Heat and Mass Transfer Effect on MHD Flow of a Viscoelastic Fluid through a Porous Medium Bounded by an Oscillating Porous Plate in Slip Flow Regime

Abstract

Unsteady flow of an eclectically conducting and incompressible viscoelastic liquid of the Walter 𝐵^′ model with simultaneous heat and mass transfer near an oscillating porous plate in slip flow regime under the influence of a transverse magnetic field of uniform strength is presented. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter, and the expressions for the velocity, temperature, concentration, skin friction 𝐶_𝑓, the heat flux in terms of the Nusselt number 𝑁_𝑢, and the rate of mass transfer in terms of the Sherwood number 𝑆_ℎ are obtained. The effects of the important flow parameters on the dynamics are discussed. Findings of the study reveal that the rarefaction parameter accelerates the fluid particles in the flow domain. Elastic parameter contributes to sudden fall of the velocity near the plate. Magnetic force contributes to greater skin friction as the time elapses. Destructive reaction reduces, whereas generative reaction enhances the concentration distribution.

1. Introduction

Free convective flow in presence of heat source has been a subject of interest of many researchers because of its possible application to geophysical sciences, astrophysical sciences, and in cosmical studies. Such flows arise either due to unsteady motion of the boundary or the boundary temperature. The study of fluctuating flow is important in the paper industry and many other technological fields. Therefore, many researchers have paid their attention towards the fluctuating flow obf viscous incompressible fluid past an infinite plate. Singh et al. [1] have analyzed the heat and mass transfer in MHD flow of a viscous fluids past a vertical plate under oscillatory suction velocity. Sharma and Singh [2] have reported the unsteady MHD-free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation. The problem of slip flow regime is very important in this era of modern science, technology, and vast ranging industrialization. In many practical applications, the particle adjacent to a solid surface no longer takes the velocity of the surface. The particle at the surface has a finite tangential velocity; it slips along the surface. The flow regime is called the slip flow regime, and its effect cannot be neglected. The fluid slippage phenomenon at the solid boundaries appear in many applications such as microchannels or nanochannels and in application where a thin film of light oils is attached to the moving plates or when the surface is coated with special coating such as thick monolayer of hydrophobic octadecyltrichlosilane, that is, lubrication of mechanical device where a thin film of lubricant is attached to the surface slipping over one another or when the surfaces are coated with special coating to minimize the friction between them. Singh and Gupta [3] have discussed the MHD-free convective flow of a viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime with mass transfer. Khandelwal and Jain [4] have analyzed the unsteady MHD flow of a stratified fluid through porous medium over a moving plate in slip flow regime. Das et al. [5] have studied the magnetohydrodynamic unsteady flow of a viscous stratified fluid through a porous medium past a porous flat moving plate in the slip flow regime with heat source. The study of heat and mass transfer problems with chemical reaction is of great practical importance to engineers and scientists because of their almost universal occurrence in many branches of science and engineering. A few representative fields of interest in which combined heat and mass transfer along with chemical reaction play an important role are chemical process industries such as food processing and polymer production. Mahapatra et al. [6] have studied effects of chemical reaction on free convection flow through a porous medium bounded by a vertical surface. Muthucumaraswamy [7] has studied effects of chemical reaction on a moving isothermal vertical surface with suction. Al-Odat and Al-Azab [8] have studied influence of chemical reaction on a transient MHD-free convection flow over a moving vertical plate.

Viscoelastic fluid flow through porous media has attracted the attention of scientists and engineers because of its importance notably in the flow of the oil through porous rocks, the extraction of energy from geothermal region, and the filtration of solids from liquids and drug permeation through human skin. The flow through porous media also occurs in the ground water hydrology, irrigation and drainage problems, absorption and filtration processes in the ground water hydrology, irrigation and drainage problems, absorption and filtration processes in chemical engineering, and soil erosion and tile drainage. Chaudhary and Jain [9] have investigated the effects of the Hall current and radiation on MHD mixed convection flow of a viscoelastic fluid past and infinite vertical plate. Sahoo et al. [10] have discussed the unsteady two-dimensional MHD flow and heat transfer of an elastic-viscous liquid past an infinite hot vertical porous surface bounded by porous medium with source/sink. Kumar and Chand [11] have studied the effect of slip conditions and the Hall current on unsteady MHD flow of a viscoelastic fluid past an infinite vertical porous plate through porous medium.

