Analytical Solution of Thermal Radiation and Chemical Reaction Effects on Unsteady MHD Convection through Porous Media with Heat Source/Sink

Abstract

This paper analytically studies the thermal radiation and chemical reaction effect on unsteady MHD convection through a porous medium bounded by an infinite vertical plate. The fluid considered here is a gray, absorbing-emitting but nonscattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The dimensionless governing equations are solved using Laplace transform technique. The resulting velocity, temperature and concentration profiles as well as the skin-friction, rate of heat, and mass transfer are shown graphically for different values of physical parameters involved.

1. Introduction

Thermal radiation of a gray fluid which is emitting and absorbing radiation in a nonscattering medium has been investigated by Ali et al. [1], Ibrahim and Hady [2], Mansour [3], Hossain et al. [4, 5], Raptis and Perdikis [6], Makinde [7], and Abdus-Sattar and Hamid Kalim [8]. All these studies have investigated the unsteady flow in a nonporous medium. From the previous literature survey about unsteady fluid flow, we observe that few papers were done in a porous medium. The radiative flows of an electrically conducting fluid with high temperature in the presence of a magnetic field are encountered in electrical power generation, astrophysical flows, solar power technology, space vehicle, nuclear engineering application, and other industrial areas. The analytical solution of unsteady MHD laminar convective flow with thermal radiation of a conducting fluid with variable properties through a porous medium in the presence of chemical reaction and heat source or sink has not been investigated.

Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in the 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, heat and mass transfer occur simultaneously. Possible applications of this type of flow can be found in many industries, for example, in the electric power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid. Many practical diffusion operations involve the molecular diffusion of a species in the presence of chemical reaction within or at the boundary. There are two types of reactions. A homogeneous reaction is one that occurs uniformly throughout a give phase. The species generation in a homogeneous reaction is analogous to internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. It can therefore be treated as a boundary condition similar to the constant heat flux condition in heat transfer. Combined heat and mass transfer with chemical reaction in geometric with and without porous media has been studied by others [9–19].

This paper deals with the study of thermal radiation and chemical reaction effects on the unsteady MHD convection through a porous medium bounded by an infinite vertical plate with heat source/sink. The governing equations are solved by Laplace transform technique. The results are obtained for velocity, temperature, concentration, skin-friction, rate of heat and mass transfer. The effects of various material parameters are discussed on flow variables and presented by graphs.

2. Formulation of the Problem

Consider the unsteady free convection flow of an incompressible viscous fluid due to heat and mass transfer through a porous medium bounded by an infinite vertical plate under the action of an external transfer magnetic field of uniform strength 𝐵₀. The fluid considered is a gray, absorbing-emitting radiation but nonscattering medium. It is assumed that there exists a homogeneous first-order chemical reaction between the fluid and species concentration. Then, by usual Boussinesq's approximation, the unsteady flow is governed by the following equations:𝜕𝑣^𝜕𝑦^𝜕𝑢_=0,(2.1)𝜕𝑡^+𝑣^𝜕𝑢^𝜕𝑦^=𝑔𝛽𝑇^−𝑇^∞+𝑔𝛽^∗𝐶^−𝐶^∞+𝜈𝜕²𝑢^𝜕𝑦^2−𝜎𝐵^2𝑜𝜌+𝐾𝜈𝑢^𝜕𝑇_,(2.2)𝜕𝑡^+𝑣^𝜕𝑇^𝜕𝑦^=𝛼𝜕²𝑇^𝜕𝑦^2+𝑄𝑇^−𝑇^∞−1𝜌𝑐_𝑝𝜕𝑞^∗𝑟𝜕𝑦^𝜕𝐶_,(2.3)𝜕𝑡^+𝑣^𝜕𝐶^𝜕𝑦^=𝐷𝜕²𝐶^𝜕𝑦^2−𝑅^∗𝐶^−𝐶^∞.(2.4) For constant and uniform suction, (2.1) integrates to𝑣^=−𝑣₀,(2.5) where the negative sign indicates that suction is towards the plate.

