Free Convection Heat and Mass Transfer MHD Flow in a Vertical Porous Channel in the Presence of Chemical Reaction

Abstract

The objective of the present study is to examine the fully developed free convective MHD flow of an electrically conducting viscous incompressible fluid in a vertical porous channel under influence of asymmetric wall temperature and concentration in the presence of chemical reaction. The heat and mass transfer coupled with diffusion-thermo effect renders the present analysis interesting and curious. The analytical solution by Laplace transform technique of partial differential equations is used to obtain the expressions for the velocity, temperature, and concentration. It is observed that under the influence of dominating mass diffusivity over thermal diffusivity with stronger Lorentz force the velocity is reduced at all points Further, low rate of thermal diffusion delays the attainment of free stream state. Flow of aqueous solution in the presence of heavier species is prone to back flow.

1. Introduction

In many transport processes and industrial applications, transfer of heat and mass occurs simultaneously as a result of combined buoyancy effects of thermal diffusion and diffusion of chemical species. Unsteady natural convection of heat and mass transfer is of great importance in designing control systems for modern free convection heat exchangers. More recently, Jha and Ajibade [1] have studied the heat and mass transfer aspect of the flow of a viscous incompressible fluid in a vertical channel considering the Dufour effect.

Soundalgekar and Akolkar [2] studied the effect of mass transfer and free convection currents on the flow past an impulsively started infinite vertical plate and observed that the presence of foreign gasses in the flow domain leads to reduce the shear stress and rate of mass transfer significantly.

In nature, flow occurs due to density differences caused by temperature as well as chemical composition gradients. Therefore, it warrants the simultaneous consideration of temperature difference as well as concentration difference when heat and mass transfer occurs simultaneously. It has been found that an energy flux can be created not only by temperature gradients but by composition gradients also. This is called Dufour effect. If, on the other hand, mass fluxes are created by temperature gradients, it is called the Soret effect.

The Soret and the Dufour effects have been found to be useful as the Soret effect is utilized for isotope separation and, in a mixture of gases of light and medium molecular weight, the Dufour effect is found to be of considerable order of magnitude such that it cannot be neglected. In view of importance of the diffusion-thermo effect Kafoussias and Williams [3] have studied the effects of thermal diffusion and diffusion-thermo on mixed free and forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Jha and Singh [4], Kafoussias [5], and Alam et al. [6, 7] have contributed significantly to this field of study. Recently, Beg et al. [8] studied chemically reacting mixed convective heat and mass transfer along inclined and vertical plates considering Soret and Dufour effects. More recently, Dursunkaya and Worek [9] have studied the diffusion-thermo and thermal diffusion effects in transient and steady natural convections from a vertical surface.

MHD flow with thermal diffusion and chemical reaction finds numerous applications in various areas such as thermo nuclear fusion, liquid metal cooling of nuclear reactions, and electromagnetic casting of metals. Abreu et al. [10] have studied the boundary layer flows with Dufour and Soret effects. Osalusi et al. [11] have worked on mixed and free convective heat and mass transfer of an electrically conducting fluid considering Dufour and Soret effects. Anghel et al. [12] and Postelnicu [13] obtained numerical solutions of chemical reacting mixed convective heat and mass transfer along inclined and vertical plates with the Soret and the Dufour effects and concluded that skin friction increases with a positive increase in the concentration-to-thermal-buoyancy ratio parameter.

The analysis of natural convection heat and mass transfer near a moving vertical plate has received much attention in recent times due to its wide application in engineering and technological processes. There are applications of interest in which combined heat and mass transfer by natural convection occurs between a moving material and ambient medium, such as the design and operation of chemical processing equipment, design of heat exchangers, transpiration cooling of a surface, chemical vapour deposition of solid layer, nuclear reactor, and many manufacturing processes like hot rolling, hot extrusion, wire drawing, continuous casting, and fiber drawing. Gebhart and Pera [14] studied the effects of mass transfer on a steady free convection flow past a semi-infinite vertical plate by the similarity method and it was assumed that the concentration level of the diffusing species in fluid medium was very low. This assumption enabled them to neglect the diffusion-thermo and thermo-diffusion effects as well as the interfacial velocity at the wall due to species diffusion. Following this assumption, Das et al. [15] investigated the effects of simultaneous heat and mass transfer on free convection flow past an infinite vertical plate under different physical situations.

Recently, the unsteady MHD heat and mass transfer free convection flow of a polar fluid past a vertical moving porous plate in a porous medium with heat generation and thermal diffusion has been studied by Saxena and Dubey [16]. Raveendra Babu et al. [17] studied diffusion-thermo and radiation effects on MHD free convective heat and mass transfer flow past an infinite vertical plate in the presence of a chemical reaction of first order. Sudhakar et al. [18] discussed chemical reaction effect on an unsteady MHD free convection flow past an infinite vertical accelerated plate with constant heat flux, thermal diffusion, and diffusion thermo.

In the present paper, the flow of an electrically conducting viscous incompressible fluid in a vertical porous channel formed by two vertical parallel porous plates in the presence of a transverse magnetic field is studied. Further, the mass transfer phenomena considered in this problem is associated with chemically reacting species. The objective of the present study is to extend the work of Jha and Ajibade [1] by incorporating the permeability and the magnetic field effect as well as chemical reaction on the flow, heat and mass transfer phenomena.

