Influence of Variable Permeability and Radiation Absorption on Heat and Mass Transfer in MHD Micropolar Flow over a Vertical Moving Porous Plate

Abstract

An analysis of unsteady free convection with heat and mass transfer flow for a micropolar fluid through a porous medium with a variable permeability bounded by a semi-infinite vertical plate in the presence of heat generation, thermal radiation, first-order chemical reaction, and the radiation absorption is reported. The plate is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts perpendicular to the porous surface which absorbs micropolar fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and nonharmonic functions. To observe physical insight and interesting aspects of the problem, the velocity, angular velocity, temperature, and concentration field are numerically studied and displayed graphically for pertinent parameters.

1. Introduction

The study of heat and mass transfer for an electrically conducting micropolar fluid past a porous plate under the influence of a magnetic field has attracted the interest of many investigators in view of its applications in many engineering problems such as magnetohydrodynamic (MHD) generators, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, and the boundary layer control in the field of aerodynamics. The theory of microfluids that was developed by Eringen [1], which includes the effect of local rotary inertia, the couple stresses, and inertial spin, satisfactorily provides a model for the non-Newtonian behavior observed in polymers, paints, lubricants, suspended fluids and blood, and so forth. He [2] extended the theory of thermomicropolar fluids and derived the constitutive laws for fluids with microstructure. An excellent review of micropolar fluids and their applications was given by Ariman et al. [3]. Gorla [4] discussed the steady-state heat transfer in a micropolar fluid flow over a semi-infinite plate, and the analysis is based on similarity variables. Kim [5] studied unsteady free convection flow of micropolar fluids through a porous medium bounded by an infinite vertical plate. Charya [6] analyzed the flow of micropolar fluid through a constricted channel. Rees and Pop [7] discussed the free convection boundary layer flow of a micropolar fluid from a vertical flat plate, While Sharma and Gupta [8] studied the effects of medium permeability on thermal convection in micropolar fluids. Kumar et al. [9] have studied the problem of fully developed free convective flow of micropolar and viscous fluids in a vertical channel. Muthu et al. [10] studied peristaltic motion of micropolar fluid in circular cylindrical tubes. Srinivasacharya et al. [11] analyzed the unsteady stokes flow of micropolar fluid between two parallel porous plates. Recently, Muthuraj and Srinivas [12] investigated fully developed MHD flow of micropolar and viscous fluids in a vertical porous space using HAM.

The combined heat and mass transfer problems with chemical reactions are of importance in many processes and, therefore, have received a considerable amount of attention in recent years. In processes, such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower, and the flow in a desert cooler, the heat and mass transfer occurs simultaneously. Das et al. [13] considered the effects of a first-order chemical reaction on the flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer. The effects of the chemical reaction and mass transfer on MHD unsteady free convection flow past an infinite/semi-infinite vertical plate were analyzed by [14–17] with constant suction and heat sink. Reddy et al. [18] studied peristaltic transport of a conducting fluid in an inclined asymmetric channel. Recently, Pal and Talukdar [19] studied the unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction.

Convection problems associated with heat sources within fluid-saturated porous media are of great practical significance, for that there are a number of practical applications in geophysics and energy-related problems, such as recovery of petroleum resources, geophysical flows, cooling of underground electric cables, storage of nuclear waste materials ground water pollution, fiber and granular insulations, solidification of costing, chemical catalytic reactors, and environmental impact of buried heat generating waste. Abo-Eldahab and El Aziz [20] investigated the problem of steady, laminar, free convection boundary layer flow of a micropolar fluid from a vertical stretching surface embedded in a nondaring porous medium in the presence of a uniform magnetic field, heat generation/absorption, and free stream velocity. Gorla et al. [21] have presented an analysis for the unsteady natural convection from a heated vertical surface placed in a micropolar fluid in the presence of internal heat generation or absorption. Raptis and Perdikis [22] analyzed the effect of a chemical reaction of an electrically conducting viscous fluid on the flow over a nonlinearly (quadratic) semi-infinite stretching sheet in the presence of a constant magnetic field which is normal to the sheet. Kesavaiah et al. [23] investigated the effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate. Recently, Vajravelu et al. [24] studied the influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum.

The role of thermal radiation in the flow heat transfer process is of major importance in the design of many advanced energy conversion systems operating at higher temperatures. Thermal radiation within these systems is usually the result of emission by hot walls and the working fluid. Bakier and Gorla [25] studied thermal radiation effect on mixed convection from horizontal surfaces in porous media. Bakier [26] reported the effect of radiation on the mixed convection flow on an isothermal vertical surface in a saturated porous medium and has obtained self-similar solution. Hossain and Takhar [27] analyzed the effect of radiation on mixed convection along a vertical plate with uniform surface temperature.