The objective of the present study is to consider the viscoelastic fluid past a vertical plate in the slip flow regime packed with uniform porous matrix in the presence of a transverse magnetic field. In this paper, we consider the problem of Singh and Gupta [3] in viscoelastic fluid of Walter’s 𝐵^′ model taking into account the effect of heat source parameter and chemical reaction parameter.

2. Formulation and Solution of the Problem

The physical configuration consists of an unsteady flow of an electrically conducting and incompressible viscoelastic liquid of Walter’s 𝐵^′ model with simultaneous heat and mass transfer near an oscillating infinite porous plate in slip flow regime with heat source and chemical reaction under the influence of a transverse magnetic field of uniform strength. The 𝑦-axis is taken along the plate in vertical direction, and 𝑥-axis is perpendicular to it. A uniform magnetic field of strength 𝐵₀ is applied in the direction of 𝑦-axis. Let 𝑢 and 𝑣 be the velocity components in 𝑥- and 𝑦-directions, respectively. As the plate is of infinite length, all the variables in the problem are functions of 𝑦 and 𝑡. Initially, the plate and fluid are at rest, and then the plate is set to an oscillatory motion. The Reynolds number is assumed to be very small, and the induced magnetic field due to the flow is neglected with respect to the applied magnetic field. The pressure 𝑝 in the fluid is assumed to be constant. If 𝑣₀ represents the suction or injection velocity at the plate, the equation of continuity is 𝜕𝑣𝜕𝑦=0. (1) Under the condition 𝑦=0, 𝑣=−𝑣₀ everywhere. We have

Flow Diagram

Now, the governing boundary layer equations of the flow field are 𝜕𝑢𝜕𝑡−𝑣₀𝜕𝑢𝜕𝜕𝑦=𝜐²𝑢𝜕𝑦²𝐵−𝜎²⁰𝑢𝜌−𝜐𝑢𝐾+𝑔𝛽𝑇−𝑇_∞+𝑔𝛽𝐶−𝐶_∞−𝐾₀𝜌𝜕³𝑢𝜕𝑡𝜕𝑦+𝑣₀𝜕³𝑢𝜕𝑦³,𝜕𝑇𝜕𝑡−𝑣₀𝜕𝑇𝜕𝜕𝑦=𝛼²𝑇𝜕𝑦²−𝑆𝑇−𝑇_∞,𝜕𝐶𝜕𝑡−𝑣₀𝜕𝐶𝜕𝜕𝑦=𝐷²𝐶𝜕𝑦²−𝐾_𝑙𝐶−𝐶_∞. (2) The first order velocity slip boundary conditions of the problem when the plate executes linear harmonic oscillations in its own plane are given by 𝑦=0∶𝑢=𝑈₀𝑒^int+𝐿₁𝜕𝑢𝜕𝑦,𝑇=𝑇_∞,𝐶=𝐶_∞,𝑦⟶∞∶𝑢⟶0,𝑇⟶𝑇_∞,𝐶⟶𝐶_∞, (3) where 𝐿₁=(2−𝑚₁)(𝐿/𝑚₁), 𝐿=𝜇(𝜋/2𝑝𝜌)^1/2 is the mean free path, and 𝑚₁ is Maxwell’s reflection coefficient.

