑞𝑟𝜕𝑦.(4) The third term on the right-hand side of the momentum equation (2) denotes thermal buoyancy effect.

By using the Rosseland approximation, Brewster [18], the radiative heat flux in the 𝑦 direction is given by 𝑞𝑟=4𝜎𝑠𝜕𝑇43𝑘𝑒𝜕𝑦,(5) where 𝜎𝑠 is the Stefan-Boltzmann constant and 𝑘𝑒 is the mean absorption coefficient, respectively.

It should be noted that by Rosseland approximation, we limit our analysis to optically thick fluids. If the temperature differences within the flow are sufficiently small, then (4) can be linearized by expanding 𝑇4 into the Taylor series about 𝑇 and neglecting higher terms to take the form: 𝑇44𝑇3𝑇3𝑇4.(6) It is assumed that the porous plate moves with constant velocity in the longitudinal direction, and the free steam velocity follows an exponentially increasing or decreasing small perturbation law. It is also assumed that the plate temperature and suction velocity vary exponentially with time.

Under these assumptions, the appropriate boundary conditions for the velocity, microrotation, temperature, and concentration fields are 𝑢=0,𝜔1=2𝜕𝑢𝜕𝑦,𝜕𝑇𝜕𝑦𝑞=𝑤𝑘1+𝜀𝐵𝑒𝑛𝑡at𝑦𝑢=0,𝑈=𝑈01+𝜀𝑒𝑛𝑡,𝜔0,(7)𝑇𝑇as𝑦,(8) where 𝐵 is a real positive constant and 𝜀 is a small parameter such that 𝜀𝐵1, 𝑇 is the free stream dimensional temperature, 𝑛 is constant, 𝜀 is small less than unity and 𝑈0 is a scale of free steam velocity. The boundary condition for microrotation variable 𝜔 describes its relationship with the surface stress.

It is clear from (1) that the suction velocity normal to the plate is either a constant or a function of time. Hence it is assumed in the form: 𝑣=𝑉01+𝜀𝐴𝑒𝑛𝑡,(9) where 𝐴 is a real constant, 𝜀 and 𝜀𝐴 are small less than unity, and 𝑉0 is the scale of suction velocity which is a nonzero positive constant. The negative sign indicates that the suction is towards the plate. Outside the boundary layer, (2) gives 1𝜌𝑑𝑝𝑑𝑥=𝑑𝑈𝑑𝑡+𝜈𝐾𝑈+𝜎𝐵20𝜌𝑈(10) In order to write the governing equations and the boundary conditions dimensionless form, the following nondimensional quantities are introduced 𝑢𝑢=𝑈0𝑣,𝑣=𝑉0𝑉,𝑦=0𝑦𝜈,𝑈=𝑈𝑈0𝜈,𝜔=𝑈0𝑉0𝜔𝑡,𝑡=𝑉20𝜈,𝜃=𝑇𝑇𝑘𝑉0𝑇𝑤𝑇𝑞𝑤𝑉,𝑗=20𝜈2𝑗,𝑛𝑛=𝑉20𝜈,Gr=𝜈𝛽𝑓𝑔𝑇𝑤𝑇𝑈0𝑉20,(11)Pr=𝜈𝜌𝐶𝑝𝑘=𝜈𝛼=𝜇𝐶𝑝𝑘,𝑅=𝑘𝑘𝑒4𝜎𝑠𝑇3.(12) Furthermore, the spin-gradient viscosity 𝛾, which gives some relationship between the coefficients of viscosity and microinertia, is defined as Λ𝛾=𝜇+2𝑗=𝜇𝑗11+2𝛽Λ,𝛽=𝜇.(13) In view of (5), (6) and (9)–(13), the governing equations (2)–(4) reduce to the following nondimensional form: 𝜕𝑢𝜕𝑡1+𝜀𝐴𝑒𝑛𝑡𝜕𝑢=𝜕𝑦𝑑𝑈𝜕𝑑𝑡+(1+𝛽)2𝑢𝜕𝑦2𝑈+Gr𝜃+𝑁𝑢+2𝛽𝜕𝜔,𝜕𝑦𝜕𝜔𝜕𝑡1+𝜀𝐴𝑒𝑛𝑡𝜕𝜔=1𝜕𝑦𝜂𝜕2𝜔𝜕𝑦2,𝜕𝜃𝜕𝑡1+𝜀𝐴𝑒𝑛𝑡𝜕𝜃=1𝜕𝑦Γ𝜕2𝜃𝜕𝑦2,(14) where 𝜂=𝜇𝑗/𝛾=2/(2+𝛽),𝑁=(𝑀+(1/𝐾)),andΓ=(1(4/(3𝑅+4)))Pr.