In many investigations, while investigating either velocity and temperature distribution or stability characteristics of the flow usually the permeability and porosity are taken as constants. All the above-mentioned studies treat permeability and thermal conductivity of the medium as uniform. It has been observed that when porosity and permeability are taken as constants, the velocity profiles are flat with a thin boundary near the walls. However, the experimental investigations of Schwartz and Smith [28] and Schertz and Bischoff [29] show that the velocity profiles exhibit channeling of velocity near the walls. This hydrodynamic channeling is attributed to the variation of porosity of the medium confined between two boundaries. Further, the measurements of porosity distribution by Benenati and Brosilow [30] in randomly packed beds established a graphical relationship between spatial dependence of porosity and radial position of the container from which we find that the porosity has a maximum value at the wall and minimum value at the center of the bed. Thus, it can be concluded that the porosity and in turn permeability are not constant in a given medium. However, there is no exact mathematical relationship between the porosity variation and the distance from the wall. Chandrasekhara and Namboodiri [31] incorporate the variable permeability concept. They indicated considerable effect on the flow velocity and temperature distributions. Murthy and Feyen [32] studied the influence of variable permeability on the dispersion of a chemically reacting solute in porous media. From the technological point of view, flow arising from temperature and material difference is applied in chemical engineering, geothermal reservoirs, aeronautics, and astrophysics. In some applications, magnetic forces are present and at other times the flow is further complicated by the presence of radiation absorption, a good example of this is in the planetary atmosphere where there is radiation absorption from nearby stars. The influence of a magnetic field on the flow of an electrically conducting viscous fluid with mass transfer and radiation absorption is also useful in planetary atmosphere research [33]. The effect of chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction was analyzed by [34, 35].

To the best of our knowledge, the problem of magnetohydrodynamics unsteady free convection flow of a micropolar fluid through a porous medium of variable permeability with radiation absorption over a vertical moving porous flat plate has remained unexplored. So the objective of present paper is to study the effects of a first-order homogeneous chemical reaction, variable permeability, and thermal radiation on the heat and mass transfer in MHD micropolar fluid flow over a vertical moving porous flat plate through a porous medium.

2. Formulation

We consider the two-dimensional unsteady flow of a laminar, incompressible micropolar fluid past a semi-infinite vertical porous moving plate embedded in a porous medium with variable permeability. The 𝑥^∗-axis is measured along the porous plate in the upward direction, and a magnetic field of uniform strength 𝐵₀ is applied in the 𝑦^∗-direction which is normal to the flow direction. Since the motion is two-dimensional and length of the plate is very large, so all the physical variables are independent of 𝑥^∗ and they are functions of normal distance 𝑦^∗ and 𝑡^∗ only. It is assumed that transversely applied magnetic field and magnetic Reynolds number are very small so that the induced magnetic field and the Hall effect are negligible. The motion of the plate as well as permeability of the porous medium and suction velocity increases exponentially with time. The wall is maintained at constant temperature 𝑇^∗_𝑤 and concentration 𝐶^∗_𝑤, higher than the ambient temperature 𝑇^∗_∞ and concentration 𝐶^∗_∞, respectively. The permeability of the porous medium is assumed to be of the form 𝐾^∗=𝐾₀(1+𝐵𝜀𝑒^{𝛿∗𝑡∗}) and suction velocity is assumed to be 𝜐^∗=−𝑉₀(1+𝜀𝐴𝑒^{𝛿∗𝑡∗}), where 𝐴,𝐵 are real positive constants and 𝜀 and 𝜀𝐴 are small values less than unity, and 𝑉₀ is the scale of suction velocity which is nonzero positive constant. The negative sign indicates that the suction is towards the plate. The fluid properties are assumed to be constants except that the influence of density variation with temperature and concentration has been considered in the body-force term. The concentration of 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 thermophysical properties are assumed to be constant.

Under these conditions, the governing equations can be written in a Cartesian frame of reference as follows: 𝜕𝑣^∗𝜕𝑦^∗=0,𝜕𝑢^∗𝜕𝑡^∗+𝜐^∗𝜕𝑢^∗𝜕𝑦^∗1=−𝜌𝜕𝑃^∗𝜕𝑥^∗+𝜐+𝜐_𝑟𝜕²𝑢^∗𝜕𝑦^∗2+𝑔𝛽_𝑓𝑇^∗−𝑇^∗_∞+𝑔𝛽_𝑐𝐶^∗−𝐶^∗_∞𝑢−𝜐^∗𝐾₀1+𝐵𝜀𝑒^{𝛿∗𝑡∗}−𝜎𝜌𝐵²₀𝑢^∗+2𝜐_𝑟𝜕𝜔^∗𝜕𝑦^∗,𝜌𝑗^∗𝜕𝜔^∗𝜕𝑡^∗+𝜐^∗𝜕𝜔^∗𝜕𝑦^∗𝜕=𝛾²𝜔^∗𝜕𝑦^∗2,𝜕𝑇^∗𝜕𝑡^∗+𝜐^∗𝜕𝑇^∗𝜕𝑦^∗=𝐾𝜌𝑐_𝑝𝜕²𝑇^∗𝜕𝑦^∗2+𝑄₀𝜌𝑐_𝑝𝑇^∗−𝑇^∗_∞+𝑄^∗_𝑙𝐶^∗−𝐶^∗_∞−1𝜌𝑐_𝑝𝜕𝑞^∗_𝑟𝜕𝑦^∗,𝜕𝑐^∗𝜕𝑡^∗+𝜐^∗𝜕𝑐^∗𝜕𝑦^∗=𝐷^∗𝜕²𝑐^∗𝜕𝑦^∗2−𝑅^∗𝐶^∗−𝐶^∗_∞. (1)(2)(3)(4)(5) The third and fourth terms on the RHS of the momentum equation (2) denote the thermal and concentration buoyancy effects, respectively. The second, third, and fourth terms on RHS of (3) denote the heat absorption, radiation absorption, and thermal radiation effects, respectively.

