MHD Heat and Mass Transfer Steady Flow of a Convective Fluid Through a Porous Plate in the Presence of Multiple Parameters Along with Dufour Effect

The simultaneous occurrence of HAMT in a flowing fluid, affecting each other creates a cross diffusion effect. The transfer of mass acquired by temperature gradient is termed as Soret effect. The transfer of heat caused by concentration effect is called as Dufour effect. Dufour and Soret effects are important phenomena in hydrology, geosciences and petrology. Diffusion is the net movement of an object from a higher concentration area to a lower concentration area. Distribution is regulated by a concentration curve. Numerous disciplines, including physics (more precisely, diffusion), chemistry, genetics, economics, and finance, regularly employ the diffusion idea. Chemical reactors are the primary focus of chemical reaction engineering, a subfield of industrial chemistry or chemical engineering. The expression frequently only refers to catalytic reaction systems when a homogeneous or heterogeneous catalyst is present in the reactor. Examples include photo catalytic surfaces, reactive separations, containers, retorts, certain fuel cells, and so forth. Additionally, it is believed that the subject of solvent effects on kinetic parameters is vital. Reactors are built using principles from several different scientific disciplines, including thermodynamics, chemical kinetics, fluid mechanics, heat and mass transport, and economics. Chemical reaction engineering seeks to synthesize each of these components in order to successfully build a chemical reactor. According to Srinivas Reddy [1], thermal diffusion and diffusion thermo effects were taken into account when analyzing the effect of chemical reactions on MHD free convection HAMT from vertical surfaces in porous media. Effects of diffusion-thermo and chemical reactions on an unstable MHD free convection flow in a micropolar fluid were examined by Sheri and Shamshuddin [2]. Krishna Murthy and Kumar [3] studied the MHD double diffusive free convection operation in addition to a normal “wavy surface embedded in a doubly stratified, fluid-saturated medium that is porous in nature after learning the results of Dufour and Soret effect” [3]. Umamaheshwar et al. [4] carefully observed the impact of chemicals on MHD fluid flow that is doubly diffusive past a rotating porous plate. Agarwalla and Ahmed [5] looked into the MHD transfer of mass flow via an inclined plate that is embedded in a penetrable medium and has variable velocity and temperature. Lorenzini et al. [6] examined MHD HAMT steady flow of a convective fluid past a permeable plate in the existence of various parameters. Kafoussias and Williams [7] investigated the impact of diffusion-thermo and thermal diffusion on mixed free-forced convective and mass transfer boundary layer flow with temperature-dependent viscosity in light of the significance of the diffusion-thermo effect. Raju et al. [8] have conducted a thorough investigation on MHD fluid with visco-elastic flow across a limitless perpendicular plate while taking into account chemical reaction and radiation reaction. Seth et al. [9] reveals the theoretical analysis of convective fluid flow under thermal diffusion, heat increase and MHD transfer in a conducting field. Krishna Murthy et al. [10] used soret and dufour effects to analyse the HAMT in two dimensional MHD free convective boundary layer flow along a straight semi infinite flat surface submerged in a thermally and mass stretched Darcy porous media. Ibrahim [11] investigated the influence of thermal radiation, heat generation and chemical reaction on unstable free convection on the unsteady free convection flow of viscous incompressible fluid through a porous medium with high porosity that is contained by a perpendicular endless moving plate. Postelnicu [12] studied the dufour and soret effects on the properties of natural convection around a vertical plane contained in a saturated porous media exposed to a magnetic field.

In the presence of a magnetic field, there are numerous instruments that measure thermal energy diffusion through mercury, electrolyte liquids, water, and air as well as absorption, thermal diffusivity and chemical effects. The effects of mass and heat transfer of a natural convection fluid flow through a plate that is porous in nature are therefore suggested to be studied in this work in the appearance of various parameters along with Dufour effect.

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Figure 1. Configuration of the problem

The flow past a porous medium is governed by the consequent equations:

Eq. (1) implies Equation of Continuity: $\nabla \cdot v=0$, (since fluid is incompressible).

Eq. (2) is Equation of Motion.

Eq. (3) implies Equation of Energy.

Eq. (4) shows Equation of Mass transfer.