The boundary conditions (7) are then given by the following dimensionless equations: 𝑢=0,𝜕𝜃𝜕𝑦=1+𝜀𝐵𝑒𝑛𝑡1,𝜔=2𝜕𝑢𝜕𝑦at𝑦=0,𝑢𝑈,𝜃0,𝜔0as𝑦.(15)

3. Solution of the Problem

In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we perform an asymptotic analysis by representing the linear velocity, microrotation, and temperature in the neighborhood of the porous plate as 𝑢=𝑢0(𝑦)+𝜀𝑒𝑛𝑡𝑢1𝜀(𝑦)+𝑂2+,𝜔=𝜔0(𝑦)+𝜀𝑒𝑛𝑡𝜔1𝜀(𝑦)+𝑂2+,𝜃=𝜃0(𝑦)+𝜀𝑒𝑛𝑡𝜃1𝜀(𝑦)+𝑂2+.(16) Substituting (16) into (14), and equating the harmonic and nonharmonic terms, and neglecting the higher-order terms of O(𝜀2), we obtain the following pairs of equations for (𝑢0,𝜔0,𝜃0) and (𝑢1,𝜔1,𝜃1): (1+𝛽)𝑢0+𝑢0𝑁𝑢0=𝑁Gr𝜃02𝛽𝜔0,(1+𝛽)𝑢1+𝑢1(𝑁+𝑛)𝑢1=(𝑁+𝑛)𝐴𝑢0Gr𝜃12𝛽𝜔1,𝜔0+𝜂𝜔0𝜔=0,1+𝜂𝜔1𝑛𝜂𝜔1=𝐴𝜂𝜔0,𝜃0+Γ𝜃0𝜃=0,1+Γ𝜃1𝑛Γ𝜃1=𝐴Γ𝜃0.(17) In the above equations, prime indicates differentiation with respect to 𝑦 only.

The corresponding boundary conditions can be written as 𝑢0=0,𝑢1=0,𝜔01=2𝑢0,𝜔11=2𝑢1,𝜃0=1,𝜃1𝑢=𝐵at𝑦=0,0=1,𝑢1=1,𝜔00,𝜔1𝜃0,00,𝜃10,as𝑦.(18) Solving (17) under the boundary conditions (18), we get 𝑢0(𝑦)=1+𝑎1𝑒2𝑦+𝑎2𝑒Γ𝑦+𝑎3𝑒𝜂𝑦,𝑢1(𝑦)=1+𝑏1𝑒1𝑦+𝑏2𝑒2𝑦+𝑏3𝑒3𝑦+𝑏4𝑒4𝑦+𝑏5𝑒Γ𝑦+𝑏6𝑒𝜂𝑦,𝜔0(𝑦)=𝑐1𝑒𝜂𝑦,𝜔1(𝑦)=𝑐2𝑒1𝑦𝐴𝜂𝑛𝑐1𝑒𝜂𝑦,𝜃01(𝑦)=Γ𝑒Γ𝑦,𝜃11(𝑦)=4𝐵+𝐴Γ𝑛𝑒4𝑦𝐴𝑛𝑒Γ𝑦,(19) where the exponential indices and coefficients are given in the appendix.

By virtue of (16), we obtain the velocity, microrotation, and temperature profiles as follows: 𝑢(𝑦,𝑡)=1+𝑎1𝑒2𝑦+𝑎2𝑒Γ𝑦+𝑎3𝑒𝜂𝑦+𝜀𝑒𝛿𝑡1+𝑏1𝑒1𝑦+𝑏2𝑒2𝑦+𝑏3𝑒3𝑦+𝑏4𝑒4𝑦+𝑏5𝑒Γ𝑦+𝑏6𝑒𝜂𝑦,𝜔(𝑦,𝑡)=𝑐1𝑒𝜂𝑦+𝜀𝑒𝛿𝑡𝑐2𝑒1𝑦𝐴𝜂𝑛𝑐1𝑒𝜂𝑦,𝜃1(𝑦,𝑡)=Γ𝑒Γ𝑦+𝜀𝑒𝛿𝑡14𝐵+𝐴Γ𝑛𝑒4𝑦𝐴𝑛𝑒Γ𝑦.(20) Knowing the velocity field in the boundary layer, we can calculate the skin-friction coefficient 𝐶𝑓 at the porous plate, which in the nondimensional form is given by 𝜏𝑤=𝜇𝜕𝑢𝜕𝑦||||𝑦=0,𝐶𝑓=𝜏𝑤𝜌𝑈0𝑉0=𝜕𝑢||||𝜕𝑦𝑦=0𝑎=11+𝑎2Γ+𝑎3𝜂𝜀𝑒𝑛𝑡𝑏11+𝑏22+𝑏33+𝑏44+𝑏5Γ+𝑏6𝜂.(21) Knowing the microrotation in the boundary layer, we can calculate the couple stress coefficient 𝐶𝑚 at the porous plate, which in the nondimensional form is given by 𝑀𝑤=𝛾𝜕𝑤𝜕𝑦||||𝑦=0,𝐶𝑚=𝑀𝑤𝜇𝑗𝑈0=𝜕𝑤||||𝜕𝑦𝑦=0=1+𝜀𝑒𝑛𝑡𝐵+𝐴Γ𝑛𝐴Γ𝑛.(22)