Under these assumptions, the appropriate boundary conditions for the velocity, angular velocity, temperature, and concentration fields are 𝑢^∗=𝑢^∗_𝑝,𝑇^∗=𝑇^∗_𝑤+𝜀𝑇^∗_𝑤−𝑇^∗_∞𝑒^{𝛿∗𝑡∗},𝜔^∗=−𝑛𝜕𝑢^∗𝜕𝑦^∗,𝐶^∗=𝐶^∗_𝑤+𝜀𝐶^∗_𝑤−𝐶^∗_∞𝑒^{𝛿∗𝑡∗}𝑢at𝑦→0,_∗→𝑈^∗_∞=𝑈₀1+𝜀𝑒^{𝛿∗𝑡∗},𝑇^∗→𝑇^∗_∞,𝐶^∗→𝐶^∗_∞,𝜔^∗→0at𝑦→∞, (6) where the variables and related quantities are defined in the nomenclature.

Outside the boundary layer, (2) gives −1𝜌𝜕𝑃^∗𝜕𝑥^∗=𝑑𝑈^∗_∞𝑑𝑡^∗+𝜐𝐾^∗𝑈^∗_∞+𝜎𝜌𝐵²₀𝑈^∗_∞. (7)

The radiative heat flux term by using the Rosseland approximations is given by 𝑞^∗_𝑟=4𝜎^∗3𝐾^∗₁𝜕𝑇^∗4𝜕𝑦^∗, (8) where 𝑇^∗4=4𝑇_∞^∗3−3𝑇_∞^∗4.

In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced: 𝑢^∗=𝑈₀𝑢,𝜐^∗=𝑉₀𝜐,𝑢^∗_𝑝=𝑈₀𝑈_𝑝,𝑦^∗=𝜐𝑦𝑉₀,𝑈^∗_∞=𝑈₀𝑈_∞,𝜔^∗=𝜔𝑈₀𝑉₀𝑉₀,𝑗^∗=𝑗𝜐²𝑉²₀,𝑡^∗=𝑡𝜐𝑉²₀,𝑇^∗=𝑇^∗_∞+𝜃𝑇^∗_𝑤−𝑇^∗_∞,𝑐^∗=𝑐^∗_∞+𝑐𝑐^∗_𝑤−𝑐^∗_∞,𝛿^∗=𝛿𝑣^∗₀𝜐,𝐾^∗=𝐾𝜐²₀𝑉²₀,𝑠_𝑐=𝜐𝐷^∗,𝐻=𝜐𝑄₀𝜌𝜐𝑐_𝑝,𝑅𝜁=^∗𝜐𝑉²₀,𝑄_𝑙=𝑄^∗_𝑙𝐶𝐶^∗_𝑤−𝐶^∗_∞𝑇^∗_𝑤−𝑇^∗_∞𝑉²₀,𝑀=𝜐𝜎𝐵²₀𝜌𝑉²₀isthemagneticfieldparameter,Gr=𝜐𝑔𝐵_𝑓𝑈₀𝑉²₀𝑇^∗_𝑤−𝑇^∗_∞isthethermalGrashofnumber,Gc=𝜐𝑔𝐵_𝑐𝑈₀𝑉²₀𝑐^∗_𝑤−𝑐^∗_∞isthesolutalGrashofnumber,Pr=𝜐𝜌𝑐_𝑝𝐾=𝜐𝛼isthePrandtlnumber,𝑅=4𝑇_∞^∗3𝜎^∗𝐾𝐾^∗₁𝐾istheradiationparameter,𝛼=𝜌𝑐_𝑝istheeffectivefluidthermaldiffusivity. (9)

Furthermore, the spin-gradient viscosity, which gives the same relationship between the coefficients of viscosity and microinertia, is defined as Γ𝛾=𝜇+2𝑗^∗=𝜇𝑗^∗𝛽1+2, (10) where 𝛽=Γ/𝜇.

In view of (7)–(10), the governing equations (2)–(5) reduce to the following dimensionless form: 𝜕𝑢−𝜕𝑡1+𝜀𝐴𝑒^𝛿𝑡𝜕𝑢=𝜕𝑦𝑑𝑢_∞+𝜕𝑑𝑡1+𝛽²𝑢𝜕𝑦²+𝑈+Gr𝜃+Gc𝐶_∞−𝑢𝐾1+𝐵𝜀𝑒^𝛿𝑡+𝑀𝑈_∞−𝑢+2𝛽𝜕𝜔,𝜕𝑦𝜕𝜔−𝜕𝑡1+𝜀𝐴𝑒^𝛿𝑡𝜕𝜔=1𝜕𝑦𝜂𝜕²𝜔𝜕𝑦²,𝜕𝜃−𝜕𝑡1+𝜀𝐴𝑒^𝛿𝑡𝜕𝜃=1𝜕𝑦Pr1+4𝑅3𝜕²𝜃𝜕𝑦²+𝐻𝜃+𝑄_𝑙𝑐,𝜕𝑐−𝜕𝑡1+𝜀𝐴𝑒^𝛿𝑡𝜕𝑐=1𝜕𝑦𝜕Sc²𝑐𝜕𝑦²−𝜁𝑐, (11)(12)(13)(14) where 𝜂=2/2+𝛽=𝜇𝑗^∗/𝛾.

