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

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^∗𝜕𝑦^∗=−1𝜌𝜕𝜕𝑥𝑝^∗∗+𝜈+𝜈_𝑟𝜕²𝑢^∗𝜕𝑦^∗2−𝜈+𝑔𝛽_𝑓𝑇−𝑇_∞𝐾^∗𝑢^∗−𝜎𝐵²⁰𝜌𝑢^∗+2𝜈_𝑟𝜕𝜔^∗𝜕𝑦^∗,(2)

angular momentum equation: 𝜌𝑗^∗𝜕𝜕𝜔𝑡^∗∗+𝑣^∗𝜕𝜔^∗𝜕𝑦^∗=𝛾𝜕²𝜔^∗𝜕𝑦^∗2,(3)

energy equation: 𝜕𝑇𝜕𝑡^∗+𝑣^∗𝜕𝑇𝜕𝑦^∗=𝛼𝜕²𝑇𝜕𝑦^∗2−1𝑘𝜕𝜕𝑦𝑞^𝑟∗.(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 𝑞_𝑟=−4𝜎^𝑠𝜕𝑇⁴3𝑘_𝑒𝜕𝑦^∗,(5) where 𝜎_𝑠 is the Stefan-Boltzmann constant and 𝑘_𝑒 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: 𝑇⁴≅4𝑇^3∞𝑇−3𝑇^4∞.(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,𝜔^∗=−12𝜕𝜕𝑦𝑢^∗∗𝜕𝑇,𝜕𝑦^∗=−𝑞^𝑤𝑘1+𝜀𝐵𝑒^{𝑛∗𝑡∗}at𝑦^∗𝑢=0,_∗⟶𝑈^∗∞𝑇⟶=𝑈₀𝑇_∞1+𝜀𝑒^{𝑛∗𝑡∗},𝜔^∗⟶0,(7)as𝑦^∗⟶∞,(8) where 𝐵 is a real positive constant and 𝜀 is a small parameter such that 𝜀𝐵≤1, 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^∗=−𝑉₀1+𝜀𝐴𝑒^{𝑛∗𝑡∗},(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 −1𝜌𝑑𝑑𝑥𝑝^∗∗=𝑑𝑈^∗∞𝑑𝑡^∗+𝜈𝐾^∗𝑈^∗∞+𝜎𝐵²⁰𝜌𝑈^∗∞(10) In order to write the governing equations and the boundary conditions dimensionless form, the following nondimensional quantities are introduced 𝑢=𝑢^∗𝑈₀,𝑣=𝑣^∗𝑉₀,𝑦=𝑉⁰𝑦^∗𝑈_∞𝜈,=𝑈^∗∞𝑈₀,𝜔=𝜈𝑈₀𝑉₀𝜔^∗,𝑡=𝑡^∗𝑉²⁰𝜃=𝑇−𝑇_∞𝜈,𝑘𝑉₀𝑇_𝑤−𝑇_∞𝑞_𝑤,𝑗=𝑉²⁰𝜈²𝑛=𝑛_∗𝑗^∗,𝑉²⁰𝜈,Gr=𝜈𝛽^𝑓𝑔𝑇_𝑤−𝑇_∞𝑈₀𝑉²⁰Pr=𝜈𝜌𝐶_𝑝,(11)𝑘=𝛼𝜈=𝜇𝑘𝐶^𝑝,𝑅=4𝜎𝑘_𝑠𝑘𝑇^𝑒3∞.(12) Furthermore, the spin-gradient viscosity 𝛾, which gives some relationship between the coefficients of viscosity and microinertia, is defined as 𝛾=𝜇+Λ2𝑗^∗=𝜇𝑗^∗1+12𝛽,𝛽=Λ𝜇.(13) In view of (5), (6) and (9)–(13), the governing equations (2)–(4) reduce to the following nondimensional form: 𝜕𝑢𝜕𝑡−1+𝜀𝐴𝑒^𝑛𝑡𝜕𝜕𝑦𝑢=𝑑𝑈𝑑𝑡^∞+(1+𝛽)𝜕²𝑢𝜕𝑦²+Gr𝜃+𝑁𝑈_∞−𝑢+2𝛽𝜕𝜔𝜕𝜔𝜕𝑦,𝜕𝑡−1+𝜀𝐴𝑒^𝑛𝑡𝜕𝜕𝜔𝑦=1𝜂𝜕²𝜔𝜕𝑦²𝜕𝜃,𝜕𝑡−1+𝜀𝐴𝑒^𝑛𝑡𝜕𝜕𝜃𝑦=Γ1𝜕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,𝜕𝜃𝑢⟶𝑈𝜕𝑦_∞=−1+𝜀𝐵𝑒^𝑛𝑡,𝜔=−21𝜕𝜕𝑦𝑢at𝑦=0,,𝜃⟶0,𝜔⟶0as𝑦⟶∞.