The initial and boundary conditions are 𝑢𝑢^=0,𝑇^=𝑇^∞,𝐶^=𝐶^∞,for𝑡≤0,_𝑢=0,𝑇^=𝑇^𝑤,𝐶^=𝐶^𝑤,for𝑦^=0,𝑡>0,_⟶0,𝑇^⟶𝑇^∞,𝐶^⟶𝐶^∞,for𝑦^⟶∞,𝑡>0.(2.6) The radiative heat flux under term by using the Rosseland approximation is given by𝑞^∗𝑟=−4𝜎^∗3𝑘^∗𝜕𝑇^4𝜕𝑦^.(2.7)

We assume that the temperature differences within the flow are sufficiently small such that 𝑇^4 may be expressed as a linear function of the temperature. This is accomplished by expanding 𝑇^4 in a Taylor series about 𝑇^∞ and neglecting higher-order terms, thus 𝑇^4 can be expressed as 𝑇^4≅4𝑇^∞3𝑇^−3𝑇^∞4.(2.8) By using (2.7) and (2.8), (2.3) reduces to𝜕𝑇^𝜕𝑡^=𝛼𝜕²𝑇^𝜕𝑦^2+𝑄𝑇^−𝑇^∞+16𝜎^∗𝑇^∞33𝑘^∗𝜌𝑐_𝑝𝜕²𝑇^𝜕𝑦^2.(2.9) To present solutions which are independent of geometry of the flow regime, we introduce the dimensionless variable as follows:𝑢=𝑢^𝑣_𝑜,𝑦=𝑦^𝑣_𝑜𝜈,𝑡=𝑡^𝑣^2𝑜𝜃=𝑇_𝜈,−𝑇^∞𝑇^𝑤−𝑇^∞,𝜙=𝐶^−𝐶^∞𝐶^𝑤−𝐶^∞.(2.10) Substituting from (2.10) into (2.2), (2.9), and (2.4), we obtain𝜕𝑢𝜕𝑡−𝜕𝜕𝑦𝑢=𝜕²𝑢𝜕𝑦²+𝐺𝑟𝜃+𝐺𝑚𝜙−11+4𝑅𝑘+𝑀𝑢,3𝜕²𝜃𝜕𝑦²−Pr𝜕𝜃𝜕𝜕𝑡−𝜕𝜕𝜃𝑦+Pr𝜂𝜃=0,₂𝜙𝜕𝑦²−Sc𝜕𝜙𝜕𝑡−𝜕𝜕𝜙𝑦−Sc𝛿𝜙=0,(2.11) where 𝐺𝑟=𝑔𝛽𝜈𝑇^𝑤−𝑇^∞𝑣^3𝑜,𝐺𝑚=𝑔𝛽^∗𝜈𝐶^𝑤−𝐶^∞𝑣^3𝑜,𝛿=𝑅^∗𝜈𝑣^2𝑜𝑘=𝐾𝜈_2𝑜,𝜈²,𝑀=𝜎𝐵^2𝑜𝜈𝜌𝑣^2𝑜,Sc=𝜈𝐷,𝜂=𝜈𝑣𝑄^2𝑜𝛿=𝑅_∗,𝜈𝑣^2𝑜,Pr=𝜌𝑐^𝑝𝜈𝑘₁=𝜈𝛼,𝑅=4𝜎^∗𝑇^∞3𝑘^∗𝑘₁.(2.12) The initial and boundary conditions in nondimensional form are 𝑢⟶𝑢=0,𝑢0=,𝜃0⟶𝜃,=𝜃01=,,0𝜙𝜙,⟶=𝜙1=,00a,ats𝑦∀𝑦=𝑦,⟶𝑡0,≤𝑡∞0>,,0𝑡,>0.(2.13) All the physical variables are defined in the nomenclature. The solutions are obtained for hydrodynamic flow field in the presence of thermal radiation, chemical reaction and heat source/sink.

3. Analytic Solution

In order to obtain the solution of the present problem, we will use the Laplace transform technique.