2. Mathematical Formulation

Let us consider the free convective heat and mass transfer MHD flow of a viscous incompressible fluid in a vertical porous channel formed by two infinite vertical parallel porous plates in the presence of chemical reaction. A magnetic field of uniform strength 𝐵₀ is applied transversely to the plate. The induced magnetic field is neglected as the magnetic Reynolds number of the flow is taken to be very small. The convection current is induced due to both the temperature and concentration differences. The flow is assumed to be in the 𝑥^′-direction which is taken to be vertically upward along the channel walls and 𝑦^′-axis is taken to be normal to the plates that are ℎ distance apart.

Under the usual Boussinesq’s approximations, the governing equations for flow are given by 𝜕𝑢^′𝜕𝑡^′𝜕=𝜈²𝑢^′𝜕𝑦^′2+𝑔𝛽𝑇^′−𝑇₀+𝑔𝛽^∗𝐶^′−𝐶₀−𝜎𝐵²₀𝑢^′𝜌−𝜈𝐾^′_𝑝𝑢^′,𝜕𝐶^′𝜕𝑡^′𝜕=𝐷²𝐶^′𝜕𝑦^′2−𝐾^′_𝑐𝐶^′−𝐶₀,𝜕𝑇^′𝜕𝑡^′𝜕=𝛼²𝑇^′𝜕𝑦^′2+𝐷₁𝜕²𝐶^′𝜕𝑦^′2, (1) where 𝜈, 𝑔, 𝛽, 𝛽^∗, 𝜎, 𝐵₀, 𝜌, 𝐷, 𝐾^′_𝑐, 𝛼, 𝐷₁, and 𝐾^′_𝑝 are kinematic viscosity, acceleration due to gravity, coefficient of thermal expansion, coefficient of mass expansion, electrical conductivity of the fluid, uniform magnetic field, density of the fluid, chemical molecular diffusivity, the dimensional chemical reaction parameter in the diffusion equation, thermal diffusivity, the dimensional coefficient of the diffusion-thermo effect, and the permeability of the medium, respectively.

Initial and boundary conditions of the problem are 𝑡^′≤0∶𝑢^′𝑦^′,𝑡^′=0,𝑇^′𝑦^′,𝑡^′=𝑇₀,𝐶^′𝑦^′,𝑡^′=𝐶₀𝑡^′𝑢>0∶^′𝑜,𝑡^′=0,𝑇^′0,𝑡^′=𝑇_𝑤,𝐶^′0,𝑡^′=𝐶_𝑤𝑢^′ℎ,𝑡^′=0,𝑇^′ℎ,𝑡^′=𝑇₀,𝐶^′ℎ,𝑡^′=𝐶₀. (2) We now introduce the following dimensionless quantities: 𝑦𝑦=^′ℎ𝑡,𝑡=^′𝜈ℎ²𝑢,𝑢=^′𝜈𝑔𝛽ℎ²𝑇_𝑤−𝑇₀,𝑃_𝑟=𝜈𝛼,𝑆_𝑐=𝜈𝐷,𝑇𝑇=^′−𝑇₀𝑇_𝑤−𝑇₀𝐶,𝐶=^′−𝐶₀𝐶_𝑤−𝐶₀,𝛽𝑁=^∗𝐶_𝑤−𝐶₀𝛽𝑇_𝑤−𝑇₀,𝐷^∗=𝐷₁𝐶_𝑤−𝐶₀𝛼𝑇_𝑤−𝑇₀,𝑀=𝜎𝐵²₀ℎ²,𝐾𝜌𝜈_𝑐=𝐾^∗_𝑐ℎ²𝜈,𝐾_𝑝=𝐾^′_𝑃ℎ². (3)(4)

Here 𝑃_𝑟, 𝑆_𝑐, 𝑁,  𝐷^∗, 𝑀, 𝐾_𝑐, and 𝐾_𝑝 are the Prandtl number, the Schmidt number, the Sustentation parameter, the Dufour number, the magnetic field parameter, the chemical reaction parameter, and the Porosity parameter, respectively.

The nondimensional form of (1) is given by 𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑦²1+𝑁𝐶+𝑇−𝑀+𝐾_𝑝𝑢,𝜕𝐶=1𝜕𝑡𝑆_𝑐𝜕²𝐶𝜕𝑦²−𝐾_𝑐𝐶,𝜕𝑇=1𝜕𝑡𝑝_𝑟𝜕²𝑇𝜕𝑦²+𝐷^∗𝑝_𝑟𝜕²𝐶𝜕𝑦². (5) Subject to boundary conditions, 𝑢()()()𝑢()()()𝑡≤0∶𝑢𝑦,𝑡=0,𝑇𝑦,𝑡=0,𝐶𝑦,𝑡=0,𝑡≤0∶0,𝑡=0,𝑇0,𝑡=1,𝐶0,𝑡=1,1,𝑡=0,𝑇1,𝑡=0,𝐶1,𝑡=0. (6)

3. Solution of the Problem

Applying Laplace transform to (5) with boundary condition (6), we get 𝑑²𝐶𝑑𝑦²−𝑆_𝑐𝑠+𝐾_𝑐𝑑𝐶=0,₂𝑇𝑑𝑦²−𝑠𝑃_𝑟𝑇=𝑆_𝑐𝐷^∗𝑠+𝐾_𝑐𝑠𝐶,𝑑𝑢=²𝑢𝑑𝑦²+𝑁𝐶+1𝑇−𝑀+𝐾_𝑝𝑢. (7)(8)

