Radiation Effects on Mass Transfer Flow through a Highly Porous Medium with Heat Generation and Chemical Reaction

The present paper is concerned to analyze the influence of the unsteady free convection flow of a viscous incompressible fluid through a porous medium with high porosity bounded by a vertical infinite moving plate in the presence of thermal radiation, heat generation, and chemical reaction.The fluid is considered to be gray, absorbing, and emitting but nonscattering medium, and Rosseland approximation is considered to describe the radiative heat flux in the energy equation. The dimensionless governing equations for this investigation are solved analytically using perturbation technique. The effects of various governing parameters on the velocity, temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in detail.


Introduction
Transport of momentum and energy in fluid-saturated porous media with low porosities are commonly described by Darcy's model for conservation of momentum and by an energy equation based on the velocity field found from this model by Kaviany [1].In contrast to rocks, soil, sand, and other media that do fall in this category, certain porous materials, such as foam metals and fibrous media, usually have high porosity.Vajravelu [2] examined the steady flow of heat transfer in a porous medium with high porosity.Raptis [3] studied mathematically the case of time varying two-dimensional natural convection heat transfer of an incompressible electrically conducting viscous fluid through a high porous medium bounded by an infinite vertical porous plate.Hong et al. [4], Chen and Lin [5], and Jaiswal and Soundalgekar [6] studied the natural convection in a porous medium with high porosity.Hiremath and Patil [7] studied the effect of free convection currents on the oscillatory flow of the polar fluid through a porous medium, which is bounded by the vertical plane surface with constant temperature.
Many processes in engineering areas occur in high temperature and consequently the radiation plays a significant role.Chandrasekhara and Nagaraju [8] examined the composite heat transfer in a variable porosity medium bounded by an infinite vertical flat plate in the presence of radiation.Yih [9] studied the radiation effects on natural convection over a cylinder embedded in porous media.Mohammadein and El-Amin [10] considered the thermal radiation effects on power law fluids over a horizontal plate embedded in a porous medium.Raptis [11] studied the heat transfer in a porous medium with high porosity in the presence of radiation.Raptis and Perdikis [12] studied unsteady flow through a porous medium with high porosity bounded by a vertical infinite plate in the presence of radiation.
In the processes such as drying, Evaporation at the surface of water body, Energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneously.Possible applications of this type of flow can be found in many industries.For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid.The study of heat and mass transfer with chemical reaction is of great practical importance to engineers and scientists because of its frequent occurrence in many branches of science and engineering.Chambré and Young [13] presented a first order chemical reaction in the neighborhood of a horizontal plate.Das et al. [14].investigated the effect of the first order homogeneous chemical reaction on the process of an unsteady flow past a vertical plate with a constant heat and mass transfer.Muthucumarswamy and Ganesan [15] studied the effect of chemical reaction and injection on the flow characteristics in an unsteady upward motion of an isothermal plate.Anand Rao and Shivaiah [16] noticed that the chemical reaction effects on an unsteady MHD free convective flow past an infinite vertical porous plate with constant suction and heat source or generation.

ISRN Computational Mathematics
The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reactions.Vajravelu and Hadjinicolaou [17] studied the heat transfer characteristics in the laminar boundary layer of a viscous fluid over a stretching sheet with viscous dissipation or frictional heating and internal heat generation.Molla et al. [18] studied the problem of natural convection flow along a vertical wavy surface with uniform surface temperature in the presence of heat generation/absorption.Alam et al. [19] considered the problem of free convective heat and mass transfer flow past an inclined semi-infinite heated surface of a steady viscous incompressible fluid in the presence of magnetic field and heat generation.Chamkha [20] investigated an unsteady convective heat and mass transfer flow past a semi-infinite porous moving plate with heat absorption.Hady et al. [21] studied the problem of free convection flow along a vertical wavy surface embedded in a saturated porous media in the presence of internal heat generation or absorption effect.Ambethkar [22] investigated the numerical solutions of heat and mass transfer effects of an unsteady MHD free convective flow past an infinite vertical plate with constant suction and heat source or sink.Mohammed Ibrahim and Bhaskar Reddy [23] studied the radiation and mass transfer effects on MHD free convection flow along a stretching surface with viscous dissipation and heat generation.
In view of the above studies, an unsteady free convective heat and mass transfer flow of a viscous incompressible radiating fluid through a porous medium with high porosity bounded by an infinite vertical moving plate is considered in the presence of heat generation and chemical reaction.It is assumed that the plate is embedded in porous medium and moves with constant velocity in the flow direction.The equations of continuity, linear momentum, energy, and diffusion, which govern the flow field, are solved by using a regular perturbation method.The behavior of the velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number has been discussed for variations in the governing parameters.