The boundary conditions of (6) are given by the following dimensionless form: 𝑢=𝑈_𝑝,𝜃=1+𝜀𝑒^𝛿𝑡,𝜔=−𝑛𝜕𝑢,𝜕𝑦𝑐=1+𝜀𝑒^𝛿𝑡at𝑦→0,𝑢=𝑈_∞→1+𝜀𝑒^𝛿𝑡,𝜃→0,𝜔→0,𝑐→0at𝑦→∞. (15)

3. Solution of the Problem

Equations (11)–(14) are coupled, nonlinear partial differential equations and these cannot be solved in closed form. However, these equations can be reduced to ordinary differential equations, which can be solved analytically. For these, we may represent the linear and angular velocities, temperature, and concentration as follows: 𝑢=𝑢₀𝑦+𝜀𝑒^𝛿𝑡𝑢₁𝑦+𝑜𝜀²,𝜔=𝜔₀𝑦+𝜀𝑒^𝛿𝑡𝜔₁𝑦+𝑜𝜀²,𝜃=𝜃₀𝑦+𝜀𝑒^𝛿𝑡𝜃₁𝑦+𝑜𝜀²,𝑐=𝑐₀𝑦+𝜀𝑒^𝛿𝑡𝑐₁𝑦+𝑜𝜀². (16) By substituting the above equations (16) into (11)–(14), 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,𝐶₀) and (𝑢₁,𝜔₁,𝜃₁,𝐶₁): 1+𝛽𝑢₀^′′+𝑢^′₀−𝑁𝑢₀=−𝑁−Gr𝜃₀−Gc𝑐₀−2𝛽𝜔^′₀,1+𝛽𝑢₁^′′+𝑢^′₁−(𝑁+𝛿)𝑢₁=−𝐿𝑢₀+𝐿−Gr𝜃₁−Gc𝑐₁−2𝛽𝜔^′₁−𝐴𝑢^′₀,𝜔₀^′′+𝜂𝜔₀𝜔=0,_1′′+𝜂𝜔^′₁−𝛿𝜂𝜔₁=−𝐴𝜂𝜔^′₀,(3+4𝑅)𝜃₀^′′+3Pr𝜃^′₀+𝐻𝜃₀=−3Pr𝑄_𝑙𝑐₀,(3+4𝑅)𝜃₁^′′+3Pr𝜃^′₁−3Pr(𝛿−𝐻)𝜃₁=−3𝐴Pr𝜃^′₀−3Pr𝑄_𝑙𝑐₁,𝑐₀^′′+Sc𝑐^′₀−𝜁Sc𝑐₀𝑐=0,_1′′+Sc𝑐^′₁−(𝛿+𝜁)Sc𝑐₁=−𝐴Sc𝑐^′₀, (17) where 𝐿=𝐵/𝐾, 𝑁=(𝑀+1/𝐾) and the primes denote differentiation with respect to 𝑦.

The corresponding boundary conditions can be written as 𝑢₀=𝑈_𝑝,𝑢₁=0,𝜔₀=−𝑛𝑢^′₀,𝜔₁=−𝑛𝑢^′₁,𝜃₀=1,𝜃₁=1,𝑐₀=1,𝑐₁𝑢=1at𝑦→0,₀=1,𝑢₁=1,𝜔₀=0,𝜔₁𝜃=0,₀=0,𝜃₁=0,𝑐₀=0,𝑐₁=0at𝑦→∞. (18)

Solving (17) under the boundary conditions (18), we obtain the linear and angular velocities, temperature, and concentration profiles as follows: 𝑢𝑦,𝑡=1+𝑎₁𝑒^−𝑅1𝑦+𝑎₂𝑒^−ℎ3𝑦+𝑎₆𝑒^−ℎ1𝑦+𝑎₅𝑒^−𝜂𝑦+𝜀𝑒^𝛿𝑡1+𝑏₁𝑒^−𝑅3𝑦+𝑏₂𝑒^−ℎ4𝑦+𝑑₁𝑒^−ℎ3𝑦+𝑑₂𝑒^−ℎ2𝑦+𝑑₃𝑒^−ℎ1𝑦+𝑏₈𝑒^−ℎ6𝑦+𝑏₁₀𝑒^−𝑅1𝑦+𝑑₄𝑒^−𝜂𝑦=𝑐,𝜔𝑦,𝑡₁₁𝑒^−𝜂𝑦+𝜀𝑒^𝛿𝑡𝑐₁₃𝑒^−ℎ6𝑦−𝐴𝜂𝑐₁₁𝛿𝑒^−𝜂𝑦=,𝜃𝑦,𝑡1−𝑧₃𝑒^−ℎ3𝑦+𝑧₃𝑒^−ℎ1𝑦+𝜀𝑒^𝛿𝑡𝑐₉𝑒^−ℎ4𝑦+𝑧₄𝑒^−ℎ3𝑦+𝑧₆𝑒^−ℎ2𝑦+𝑧₈𝑒^−ℎ1𝑦,𝑐𝑦,𝑡=𝑒^−ℎ1𝑦+𝜀𝑒^𝛿𝑡1−𝑧₃𝑒^−ℎ2𝑦+𝑧₂𝑒^−ℎ1𝑦, (19) where the expressions for the constants are given in the appendix.

The skin friction, Nusselt number, and Sherwood number are important physical parameters for this type of boundary layer.