The boundary condition (6) becomes 𝐶()=10,𝑠𝑠,𝐶(1,𝑠)=0,𝑇()=10,𝑠𝑠,𝑇(1,𝑠)=0,𝑢(0,𝑠)=0,𝑢(1,𝑠)=0. (9) Applying inverse Laplace transform to (7) and (8) subject to boundary condition (9), following Debanath and Bhatta [19], and using the formula 𝐿⁻¹𝑒^{√−𝑏𝑠+𝑎}𝑠=12𝑒^{√−𝑏𝑎}𝑏erfc2√𝑡−√𝑎𝑡+𝑒^𝑏√𝑎𝑏erfc2√𝑡+√𝑎𝑡. (10) The solutions of (5) subject to the boundary conditions (6) are obtained by Laplace transform technique.

Case 1 (𝑃_𝑟≠1, 𝑆_𝑐≠1). The solutions of (5) subject to boundary conditions (6) are given by 𝐶=^∞_𝑛=0𝑓𝐾_𝑐,𝑎_𝑛𝑆_𝑐,1,0,𝑡−𝑓𝐾_𝑐,𝑏_𝑛𝑆_𝑐,1,0,𝑡𝑇=𝐴₁^∞_𝑛=0𝑓0,𝑎_𝑛𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝑓0,𝑏_𝑛𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝐴₁^∞_𝑛=0𝑓𝐾_𝑐,𝑎_𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐾,𝑡−𝑓_𝑐,𝑏_𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐+𝐷,𝑡_∗+1^∞_𝑛=0𝑓0,𝑎_𝑛𝑃_𝑟,1,0,𝑡−𝑓0,𝑏_𝑛𝑃_𝑟,1,0,𝑡−𝐷^∗𝐶𝑢=𝐴₂^∞_𝑛=0𝑓1𝑀+𝐾_𝑝,𝑎_𝑛,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝1,𝑡−𝑓𝑀+𝐾_𝑝,𝑏_𝑛,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝,𝑡−𝐴₂^∞_𝑛=0𝑓𝐾_𝑐,𝑎_𝑛𝑆_𝑐,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝𝐾,𝑡−𝑓_𝑐,𝑏_𝑛𝑆_𝑐,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝,𝑡+𝐴₃^∞_𝑛=0𝑓1𝑀+𝐾_𝑝,𝑎_𝑛,1−𝑃_𝑟1,−𝑀−𝐾_𝑝1,𝑡−𝑓𝑀+𝐾_𝑝,𝑏_𝑛,1−𝑃_𝑟1,−𝑀−𝐾_𝑝,𝑡−𝐴₃^∞_𝑛=0𝑓𝐾_𝑐,𝑎_𝑛𝑃_𝑟,1−𝑃_𝑟1,−𝑀−𝐾_𝑝𝐾,𝑡−𝑓_𝑐,𝑏_𝑛𝑃_𝑟,1−𝑃_𝑟1,−𝑀−𝐾_𝑝,𝑡+2𝐴₄^∞_𝑛=0𝑓1𝑀+𝐾_𝑝,𝑎_𝑛,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1,𝑡−𝑓𝑀+𝐾_𝑝,𝑏_𝑛,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝐴₄^∞_𝑛=0𝑓𝐾_𝑐,𝑎_𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐾,𝑡−𝑓_𝑐,𝑏_𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝐴₄^∞_𝑛=0𝑓0,𝑎_𝑛𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝑓0,𝑏_𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐+𝐴,𝑡₅−𝐴₆^∞_𝑛=0𝑓1𝑀+𝐾_𝑝,𝑎_𝑛1,1,0,𝑡−𝑓𝑀+𝐾_𝑝,𝑏_𝑛,1,0,𝑡+𝐴₆^∞_𝑛=0𝑓0,𝑎_𝑛𝑃_𝑟,1,0,𝑡−𝑓0,𝑏_𝑛𝑃_𝑟,1,0,𝑡, (11) where 𝑎_𝑛=2𝑛+𝑦 and 𝑏_𝑛=2𝑛+2−𝑦.
The functions 𝑓 and erfc are given in Appendix A. The constants 𝐴₁, 𝐴₂, 𝐴₃, 𝐴₄, 𝐴₅, and 𝐴₆ are given in Appendix B.

The rate of mass transfer (Sh), the rate of heat transfer (Nu), and the skin function (𝜏) at the walls of the channel are obtained as follows:

Sherwood Number (𝑆ℎ). ConsiderSh₀=𝜕𝐶𝜕𝑦_𝑦=0=−𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,2𝑛𝑆_𝑐𝐾,1,0,𝑡+𝑔_𝑐,(2𝑛+2)𝑆_𝑐,1,0,𝑡,Sh₁=𝜕𝐶𝜕𝑦_𝑦=1=2𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,(2𝑛+1)𝑆_𝑐,1,0,𝑡. (12)