Applying the Laplace transform to the system of (2.11), and the boundary conditions (2.13), we get𝜕²𝜃𝜕𝑦²+𝐹₂𝜕𝜃𝜕𝑦−𝐹²𝜕(𝑠−𝜂)𝜃=0,(3.1)₂𝜙𝜕𝑦²+Sc𝜕𝜙𝜕𝑦−Sc(𝑠+𝛿)𝜕𝜙=0,(3.2)₂𝑢𝜕𝑦²+𝜕𝑢𝜕𝑦−𝑠+𝑀^𝑢+𝐺𝑟𝜃+𝐺𝑚𝜙=0,(3.3) where,𝑀^=1𝑘+𝑀,𝐹²=Pr(1+(4/3)𝑅),(3.4)𝑠 is Laplace transformation parameter, 𝑢, 𝜃, and 𝜙 are Laplace transformation of 𝑢, 𝜃, and 𝜙, respectively, 𝑢=0,𝜃=𝜙=1𝑠at𝑦=0,𝑡>0,𝑢=𝜃=𝜙=0,as𝑦⟶∞,𝑡>0.(3.5) Solving the system of (3.1)–(3.3), with the help of the result in (3.5), we get𝜃(𝑦,𝑠)=1𝑠Exp𝜆^𝜃𝑦,(3.6)𝜙(𝑦,𝑠)=1𝑠Exp𝜆^𝜙𝑦,(3.7)𝑢(𝑦,𝑠)=𝑢₁(𝑦,𝑠)+𝑢₂(𝑦,𝑠)+𝑢₃(𝑦,𝑠)+𝑢₄(𝑦,𝑠),(3.8) where𝑢₁(𝑦,𝑠)=−𝐺𝑟𝑠𝜆^2𝜃+𝑒𝜆^𝜆𝜃𝑦_𝜃−𝐵₄=−𝐺𝑠𝑟𝑒^𝜆𝜃𝑦⎡⎢⎢⎢⎣(𝛼𝑠+𝛾)−𝛽𝑠+𝐵²¹𝛼²(𝑠+𝑊)²+𝑄²⎤⎥⎥⎥⎦,𝑢₂(𝑦,𝑠)=−𝐺𝑚𝑠𝜆^2𝜙+𝑒𝜆^𝜆𝜙𝑦_𝜙−𝐵₄=−𝐺𝑠𝑚𝑒^𝜆𝜙𝑦⎡⎢⎢⎢⎣𝛼₁𝑠+𝛾₁−𝛽₁𝑠+𝐵²²𝛼²¹𝑠+𝑊¹²+𝑄²¹⎤⎥⎥⎥⎦,𝑢₃(𝑦,𝑠)=𝐺𝑟𝑠𝜆^2𝜃+𝑒𝜆^𝜆𝑢𝑦_𝜃−𝐵₄=𝐺𝑠𝑟𝑒^𝜆𝑢𝑦⎡⎢⎢⎢⎣(𝛼𝑠+𝛾)−𝛽𝑠+𝐵²¹𝛼²(𝑠+𝑊)²+𝑄²⎤⎥⎥⎥⎦,𝑢₄(𝑦,𝑠)=𝐺𝑚𝑠𝜆^2𝜙+𝑒𝜆^𝜆𝑢𝑦_𝜙−𝐵₄=𝐺𝑠𝑚𝑒^𝜆𝑢𝑦⎡⎢⎢⎢⎣𝛼₁𝑠+𝛾₁−𝛽₁𝑠+𝐵²²𝛼²¹𝑠+𝑊¹²+𝑄²¹⎤⎥⎥⎥⎦,(3.9) where𝐵²¹=𝐹²4−𝜂,𝜆^𝜃=−𝐹²2−√𝐹₂𝑠+𝐵²¹,𝐵²²=Sc𝜆4+𝛿,_𝜙=−Sc2−√Sc𝑠+𝐵²²,𝐵²³=14+𝑀^,𝜆_𝑢=−12−𝛼=𝐹₂𝑠+𝐵²³,−1,𝛽=𝛼√𝐹₂,𝛾=𝐹²2𝛼−𝐹²𝜂−𝑀^,𝑊=2𝛼𝛾−𝛽²2𝛼²𝑄=,𝛾²−𝛽²𝐵²¹𝛼²−𝑊²,𝑄₁=⎷𝛾²¹−𝛽²¹𝐵²²𝛼²¹𝛽−𝑊²¹,𝛼₁=Sc−1,₁=𝛼₁√Sc,𝛾₁=Sc²2+Sc𝛿−21−𝑀^,𝑊₁=2𝛼¹𝛾₁−𝛽²¹2𝛼²¹,𝐵₄=𝑆+𝑀^.(3.10) The inverse Laplace transformation of (3.6) is𝜃(𝑦,𝑡)=𝑒^{−(𝑦/2)𝐹2}2𝑒^{𝑦√𝐹2√𝐹2/4−𝜂}erf𝑐𝑦2𝐹₂𝑡+𝐵¹+𝑒_{√−𝑦𝐹2√𝐹2/4−𝜂}√𝑡erf𝑐𝑦2𝐹₂𝑡−𝐵¹√𝑡.(3.11) The analytic solution of (3.2) can be obtained by taking the inverse transforms of (3.7). So, the solution of the problem for the concentration 𝜙(𝑦,𝑡) for𝑡>0𝜙(𝑦,𝑡)=𝑒^{−(𝑦/2)Sc}2𝑒^{𝑦√Sc𝐵2}erf𝑐𝑦2Sc𝑡+𝐵²√𝑡+𝑒^{√−𝑦Sc𝐵2}erf𝑐𝑦2Sc𝑡−𝐵²√𝑡.(3.12) Similarly, the general solution of (3.3) can be obtained by taking the inverse Laplace transform of (3.8). The expressions for velocity, 𝑢, velocity gradient, 𝜕𝑢/𝜕𝑦|_𝑦=0, temperature gradient, 𝜕𝜃/𝜕𝑦|_𝑦=0 and concentration gradient, 𝜕𝜙/𝜕𝑦|_𝑦=0 are shown in Appendix A.