4. Results and Discussion

The formulation of the problem that accounts for the effects of MHD and radiation on flow of an incompressible, micropolar fluid along a vertical porous plate has been carried out in the preceding sections. This enables us to carry out the numerical computations for the velocity, microrotation, and temperature for various values of the flow and material parameters. In the present study the following default parameter values are adopted for computations: 𝛽=0.2, 𝑡=1.0, 𝑛=0.1, 𝜀=0.001, 𝐴=0.5, Gr=2.0, 𝑀=2.0, 𝐾=1.0, Pr=0.71, 𝑅=2.0, and 𝐵=0.1. All graphs therefore correspond to these values unless specifically indicated on the appropriate graph. Also, the boundary condition for 𝑦 is replaced by where 𝑦max is a sufficiently large value of 𝑦 where the velocity profile 𝑢 approaches the relevant free stream velocity. We choose 𝑦max=8 and a step size Δ𝑦=0.001.

The effect of viscosity ratio 𝛽 on the translational velocity and microrotation profiles across the boundary layer are presented in Figure 1. It is noteworthy that the velocity distribution is greater for a Newtonian fluid (𝛽=0) with given parameters, as compared with micropolar fluids until its peak value reaches. The translational velocity shows a decelerating nature near the porous plate as 𝛽-parameter increases, and then decays to the relevant free stream velocity. In addition, the magnitude of microrotation at the wall is decreased as 𝛽-parameter increases. However, the distributions of microrotation across the boundary layer do not show consistent variations with increment of 𝛽-parameter.

fig1
Figure 1: Velocity and microrotation profiles for various values of 𝛽.

The translational velocity and the microrotation profiles against spanwise coordinate 𝑦 for different values of Grashof number Gr are described in Figure 2. The thermal Grashof number signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force. It is observed that an increase in Gr leads to a rise in the values of velocity, but decreases due to microrotation. Here the positive values of Gr corresponds to a cooling of the surface by natural convection. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the porous plate as Gr increases, and then decays to the free stream velocity.

fig2
Figure 2: Velocity and microrotation profiles for various values of Gr.

For different values of the magnetic field parameter 𝑀, the translational velocity and microrotation profiles are plotted in Figure 3. It is obvious that the effect of increasing values of 𝑀-parameter results in a decreasing velocity distribution across the boundary layer. The results also show that the magnitude of microrotation on the porous plate is decreased as 𝑀-parameter increases.

fig3
Figure 3: Velocity and microrotation profiles for various values of 𝑀.

For various values of the permeability parameter 𝐾, the profiles of the translational velocity and the microrotation across the boundary layer are shown in Figure 4. Clearly as 𝐾 increases, although the corresponding profiles of the microrotation is difficult to show clearly due to very little variation, the translational velocity and the microrotation profiles tend to increase.

fig4
Figure 4: Velocity and microrotation profiles for various values of 𝐾.

Figure 5 shows the translational velocity and the temperature profiles across the boundary layer for different values of Prandtl number Pr. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. The results show the effect of increasing values Pr on decreasing the translational velocity at the wall and then approach to the free stream boundary layer conditions. Typical variations of the temperature profiles along the spanwise coordinate 𝑦 are shown Figure 5(b) for different values of Prandtl number Pr. The results show that an increase of Prandtl number results in decreasing the thermal boundary layer thickness and more uniform temperature distribution across 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 surface more rapidly than for higher values of Pr. Hence, the boundary layer is thicker and rate of heat transfer is reduced, as gradients have been reduced.

fig5
Figure 5: Velocity and temperature profiles for various values of Pr.

For various values of the radiation parameter 𝑅, the velocity and temperature profiles are plotted in Figure 6. It is obvious that an increase in the radiation parameter 𝑅 results in decreasing velocity and temperature within the boundary layer, as well as a decreased thickness of the velocity and temperature boundary layers. This is because the large 𝑅-values correspond to an increased dominance of conduction over radiation thereby decreasing buoyancy force (thus, vertical velocity) and thickness of the thermal and momentum boundary layers.

fig6
Figure 6: Velocity and temperature profiles for various values of 𝑅.