Nusselt Number (𝑁𝑢). ConsiderNu₀=−𝜕𝑇𝜕𝑦_𝑦=0=𝐴₁𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,2𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐾,𝑡+𝑔_𝑐,(2𝑛+2)𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡,−𝐴₁𝑃_𝑟^∞_𝑛=0𝑔0,2𝑛𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐(),𝑡+𝑔0,2𝑛+2𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐−𝐷,𝑡_∗+1𝑃_𝑟^∞_𝑛=0𝑔0,2𝑛𝑃_𝑟(),1,0,𝑡+𝑔0,2𝑛+2𝑃_𝑟,1,0,𝑡−𝐷^∗Sh₀,Nu₁=𝜕𝑇𝜕𝑦_𝑦=1=2𝐴₁𝑃_𝑟^∞_𝑛=0𝑔0,(2𝑛+1)𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−2𝐴₁𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,(2𝑛+1)𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐷,𝑡+2^∗+1𝑃_𝑟^∞_𝑛=0𝑔0,(2𝑛+1)𝑃_𝑟,1,0,𝑡−𝐷^∗Sh₁. (13)

Skin Friction (𝜏). Consider𝜏₀=−𝜕𝑢𝜕𝑦_𝑦=0=𝐴₂^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝1,𝑡+𝑔𝑀+𝐾_𝑝,2𝑛+2,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝,𝑡−𝐴₂𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,2𝑛𝑆_𝑐,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝𝐾,𝑡+𝑔_𝑐,(2𝑛+2)𝑆_𝑐,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝,𝑡+𝐴₃^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛,1−𝑃_𝑟1,−𝑀−𝐾_𝑝1,𝑡+𝑔𝑀+𝐾_𝑝,2𝑛+2,1−𝑃_𝑟,1−𝑀−𝐾_𝑝,𝑡−𝐴₃𝑃_𝑟^∞_𝑛=0𝑔𝐾_𝑐,2𝑛𝑃_𝑟,1−𝑃_𝑟1−𝑀−𝐾_𝑝𝐾,𝑡+𝑔_𝑐,(2𝑛+2)𝑃_𝑟,1−𝑃_𝑟1−𝑀−𝐾_𝑝,𝑡+2𝐴₄^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1,𝑡+𝑔𝑀+𝐾_𝑝,2𝑛+2,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝐴₄𝑆_𝐶^∞_𝑛=0𝑔𝐾_𝑐,2𝑛𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐾,𝑡+𝑔_𝑐,(2𝑛+2)𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡−𝐴₄𝑃_𝑟^∞_𝑛=0𝑔0,2𝑛𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐(),𝑡+𝑔0,2𝑛+2𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐+𝐴,𝑡₅−𝐴₆^∞_𝑛=0𝑔1𝑀+𝐾_𝑝1,2𝑛,1,0,𝑡+𝑔𝑀+𝐾_𝑝,2𝑛+2,1,0,𝑡+𝐴₅Sh₀+𝐴₆𝑃_𝑟^∞_𝑛=0𝑔0,2𝑛𝑃_𝑟(),1,0,𝑡+𝑔0,2𝑛+2𝑃_𝑟𝜏,1,0,𝑡₁=−𝜕𝑢𝜕𝑦_𝑦=1=−2𝐴₂^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛+1,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐1−𝑀−𝐾_𝑝,𝑡+2𝐴₂𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,(2𝑛+1)𝑆_𝑐,1−𝑆_𝑐,𝑆_𝑐𝐾_𝑐−𝑀,𝑡−2𝐴₃^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛+1,1−𝑃_𝑟1,−𝑀−𝐾_𝑝,𝑡+2𝐴₃𝑃_𝑟^∞_𝑛=0𝑔𝐾_𝑐,(2𝑛+1)𝑃_𝑟,1−𝑃_𝑟−1,−𝑀𝐾_𝑝,𝑡−4𝐴₄^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛+1,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡+2𝐴₄𝑆_𝑐^∞_𝑛=0𝑔𝐾_𝑐,(2𝑛+1)𝑆_𝑐,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐,𝑡+2𝐴₄𝑃_𝑟^∞_𝑛=0𝑔0,(2𝑛+1)𝑃_𝑟,𝑃_𝑟−𝑆_𝑐,𝑆_𝑐𝐾_𝑐𝐴,𝑡−2₅−𝐴₆^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛+1,1,0,𝑡−2𝐴₆𝑃_𝑟^∞_𝑛=0𝑔0,(2𝑛+1)𝑃_𝑟,1,0,𝑡+𝐴₅Sh₁, (14) where the function 𝑔 is given in Appendix A.

The mass flux (volumetric flow rate) for the problem is given by 𝑄=¹₀𝑢𝑑𝑦. (15) This is computed by using the trapezoidal rule for numerical integration.

Case 2 (𝑃_𝑟=𝑆_𝑐=1). The solutions of (5) subject to boundary conditions (6) are given by 𝐶=^∞_𝑛=0𝑓0,𝑎_𝑛,1,0,𝑡−𝑓0,𝑏_𝑛𝐷,1,0,𝑡𝑇=^∗1−𝑒^{−𝐾𝑐𝑡}2𝐾_𝑐𝑡√𝜋𝑡^∞_𝑛=0𝑎_𝑛−𝑎exp^2𝑛4𝑡−𝑏_𝑛−𝑏exp^2𝑛+𝐷4𝑡_∗+1^∞_𝑛=0𝑓0,𝑎_𝑛,1,0,𝑡−𝑓0,𝑏_𝑛,1,0,𝑡−𝐷^∗𝐶𝑢=𝐴₈𝐷𝐶+^∗2𝑡√𝜋𝑡exp−𝐾_𝑐𝑡𝐾_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝+1𝑀+1/𝐾_𝑝𝐾_𝑐−exp(−𝑀𝑡)𝑀+1/𝐾_𝑝𝐾_𝑐−𝑀−1/𝐾_𝑝×^∞_𝑛=0𝑎_𝑛−𝑎exp^2𝑛4𝑡−𝑏_𝑛−𝑏exp^2𝑛4𝑡+𝐴₆^∞_𝑛=0𝑓0,𝑎_𝑛,1,0,𝑡−𝑓0,𝑏_𝑛,1,0,𝑡, (16) where 𝐴₇ and 𝐴₈ are given in Appendix B.