On introducing the following nondimensional quantities: 𝑦^∗=𝑈₀𝜐𝑦,𝑢^∗=𝑢𝑈₀,𝑡^∗=𝑈²₀𝜐𝜃𝑡,_∗=𝑇−𝑇_∞𝑇_𝑤−𝑇_∞,𝜙^∗=𝐶−𝐶_∞𝐶_𝑤−𝐶_∞,𝑣₀^∗=𝑣₀𝑈₀,𝑛^∗=𝜐𝑈²₀𝐾𝑛,_𝑝=𝐾𝑈²₀𝜐𝐿,𝑅=¹𝑈₀𝜐,𝑆^∗=𝜐𝑆𝑈²₀,𝐾_𝑐=𝜐𝐾_𝑙𝑈²₀,𝑀=𝜎𝐵²₀𝜐𝜌𝑈²₀,𝐺_𝑟=𝑔𝛽𝜐𝑇_𝑤−𝑇_∞𝑈³₀,𝐺_𝑚=𝑔𝛽𝜐𝐶_𝑤−𝐶_∞𝑈³₀,𝑅_𝑐=𝑈²₀𝐾₀𝜌𝜐²,𝜐Pr=𝛼,𝑆_𝑐=𝜐𝐷, (4) in (2) and dropping the asterisks, we have 𝜕𝑢𝜕𝑡−𝑣₀𝜕𝑢=𝜕𝜕𝑦²𝑢𝜕𝑦²−𝑅_𝑐𝜕³𝑢𝜕𝑡𝜕𝑦³+𝑣₀𝜕³𝑢𝜕𝑦³−𝑀²+1𝐾_𝑝𝑢+𝐺_𝑟𝜃+𝐺_𝑚𝜙,Pr𝜕𝜃𝜕𝑡−𝑣₀𝜕𝜃=𝜕𝜕𝑦²𝜃𝜕𝑦²−Pr𝑆𝜃,𝑆_𝑐𝜕𝜙𝜕𝑡−𝑣₀𝜕𝜙=𝜕𝜕𝑦²𝜙𝜕𝑦²−𝐾_𝑐𝑆_𝑐𝜙, (5)(6)(7) with boundary conditions 𝑦=0∶𝑢=𝑒^int+𝑅𝜕𝑢𝜕𝑦,𝜃=1,𝜙=1,𝑦⟶∞∶𝑢⟶0,𝜃⟶0,𝜙⟶0. (8)

Equation (5) is of third order, and two boundary conditions are available. Due to inadequate boundary condition, a perturbation method has been applied with 𝑅_𝑐<1 as the perturbation parameter. This assumption is quite consistent as the model under consideration is valid only for slightly elastic fluid.

Consider the following: 𝑢=𝑢₀+𝑅_𝑐𝑢₁+𝑂𝑅_𝑐²,𝜃=𝜃₀+𝑅_𝑐𝜃₁+𝑂𝑅_𝑐²,𝜙=𝜙₀+𝑅_𝑐𝜙₁+𝑂𝑅_𝑐². (9) Substituting (9) in (5)–(7) and equating like powers of 𝑅_𝑐, we get the following.