(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 𝜔𝑢==𝜔𝑢₀₀(𝑦)+𝜀𝑒^𝑛𝑡𝑢₁(𝑦)+𝑂𝜀²+⋯,𝜃=𝜃₀(𝑦)+𝜀𝑒^𝑛𝑡𝜔₁(𝑦)+𝑂𝜀²+⋯,(𝑦)+𝜀𝑒^𝑛𝑡𝜃₁(𝑦)+𝑂𝜀²+⋯.(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+(𝛽1)+𝑢_1𝛽)𝑢^0+𝑢^0−𝑁𝑢₀=−𝑁−Gr𝜃₀−2𝛽𝜔^0,+𝑢^1−(𝑁+𝑛)𝑢₁=−−𝐴(𝑁𝑢_0+𝑛)𝜔_0−Gr𝜃₁−2𝛽𝜔^1,𝜔+𝜂𝜔^0=0,_1+𝜂𝜔^1−𝜃𝑛𝜂𝜔₁=−𝐴𝜂𝜔^0,𝜃^0+Γ𝜃^0=0,_1+Γ𝜃^1−𝑛Γ𝜃₁=−𝐴Γ𝜃^0.(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,𝜔₀=−12𝑢^0,𝜔₁=−12𝑢^1𝜃_0,𝑢=−1,𝜃^1=−𝐵at𝑦=0,₀=𝜃1,𝑢₁=1,𝜔₀⟶0,𝜔₁⟶0,₀⟶0,𝜃₁⟶0,as𝑦⟶∞.(18) Solving (17) under the boundary conditions (18), we get 𝑢𝑢₀₁(𝑦)=1+𝑎₁𝑒^−ℎ2𝑦+𝑎₂𝑒^−Γ𝑦+𝑎₃𝑒^−𝜂𝑦,(𝑦)=1+𝑏₁𝑒^−ℎ1𝑦+𝑏₂𝑒^−ℎ2𝑦+𝑏₃𝑒^−ℎ3𝑦+𝑏𝜔₄𝑒^−ℎ4𝑦₀+𝑏₅𝑒^−Γ𝑦+𝑏₆𝑒^−𝜂𝑦,𝜔₁(𝑦)=𝑐₁𝑒^−𝜂𝑦,(𝑦)=𝑐₂𝑒^−ℎ1𝑦−𝐴𝜂𝑛𝑐¹𝑒^−𝜂𝑦,𝜃₀(𝑦)=1Γ𝑒^−Γ𝑦𝜃₁,(𝑦)=1ℎ₄𝐵+𝐴𝑛Γ𝑒^−ℎ4𝑦−𝐴𝑛𝑒^−Γ𝑦,(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+𝑎₁𝑒^−ℎ2𝑦+𝑎₂𝑒^−Γ𝑦+𝑎₃𝑒^−𝜂𝑦+𝜀𝑒^𝛿𝑡1+𝑏₁𝑒^−ℎ1𝑦+𝑏₂𝑒^−ℎ2𝑦+𝑏₃𝑒^−ℎ3𝑦𝜔(𝑦,𝑡)=𝑐₁+𝑏₄𝑒^−ℎ4𝑦+𝑏₅𝑒^−Γ𝑦+𝑏₆𝑒^−𝜂𝑦,𝑒^−𝜂𝑦+𝜀𝑒^𝛿𝑡𝑐²𝑒^−ℎ1𝑦−𝐴𝜂𝑛𝑐¹𝑒^−𝜂𝑦𝜃(𝑦,𝑡)=1,Γ𝑒^−Γ𝑦+𝜀𝑒^𝛿𝑡ℎ1₄𝐵+𝐴𝑛Γ𝑒^−ℎ4𝑦−𝐴𝑛𝑒^−Γ𝑦.(20) Knowing the velocity field in the boundary layer, we can calculate the skin-friction coefficient 𝐶_𝑓 at the porous plate, which in the nondimensional form is given by 𝜏^∗𝑤=𝜇𝜕𝑢^∗𝜕𝑦^∗||||_𝑦∗=0𝐶_𝑓,=𝜏^∗𝑤𝜌𝑈₀𝑉₀=𝜕𝑢𝜕𝑦||||_𝑦=0−𝜀𝑒_𝑛𝑡=−𝑎₁ℎ₁+𝑎₂Γ+𝑎₃𝜂𝑏₁ℎ₁+𝑏₂ℎ₂+𝑏₃ℎ₃+𝑏₄ℎ₄+𝑏₅Γ+𝑏₆𝜂.(21) Knowing the microrotation in the boundary layer, we can calculate the couple stress coefficient 𝐶_𝑚 at the porous plate, which in the nondimensional form is given by 𝑀_𝑤=𝛾𝜕𝑤^∗𝜕𝑦^∗||||_𝑦∗=0𝐶_𝑚,=𝑀^𝑤𝜇𝑗𝑈₀=𝜕𝑤𝜕𝑦||||_𝑦=0=1+𝜀𝑒^𝑛𝑡𝐵+𝐴Γ𝑛−𝐴𝑛Γ.(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 𝑦_max is a sufficiently large value of 𝑦 where the velocity profile 𝑢 approaches the relevant free stream velocity. We choose 𝑦_max=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.