Numerical values for functions proportional to shear stress and wall couple stress are given in Table 1. It is noted that both the skin-friction coefficient and the wall couple stress coefficient decreases as Pr increases. From Table 1, it is noticed that as Gr increases, the skin friction and wall couple stress coefficient increases. It is obvious that an increase in the magnetic field parameter or the radiation parameter reduces the skin friction and wall couple stress coefficient.

tab1
Table 1: Values of 𝐶𝑓 and 𝐶𝑚 for various values of Gr, 𝑀, Pr, and 𝑅 with 𝛽=0.2, 𝑡 = 1.0, 𝑛 = 0.1, 𝜀=0.001, 𝐴=0.5, 𝐾=1.0, and 𝐵=0.1.

5. Conclusions

An analysis is presented for the problem of unsteady two-dimensional laminar flow of a viscous incompressible micropolar fluid past a vertical porous plate in the presence of a transverse magnetic field and thermal radiation with variable heat flux. Numerical results are presented to illustrate the details of the MHD convective radiative flow and heat transfer characteristics and their dependence on the fluid properties and flow conditions. We may conclude that the translational velocity across the boundary layer and the magnitude of microrotation at the wall are decreased with increasing values of 𝑀 and 𝐾. Also, the translational velocity across the boundary layer and temperature at the wall are decreased with increasing values of 𝑀, Pr, and 𝑅, while they show opposite trends with increasing values of Gr.

Appendix

The exponential indices in (19) are defined by 1=𝜂21+1+4𝑛𝜂,2=12(1+𝛽)1+,1+4𝑁(1+𝛽)3=12(1+𝛽)1+,1+4(𝑁+𝑛)(1+𝛽)4=Γ21+1+4𝑛Γ,(A.1) and the coefficients are given by 𝑎1=1𝑎2𝑎3,𝑎2=(Gr/Γ)(1+𝛽)Γ2,𝑎Γ𝑁3=2𝛽𝜂(1+𝛽)𝜂2𝑐𝜂𝑁1,𝑏1=2𝛽1𝑐2(1+𝛽)211,𝑏(𝑁+𝑛)2=𝐴2𝑎1(1+𝛽)222,𝑏(𝑁+𝑛)3=𝑘2𝑏1,𝑏4=Gr/4(𝐵+(𝐴Γ/𝑛))(1+𝛽)244,𝑏(𝑁+𝑛)5=𝐴𝑎2Γ+(Gr𝐴/𝑛)(1+𝛽)Γ2,𝑏Γ(𝑁+𝑛)6=𝐴𝑎3𝜂2𝛽𝜂2𝑐1/𝑛(1+𝛽)𝜂2,𝑘𝜂(𝑁+𝑛)1=𝐴𝜂𝑛𝑐1+12𝑏22+𝑏44+𝑏5Γ+𝑏6𝜂,𝑘2=1+𝑏2+𝑏4+𝑏5+𝑏6,𝑐1=(1+𝛽)𝜂2𝜂𝑁𝜂2𝜂𝑁+𝛽2𝜂𝑎2Γ2𝑎2+12,𝑐2=(1+𝛽)211𝑁211𝑁+𝛽13𝑘1+12𝑘23.(A.2)

Nomenclature

𝐴:Suction  velocity  parameter
𝐵0:Magnetic flux density
𝐵:Real positive constant
𝐶𝑓:Skin-friction coefficient
𝐶𝑚:Couple stress coefficient
𝑐𝑝:Specific heat at constant pressure
𝑔:Acceleration due to gravity
Gr:Thermal Grashof number
𝑗:Microinertia per unit mass
𝑘:Thermal conductivity
𝑀:Magnetic field parameter
𝑛:Dimensionless exponential index
𝑝:Pressure
Pr:Prandtl number
𝑞𝑟:The radiation heat flux
𝑅:Radiation parameter
𝑡:Time
𝑇:Temperature
𝑢,𝑣:Components of velocities along and perpendicular to the plate
𝑈0:Scale of free stream velocity
𝑉0:Scale of suction velocity
𝑥,𝑦:Distances of along and perpendicular to the plate.
Greek Symbols
𝛼:Fluid thermal diffusivity
𝛽:Ratio of vertex viscosity and dynamic viscosity
𝛽𝑓:Coefficient of volumetric expansion of the working fluid
𝛾:Spin gradient viscosity
𝛿:Scalar constant
𝜀:Scalar constant  (≪1)
𝜃:Dimensionless temperature
Λ:Coefficient of vertex  (microrotation) viscosity
𝜇:Fluid dynamic viscosity
𝜌:Fluid density
𝜎:Electrical conductivity
𝜈:Fluid kinematic viscosity
𝜈𝑟:Fluid dynamic rotational viscosity
𝜏:Friction coefficient
𝜔:Angular velocity vector.
Subscripts
𝑤:Wall condition
:Free steam condition.
Superscripts
():Differentiation with respect to 𝑦
:Dimensional properties.