In this case, the rate of mass transfer (Sh), the rate of heat transfer (Nu), and the skin frictions (𝜏) on the walls of the channel are obtained as follows:

Sherwood Number. ConsiderSh₀=−𝜕𝐶𝜕𝑦_𝑦=0=−^∞_𝑛=0𝑔𝐾_𝑐,2𝑛,1,0,𝑡−𝑔𝐾_𝑐,2𝑛+2,1,0,𝑡,Sh₁=−𝜕𝐶𝜕𝑦_𝑦=1=2^∞_𝑛=0𝑔𝐾_𝑐,2𝑛+1,1,0,𝑡. (17)

Nusselt Number. ConsiderNu₀=−𝜕𝑇𝜕𝑦_𝑦=0𝐷=−^∗1−exp−𝐾_𝑐𝑡2𝐾_𝑐𝑡√×𝜋𝑡_∞𝑛=0−𝑛exp²𝑡−()+exp𝑛+1²𝑡−2𝑡^∞_𝑛=0𝑛²−𝑛exp²𝑡+(𝑛+1)²−()exp𝑛+1²𝑡−𝐷^∗+1^∞_𝑛=0𝑔(0,2𝑛,1,0,𝑡)+𝑔(0,2𝑛+2,1,0,𝑡)+𝐷^∗Sh₀,Nu₁=−𝜕𝑇𝜕𝑦_𝑦=0=𝐷^∗1−exp−𝐾_𝑐𝑡2𝐾_𝑐𝑡√×2𝜋𝑡_∞𝑛=0−()exp2𝑛+1²−14𝑡𝑡^∞_𝑛=0(2𝑛+1)²−()exp2𝑛+1²𝐷4𝑡+2^∗+1^∞_𝑛=0𝑔(0,2𝑛+1,1,0,𝑡)−𝐷^∗Sh₁. (18)

Skin Friction. Consider𝜏₀=−𝜕𝑢𝜕𝑦_𝑦=0=𝐴₈Sh₀+𝐷^∗2𝑡√×𝜋𝑡exp−𝐾_𝑐𝑡𝐾_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝+1𝐾_𝑐𝑀+1/𝐾_𝑝−exp(−𝑀𝑡)𝑀+1/𝐾_𝑝𝐾_𝑐−𝑀−1/𝐾_𝑝×^∞_𝑛=0−𝑛exp²𝑡−()+exp𝑛+1²𝑡−2𝑡^∞_𝑛=0𝑛²−𝑛exp²𝑡+(𝑛+1)²−()exp𝑛+1²𝑡+𝐴₆^∞_𝑛=0𝑔(0,2𝑛,1,0,𝑡)+𝑔(0,2𝑛+2,1,0,𝑡)−𝐴₇^∞_𝑛=0𝑔1𝑀+𝐾_𝑝1,2𝑛,1,0,𝑡+𝑔𝑀+𝐾_𝑝,𝜏,2𝑛+2,1,0,𝑡₁=−𝜕𝑢𝜕𝑦_𝑦=1=𝐴₈Sh₁+𝐷^∗𝑡√×𝜋𝑡exp−𝑀−1/𝐾_𝑝𝑡𝑀+1/𝐾_𝑝𝐾_𝑐−𝑀−1/𝐾_𝑝−1𝐾_𝑐𝑀+1/𝐾_𝑝−exp−𝐾_𝑐𝑡𝐾_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝×^∞_𝑛=0−()exp2𝑛+1²𝑡()−22𝑛+1²𝑡−()exp2𝑛+1²𝑡−2𝐴₆^∞_𝑛=0𝑔(0,2𝑛+1,1,0,𝑡)×2𝐴₇^∞_𝑛=0𝑔1𝑀+𝐾_𝑝,2𝑛+1,1,0,𝑡. (19)(20)

Steady State. Setting 𝜕𝑢/𝜕t=0, 𝜕𝑇/𝜕t=0 and 𝜕𝐶/𝜕t=0 in (5), the steady state of the problem is obtained as 𝑑²𝑢𝑑𝑦²1+𝑁𝐶+𝑇−𝑀+𝐾_𝑝𝑑𝑢=0,₂𝐶𝑑𝑦²−𝑆_𝑐𝐾_𝑐𝑑𝐶=0,₂𝑇𝑑𝑦²+𝐷^∗𝑑²𝐶𝑑𝑦²=0. (21)

The solutions of (21) subject to boundary conditions (6) are given by 𝐶=sinh𝑆_𝑐𝐾_𝑐1−𝑦sinh𝑆_𝑐𝐾_𝑐,𝑇=𝐷^∗+11−𝑦−𝐷^∗𝐶,𝑢=𝐴₉sinh𝑀+1/𝐾_𝑝1−𝑦sinh𝑀+1/𝐾_𝑝+𝐴₅𝐶+𝐴₆1−𝑦, (22) where 𝐴₉ is given in Appendix B.