4. Numerical Results and Discussion

In order to get physical insight into the problem, the numerical calculations are carried out to study the variations in velocity 𝑢, temperature 𝜃 and concentration 𝜙. The variation in skin-friction (shear stress at the wall), rate of heat and mass transfer are computed. These variations involve the effects of time 𝑡, heat generation parameter 𝜂, chemical reaction parameter𝛿, Schmidt number Sc, Prandtl number Pr, thermal radiation parameter 𝑅, magnetic field parameter 𝑀 and permeability parameter 𝑘. The values of Prandtl number Pr are chosen to be 3, 7, and 10. The values of Schmidt number Sc are chosen to be 0.22, 0.62, and 0.78, which represent hydrogen, water vapour and ammonia, respectively.

Representative velocity, temperature and concentration profiles across the boundary layer for different values of the dimensionless time 𝑡 are presented in Figure 1. As the dimensionless time increases, the velocity, temperature and concentration profiles increase.

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Figure 1 

Velocity, temperature and concentration profiles against 𝑦 for various values of 𝑡 with Pr=10, 𝑀=0.2, 𝜂=−2, 𝑘=1, Sc=0.22, 𝐺𝑟=𝐺𝑚=5, 𝛿=0.1, and 𝑅=0.1.

Figure 2 describe the behavior of velocity and temperature profiles across the boundary layer for different values of the heat generation parameter 𝜂. As the heat source parameter 𝜂 increases, the velocity and temperature profiles increase. The volumetric heat source term may exert a strong influence on the heat transfer and as a consequence, also on the fluid flow.

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Figure 2 

Velocity and temperature profiles against 𝑦for various values of 𝜂 with 𝑡=0.1, Pr=10, 𝑀=0.2, 𝑘=1, Sc=0.22, 𝐺𝑟=𝐺𝑚=5, 𝛿=0.1, and 𝑅=0.1.

Figure 3 shows the velocity and concentration profiles for different values of chemical reaction parameter. As the chemical reaction parameter increases, the velocity increases but the concentration profile decreases.

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Figure 3 

Velocity and concentration profiles against 𝑦for various values of 𝛿 with 𝑡=0.1, Pr=10, 𝑀=0.2, 𝑘=1, 𝜂=−2, Sc=0.22, 𝐺𝑟=𝐺𝑚=5, and 𝑅=0.1.

Figure 4 displays the effects of Schmidt number on the velocity and concentration profiles, respectively. As the Schmidt number increases, the velocity and concentration decrease. Reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers. These behaviors are evident from Figure 4.

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Figure 4 

Velocity and concentration profiles against 𝑦for various values of Sc with 𝑡=0.1, Pr=10, 𝑀=0.2, 𝑘=1, 𝜂=−2, 𝐺𝑟=𝐺𝑚=5, and 𝑅=0.1.

Figure 5 illustrates the velocity and temperature profiles for different values of Prandtl number. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing velocity. Also, it is shown that an increase in the Prandtl number results tend to a decreasing of the thermal boundary layer and in general it lowers the average temperature through the boundary layer. The reason is that, the 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.

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Figure 5 

Velocity and temperature profiles against 𝑦for various values of Pr with 𝑡=0.1, Sc=0.22, 𝑀=0.2, 𝑘=1, 𝜂=−2, 𝐺𝑟=𝐺𝑚=5, and 𝑅=0.1.