(a)

(b)

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.

(a)

(b)

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.

(a)

(b)

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.

(a)

(b)

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.

(a)

(b)

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.

(a)

(b)

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.

Table 1 

Values of 𝐶_𝑓 and 𝐶_𝑚 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 ℎ₁=𝜂21+1+4𝑛ℎ₂𝜂,=12(1+𝛽)1+√ℎ1+4𝑁(1+𝛽)₃,=12(1+𝛽)1+√ℎ1+4(𝑁+𝑛)(1+𝛽)₄,=Γ21+1+4𝑛Γ,(A.1) and the coefficients are given by 𝑎₁=−1−𝑎₂−𝑎₃,𝑎₂=−(Gr/Γ)𝑎(1+𝛽)Γ²−Γ−𝑁₃,=2𝛽𝜂(1+𝛽)𝜂²−𝜂−𝑁𝑐¹𝑏₁,=2𝛽ℎ¹𝑐₂𝑏(1+𝛽)ℎ²¹−ℎ₁−(𝑁+𝑛)₂,=𝐴ℎ²𝑎₁(1+𝛽)ℎ²²−ℎ₂−(𝑁+𝑛),𝑏³𝑏₄=𝑘₂−𝑏₁,=−Gr/ℎ⁴(𝐵+(𝐴Γ/𝑛))𝑏(1+𝛽)ℎ²⁴−ℎ₄−(𝑁+𝑛)₅,=𝐴𝑎²Γ+(Gr𝐴/𝑛)𝑏(1+𝛽)Γ²−Γ−(𝑁+𝑛)₆,=𝐴𝑎³𝜂−2𝛽𝜂²𝑐₁/𝑛𝑘(1+𝛽)𝜂²−𝜂−(𝑁+𝑛)₁,=𝐴𝜂𝑛𝑐¹+12𝑏²𝑘₂ℎ₂+𝑏₄ℎ₄+𝑏₅Γ+𝑏₆𝜂,𝑐₁=−1+𝑏₂+𝑏₄+𝑏₅+𝑏₆,=(1+𝛽)𝜂²−𝜂−𝑁𝜂²−𝜂−𝑁+𝛽ℎ₂𝜂𝑎²Γ−ℎ₂𝑎₂+1𝑐₂2,=(1+𝛽)ℎ²¹−ℎ₁−𝑁ℎ²¹−ℎ₁−𝑁+𝛽ℎ₁ℎ₃𝑘¹+12𝑘²ℎ₃.(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.