3.1. Skin-Friction

Knowing the velocity field in the boundary layer, we can now calculate the skin-friction at the plate which is given by 𝜏^∗_𝑤=𝜇+Γ𝜕𝑢^∗𝜕𝑦^∗+Γ𝜔^∗_𝑦∗=0, (20) and in the dimensionless form, we obtain 𝑐_𝑓=2𝜏^∗_𝑤𝜌𝑉₀𝑈₀=21+(1−𝑛)𝛽𝑢^′(0). (21) The couple stress coefficient (𝐶_𝑚) at the plate is written as 𝑀_𝑤=𝛾𝜕𝜔^∗𝜕𝑦^∗_𝑦∗=0, (22) and in the dimensionless from, we obtain 𝑐_𝑚=𝑀_𝑤𝜇𝑗𝑈₀=𝛽1+2𝜔^′(0). (23)

3.2. Nusselt Number

Knowing the temperature field, it is interesting to study the effect of the free convection and thermal radiation on the rate of heat transfer 𝑞^∗_𝑤, and this is given by 𝑞^∗_𝑤=−𝑘𝜕𝑇^∗𝜕𝑦^∗−4𝜎^∗3𝐾^∗₁𝜕𝑇^∗4𝜕𝑦^∗_𝑦∗=0. (24) In view of (8), the above equation becomes 𝑞^∗_𝑤=−𝑘+16𝜎^∗𝑇^∗33𝐾^∗₁𝜕𝑇^∗𝜕𝑦^∗_𝑦∗=0. (25) This is written in dimensionless from as 𝑞^∗_𝑤=−𝑘𝑇^∗_𝑤−𝑇^∗_∞𝑣₀𝜐1+4𝑅3𝜕𝜃𝜕𝑦_𝑦=0. (26) The dimensionless local surface heat flux (i.e., Nusselt number) is obtained as Nu_𝑥=𝑥𝑞^∗_𝑤𝑘𝑇^∗_𝑤−𝑇^∗_∞, (27) where Nu_𝑥Re_𝑥⁻¹=−1+4𝑅3𝜕𝜃𝜕𝑦_𝑦=0,Re_𝑥=𝑣₀𝑥𝜐. (28)

3.3. Sherwood Number

The definition of the local mass flux and the local Sherwood number are, respectively, given by 𝑗_𝑤=−𝐷^∗𝜕𝑐^∗𝜕𝑦^∗_𝑦∗=0,Sh_𝑥=𝑥𝑗_𝑤𝐷^∗𝑐^∗_𝑤−𝑐^∗_𝑤 (29) and with the help of these equations, one can write Sh_𝑥Re_𝑥⁻¹=−𝜕𝑐𝜕𝑦_𝑦=0. (30)

4. Results and Discussions

The formulation of the problem that accounts for the effects of chemical reaction and thermal radiation on MHD convective flow and mass transfer of an incompressible, micropolar fluid along a semi-infinite vertical porous moving plate in a porous medium with variable permeability in the presence of heat generation has been performed in the preceding sections. The governing equations of the flow field were solved analytically, using a perturbation method, and the expressions for the velocity, microrotation, temperature, concentration, skin-friction, Nusselt number, and Sherwood number were obtained. In order to get a physical insight of the problem, the above physical quantities are computed numerically and presented graphically for different values of the governing parameters: thermal Grashof number Gr, solutal Grashof number Gm, magnetic field parameter 𝑀, Schmidt number Sc, heat absorption parameter 𝐻, chemical reaction parameter 𝜁, and thermal radiation parameter 𝑅, while the values of some of the physical parameters are taken as constants such as 𝑡=1, 𝛿=0.01, 𝜀=0.01, 𝐵=0.1 and 𝐴=0.1, in all the tables and figures. In the present study, the boundary conditions for 𝑦→∞ are replaced by identical ones at 𝑦_max which 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.

In order to verify the accuracy of the present work, particular results are compared with those available in the literature through Table 1. The numerical values of skin-friction, Nusselt number, and Sherwood number obtained in the present case are compared with those of Kim [5] and Mohamed and Abo-Dahab [36] for different values of Prandtl number (Pr) and fixed values of remaining parameters. It is clearly seen that there is an excellent agreement between the respective research of our study and those of Kim [5] and Mohamed and Abo-Dahab [36]. Further, it is found from Table 2 that skin-friction decreases with the increase in magnetic parameter 𝑀. Also no effect of 𝑀 is seen on Sherwood number Sh as noted in [36].

Table 1 

Effect of Pr on 𝐶_𝑓, Nu and Sh for 𝑅=0.1, 𝑛=0.5, 𝐾=2, 𝐻=0.1, 𝜁=0.2, 𝛽=0.5, Sc=0.2, Pr=1, Gr=2, 𝑈_𝑝=0.5, 𝑄_𝑙=0.01, 𝐵=0.01, and Gc = 2.

Table 2 

Effect of 𝑀 on 𝐶_𝑓 and Sh for 𝑅=0.1, 𝑛=0.5, 𝐾=2, 𝐻=0.1, 𝜁=0.2, 𝛽=0.5, Sc=0.2, Pr=1, Gr=2, 𝑈_𝑝=0.5, 𝑄_𝑙=0.01, 𝐵=0.01, and Gc=2.

For different values of the magnetic field parameter 𝑀, the translational velocity and microrotation profiles are plotted in Figure 1. It is obvious that the effect of increasing values of the parameter 𝑀 results in a decreasing velocity distribution across the boundary layer. The effect of magnetic field is more prominent at the point of peak value; that is, the peak value drastically decreases with the increases in the value of magnetic field because the presence of 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 results also show that the magnitude of the microrotation on the porous plate increases as 𝑀 increases.

(a)

(b)

Figure 1 

Velocity and microrotation distribution.