$\frac{\partial v^{\prime}}{\partial y^{\prime}}=0$                                (1)

$\begin{aligned} & v^{\prime} \frac{\partial u^{\prime}}{\partial y^{\prime}}=v \frac{\partial^2 u^{\prime}}{\partial y^{\prime 2}}+g \beta\left(T^{\prime}-T_{\infty}^{\prime}\right)+g \beta *\left(C^{\prime}-C_{\infty}{ }^{\prime}\right)  -\frac{\sigma B_0{ }^2}{\rho} u^{\prime}-\frac{v u^{\prime}}{K_p}-\frac{\partial p}{\partial x}\left(\frac{1}{\rho}\right)\end{aligned}$                  (2)

$\begin{aligned} & v^{\prime} \frac{\partial T^{\prime}}{\partial y^{\prime}}=\frac{k}{\rho C_p} \frac{\partial^2 T^{\prime}}{\partial y^{\prime 2}}+\frac{v}{C_p}\left(\frac{\partial u^{\prime}}{\partial y^{\prime}}\right)^2-\frac{1}{\rho C_p} \frac{\partial q_r^{\prime}}{\partial y^{\prime}}  +\frac{\sigma B_0^2}{\rho C_p} u^{\prime}-\frac{Q_1}{\rho C_p}\left(T^{\prime}-T_{\infty}\right)+\frac{D_{K_t}}{\rho C_p C_s} \frac{\partial^2 C^{\prime}}{\partial y^{\prime 2}}\end{aligned}$                             (3)

$v^{\prime} \frac{\partial C^{\prime}}{\partial y^{\prime}}=D \frac{\partial^2 C^{\prime}}{\partial y^{\prime 2}}-K\left(C^{\prime}-C_{\infty}^{\prime}\right)$                             (4)

The boundary conditions (B.C) are:

$\begin{aligned} & u^{\prime}=0, T^{\prime}=T_w, C^{\prime}=C_w a t y^{\prime}=0 \\ & u^{\prime} \rightarrow 0, T^{\prime} \rightarrow T_{\infty}, C^{\prime} \rightarrow C_{\infty} a s y^{\prime} \rightarrow \infty\end{aligned}$                                   (5)

The Eq. (1) shows,

$v^{\prime}=$ constant $=-v_0\left(v_0>0\right)$                             (6)

The dimension-less quantities are as mentioned below:

$\begin{aligned} & u=\frac{u^{\prime}}{v_0}, y=\frac{v_0 y^{\prime}}{v}, \theta=\frac{T^{\prime}-T_{\infty}{ }^{\prime}}{T_w-T_{\infty}^{\prime}}, \varphi=\frac{C^{\prime}-C_{\infty}^{\prime}}{C_w-C_{\infty}{ }^{\prime}}, \\ & P r=\frac{\mu C_p}{K}, S c=\frac{v}{D}, M=\frac{\sigma B_0^2 v}{\rho v_0{ }^2}, \\ & G r=\frac{v g \beta\left(T_w-T_{\infty}{ }^{\prime}\right)}{v_0{ }^3}, G m=\frac{v g \beta *\left(C_w-C_{\infty}{ }^{\prime}\right)}{v_0^3}, \\ & E=\frac{v_0{ }^2}{C_p\left(T_w-T_{\infty}{ }^{\prime}\right)}, K=\frac{v_0^2 K_p}{v^2}, K_0=\frac{v K c}{v_0^2}, \\ & F=\frac{4 I_1 v^2}{K v_0^2}, Q=\frac{Q_1 v^2}{K v_0^2}, D_f=\frac{D_{K T}\left(C_w-C_{\infty}{ }^{\prime}\right)}{C_s v\left(T_w-T_{\infty}{ }^{\prime}\right)}\end{aligned}$                                  (7)

The dimension less form of the governing Eqns. (2)-(4) are written in the following form as:

$u^{\prime \prime}+u^{\prime}=-G r \theta-G m \varphi+M_1 u$                             (8)

$\begin{aligned} \theta^{\prime \prime}+\operatorname{Pr} \theta^{\prime}-(F & +Q) \theta=-\operatorname{Pr} u^{\prime}2 E-\operatorname{Pr} M u^2 E+D f \varphi^{\prime \prime} \operatorname{Pr}\end{aligned}$                               (9)

$\varphi^{\prime \prime}+S c \varphi^{\prime}-S c K_0 \varphi=0$                                         (10)

where, $M_1=M+1 / K$.