For different values of the radiation parameter 𝑅, the velocity and temperature profiles are shown in Figure 6. It is noticed that an increase in the radiation parameter results an increase in the velocity and temperature within the boundary layer, also it increases the thickness of the velocity and temperature boundary layers.

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Figure 6 

Velocity and temperature profiles against 𝑦for various values of 𝑅 with 𝑡=0.1, Pr=10, 𝑀=0.2, 𝑘=1, 𝜂=−2, 𝐺𝑟=𝐺𝑚=5.0, and Sc=0.22.

The velocity profiles for different values of the magnetic field parameter and dimensionless permeability are shown in Figure 7. It is clear that the velocity decreases with increasing of the magnetic field parameter. It is because that the application of transverse magnetic field will result a restrictive type force (Lorentz's force), similar to drag force which tends to restrictive the fluid flow and thus reducing its velocity. The presence of porous media increases the resistance flow resulting in a decrease in the flow velocity. This behavior is depicted by the decrease in the velocity as permeability decreases and when 𝑘→∞ (i.e., the porous medium effect is vanishes) the velocity is greater in the flow field. These behaviors are shown in Figure 7.

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Figure 7 

Velocity profiles against 𝑦for various valuesof 𝑀 and 𝑘 with 𝑡=0.1, Pr=10, 𝑀=0.2, Sc=0.22, 𝑅=0.1, and 𝜂=−2.

Figure 8 displays the effect of Sc, 𝑡, 𝑀, 𝑘, 𝛿, and Pr on shear stress 𝑢^(0,𝑡) with respect to radiation parameter 𝑅, it is obvious that there is a slight changes in shear stress, also, it is seen that shear stress increases with an increasing of 𝑘 and 𝛿 but decreases with an increasing values of Sc, 𝑡, 𝑀, and Pr.

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Figure 8 

Variation of shear stress 𝑢(0,𝑡) against 𝑅for various values of Sc, 𝑡, 𝑀, 𝑘, 𝛿, and Prwith constant Pr=10, 𝑀=0.2, 𝑘=1, Sc=0.22, 𝐺𝑚=𝐺𝑟=5, and 𝜂=−2.

Figure 9 shows the influence of time, heat source parameter and the radiation parameter on the negative values of gradient temperature (i.e., −𝜃^(0,𝑡))with respect to the Prandtl number, it is seen that the increasing values of the time, heat source parameter and radiation parameter tend to decreasing in the negative values of the temperature gradient, also, it increases with the increasing of Pr.

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Figure 9 

Variation of −𝜃(0,𝑡) for various values of 𝑡, 𝜂, and 𝑅 with Pr=10, 𝑀=0.2, 𝑘=1, Sc=0.22, 𝐺𝑟=𝐺𝑚=5.0, and 𝜂=−2.

Figure 10 displays the influence of time and chemical reaction parameter on the negative values of concentration gradient (i.e., −𝜙^(0,𝑡)) respect the Schmidt number, it is concluded that it increases with the increasing of Sc and chemical reaction 𝛿 but decreases with an increase of 𝑡.

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Figure 10 

Variation of −𝜙(0,𝑡) for various values of 𝑡 and 𝛿 with Pr=10, 𝑀=0.2, 𝑘=1, Sc=0.22, 𝐺𝑟=𝐺𝑚=5.0, and𝜂=−2.

5. Conclusion

A mathematical model has been presented for analytically studies the thermal radiation and chemical reaction effect on unsteady MHD convection through a porous medium bounded by an infinite vertical plate. The fluid considered here is a gray, absorbing-emitting but nonscattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The dimensionless governing equations are solved using Laplace transform technique. The resulting velocity, temperature and concentration profiles as well as the skin-friction, rate of heat and mass transfer are shown graphically for different values of physical parameters involved. It has been shown that the fluid is accelerated, that is, velocity (𝑢) is increased with an increasing values of time (𝑡), chemical reaction parameter (𝛿), Prandtl number (Pr), radiation parameter (𝑅), while they show opposite tends with an increasing values of heat source parameter (𝜂), Schmidt number (Sc) and magnetic field parameter (𝑀). Also, it is shown that velocity (𝑢) does not affected with the various values of the dimensionless permeability 𝑘. We conclude that, the negative temperature gradient (−𝜃^(0,𝑡)) increases as Pr increases but decreases as 𝑡, 𝜂, and 𝑅increase. Finally, we obvious that, the negative concentration gradient (−𝜙^(0,𝑡)) increases as Sc and 𝛿 increases but decreases as 𝑡increase.