Figure 2 illustrates the variation of velocity and microrotation distribution across the boundary layer for several values of plate moving velocity 𝑈_𝑝 in the direction of the fluid flow, respectively. Although we have different initial plate moving velocities, the velocity decreases to the constant values for given material parameters. The results also show that the magnitude of microrotation on porous plate increases as 𝑈_𝑝 increases.

(a)

(b)

Figure 2 

Velocity and microrotation distribution.

For various values of the permeability 𝐾, the profiles of translational velocity and the microrotation across the boundary layer are shown in Figure 3. It is clear that as 𝐾 increase, the translational velocity profiles across the boundary layer increases, but the magnitude of microrotation decreases with an increase of 𝐾-parameter.

(a)

(b)

Figure 3 

Velocity and microrotation distribution.

Figure 4 presents the velocity and microrotation profiles for different values of the Prandtl number (Pr). The results show that the effect of increasing values of Pr results in a decrease of the translational velocity, while the reverse effect is observed for the microrotation profiles.

(a)

(b)

Figure 4 

Velocity and microrotation distribution.

For various values of the Schmidt number (Sc), the profiles of translational velocity and the microrotation across the boundary layer are shown in Figure 5. It is clear that as Sc increase, the translational velocity profiles across the boundary layer decreases. The results also show that the magnitude of microrotation on porous plate increases as Sc increases.

(a)

(b)

Figure 5 

Velocity and microrotation distribution.

For various values of 𝑄_𝑙, the profiles of translational velocity and the microrotation across the boundary layer are shown in Figure 6. It is clear from the graph that the translational velocity increases with an increase of 𝑄_𝑙 which is due to the fact that when heat is absorbed the buoyancy force which accelerated the flow rate. The results also show that the magnitude of microrotation on porous plate decreases as 𝑄_𝑙 increases.

(a)

(b)

Figure 6 

Velocity and microrotation distribution.

For various values of radiation parameter 𝑅, the translational velocity and microrotation profiles are plotted in Figure 7. It is obvious that with increasing values of 𝑅-parameter, the velocity distribution across the boundary layer increases. The results also show that the magnitude of microrotation on the porous plate is decreased as 𝑅-parameter increases.

(a)

(b)

Figure 7 

Velocity and microrotation distribution.

The effects of heat generation 𝐻 on the translational velocity and microrotation profiles across the boundary layer are displayed in Figure 8. It is shown that the translational velocity across the boundary layer increases with an increasing of 𝐻. The results also show that the magnitude of microrotation decreases with an increasing of 𝐻.

(a)

(b)

Figure 8 

Velocity and Microrotation distribution.

Figure 9 shows the translational velocity and the microrotation profiles across the boundary layer for various values of the chemical reaction parameter 𝜁. The results display that with an increase of 𝜁, the translational velocity decreases, but the magnitude of microrotation increases as 𝜁 decreases.

(a)

(b)

Figure 9 

Velocity and microrotation distribution.

The effects of viscosity ratio 𝛽 on the translational velocity and the microrotation profiles across the boundary layer are presented in Figure 10. It is clear that the velocity distribution is greater for a Newtonian fluid (𝛽=0) with the fixed flow and material parameters, as compared with micropolar fluids. The results display that with an increase of viscosity ratio 𝛽, the translational velocity decreases, but the magnitude of microrotation increases as viscosity ratio 𝛽 increases. The translational velocity and the microrotation profiles against spanwise coordinate 𝑦 for different values of thermal Grashof number (Gr) are displayed in Figure 11. It is observed that an increase in Gr leads to an increase in the values of velocity and a decrease in the magnitude of microrotation. The translational velocity and the microrotation profiles against spanwise coordinate 𝑦 for different values of solutal Grashof number (Gc) are displayed in Figure 12. It is observed that an increase in Gc leads to an increase in the values of velocity and a decrease in the magnitude of microrotation.

(a)

(b)

Figure 10 

Velocity and microrotation distribution.

(a)

(b)

Figure 11 

Velocity and microrotation distribution.

(a)

(b)

Figure 12 

Velocity and microrotation distribution.

Figure 13 shows the effect of 𝑛-parameter, which is related to microrotation vector and shear stress, on the translational velocity and the microrotation profiles. It is observed that the magnitude of velocity and microrotation increases with an increase of 𝑛-parameter, but the magnitude of microrotation decreases with an increase of 𝑛-parameter.

(a)

(b)

Figure 13 

Velocity and microrotation distribution.

Figures 14(a) and 14(b) represent the temperature profiles for different values of thermal radiation parameter 𝑅 and prandtl number (Pr) in the boundary layer. These figures indicate that the effect of thermal radiation is to enhance heat transfer because of the fact that thermal boundary layer thickness increases with an increase in the thermal radiation. Thus, it is pointed out that the radiation should be minimized to have the cooling process at a faster rate. But the temperature decreases with increasing the values of Prandtl number (Pr) in the boundary layer. From this plot, it is evident that the temperature in the boundary layer falls very quickly for large value of the Prandtl number because of the fact that thickness of the boundary layer decreases with an increase in the value of the Prandtl number.

(a)

(b)

(c)

(d)

Figure 14 

Temperature distribution.

Typical variations of the temperature profiles along the spanwise coordinate 𝑦 are shown in Figure 14(c) for various values of 𝑄_𝑙. The results show that the temperature profiles decrease with increasing 𝑄_𝑙. The temperature profiles with respect to spanwise coordinate 𝑦 for various values of 𝜁 are shown in Figure 14(d). The results show that as the chemical reaction parameter 𝜁 increase, the thermal boundary layer increases.