B.C are:

$\begin{aligned} & u=0, \theta=1, \varphi=1 \text { aty }=0 \\ & u \rightarrow 0, \theta \rightarrow 0, \varphi \rightarrow 0.5 \text { asy } \rightarrow \infty\end{aligned}$                          （11）

In this work, a detailed study in velocity, temperature and concentration field concerning to the effects of MHD convective flow on HAMT of a viscous, conducting, incompressible fluid through a porous and infinite plate is considered. Along with this, the effects of magnetic field, radiation, chemical effects and absorption are also taken into account. Two versions of ‘Gr’ are presented numerically during heating and cooling of the plate. The Prandtl number (Pr) values are chosen as 0.025, 0.71, 1 and 7, which imply mercury, air, electrolytic friction, water, heat and mass transfer respectively. At 20℃ temperature, 1 atmospheric pressure and at the choice of Eckert number as 0.02 the mercury level is taken into consideration. The values of Schmidt number Sc are taken to be 0.22, 0.30, 0.60 and 0.78 that represents hydrogen, helium, water vapour and ammonia respectively. The values Gr=5, Gm=5, K₀=1, K=1, M=2, F=1, Q=1 is chosen for consideration. By the aid of following figures, the obtained results are explained.

On choosing different values of the parameters, the outcome of parameters on velocity, temperature and concentration can be studied. The influence of parameters on SF and Nusselt number are given in Tables 1 and 2.

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Figure 2. Mon VP (Gr=5)

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Figure 3. M on VP (Gr=-5.0)

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Figure 4. Sc on VP (Gr=5)

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Figure 5. Sc on VP (Gr=-5.0)

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Figure 6. K on VP (Gr=5)

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Figure 7. K on VP (Gr=-5.0)

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Figure 8. Gm on VP (Gr=5)

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Figure 9. Gm on VP (Gr=-5.0)

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Figure 10. Df on VP (Gr=5)

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Figure 11. Df on VP (Gr=-5.0)

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Figure 12. Pr on TP (Gr=5)

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Figure 13. Pr on TP (Gr=-5.0)

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Figure 14. F on TP (Gr=5)

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Figure 15. F on TP (Gr=-5.0)

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Figure 16. Q on TP (Gr=5)

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Figure 17. Q on TP (Gr=-5.0)

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Figure 18. K₀ on CP (Gr=5)

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Figure 19. K₀ on CP (Gr=-5.0)

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Figure 20. Sc on CP (Gr=5)

Figure 2 to Figure 5 imply the effect of M and Sc on the velocity profile (VP) of fluid for Pr = 0.71 and Pr = 7. They show velocity drops with the increase in the values of M and Sc. The application of magnetic field repels fluid transportation; hence velocity diminishes if M boosts up. The effect of K and Gm for Pr = 0.71 and Pr = 7 for fluid VP are depicted in Figure 6 to Figure 9. The figures expose that velocity rises with the boom in the values of K and Gm. Figure 10 exhibits VP falls with rise in Df for cooled plate and reverse effect is produced in Figure 11 for heated plate. The influence of Pr, F and Q on temperature profiles (TP) are depicted in Figure 12 to Figure 17 for cooled and heated plates. The figures show temperature lowers with enlargement in the values of Pr, F and Q. As Pr increases, thermal boundary layer thickness reduces, hence temperature falls for the rise in value of Pr. Figure 18 to Figure 21 exhibit the consequence of K₀ and Sc on concentration profile (CP) for Gr > 0 and Gr < 0. They exhibit that with the boost in K₀ and Sc, concentration reduces. The molecular diffusivity decreases as Sc value rises; hence concentration diminishes for enlarged Sc values.

The influences of Gm, M, K, Df, F and Q on SF and heat transfer rate (Nu) are validated in Table 1 and Table 2.

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Figure 21. Sc on CP (Gr=-5.0)

Table 1. Variations in SF

Table 2. Variations in Nusselt number

The results of the present study can be applied effectively to the development of many chemical engineering processes, including aeration, material vaporisation and condensation, crystal evolution and sublimation.