In this case, the Sherwood number (Sh), the Nusselt number (Nu), and the skin friction (𝜏) are given by the following:

Sherwood Number (𝑆ℎ). ConsiderSh₀=𝑆_𝑐𝐾_𝑐coth𝑆_𝑐𝐾_𝑐,Sh₁=𝑆_𝑐𝐾_𝑐cosech𝑆_𝑐𝐾_𝑐. (23)

Nusselt Number (𝑁𝑢). ConsiderNu₀=𝐷^∗+1+𝐷^∗Sh₀,Nu₁=−𝐷^∗+1−𝐷^∗Sh₁. (24)

Skin Friction (𝜏). Consider𝜏₀=−𝐴₉1𝑀+𝐾_𝑝coth1𝑀+𝐾_𝑝+𝐴₅Sh₀−𝐴₆,𝜏₁=𝐴₉𝑀+1/𝐾_𝑝sinh𝑀+1/𝐾_𝑝−𝐴₅Sh₁+𝐴₆. (25)

Mass Flux. Consider𝑄=𝐴₉1𝑀+𝐾_𝑝cosh𝑀+1/𝐾_𝑝−1sinh𝑀+1/𝐾_𝑝+𝐴₅𝑆_𝑐𝐾_𝑐cosh𝑆_𝑐𝐾_𝑐−1sinh𝑆_𝑐𝐾_𝑐+𝐴₆2. (26)

4. Results and Discussion

Computations have been carried out by assigning the values to the pertinent parameters characterising the fluids of practical interest. The flow phenomenon is characterized by magnetic parameter (𝑀), chemical reaction parameter (𝐾_𝑐), sustentation parameter (𝑁), Dufour number (𝐷^∗), Schmidt number (𝑆_𝑐), Prandtl number (𝑃_𝑟), and porosity parameter (𝐾_𝑝). The effects of various parameters on velocity, temperature, and concentration profiles are shown graphically and in tabulated form for Sherwood number (Sh), Nusselt number (Nu), and skin friction (𝜏).

The particular case without magnetic field and porosity (𝑀=0, 𝐾_𝑝→∞) can be obtained from (8) which is in good agreement with the work reported earlier [1].

Another interesting point to note is that diffusion in aqueous solution, that is, for higher values of 𝑆_𝑐 gives rise to oscillations in the velocity as well as temperature field in the presence of chemical reaction parameter (𝐾_𝑐).

Figure 1 depicts the velocity profile for various values of pertinent parameters characterising flow fields. The common feature of the profiles is parabolic. This is evident from curves II and X (𝑡=0.2, Curve II; the steady state curve X) in case of steady state as well as 𝑡≤0.3; the velocity profiles remain positive throughout the flow field. This is evident from curves II and X (𝑡=0.2, Curve II; the steady state curve X). Thus it is concluded that the time span plays a vital role for engendering back flow (Curve I and Curve II).

Figure 1 

Velocity profile.

Further, magnetic parameter (𝑀), chemical reaction parameter (𝐾_𝑐), sustentation parameter (𝑁), and porosity parameter (𝐾_𝑝) decrease the velocity |𝑢| at all points. In all other cases such as 𝑆_𝑐, 𝑃_𝑟, and 𝐷^∗, the reverse effect is observed. If the mass diffusivity becomes greater than the thermal diffusivity (i.e., 𝑁, the sustentation parameter >1.0) then the velocity decreases. Moreover, with the stronger magnetic field Lorentz force also reduces the velocity field which is in conformity with the result of Cramer and Pai [20].

An increase in 𝑃_𝑟 leads to decrease in the velocity which suggests that low rate of thermal diffusion leads to increase in the velocity boundary layer thickness.

Figure 2 depicts the velocity distribution in the absence of chemical reaction. It is noteworthy to observe that the back flow occurs for higher values of 𝑆_𝑐; that is, 𝑆_𝑐=150 and 𝑆_𝑐=200 and the permeability of the medium representing the diffusion in aqueous solution in the presence of constant magnetic field and Dufour effect. Further, it is to mention that for 𝑆_𝑐<1.0 and 𝐾_𝑐=0.0, no velocity profile could be graphed/presented. Further, it is to note that heavier species that is, with increasing 𝑆_𝑐 lead to flow reversal. It is most interesting to note that in the presence of destructive reaction that is, 𝐾_𝑐>0, higher value of 𝑆_𝑐 leads to oscillatory flow and, in case of diffusion in the aqueous solution, it leads to back flow. Thus, this suggests that the back flow is prevented by diffusing lighter species which is in conformity with the results reported earlier by Pop [21], Hossain and Mohammad [22], and Rath et al. [23].

Figure 2 

Velocity profile for 𝑡=0.4, 𝑀=0.5, 𝑁=2, 𝑃𝑟=7, and 𝐷∗=2.

It is also remarked that when 𝑆_𝑐 varies from 0 to 5 the back flow is prevented in the absence of chemical reaction. But in the presence of heavier species the sharp fall of velocity is well marked.

Figure 3 exhibits the steady state velocity profiles indicating no back flow irrespective of higher or lower value of mass transfer coefficient and Dufour effect.

Figure 3 

Velocity profile (steady state).