The effect of the reaction rate parameter 𝜁 on the species concentration profiles for generative chemical reaction is shown in Figure 15(a). It is noticed from the graph that there is a marked effect with increasing the value of the chemical reaction rate parameter 𝜁 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 reaction rate parameter. Further, it is observed that increasing the value of the chemical reaction decreases the concentration of species in the boundary layer; this is due to the fact that boundary layer decreases with increase in the value of 𝜁 chemical reaction in this system result in the consumption of the chemical and hence result in decreasing concentration profile. Similar facts are seen in the case when Schmidt number is increased as noted in Figure 15(b). It may also be observed from this figure that the effect of Schmidt number (Sc) on concentration distribution decreases slowly for higher values of Sc. It is expected that the mass transfer rate increases as 𝜁 increases with all other parameters fixed; that is, an increase in the Schmidt number (Sc) produces a decrease in the concentration boundary layer thickness, associated with the reduction in the concentration profiles. Physically, the increase in the value of Sc means the decrease of molecular diffusion 𝐷. Hence, the concentration of the species is higher for smaller values of Sc and lower for larger values of Sc.

(a)

(b)

Figure 15 

Concentration distribution.

5. Conclusions

The following conclusions are drawn from the present study.(1) The effect of increasing magnetic parameter 𝑀 is to retard the transient velocity of the flow field at all points, while a growing Grashof number (Gr) for heat transfer or heat absorption parameter accelerates the transient velocity of the flow field at all points. (2) A growing magnetic parameter 𝑀 or Prandtl number (Pr) decelerates the transient temperature of the flow field at all points, while a growing permeability parameter 𝐾 or heat source parameter 𝑆 reverses the effect. (3) The effect of increasing magnetic parameter 𝑀 is to reduce the skin-friction at the wall, while a growing permeability parameter 𝐾 or heat source parameter 𝑆 reverses the effect. (4) A growing Prandtl number (Pr) or magnetic parameter 𝑀 increases the magnitude of the rate of heat transfer at the wall. On the other hand, a growing heat source 𝑆 parameter reverses the effect.(5) Skin-friction decreases when, magnetic field parameter is increased.

Appendix

Consider ℎ₁=𝑠_𝑐21+1+4𝜁𝑠_𝑐,ℎ₂=𝑠_𝑐21+4()1+𝛿+𝜁𝑠_𝑐,ℎ₃=3Pr2(3+4𝑅)1+()1−4𝐻3+4𝑅,ℎ3Pr₄=3Pr2(3+4𝑅)1+4()1+𝛿−𝐻)(3+4𝑅,ℎ3Pr₆=𝜂21+()1+4𝛿𝜂,𝑅₁=121+𝛽1+(),𝑅1+𝑁+𝛿1+𝛽₃=121+𝛽1+(,𝑧1+4𝑁+𝛿)1+𝛽₂=𝐴𝑠_𝑐ℎ₁ℎ²₁−ℎ₁𝑠_𝑐−(𝛿+𝜁)𝑠_𝑐,𝑧₃=−3Pr𝑄_𝑙(3+4𝑅)ℎ²₁−3Prℎ₁,𝑧+3Pr𝐻₄=3Prℎ₃1−𝑧₃(3+4𝑅)ℎ²₃−3Prℎ₃(),𝑧−3Pr𝛿−𝐻₅=3Prℎ₁𝑧₃(3+4𝑅)ℎ²₁−3Prℎ₁(),𝑧−3Pr𝛿−𝐻₆=−3Pr𝑄_𝑙1−𝑧₂(3+4𝑅)ℎ²₂−3Prℎ₂(),𝑧−3Pr𝛿−𝐻₇=−3Pr𝑄_𝑙𝑧₂(3+4𝑅)ℎ²₁−3Prℎ₁(),𝑎−3Pr𝛿−𝐻₂=−Gr1−𝑧₃1+𝛽ℎ²₃−ℎ₃,𝑎−𝑁₃=−Gr𝑧₃1+𝛽ℎ²₁−ℎ₁,𝑎−𝑁₄=−Gc1+𝛽ℎ²₁−ℎ₁,−𝑁𝑃=2𝛽𝜂1+𝛽𝜂²,𝑎−𝜂−𝑁₆=𝑎₃+𝑎₄,𝑛𝐷=1+𝑛𝑅₁𝑃,𝑐−𝜂₁₁=𝐷𝑈_𝑝−1𝑅₁+𝑎₂ℎ₃−𝑅₁+𝑎₆ℎ₁−𝑅₁,𝑎₅=𝑃^∗𝑐₁₁;𝑎₁=𝑈_𝑝−1−𝑎₂−𝑎₆−𝑎₅,𝑧₈=𝑧₅+𝑧₇,𝑐₉=1−𝑧₄−𝑧₆−𝑧₈,𝑏₂=−Gr𝑐₉1+𝛽ℎ²₄−ℎ₄−(),𝑏𝑁+𝛿₃=−Gr𝑧₄1+𝛽ℎ²₃−ℎ₃−(),𝑏𝑁+𝛿₄=−Gr𝑧₆1+𝛽ℎ²₂−ℎ₂−(),𝑏𝑁+𝛿₅=−Gr𝑧₈1+𝛽ℎ²₁−ℎ₁−(),𝑏𝑁+𝛿₆=−Gc1−𝑧₂1+𝛽ℎ²₂−ℎ₂−(),𝑏𝑁+𝛿₇=−Gc𝑧₂1+𝛽ℎ²₁−ℎ₁−(),𝐷𝑁+𝛿₁=−2𝛽𝐴𝜂²𝑐₁₁𝛿,𝑏₉=𝐷₁1+𝛽𝜂²(),𝑏−𝜂−𝑁−𝛿₁₀=𝐴𝑅₁𝑎₁1+𝛽𝑅²₁−𝑅₁−(),𝑏𝑁+𝛿₁₁=𝐴ℎ₃𝑎₂1+𝛽ℎ²₃−ℎ₃−(),𝑏𝑁+𝛿₁₂=𝐴ℎ₁𝑎₆1+𝛽ℎ²₁−ℎ₁−(),𝑏𝑁+𝛿₁₃=𝐴𝜂𝑎₅1+𝛽𝜂²(),𝑑−𝜂−𝑁+𝛿₁=𝑏₃+𝑏₁₁,𝑑₂=𝑏₄+𝑏₆,𝑑₃=𝑏₅+𝑏₇+𝑏₁₂,𝑑₄=𝑏₉+𝑏₁₃,𝑄=2𝛽ℎ₆1+𝛽ℎ²₆−ℎ₆−(),𝑘𝑁−𝛿₁=𝐴𝜂𝑐₁₁𝛿+𝑛𝑏₂𝑏₄−𝑅₃+𝑑₁ℎ₃−𝑅₃+𝑑₂ℎ₂−𝑅₃+𝑑₃ℎ₁−𝑅₃+𝑑₄𝜂−𝑅₃+𝑏₁₀𝑅₁−𝑅₃−𝑅₃,𝑐₁₃=11+𝑛𝑅₃−ℎ₆𝑄∗𝜅₁,𝑏₈=𝑄𝑐₁₃,𝑏₁=−1+𝑏₂+𝑑₁+𝑑₂+𝑑₃+𝑑₄+𝑏₈+𝑏₁₀. (A.1)

