Effects of Thermodiffusion and Chemical Reaction on Magnetohydrodynamic-Radiated Unsteady Flow Past an Exponentially Accelerated Inclined Permeable Plate Embedded in a Porous Medium

Prabhakar Reddy, B.; Simba, M. H.; Hugo, Alfred

doi:https://doi.org/10.1155/2023/9342174

International Journal of Chemical Engineering

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Abstract Introduction Results and Discussion Conclusions Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9342174 | https://doi.org/10.1155/2023/9342174

Effects of Thermodiffusion and Chemical Reaction on Magnetohydrodynamic-Radiated Unsteady Flow Past an Exponentially Accelerated Inclined Permeable Plate Embedded in a Porous Medium

B. Prabhakar Reddy,¹M. H. Simba,¹and Alfred Hugo¹

Academic Editor: Pedro Castano

Received07 Mar 2022

Revised01 Jun 2022

Accepted12 Apr 2023

Published25 Apr 2023

Abstract

A finite difference computational study is conducted to assess the influence of thermodiffusion and chemical reaction on unsteady free convective radiated magnetohydrodynamic flow past an exponentially accelerated inclined permeable plate embedded in a saturated porous medium of uniform permeability with variable temperature and concentration. The governing nondimensional set of coupled nonlinear partial differential equations with related initial and boundary conditions are solved numerically by using the accurate and efficient DuFort–Frankel’s explicit finite difference method. The physical features of fluid flow, heat, and mass transfer under the influence of the magnetic field, angle of inclination, plate acceleration, radiation, heat source/sink, thermodiffusion, chemical reaction, and time are scrutinized by plotting graphs and then discussed in detail. It was found that the effective magnetic field and angle of inclination tend to decline the fluid motion, whereas the reverse result is detected by increasing the porosity parameter and plate acceleration. The velocity and temperature of the fluid lessen with increasing the radiation parameter. The effect of thermodiffusion raises the fluid velocity and concentration, whereas a chemical reaction has the opposite impact. The Nusselt number increases with increasing the radiation parameter and time. Increasing chemical reaction and time causes to improve the Sherwood number. This kind of problem finds momentous industrial applications such as food processing, polymer production, inclined surfaces in a seepage flow, and design of fins.

1. Introduction

The study of thermal radiation effect on the boundary-layer flow with heat and mass transfer is of immense curiosity due to its significant applications in many industrial and space technologies such as glass production and furnace design, space vehicle, propulsion systems, comical flight aerodynamics rocket, and spacecraft re-entry aerodynamics which operate at high temperatures, ship compressors, and solar radiation. Due to these imperative applications, the researchers [1, 2] analyzed the radiation effect on MHD flow under different conditions. Pattnaik et al. [3] discussed radiation and mass transfer effects on MHD-free convective flow through a porous medium past an exponentially accelerated vertical plate with variable temperature. Subsequently, the significant research investigations [4–8] richly scrutinized the effect of radiation by considering dissimilar fluids under different circumstances. Veera Krishna et al. [9] recently studied radiation effects on free convective rotating MHD flow through a saturated porous medium over an exponentially accelerated plate by considering Hall and ion slip effects. Veera Krishna et al. [10] examined radiation effects on the MHD flow of second-grade fluid through a porous medium over a semi-infinite vertical stretching sheet. Veera Krishna et al. [11] discussed radiation absorption effects by considering the convective flow of MHD nanofluids through a vertically travelling absorbent plate. Veera Krishna et al. [12] analyzed thermal radiation effects on oscillatory rotating MHD micropolar liquid in the presence of chemical reaction and Hall and ion slip effects.

The chemical reaction is the process by means of which one set of chemical substances is transported into another. In addition, the process of chemical reaction takes place between the fluid and foreign mass. The study of heat and mass transfer problems by considering chemical reaction effects into account originate plentiful applications in several chemical and hydrometallurgical industries, such as the production of glassware and ceramics, food processing, undergoing endothermic or exothermic chemical reaction, and catalytic chemical reactors, etc. Sugunamma et al. [13] studied inclined magnetic field and chemical reaction effects on flow over a semi-infinite vertical porous plate through a porous medium. Ali et al. [14] presented chemical reaction effects on heat and mass transfer-free convection MHD flow past a vertical plate embedded in a porous medium. Mythreye et al. [15] analyzed chemical reaction effects on unsteady convective MHD heat and mass transfer past a semi-infinite vertical permeable moving plate in the presence of heat absorption. Seth and Sarkar [16] studied hydromagnetic natural convection flow with an induced magnetic field and nth-order chemical reaction of a heat-absorbing fluid past an impulsively moving vertical plate with ramped temperature. Tripathy et al. [17] discussed the chemical reaction effect on MHD-free convective surface over a moving vertical plate through a porous medium. Agarwalla and Ahmed [18] investigated chemical reaction effects on MHD flow past an inclined plate with variable temperature and plate velocity embedded in a porous medium. Subsequently, the research studies [19-23] considered dissimilar fluids to discuss the impact of chemical reaction under different conditions. Recently, Veera Krishna et al. [24, 25] studied a chemical reaction impact by considering the MHD flow of Casson hybrid nanofluid and heat-generating/absorbing second-grade fluid.