It is hoped that the results obtained here will not only provide useful information for applications, but also serve as a complement to the previous studies.

Appendix

A.

The inverse laplace transformation of (3.8) is𝑢(𝑦,𝑡)=𝑢₁(𝑦,𝑡)+𝑢₂(𝑦,𝑡)+𝑢₃(𝑦,𝑡)+𝑢₄(𝑦,𝑡),(A.1) where 𝑢₁(𝑦,𝑡)=−𝐺𝑟𝛼²𝛼^𝑡0𝜃(𝑦,𝑢)𝑒^{−𝑊(𝑡−𝑢)}cos𝑄(𝑡−𝑢)𝑑𝑢+𝛾−𝛼𝑊×_𝑡0𝑄−𝛽𝜃𝐵(₂₁𝑦,𝑢)𝑒^{−𝑊(𝑡−𝑢)}sin𝑄(𝑡−𝑢)𝑑𝑢−𝛽^𝑡0𝜃(𝑦,𝑢)𝐺₁(𝑡−𝑢)𝑑𝑢+𝛽−2𝑊^𝑡0𝜃(𝑦,𝑢)^0𝑡−𝑢𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑢−𝜏)}cos𝑄(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢𝑄𝑊𝐵²¹×−2𝑊+𝑊²+𝑄²_𝑡0𝑢𝜃(𝑦,𝑢)^0𝑡−𝑢𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑢−𝜏)}sin𝑄(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢₂,(𝑦,𝑡)=−𝐺𝑚𝛼²¹𝛼¹^𝑡0𝜙(𝑦,𝑢)𝑒^{−𝑊1(𝑡−𝑢)}cos𝑄₁(𝑡−𝑢)𝑑𝑢+𝛾¹−𝛼₁𝑊₁𝑄₁×^𝑡0𝜙(𝑦,𝑢)𝑒^{−𝑊1(𝑡−𝑢)}sin𝑄₁(𝑡−𝑢)𝑑𝑢−𝛽₁^𝑡0−𝛽₁𝜙(𝑦,𝑢)𝐺₂(𝑡−𝑢)𝑑𝑢+𝛽𝐵²²−2𝑊₁^𝑡0𝜙(𝑦,𝑢)^0𝑡−𝑢𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑢−𝜏)}cos𝑄₁(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢₁𝑄₁𝑊₁𝐵²²×−2𝑊₁+𝑊²¹+𝑄²¹^𝑡0𝜙(𝑦,𝑢)_0𝑡−𝑢𝑢𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑢−𝜏)}sin𝑄₁(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢₃,(𝑦,𝑡)=𝐺𝑟𝛼²𝛼^𝑡0𝜃₁(𝑦,𝑢)𝑒^{−𝑊(𝑡−𝑢)}cos𝑄(𝑡−𝑢)𝑑𝑢+𝛾−𝛼𝑊𝑄₁×−𝛽^𝑡0𝜃𝐵₁₂₁(𝑦,𝑢)𝑒^{−𝑊(𝑡−𝑢)}sin𝑄(𝑡−𝑢)𝑑𝑢−𝛽^𝑡0𝜃₁(𝑦,𝑢)𝐺₁(𝑡−𝑢)𝑑𝑢+𝛽−2𝑊^𝑡0𝜃₁(𝑦,𝑢)^0𝑡−𝑢𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑢−𝜏)}cos𝑄(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢𝑄𝑊𝐵²¹×−2𝑊+𝑊²+𝑄²^𝑡0𝜃₁(𝑦,𝑢)_0𝑡−𝑢𝑢𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑢−𝜏)}sin𝑄(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢₄,(𝑦,𝑡)=𝐺𝑚𝛼²¹𝛼¹^𝑡0𝜃₁(𝑦,𝑢)𝑒^{−𝑊1(𝑡−𝑢)}cos𝑄₁(𝑡−𝑢)𝑑𝑢+𝛾¹−𝛼₁𝑊₁𝑄₁×^𝑡0𝜃₁(𝑦,𝑢)𝑒^{−𝑊1(𝑡−𝑢)}sin𝑄₁(𝑡−𝑢)𝑑𝑢−𝛽₁^𝑡0−𝛽₁𝜃₁(𝑦,𝑢)𝐺₂(𝑡−𝑢)𝑑𝑢+𝛽𝐵²²−2𝑊₁^𝑡0𝜃₁(𝑦,𝑢)^0𝑡−𝑢𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑢−𝜏)}cos𝑄₁(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢₁𝑄₁𝑊₁𝐵²²×−2𝑊₁+𝑊²¹+𝑄²¹^𝑡0𝜃₁(𝑦,𝑢)_0𝑡−𝑢𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑢−𝜏)}sin𝑄₁(𝑡−𝑢−𝜏)𝑑𝜏𝑑𝑢,(A.2) where 𝐺₁(𝑡)=1𝐵₁𝐺₂erf𝐵₁√𝑡,(𝑡)=1𝐵₂𝜃₁erf𝐵₂√𝑡,(𝑦,𝑡)=12𝑒^−𝑦/2𝑒^𝑦𝐵3erf𝑐𝑦2√𝑡+𝐵³√𝑡+𝑒^−𝑦𝐵3erf𝑐𝑦2√𝑡−𝐵³√𝑡.