References

  1. A. C. Eringen, “Theory of micropolar fluids,” Journal of Mathematics and Mechanics, vol. 16, pp. 1–18, 1966.
  2. A. C. Eringen, “Theory of thermomicrofluids,” Journal of Mathematical Analysis and Applications, vol. 38, no. 2, pp. 480–496, 1972. View at Scopus
  3. G. Ahmadi, “Self-similar solution of imcompressible micropolar boundary layer flow over a semi-infinite plate,” International Journal of Engineering Science, vol. 14, no. 7, pp. 639–646, 1976. View at Scopus
  4. J. Peddieson and R. P. McNitt, “Boundary layer theory for micropolar fluid,” Recent Advances in Engineering Science, vol. 5, pp. 405–426, 1970.
  5. H. S. Takhar and V. M. Soundagekhar, “Flow and heat transfer of a micropolar fluid past a porous plate,” Indian Journal of Pure Applied Mathematics, vol. 16, no. 5, pp. 552–558, 1985.
  6. R. S. R. Gorla, “Combined forced and free convection in micropolar boundary layer flow on a vertical flat plate,” International Journal of Engineering Science, vol. 26, no. 4, pp. 385–391, 1988. View at Scopus
  7. A. Raptis, “Flow of a micropolar fluid past a continuously moving plate by the presence of radiation,” International Journal of Heat and Mass Transfer, vol. 41, no. 18, pp. 2865–2866, 1998. View at Publisher · View at Google Scholar · View at Scopus
  8. R. C. Sharma and U. Gupta, “Thermal convection in micropolar fluids in porous medium,” International Journal of Engineering Science, vol. 33, no. 13, pp. 1887–1892, 1995. View at Scopus
  9. Y. J. Kim, “Unsteady convetion flow of micropolar fluids past a vertical porous plate embedded in a porous medium,” Acta Mechanica, vol. 148, no. 1–4, pp. 105–116, 2001. View at Publisher · View at Google Scholar · View at Scopus
  10. Y. J. Kim, “Unsteady MHD convection flow of polar fluids past a vertical moving porous plate in a porous medium,” International Journal of Heat and Mass Transfer, vol. 44, no. 15, pp. 2791–2799, 2001. View at Publisher · View at Google Scholar · View at Scopus
  11. V. M. Soundalgekar, “Free convection effects on the oscillatory flow past an infinite, vertical, porous plate with constant suction,” Proceedings of the Royal Society A, vol. 333, pp. 25–36, 1973.
  12. A. A. Raptis, “Flow through a porous medium in the presence of a magnetic field,” International Journal of Energy Research, vol. 10, no. 1, pp. 97–100, 1986. View at Scopus
  13. M. A. El-Hakiem, A. A. Mohammadein, S. M. M. El-Kabeir, and R. S. R. Gorla, “Joule heating effects on magnetohydrodynamic free convection flow of a micropolar fluid,” International Communications in Heat and Mass Transfer, vol. 26, no. 2, pp. 219–227, 1999. View at Publisher · View at Google Scholar · View at Scopus
  14. M. M. Rahman and M. A. Sattar, “Magnetohydrodynamic convective flow of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption,” Journal of Heat Transfer, vol. 128, no. 2, pp. 142–152, 2006. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Raptis, “Flow of a micropolar fluid past a continuously moving plate by the presence of radiation,” International Journal of Heat and Mass Transfer, vol. 41, no. 18, pp. 2865–2866, 1998. View at Publisher · View at Google Scholar · View at Scopus
  16. E. M. Abo-Eldahab and A. F. Ghonaim, “Radiation effect on heat transfer of a micropolar fluid through a porous medium,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 500–510, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. J. Kim and A. G. Fedorov, “Transient mixed radiative convection flow of a micropolar fluid past a moving, semi-infinite vertical porous plate,” International Journal of Heat and Mass Transfer, vol. 46, no. 10, pp. 1751–1758, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. M. Q. Brewster, Thermal Radiative Transfer and Properties, John Wiley & Sons, New York, NY, USA, 1992.