On careful observation, it is revealed that for higher values of Dufour number 𝐷^∗ (curve VI), sustentation parameter 𝑁 (Curve VIII), magnetic parameter 𝑀 (Curve V), and porosity parameter 𝐾_𝑝 (Curve IV) steady state velocity is effected significantly whereas variation in reaction parameter and Schmidt number produces no change. Thus, it is important to note that mass transfer phenomena with chemical reaction does not affect the steady flow significantly but increase in 𝑀, 𝑁, 𝐷^∗, and 𝐾_𝑝, decrease the velocity, increase the velocity, increase the velocity, and decrease the velocity, respectively. It may be inferred that magnetic parameter, porosity parameter, and Dufour effect produce the same effect irrespective of steady or unsteady state except for sustentation parameter.

Figures 4, 5, and 6 present the temperature distribution. It is seen that for an increase in destructive reaction parameter (𝐾_𝑐>0) temperature increases whereas increase in 𝑃_𝑟, 𝑡, and 𝑆_𝑐 reduces the temperature at all points for 𝑃_𝑟=7.0 (water) when 𝑦<0.7. It is noted that, the slow rate of diffusion leaves the heat energy spread in the fluid mass and it is enhanced for heavier species.

Figure 4 

Temperature profile for 𝐷∗=2.0, 𝑃𝑟=7.0 (water), and 𝑃𝑟=0.71 (air).

Figure 5 

Temperature profile (𝑃𝑟=1, 𝑆𝑐=1).

Figure 6 

Temperature profile (steady state).

It is observed in case of 𝑃_𝑟=0.71 (air) that the temperature falls sharply at all points with almost linear distribution regardless of other parameters. Thus, it may be pointed out that thermal diffusivity property of the fluid plays a vital role in controlling the temperature distribution and hence contributing to thermal boundary layer thickness.

Figure 5 also exhibits the temperature distribution when 𝑃_𝑟=1 and 𝑆_𝑐=1 which means that thermal diffusivity and mass diffusivity are at par. This contributes to the uniform variation/decrease in temperature. Increase in reaction parameter gives rise to increase in temperature whereas allowing for a large span of time reduces it.

Figure 6 presents the nonlinearity distribution of temperature due to high value of 𝐷^∗, that is, for large Dufour effect. Thus, sharp rise and fall of temperature is due to low diffusivity and high Dufour effect.

Figures 7, 8 and 9 exhibit the concentration variation in the flow domain. Sharp fall of concentration is indicated in Figures 7 and 9 along with mass absorption near the plate for very high value of 𝑆_𝑐 (i.e., 𝑆_𝑐=617). This means that heavier species give rise to sharp fall of concentration accelerating the process of mass diffusion whereas in case of lighter species the variation is smooth.

Figure 7 

Concentration profile.

Figure 8 

Concentration profile (𝑆𝑐=1).

Figure 9 

Concentration profile (steady state).

Skin friction plays an important role in flow characteristics. It measures the frictional forces encountered at the solid surfaces due to the motion of the fluid. One striking feature of the entries of the skin friction in Table 1, in case of unsteady motion, is that the skin frictions bear same sign in case of lower and upper plate. Further, on careful observation it is revealed that at the lower plate all are negative for 𝑡=0.4 except for the case when 𝑡=0.2. Thus, it may be concluded that time span plays an important role to modify the frictional drag due to shear stress at the plates.

Table 1 

Skin friction (unsteady case).

Moreover, from (18)-(19) it is clear that the Prandtl number (𝑃_𝑟) has no role to play to affect the velocity, temperature, and concentration fields in the steady flow and hence the skin friction. In case of unsteady flow, as 𝑃_𝑟 increases, the shearing stress increases at both the plates.

Considering Tables 1 and 2 for both steady and unsteady cases, it is concluded that an increase in magnetic parameter, porosity parameter, and sustentation parameter leads to reduce the magnitude of frictional drag at both plates. Moreover, an increase in 𝑆_𝑐, 𝐾_𝑐, and 𝐷^∗ leads to enhance the frictional drag at the plates for steady and unsteady flow.

Table 2 

Skin friction (steady case).

Physically, we may interpret the above results as follows.

Lorentz force and sustentation parameter, that is, ratios of buoyancy effects due to temperature and concentration differences reduce the frictional drag which agrees well with the established result reported earlier. Moreover, presence of heavier species and exothermic reaction and Dufour effect enhance the magnitude of the shearing stress at both plates.

From Table 3 it is seen that an increase in 𝑡, 𝑆_𝑐, and 𝐾_𝑐 leads to decrease the Nusselt number at both plates. One interesting point is that rate of heat transfer remains negative for all the parameters at both the plates for aqueous solution 𝑃_𝑟=7.0 but in case of air, that is, 𝑃_𝑟=0.71, Nusselt number is positive at the lower plate and negative at the upper plate.

Table 3 

Nusselt number.

Table 4 presents the variation of mass transfer for various values of 𝑡, 𝑆_𝑐, and 𝐾_𝑐. It is seen that an increase in chemical reaction parameter 𝐾_𝑐 increases the mass transfer at the lower plate and decreases it at the other. Further, it is to note that Sherwood number at both the plates increases as 𝑆_𝑐 increases, that is, for heavier species mass transfer increases at the plates. The effect of increase in time span is to reduce the rate of mass transfer at both plates.

Table 4 

Sherwood number.