(A.3)𝜃^(0,𝑡)=−𝐹²2−𝐵¹√𝐹₂erf𝐵₁√𝑡−𝐹₂𝜋𝑡𝑒^{−𝐵21𝑡},(A.4)𝜙^(0,𝑡)=−Sc2−𝐵²√Scerf𝐵₂√𝑡−Sc𝜋𝑡𝑒^{−𝐵22𝑡},(A.5)𝜕𝑢(𝑦,𝑡)𝜕𝑦||||_𝑦=0=⁴_𝑖=1𝜕𝜕𝑦𝑢^𝑖𝜕𝑢₁(𝑦,𝑡)||_𝑦=0,𝑖=1,2,3,4,𝜕𝑦||||_𝑦=0=−𝐺𝑟𝛼²𝛼^√𝑡0𝜃^(0,𝑥)𝑒^{−𝑊(𝑡−𝑥2)}cos𝑄𝑡−𝑥²𝑑𝑥+𝛾−𝛼𝑊×_√𝑡0𝑄−𝛽𝐵𝜃^₂₁(0,𝑥)𝑒^{−𝑊(𝑡−𝑥2)}sin𝑄𝑡−𝑥²𝑑𝑥−𝛽^√𝑡0𝜃^(0,𝑥)𝐺₁𝑡−𝑥²𝑑𝑥−2𝑊^√𝑡0Θ^(0,𝑥)^𝑡−𝑥2₀𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑥2−𝜏)}×+c𝛽os𝑄𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥𝑄𝑊𝐵²¹×−2𝑊+𝑊²+𝑄²^√𝑡0𝜃^(0,𝑥)_{𝑡−𝑥20}𝜕𝑢₂𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑥2−𝜏)}sin𝑄𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥,𝜕𝑦||||_𝑦=0=−𝐺𝑚𝛼²¹𝛼¹^√𝑡0𝜙^(0,𝑥)𝑒^{−𝑊1(𝑡−𝑥2)}cos𝑄₁𝑡−𝑥²𝑑𝑥+𝛾¹−𝛼₁𝑊₁𝑄₁×^√𝑡0𝜙^(0,𝑥)𝑒^{−𝑊1(𝑡−𝑥2)}sin𝑄₁𝑡−𝑥²𝑑𝑥−𝛽₁^√𝑡0−𝛽₁𝜙^(0,𝑥)𝐺₂𝑡−𝑥²𝑑𝑥𝐵²²−2𝑊₁^√𝑡0𝜙^(0,𝑥)^𝑡−𝑥2₀𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑥2−𝜏)}+×c𝛽os𝑄₁𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥₁𝑄₁𝑊₁𝐵²²×−2𝑊₁+𝑊²¹+𝑄²¹^√𝑡0𝜙^(0,𝑥)_{𝑡−𝑥20}𝜕𝑢₃𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑥2−𝜏)}sin𝑄₁𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥,𝜕𝑦||||_𝑦=0=𝐺𝑟𝛼²𝛼^√𝑡0𝜃^1(0,𝑥)𝑒^{−𝑊(𝑡−𝑥2)}cos𝑄𝑡−𝑥²𝑑𝑥+𝛾−𝛼𝑊×_√𝑡0𝑄−𝛽𝐵𝜃^₂₁(0,𝑥)𝑒^{−𝑊(𝑡−𝑥2)}sin𝑄𝑡−𝑥²𝑑𝑥−𝛽^√𝑡0𝜃^(0,𝑥)𝐺₁𝑡−𝑥²𝑑𝑥−2𝑊^√𝑡0𝜃^(0,𝑥)^𝑡−𝑥2₀𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑥2−𝜏)}×+c𝛽os𝑄𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥𝑄𝑊𝐵²¹×−2𝑊+𝑊²+𝑄²^√𝑡0𝜃^(0,𝑥)_{𝑡−𝑥20}𝜕𝑢₄𝐺₁(𝜏)𝑒^{−𝑊(𝑡−𝑥2−𝜏)}sin𝑄𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥,𝜕𝑦||||_𝑦=0=𝐺𝑚𝛼²¹𝛼¹^√𝑡0𝜃^1(0,𝑥)𝑒^{−𝑊1(𝑡−𝑥2)}cos𝑄₁𝑡−𝑥²𝑑𝑥+𝛾¹−𝛼₁𝑊₁𝑄₁×^√𝑡0𝜃^1(0,𝑥)𝑒^{−𝑊1(𝑡−𝑥2)}sin𝑄₁𝑡−𝑥²𝑑𝑥−𝛽₁^√𝑡0−𝛽₁𝜃^1(0,𝑥)𝐺₂𝑡−𝑥²𝑑𝑥𝐵²²−2𝑊₁^√𝑡0𝜃^1(0,𝑥)^𝑡−𝑥2₀𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑥2−𝜏)}+×c𝛽os𝑄₁𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥₁𝑄₁𝑊₁𝐵²²×−2𝑊₁+𝑊²¹+𝑄²¹^√𝑡0𝜃^1(0,𝑥)_{𝑡−𝑥20}𝜃𝐺₂(𝜏)𝑒^{−𝑊1(𝑡−𝑥2−𝜏)}sin𝑄₁𝑡−𝑥²−𝜏𝑑𝜏𝑑𝑥_,𝜙_(0,𝑥)=𝑥𝜃^(0,𝑡)||_𝑡=𝑥2,Θ_1(0,𝑥)=𝑥𝜙^(0,𝑡)||_𝑡=𝑥2,(0,𝑡)=−12−𝐵³erf𝐵₃√𝑡−1𝜋𝑡𝑒^{−𝐵23𝑡}Θ_1,(0,𝑥)=𝑥Θ^1(0,𝑡)||_𝑡=𝑥2.(A.6)