Zeroth order equations: 𝜕𝑢₀𝜕𝑡−𝑣₀𝜕𝑢₀=𝜕𝜕𝑦²𝑢₀𝜕𝑦²−𝑀²+1𝐾_𝑝𝑢₀+𝐺_𝑟𝜃₀+𝐺_𝑚𝜙₀,Pr𝜕𝜃₀𝜕𝑡−𝑣₀𝜕𝜃₀=𝜕𝜕𝑦²𝜃₀𝜕𝑦²−Pr𝑆𝜃₀,𝑆_𝑐𝜕𝜙₀𝜕𝑡−𝑣₀𝜕𝜙₀=𝜕𝜕𝑦²𝜙₀𝜕𝑦²−𝐾_𝑐𝑆_𝑐𝜙₀, (10) first order equations: 𝜕𝑢₁𝜕𝑡−𝑣₀𝜕𝑢₁=𝜕𝜕𝑦²𝑢₁𝜕𝑦²−𝑀²+1𝐾_𝑝𝑢₁+𝐺_𝑟𝜃₁+𝐺_𝑚𝜙₁−𝑣₀𝜕³𝑢₁𝜕𝑦³−𝜕³𝑢₀𝜕𝑡𝜕𝑦²,Pr𝜕𝜃₁𝜕𝑡−𝑣₀𝜕𝜃₁=𝜕𝜕𝑦²𝜃₁𝜕𝑦²−Pr𝑆𝜃₁,𝑆_𝑐𝜕𝜙₁𝜕𝑡−𝑣₀𝜕𝜙₁=𝜕𝜕𝑦²𝜙₁𝜕𝑦²−𝐾_𝑐𝑆_𝑐𝜙₁. (11) The corresponding boundary conditions are 𝑦=0∶𝑢₀=𝑒^int+𝑅𝜕𝑢₀𝜕𝑦,𝑢₁=0,𝜃₀𝜃=1,₁=1,𝜙₀=1,𝜙₁=1,𝑦⟶∞∶𝑢₀⟶0,𝑢₁⟶0,𝜃₀𝜃⟶0,₁⟶0,𝜙₀⟶0,𝜙₁⟶0. (12) In order to reduce the system of partial differential equations (10)–(11) to a system of ordinary differential equations, we further introduce 𝑢₀𝑦,𝑡=𝑢₀₀𝑦+𝑢₀₁𝑦𝑒^int,𝑢₁𝑦,𝑡=𝑢₁₀𝑦+𝑢₁₁𝑦𝑒^int,𝜃₀𝑦,𝑡=𝜃₀₀𝑦+𝜃₀₁𝑦𝑒^int,𝜃₁𝑦,𝑡=𝜃₁₀𝑦+𝜃₁₁𝑦𝑒^int,𝜙₀𝑦,𝑡=𝜙₀₀𝑦+𝜙₀₁𝑦𝑒^int,𝜙₁𝑦,𝑡=𝜙₁₀𝑦+𝜙₁₁𝑦𝑒^int. (13) Substituting (13) into (10)–(11) and equating the harmonic and nonharmonic terms, we obtain 𝑢^′′₀₀+𝑣₀𝑢^′₀₀−𝑀²+1𝐾_𝑝𝑢₀₀=−𝐺_𝑟𝜃₀₀−𝐺_𝑚𝜙₀₀,𝑢^′′₀₁+𝑣₀𝑢^′₀₁−𝑀²+1𝐾_𝑝+𝑖𝑛𝑢₀₁=−𝐺_𝑟𝜃₀₁−𝐺_𝑚𝜙₀₁,𝑢^′′₁₀+𝑣₀𝑢^′₁₀−𝑀²+1𝐾_𝑝𝑢₁₀=−𝐺_𝑟𝜃₁₀−𝐺_𝑚𝜙₁₀+𝑣₀𝑢^′′′₀₀,𝑢^′′₁₁+𝑣₀𝑢^′₁₁−𝑀²+1𝐾_𝑝+𝑖𝑛𝑢₁₁,=−𝐺_𝑟𝜃₁₁−𝐺_𝑚𝜙₁₁+𝑣₀𝑢^′′′₀₁+𝑖𝑛𝑢^′′₀₁,𝜃^′′₀₀+𝑣₀Pr𝜃^′₀₀−Pr𝑆𝜃₀₀𝜃=0,_′′01+𝑣₀Pr𝜃^′₀₁−(𝑖𝑛Pr+Pr𝑆)𝜃₀₁𝜃=0,_′′10+𝑣₀Pr𝜃^′₁₀−Pr𝑆𝜃₁₀𝜃=0,_′′11+𝑣₀Pr𝜃^′₁₁−(𝑖𝑛Pr+Pr𝑆)𝜃₁₁𝜙=0,_′′00+𝑣₀𝑆_𝑐𝜙^′₀₀−𝐾_𝑐𝑆_𝑐𝜙₀₀𝜙=0,_′′01+𝑣₀𝑆_𝑐𝜙^′₀₁−𝑖𝑛𝑆_𝑐+𝐾_𝑐𝑆_𝑐𝜙₀₁𝜙=0,_′′10+𝑣₀𝑆_𝑐𝜙^′₁₀−𝐾_𝑐𝑆_𝑐𝜙₁₀𝜙=0,_′′11+𝑣₀𝑆_𝑐𝜙^′₁₁−𝑖𝑛𝑆_𝑐+𝐾_𝑐𝑆_𝑐𝜙₁₁=0, (14) with boundary conditions 𝑦=0∶𝑢₀₀=𝑅𝜕𝑢₀₀𝜕𝑦,𝑢₀₁=1+𝑅𝜕𝑢₀₁,𝑢𝜕𝑦₁₀=0,𝑢₁₁=0,𝜃₀₀𝜃=1,₀₁=0,𝜃₁₀=1,𝜃₁₁𝜙=0,₀₀=1,𝜙₀₁=0,𝜙₁₀=0,𝜙₁₁=0,𝑦⟶∞∶𝑢₀₀⟶0,𝑢₀₁⟶0,𝑢₁₀𝑢⟶0,₁₁⟶0,𝜃₀₀⟶0,𝜃₀₁𝜃⟶0,₁₀⟶0,𝜃₁₁⟶0,𝜙₀₀𝜙⟶0,₀₁⟶0,𝜙₁₀⟶0,𝜙₁₁⟶0. (15) The solutions of (14) applying boundary conditions (15) are 𝑢₀₀=𝐴₁𝑒^−𝑎1𝑦+𝐴₂𝑒^−𝑎2𝑦+𝐴₃𝑒^−𝑎3𝑦,𝑢₀₁=𝐴₄𝑒^−𝑎4𝑦,𝑢₁₀=𝐴₅𝑒^−𝑎1𝑦+𝐴₆𝑒^−𝑎2𝑦+𝐴₆𝑒^−𝑎3𝑦,𝑢₁₁=𝐴₈𝑒^−𝑎4𝑦,𝜃₀₀=𝑒^−𝑎1𝑦,𝜃₀₁=𝜃₁₀=𝜃₁₁𝜙=0,₀₀=𝑒^−𝑎2𝑦,𝜙₀₁=𝜙₁₀=𝜙₁₁=0. (16) Hence, the velocity, temperature, and concentration of the flow field are 𝑢=𝐴₁𝑒^−𝑎1𝑦+𝐴₂𝑒^−𝑎2𝑦+𝐴₃𝑒^−𝑎3𝑦+𝐴₄𝑒^−𝑎4𝑦(cos𝑛𝑡+𝑖sin𝑛𝑡)+𝑅_𝑐𝐴₅𝑒^−𝑎1𝑦+𝐴₆𝑒^−𝑎2𝑦+𝐴₇𝑒^−𝑎3𝑦+𝐴₈𝑒^−𝑎4𝑦(cos𝑛𝑡+𝑖sin𝑛𝑡),𝜃=𝑒^−𝑎1𝑦,𝜙=𝑒^−𝑎2𝑦. (17) The skin friction at the plate is given by 𝐶_𝑓=𝜏_𝑥𝑦𝑈²₀=𝜕𝑢𝜕𝑦−𝑅_𝑐𝜕²𝑢𝜕𝑡𝜕𝑦+𝑣₀𝜕²𝑢𝜕𝑦²_𝑦=0, (18) where 𝜏_𝑥𝑦=𝜕𝑢−𝐾𝜕𝑦𝜌⁰𝜕²𝑢𝜕𝑡𝜕𝑦+𝑣₀𝜕²𝑢𝜕𝑦². (19) The rate of heat transfer, that is, the heat flux at the plate in terms of the Nusselt number, is given by 𝑁_𝑢=−𝜕𝜃𝜕𝑦_𝑦=0=𝑎₁. (20) The rate of mass transfer at the plate in terms of the Sherwood number is given by 𝑆_ℎ=−𝜕𝜙𝜕𝑦_𝑦=0=𝑎₂. (21)