Tables 5 and 6 show the effects of various parameters on mass flux both for unsteady and steady cases, respectively. From Table 5, it is observed that mass flux increases with the increase in the values in 𝑆_𝑐, 𝐾_𝑐, 𝑃_𝑟, and 𝐷^∗ but the reverse effect is observed in case of 𝑡, 𝑀, 𝑁, and 𝐾_𝑝 when the motion is unsteady. Hence, it is concluded that in case of unsteady motion more flux fluid was experienced with heavier species having low thermal diffusivity and increasing Dufour effect in the presence of constructive chemical reaction.

Table 5 

Mass flux (unsteady case).

Table 6 

Mass flux (steady case).

Now, if we analyse the steady case,we observe that more flux is measured with higher value of 𝑁 and 𝐷^∗ whereas the reverse effect is observed in case of 𝑀 and 𝐾_𝑝. In case of 𝑆_𝑐 and 𝐾_𝑐  there is no significant effect on mass flux.

Comparing both cases steady and unsteady motion, it is observed that an increase in 𝐾_𝑐 and 𝐷^∗ gives rise to higher flux whereas magnetic field yields less amount of flux.

5. Conclusion

(i)Under the influence of dominating mass diffusivity over thermal diffusivity with stronger Lorentz force the velocity is reduced at all points of the channel.(ii)Low rate of thermal diffusion leads to increase the thickness of boundary layer.(iii)Diffusion of heavier species leads to flow reversal. It is well marked in case of aqueous solution. Flow of aqueous solution in the presence of heavier species is prone to back flow.(iv)Destructive reaction in the presence of heavier species leads to oscillatory flow.(v)In case of steady state, no back flow occurs irrespective of high or low value of mass transfer coefficient and Dufour effects.(vi)Sharp rise and fall of temperature is the outcome of low diffusivity and high Dufour effects.(vii)Dufour effect and chemical reaction rate in the presence of heavier species enhance the frictional drag.(viii)Heat transfer bears same sign at both the plates in the presence of aqueous solution but in case of air it is of opposite sign.(ix)Mass transfer increases at both plates due to the presence of heavier species but it reduces with increasing time span.(x)In case of unsteady motion more flux of fluid is experienced due to heavier species with low thermal diffusivity. Dufour effect enhances the flux both in steady and unsteady motion.

Appendices

A.

The complementary error function is given by (x)=2erfc√𝜋^∞_𝑥𝑒^−𝜆2𝑓𝑥𝑑𝜆,₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅=12𝑥exp⁴𝑥₅𝑥₃×exp−𝑥₂𝑥₄𝑥₃+𝑥₁𝑥×erfc²2√𝑥₅−𝑥₄𝑥₃+𝑥₁𝑥₅+exp𝑥₂𝑥₄𝑥₃+𝑥₁𝑥×erfc²2√𝑥₅+𝑥₄𝑥₃+𝑥₁𝑥₅,𝑔𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅=12𝑥₄𝑥₃+𝑥₁𝑥exp⁴𝑥₅𝑥₃×exp𝑥₂𝑥₄𝑥₃+𝑥₁𝑥×erfc²2√𝑥₅+𝑥₄𝑥₃+𝑥₁𝑥₅−exp−𝑥₂𝑥₄𝑥₃+𝑥₁𝑥×erfc²2√𝑥₅−𝑥₄𝑥₃+𝑥₁𝑥₅−2√𝜋−𝑥exp²³4𝑥₅+𝑥₁𝑥₅. (A.1)

B.

𝐴₁=−𝐷^∗𝑆_𝑐𝑃_𝑟−𝑆_𝑐+𝑃_𝑟−𝑆_𝑐,𝐴₂=−1−𝑆_𝑐𝑆_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝×𝑆𝑁+^𝑐𝐷^∗𝐾_𝑐−𝑀−1/𝐾_𝑝𝑆_𝑐𝐾_𝑐1−𝑃_𝑟+𝑀+1/𝐾_𝑝𝑃_𝑟−𝑆_𝑐,𝐴₃=1−𝑃_𝑟𝑀+1/𝐾_𝑝×𝑆1+^𝑐𝐷^∗𝐾_𝑐−𝑀−1/𝐾_𝑝𝑆_𝑐𝐾_𝑐1−𝑃_𝑟+𝑀+1/𝐾_𝑝𝑃_𝑟−𝑆_𝑐,𝐴₄=𝐷^∗𝑃_𝑟𝑃_𝑟−𝑆_𝑐𝑆_𝑐𝐾_𝑐1−𝑃_𝑟+𝑀+1/𝐾_𝑝𝑃_𝑟−𝑆_𝑐,𝐴₅=𝐷^∗−𝑁𝑆_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝,𝐴₆=𝐷^∗+1𝑀+1/𝐾_𝑝,𝐴₇=𝐾_𝑐𝐷^∗+1−𝑀+1/𝐾_𝑝(𝑁+1)𝑀+1/𝐾_𝑝𝐾_𝑐−𝑀−1/𝐾_𝑝,𝐴₈=𝐷^∗−𝑁𝐾_𝑐−𝑀−1/𝐾_𝑝,𝐴₉=−𝑆_𝑐𝐾_𝑐𝐷^∗+1+𝑀+1/𝐾_𝑝(𝑁+1)𝑀+1/𝐾_𝑝𝑆_𝑐𝐾_𝑐−𝑀−1/𝐾_𝑝. (B.1)