Nomenclatures

𝐵0:	Magnetic induction
𝐶∞:	Concentration of the fluid for away from the plate 𝐶 in the free stream
𝐶:	Concentration of the fluid
𝐷:	Chemical molecular diffusivity
𝑔:	Gravitational acceleration
𝐺𝑚:	Solutal Grashof number
𝐺𝑟:	Grashof number
𝐾:	Permeability of the porous medium
𝑘:	Dimensionless permeability
𝑘1:	Thermal conductivity of the fluid
𝑘∗:	Mean absorption coefficient
𝑀:	Magnetic field parameter
Pr:	Prandtl number
𝑞∗𝑟:	Radiative heat flux
𝑄:	Hea tsource/sink coefficient
𝑅:	Radiation parameter
𝑅∗:	First-order chemical reaction rate
Sc:	Schmidt number
𝑡:	Dimensionless time
𝑡:	Dimensional time
𝑇:	Temperature of the fluid
𝑇𝑤, 𝐶𝑤∶	Surface temperature and concentration
𝑇∞:	Temperature of the fluid for away from the plate (in the free stream)
𝑢:	Dimensionless velocity
𝑢, 𝑣:	Components of dimensional velocities along 𝑥 and 𝑦 directions
𝑥, 𝑦:	Dimensional distances along and perpendicular to the plate
𝑦:	Nondimensional distance
𝛼:	Thermal diffusivity
𝛽:	Coefficient of thermal expansion
𝛽∗:	Coefficient of expansion with concentration
𝜂:	Heat source parameter
𝜃:	Dimensionless temperature
𝜈:	Kinematic coefficient of viscosity
𝛿:	Chemical reaction parameter
𝜌:	Fluid density
𝜎:	Electrical conductivity of the fluid
𝜎∗:	Stefan-Boltzmann flux
𝜙:	Dimensionless concentration.