3. Results and Discussion

The problem of unsteady flow of an electrically conducting and incompressible viscoelastic liquid of Walter’s 𝐵^′ model with heat and mass transfer near an oscillating infinite porous plate in slip flow regime with heat source and chemical reaction parameter under the influence of a transverse magnetic field of uniform strength has been considered. The effects of the flow parameters such as the Prandtl number Pr, porosity parameter 𝐾_𝑝, magnetic parameter 𝑀, elastic parameter 𝑅_𝑐, heat source parameter 𝑆, chemical reaction parameter 𝐾_𝑐, thermal Grashof number 𝐺_𝑟, the mass Grashof number 𝐺_𝑚, the Scmidt number 𝑆_𝑐, suction parameter 𝑣₀, and rarefaction parameter 𝑅 on the velocity field have been studied analytically and presented with the help of Figures 1–4. The effects of the flow parameters on the temperature field and concentration distribution have been presented in Figures 5 and 6, respectively. Further, the effects of the flow parameters on the skin friction, heat flux, and rate of mass transfer have been discussed with the help of Tables 1–3. For numerical computation, the values of 𝐺_𝑟 are taken positive. This indicates that the study has been carried out under the influence of the cooling of the plate. Also, we have taken 𝑛𝑡=𝜋/2. The interesting aspect of the problem is to study the combined effect of the flow parameters with that of the first order velocity slip boundary condition when the plate executes linear harmonic oscillation in its own plane.

Table 1 

Skin friction (𝐶_𝑓).

Table 2 

The Nusselt number (𝑁_𝑢).

Table 3 

The Sherwood number (𝑆_ℎ).

Figure 1 

Effect of Pr, 𝐾𝑝 and 𝑀 on velocity profile.

Figure 2 

Effect of 𝑅𝑐, 𝑆, and 𝐾𝑐 on velocity profile.

Figure 3 

Effect of 𝐺𝑟, 𝐺𝑚, and 𝑆𝑐 on velocity profile.

Figure 4 

Effect of 𝑣0 and 𝑅 on velocity profile.