Mathematical Analysis
An unsteady two-dimensional laminar free convective mass transfer flow of a viscous incompressible fluid through a highly porous medium past an infinite vertical moving porous plate in the presence of thermal radiation, heat generation, and chemical reaction is considered.The fluid and the porous structure are assumed to be in local thermal equilibrium.It is also assumed that there is radiation only from the fluid.The fluid is a gray, emitting, and absorbing, but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation.A homogeneous first order chemical reaction between fluid and the species concentration is considered, in which the rate of chemical reaction is directly proportional to the species concentration.All the fluid properties are assumed to be constant except that the influence of the density variation with temperature is considered only in the body force term (Boussinesq's approximation).The   -axis is chosen along the plate in the direction opposite to the direction of gravity and the   -axis is taken normal to it.Since the flow field is of extreme size, all the variables are functions of   and   only.Hence, under the usual Boussinesq's approximation, the equations of mass, linear momentum, energy, and diffusion are continuity equation: momentum equation: energy equation: diffusion equation: where   ,   , and   are the dimensional distances along and perpendicular to the plate and dimensional time, respectively;   , and   the components of dimensional velocities along   and   directions, respectively;   and   the dimensional concentration and temperature of the fluid, respectively;  the fluid density;  the kinematic viscosity;   the specific heat at constant pressure;  the heat capacity ratio;  the acceleration due to gravity;  and  * the volumetric coefficient of thermal and concentration expansion;   the permeability of the porous medium;  the porosity;  the molecular diffusivity;    the chemical reaction parameter; and  the fluid thermal conductivity.The third and fourth terms on the right hand side of the momentum equation ( 2) denote the thermal and concentration buoyancy effects, respectively, and the fifth term represents the bulk matrix linear resistance, that is, Darcy term.Also, the second term on the right hand side of the energy equation ( 3) represents thermal radiation.The radiative heat flux term by using the Rosseland approximation (Brewster [24]) is given by where   is the Stefan-Boltzmann constant and   is the mean absorption coefficient.It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids.If temperature differences within the flow are sufficiently small, then ( 6) can be linearised by expanding   4 into the Taylor series about  ∞ , which after neglecting higher order terms takes the form It is assumed that the permeable plate moves with constant velocity in the direction of fluid flow.It is also assumed that the plate temperature and concentration are exponentially varying with time.Under these assumptions, the appropriate boundary conditions for the velocity, temperature, and concentration fields are where    is the wall dimensional velocity;    and    are the wall dimensional concentration and temperature, respectively;   ∞ ,   ∞ , and   ∞ are the free stream dimensional velocity, concentration, and temperature, respectively; and   is the constant.
It is clear from (1) that the suction velocity normal to the plate is either a constant or a function of time.Hence the suction velocity normal to the plate is taken as where  0 is a 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 In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced: In view of ( 5)-( 10), ( 2)-( 4) reduce to the following nondimensional form: where Γ = (1 − 4/(3 + 4))Pr and Gr, Gc, Pr, , Sc, , , and   are the thermal Grashof number, solutal Grashof number, Prandtl number, radiation parameter, Schmidt number, heat generation parameter, permeability of the porous medium, and chemical reaction parameter, respectively The corresponding dimensionless boundary conditions are