Nomenclature

𝑀∶	The local magnetic field parameter
𝑛:	The parameter related to microrotation vector and shear stress
𝑁:	The model parameter
Nu:	Nusselt number
𝑃∗:	The pressure, kg m−1s−2
Pr:	The fluid Prandtl number
𝑄0:	The heat generation coefficient, Wm−3K−1
𝑞∗𝑟:	The radiative heat flux, Wm−2
𝑅∗:	The reaction rate constant,J
𝑅:	The radiation parameter
Re:	The Reynolds number
Sc:	The generalized Schmidt number
Sh:	Sherwood number
𝑄𝑙:	The radiation absorption number
𝑡∗:	The time, s
𝑡:	The dimensionless time
𝑇∗:	The temperature in the boundary layer, K
𝑇∗∞:	The temperature far away from the plate, K
𝑇∗𝑤:	The temperature at the wall, K
𝑢:	The dimensionless velocity
𝑢∗,𝑣∗:	The velocities along and perpendicular to the plate, ms−1
𝑈0:	The scale of free stream velocity
𝑈∗∞:	The free stream velocity, ms−1
𝑢∗𝑝:	The wall dimensional velocity, ms−1
𝑈𝑝:	The wall dimensionless velocity
𝑉0:	The scale of suction velocity
𝑥∗and𝑦∗:	The distances along and perpendicular to the plate, respectively, m
𝑦:	The dimensionless spanwise coordinate normal to the plate
𝛾:	The spingradient velocity, kg ms−1
𝛿:	The scalar constant
𝜀:	The scalar constant
𝜃:	The dimensionless temperature
𝜁:	The chemical reaction parameter
𝜂:	The scalar constant
𝜇:	The fluid dynamic viscosity, kgm−1s−1
𝜐:	The fluid kinematic rotational viscosity, m2s−1
𝜐𝑟:	The kinematic rotational viscosity, m2s−1
𝜌:	Fluid density, kg m−3
𝜎:	The electrical conductivity of the fluid, Sm−1
𝜎∗:	The Stefan-Boltzmann constant, Wm−2K−4
𝜔:	Angular velocity vector, ms−1
Γ:	The coefficient of gyro-viscosity (or vortex viscosity), kgm−1s−1
𝐵0:	The magnetic induction, tesla
𝐶∗:	The species concentration, molm−3
𝐶∗𝜔:	The surface concentration, molm−3
𝐶∗∞:	The species concentration far from the surface, molm−3
𝐶𝑓:	The skin-friction coefficient
𝐶𝑚:	The couple stress coefficient
𝐶𝑝:	The specific heat at constant  pressure, Jkg−1K−1
𝐷∗:	The chemical molecular diffusivity, m2s−1
Gm:	The solutal Grashof number
Gr:	Grashof number
𝑔:	The acceleration due to the gravity, ms−2
𝐻:	The dimensionless heat generation/absorption coefficient
𝑗∗:	The microinertia per unit mass, m2
𝐾:	The permeability of the porous media, m2
𝐾∗1:	The mean absorption coefficient, m−1
𝐾:	The dimensionless permeability
𝑘:	The thermal conductivity of the fluid, Wm−1K−1
𝛼:	The fluid thermal diffusivity, m2s−1
𝛽:	The ratio of vortex viscosity and dynamic viscosity
𝛽𝑐:	Volumetric coefficient of thermal expansion,K−1
𝛽𝑓:	Volumetric coefficient of expansion with concentration,K−1
𝐵:	The scalar constant.