Figure 5 

Effect of Pr, 𝑆, and 𝑣0 on temperature profile.

Figure 6 

Effect of 𝑆𝑐, 𝐾𝑐, and 𝑣0 on concentration profile.

Figure 1 shows the effect of the Prandtl number (Pr), permeability parameter (𝐾_𝑝) and magnetic parameter (𝑀) on velocity profile. For this figure, we have taken that 𝑆=1, 𝐺_𝑟=5, 𝐺_𝑚=5, 𝑆_𝑐=0.24, 𝑅_𝑐=0.5, 𝐾_𝑐=2, 𝑅=0.2, and 𝑣₀=2. It is observed that the increase in the Prandtl number as well as permeability parameter decreases the velocity of the flow field, whereas increase in magnetic parameter increases it. Since Prandtl number is the ratio of kinematic viscosity to thermal diffusivity, so as Pr increases, the kinematic viscosity of the fluid dominates the thermal diffusivity of the fluid which leads to decreasing the velocity of the flow field. The application of transverse magnetic field sets up the Lorentz force, which enhances the fluid velocity.

Figure 2 shows the effect of elastic parameter (𝑅_𝑐), heat source parameter (𝑆), and chemical reaction (𝐾_𝑐) parameter on velocity frofile. For this figure, we have taken that Pr=0.71, 𝐾_𝑝=1, 𝑀=2, 𝐺_𝑟=5, 𝐺_𝑚=5, 𝑆_𝑐=0.24, 𝑅=0.2, and 𝑣₀=2. It is observed that the velocity of the flow field decreases due to the presence of elastic parameter, chemical reaction parameter, and heat source parameter. For 𝑅_𝑐=𝑆=𝐾_𝑐=0, the present work agrees with the work of Singh and Gupta [3].

Figure 3 depicts the effect of the thermal Grashof number (𝐺_𝑟), the mass Grashof number (𝐺_𝑚), and the Schmidt number (𝑆_𝑐), on velocity frofile. For this figure, we have taken that Pr=0.71, 𝑆=1, 𝐾_𝑝=1, 𝑀=2, 𝑅_𝑐=0.5, 𝐾_𝑐=2, 𝑅=0.2, and 𝑣₀=2. It is observed that for the heavier species, that is, with increasing 𝑆_𝑐, the velocity decreases. The velocity of the flow field decreases due to the increase in the thermal Grashof number. Moreover, buoyancy effect (𝐺_𝑚) due to mass transfer enhances the velocity.

Figure 4 depicts the effect of suction parameter (𝑣₀) and rarefaction parameter (𝑅) on velocity frofile. For this figure, we have taken that Pr=0.71, 𝑆=1, 𝐾_𝑝=1, 𝑀=2, 𝑅_𝑐=0.5, 𝐾_𝑐=2, 𝐺_𝑟=5, 𝐺_𝑚=5, and 𝑆_𝑐=0.24. It is observed that the velocity of the flow field decreases due the presence of suction parameter, but the reverse effect is observed due the presence of the rarefaction parameter.

Figure 5 shows the effect of the Prandtl number, heat source parameter, and suction parameter on the temperature of the flow field. It is observed that the temperature of the flow field diminishes as the Prandtl number increases. This is consistent with the fact that the thermal boundary layer thickness decreases with increasing Prandtl number. Presence of heat source reduces the temperature of the flow field. This may happen due the elastic property of the fluid. It is observed that temperature of the flow field diminishes as the suction parameter increases.

Figure 6 depicts the effect of the Schmidt number, chemical reaction parameter, and suction parameter on concentration distribution. The concentration distribution decreases at all points of the flow field with the increase in the Schmidt number. This shows that the heavier diffusing species have a greater retarding effect on the concentration distribution of the flow field. It is observed that a destructive reaction (𝐾_𝑐>0) reduces the concentration distribution, whereas a generative reaction (𝐾_𝑐=0) enhances it. Also, it is observed that presence of suction parameter diminishes the concentration distribution.

The skin friction is an important phenomenon which characterizes the frictional drag at the solid surface. From Table 1, it is observed that the skin friction decreases with the increase in all the forcing forces, but it is interesting to note that the skin friction increases with the increase in magnetic parameter.

From Table 2, it is to note that all the entries are positive. It is seen that the Prandtl number (Pr), heat source (𝑆) and suction parameter (𝑣₀) increase the rate of heat transfer at the surface of the plate.