Solution of the Problem
Equations ( 11) are coupled, nonlinear partial differential equations and these cannot be solved in closed form.However, these equations can be reduced to a set of ordinary differential equations, which can be solved analytically.This can be done by representing the velocity, temperature, and concentration of the fluid in the neighborhood of the plate as Substituting ( 13) in (11), equating the harmonic and nonharmonic terms, and neglecting the higher order terms of 0() 2 , we obtain where the prime denotes ordinary differentiation with respect to .The corresponding boundary conditions can be written as Solving ( 14) subject to boundary conditions (15), we obtain the velocity, temperature, and concentration distributions in the boundary layer as

𝜃 (𝑦, 𝑡) = 𝑒
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 flow.These parameters can be defined and determined as follows.
Knowing the velocity field, the skin-friction at the plate can be obtained, which in non-dimensional form is given by Knowing the temperature field, the rate of heat transfer coefficient can be obtained, which in the non-dimensional form, in terms of the Nusselt number, is given by Knowing the concentration field, the rate of mass transfer coefficient can be obtained, which in the non-dimensional form, in terms of the Sherwood number, is given by where Re  =  0 / is the local Reynolds number.

Results and Discussion
In the preceding section, the problem of an unsteady free convective flow of a viscous incompressible, thermally radiating, and chemically reacting fluid past a semi-infinite plate in the presence of heat generation was formulated and solved by means of a perturbation method.The expressions for the velocity, temperature, and concentration were obtained.To illustrate the behavior of these physical quantities, numerical values of these quantities were computed with respect to the variations in the governing parameters, namely, the thermal Grashof number Gr, the solutal Grashof number Gc, Prandtl number Pr, Schmidt number Sc, the radiation parameter , the permeability of the porous medium , the heat generation parameter , and the chemical reaction parameter   .In the present study the following default parametric values are adopted: Gr = 2.0, Gc = 2.0,  = 5.0,  = 1.4,Sc = 0.2,  = 5.0,   = 2.0,  = 0.1, Pr = 0.71,   = 0.4,  = 0.5,  = 1.0,  = 0.1, and  = 0.01.All the graphs and tables therefore correspond to these values unless specifically indicated on the appropriate graph.Figure 1 presents the typical velocity profiles in the boundary layer for various values of the thermal Grashof number Gr.The thermal Grashof number Gr signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer.It is observed that an increase in Gr leads to a rise in the values of velocity due to enhancement of thermal buoyancy force.Here the positive values of Gr correspond to cooling of the surface.It is observed that velocity increases rapidly near the wall of the porous plate as Gr increases and then decays to the free stream velocity.
For the case of different values of the solutal Grahof number Gc, the velocity profiles in the boundary layer are shown in Figure 2. The solutal Grashof number Gc defines the ratio of the species buoyancy force to the viscous hydrodynamic force.As expected, as Gc increases, the fluid velocity increases and the peak value is more distinctive maximum value in the vicinity of the plate and then decreases properly to approach the free stream value.Figure 3 shows the velocity profiles for different values of the permeability of the porous medium .Clearly, as  increases the velocity tends to increase.
For different values of the radiation parameter  the velocity and temperature profiles are plotted in Figures 4(a) and 4(b).The radiation parameter  defines the relative contribution of conduction heat transfer to thermal radiation transfer.It is obvious that an increase in the radiation parameter  results in a decrease in the velocity and temperature within the boundary layer, as well as decreased thickness of the velocity and temperature boundary layers.
Figures 5(a  , as expected, the numerical results show that an increase in the Prandtl number results in a decrease of the thermal boundary layer and in general lower average temperature with in the boundary layer.The reason is that smaller values of Pr are equivalent to increase in the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly for higher values of Pr.Hence in the case of smaller Prandtl numbers the thermal boundary layer is thicker and the rate of heat transfer is reduced. Figures 6(a) and 6(b) display the effects of the Schmidt number Sc on velocity and concentration, respectively.The Schmidt number Sc embodies the ratio of the momentum to the mass diffusivity.The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers.As the Schmidt number increases the concentration decreases.This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity.The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the  The influences of chemical reaction parameter   on the velocity and concentration across the boundary layer are presented in Figures 7(a) and 7(b).It is seen that the velocity as well as concentration across the boundary layer decreases with an increase in the chemical reaction parameter   .
Figures 8(a) and 8(b) depict the effect of heat generation parameter  on the velocity and temperature.It is noticed that the velocity as well as temperature across the boundary layer increases with an increase in the heat generation parameter .
Tables 1-7 show the effects of the thermal Grashof number Gr, solutal Grashof number Gc, radiation parameter , Prandtl number Pr, Schmidt number Sc, chemical reaction parameter   , and heat generation parameter  on the skin friction coefficient   , Nusselt number Nu, and the Sherwood number Sh.From Tables 1 and 2, it is observed that as Gr or Gc increases, the skin-friction coefficient increases.From Table 3, it can be seen that as the radiation parameter

Conclusions
The problem of unsteady, two-dimensional laminar free convective mass transfer flow of a viscous incompressible fluid through a highly porous medium past an infinite vertical moving porous plate in the thermal radiation, heat generation, and chemical reaction has been studied.The nondimensional governing equations were solved by perturbation technique.Numerical results are presented to illustrate the details of the flow and heat transfer characteristics and their dependence on the material parameters.We observe that the velocity increases as the thermal Grashof number Gr, solutal Grashof number Gc, permeability of the porous medium , or heat generation parameter  increases, while it decreases as the Prndtl number Pr, radiation parameter R, Schmidt number Sc, or the chemical reaction parameter   .It is observed that the temperature decreases as the Prandtl number Pr or radiation parameter  increases, while it increases as the heat generation parameter  increases.The concentration decreases as the Schmidt number Sc or the chemical reaction parameter   increases.

Appendix
One has

4 Figure 1 :
Figure 1: Velocity profiles for different values of Gr.

Figure 5 (
Figure 5(b), as expected, the numerical results show that an increase in the Prandtl number results in a decrease of the thermal boundary layer and in general lower average temperature with in the boundary layer.The reason is that smaller values of Pr are equivalent to increase in the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly for higher values of Pr.Hence in the case of smaller Prandtl numbers the thermal boundary layer is thicker and the rate of heat transfer is reduced.Figures6(a) and 6(b) display the effects of the Schmidt number Sc on velocity and concentration, respectively.The Schmidt number Sc embodies the ratio of the momentum to the mass diffusivity.The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers.As the Schmidt number increases the concentration decreases.This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity.The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the

Figure 4 :
Figure 4: (a) Velocity profiles for different values of.(b) Temperature profiles for different values of .

Figure 5 :
Figure 5: (a) Velocity profiles for different values of Pr.(b) Temperature profiles for different values of Pr.

Figure 6 :
Figure 6: (a) Velocity profiles for different values of Sc.(b) Concentration profiles for different values of Sc.

Figure 7 :
Figure 7: (a) Velocity profiles for different values of   .(b) Concentration profiles for different values of   .

Figure 8 :
Figure 8: (a) Velocity profiles for different values of .(b) Temperature profiles for different values of .

Table 1 :
Effects of Gr on skin-friction   .

Table 4 ,
it is found that an increase in Pr leads to a decrease in the skin-friction and an increase in the Nusselt number.From Table5, it is observed that, as the Schmidt number increases, the skin-friction decreases and the Sherwood number increases.From Table6, it is seen that, as the chemical reaction parameter   increases, the skin-friction decreases and the Sherwood number increases.
From Table7, it is observed that, as the heat generation parameter increases, the skin-friction increases and the Nusselt number decreases.

Table 2 :
Effects of Gc on skin-friction   .

Table 3 :
Effects of R on skin-friction   and Nusselt number Nu Re −1  .

Table 4 :
Effects of Pr on skin-friction   and Nusselt number

Table 5 :
Effects of Sc on skin-friction   and Sherwood number Sh Re −1  .

Table 6 :
Effects of   on skin-friction   and Sherwood number Sh Re −1  .

Table 7 :
Effects of Q on skin-friction   and Nusselt number Nu Re −1  .