From Table 3 it is to note that all the entries are positive. It is observed that Schmidt number (𝑆_𝑐), chemical reaction parameter (𝐾_𝑐), and suction parameter (𝑣₀) increase the rate of mass transfer at the surface of the plate.

4. Conclusion

A theoretical study of unsteady MHD incompressible viscoelastic liquid of Walter’s 𝐵^′ model with heat and mass transfer near an oscillating infinite porous plate in slip flow regime under the influence of a transverse magnetic field of uniform strength is considered. Some of the important findings of the problem are given in the following.(i)Presence of the Prandtl number decreases the velocity of the flow field, whereas presence magnetic field increases it.(ii)The velocity of the flow field decreases suddenly near the plate due to the presence of elastic parameter.(iii)The velocity of the flow field decreases due to the increase in the thermal Grashof number.(iv)Thermal boundary layer thickness decreases with increasing the Prandtl number.(v)Heavier diffusing species have a greater retarding effect on the concentration distribution.

Appendix

Consider the following: 𝑎₁=12𝑣₀Pr+𝑣₀Pr²,𝑎+4𝑆Pr₂=12𝑣₀𝑆_𝑐+𝑣₀𝑆_𝑐²+4𝐾_𝑐𝑆_𝑐,𝑎₃=12𝑣²₀+𝑣²₀+4𝑄,𝑄=𝑀²+1𝐾_𝑝,𝑎₄=12𝑣²₀+𝑣²₀(,𝐴+4𝑄+𝑖𝑛)₁=−𝐺_𝑟𝑎²₁−𝑎₁𝑣₀−𝑄,𝐴₂=−𝐺_𝑟𝑎²₂−𝑎₂𝑣₀,𝐴−𝑄₃=−11+𝑎₃𝑅𝑎₁𝑅+1𝐴₁+𝑎₂𝑅+1𝐴₂,𝐴₄=11+𝑅𝑎₄,𝐴₅=𝑎³₁𝐴₁𝑣₀𝑎²₁−𝑎₁𝑣₀,𝐴−𝑄₆=𝑎³₂𝐴₂𝑣₀𝑎²₂−𝑎₂𝑣₀−𝑄,𝐴₇=𝐴₅+𝐴₆−𝑣₀𝑎³₃𝐴₃𝑦𝑣₀−2𝑎₃,𝐴₈=−𝑣₀𝑎³₄𝐴₄+𝑖𝑛𝑎²₄𝐴₄𝑦𝑣₀−2𝑎₄. (A.1)

Nomenclature

𝑥,𝑦:	Coordinate axes
𝑢,𝑣:	Velocity components in 𝑥- and 𝑦-directions
𝑡:	Time variable
𝜇:	Dynamic viscosity
𝜐:	Kinematic viscosity
𝛼:	Thermal conductivity
𝑝:	Pressure
𝑔:	Acceleration due to gravity
𝛽:	Coefficient of volume expansion
𝛽∶	Coefficient of volume expansion with concentration
𝑈0:	Reference velocity
𝑇:	Dimensional temperature
𝜃:	Nondimensional temperature
𝐶:	Dimensional concentration
𝜙:	Nondimensional concentration
Pr:	The Prandtl number
𝐺𝑟:	The thermal Grashof number
𝐺𝑚:	The mass Grashof number
𝑆𝑐:	The Schmidt number
𝑀:	Magnetic parameter
𝑆:	Heat source/sink parameter
𝐾𝑙:	Dimensional chemical reaction parameter
𝐾𝑐:	Nondimensional chemical reaction parameter
𝑇𝑤:	Temperature at the wall
𝑇∞:	Temperature far away from the wall
𝐶𝑤:	Concentration at the wall
𝐶∞:	Concentration far away from the wall
𝜎:	Electric conductivity
𝐵0:	Uniform magnetic field
𝑣0:	Suction/injection velocity
𝐷:	Mass diffusion
𝜌:	Density
𝐾:	Dimensional porosity parameter
𝐾𝑝:	Nondimensional porosity parameter
𝐾0:	Dimensional elastic parameter
𝑅𝑐:	Nondimensional elastic parameter
𝑛:	Frequency of oscillation.

Acknowledgment

The author wish to express his special thanks to Professor G. C. Dash, S “O” A University, Bhubaneswar, Odisha, India for his valuable suggesion and constant encouragement to complete the work.