In many studies mentioned earlier, the consequence of thermodiffusion on fluid transport may be ignored based on a smaller order of enormity. However, an exciting macroscopically physical phenomenon in fluid mechanics known as thermodiffusion (Soret) effects has significance in thermal diffusion processes. When the combined buoyancy effects happen due to the variation of density with temperature under natural convection, the diffusion of species will be influenced by the temperature. Mohamed [26] studied the Soret effects on double-diffusive convection-radiation interaction on unsteady MHD flow over a vertical moving plate with heat generation. Devi and Raj [27] discussed thermodiffusion effects on unsteady hydromagnetic free convection flows with heat and mass transfer past a moving vertical plate with time-dependent suction and heat source in a slip regime. Pal and Talukdar [28, 29] analyzed the thermodiffusion effect on MHD flow through the vertical surface. The research works [30–32] scrutinized the thermodiffusion effect on unsteady MHD flow over a moving vertical porous surface. Ali et al. [33] inspected the Soret–Dufour effect on the hydromagnetic flow of viscoelastic fluid over a porous oscillatory stretching sheet in the presence of thermal radiation. Pal and Biswas [34] studied the Soret effects on chemically reacting mixed convection MHD oscillatory flow of Casson fluid with thermal radiation and viscous dissipation. Kataria and Patel [35] investigated the Soret effects on the heat-generating MHD Casson fluid flow past an oscillating vertical plate embedded in a porous medium. Prabhakar Reddy and Sunzu [36] inspected Dufour and Soret effects on unsteady MHD-free convective flow of viscous incompressible fluid past an infinite vertical porous plate in the presence of radiation. Prabhakar Reddy [37] investigated thermodiffusion effects on radiating and reacting MHD convective heat-absorbing fluid past an exponentially accelerated vertical porous plate with ramped temperature in the presence of Hall current. Prabhakar Reddy and Hugo [38] discussed the Soret effect on radiating and reacting unsteady MHD flow past an exponentially accelerated inclined porous plate.

Therefore, all the abovementioned studies forced us to conduct the study of MHD unsteady-free convective flow past an exponentially accelerated inclined permeable plate embedded in a saturated porous medium of uniform permeability in the presence of chemical reaction, radiation, and thermodiffusion effects as this study is not being discussed yet. This kind of problem is essential in the areas of nuclear power plants, MHD pumps, food processing, polymer production, and also in the processes of isotope separation of medium molecular weighted gases ( air) and mixture between gases with very light molecular weight The main goal of this investigation is to fill the gap by extending the earlier published work of Pattnaik et al. [3], especially by incorporating thermodiffusion effect, which should increase the applicability of the study that constitutes the novelty of the current work.

2. Mathematical Analysis

We consider an unsteady MHD radiating-free convective flow of a viscous, incompressible fluid past an exponentially accelerated inclined plate embedded in a saturated porous medium with variable temperature and concentration, considering the thermodiffusion effect. In the coordinate system, the axis is taken along the plate, and the axis is taken normal to the plate. The plate is inclined to a vertical direction by an angle , as shown in Figure 1. A uniform magnetic field of strength is applied in the direction normal to the plate. Initially, the plate and surrounding fluid are at the same temperature and concentration At the time the temperature and species concentration level are raised or lowered linearly with time and the plate is exponentially accelerated with a velocity in its plane. A homogeneous first-order chemical reaction rate is applied between the diffusing species and the fluid. The magnetic Reynolds number of the flow is minimal, so the induced magnetic field is ignored. Following the suggestions by Kumar and Varma [1] and Pattnaik et al. [3], the equations of momentum, energy, and species concentration of the flow under the usual boundary layer approximations are given by the following expressions:

The associated initial and boundary conditions are as follows:

Here, indicates fluid velocity indicates acceleration due to gravity indicates fluid temperature indicates constant temperature of the plate indicates temperature of the fluid far away from the plate indicates fluid concentration indicates concentration of the fluid near the plate indicates concentration of the fluid far away from the plate indicates volumetric coefficient of thermal expansion indicates volumetric coefficient of concentration expansion indicates magnetic induction indicates permeability of the porous medium, indicates constant heat source, indicates thermal conductivity indicates kinematic viscosity indicates fluid viscosity indicates fluid density indicates electrical conductivity indicates specific heat at constant pressure indicates radiation heat flux indicates chemical reaction constant, indicates chemical molecular diffusivity indicates mean fluid temperature, indicates thermal diffusion ratio, indicates accelerating parameter, and indicates time .

The fluid is considered a gray, absorbing-emitting radiation but a nonscattering medium in the analysis. The local gradient for the case of an optically thin gray gas following the study by England and Emery [39] is expressed as follows:where and are, respectively, the Stefan–Boltzmann constant and absorption coefficient. Assuming slight temperature differences within the flow, the term is expressed as the linear function of temperature. Using the Taylor series, expanding about a free stream temperature and neglecting higher order terms, we obtain the following expression:

On substitution of equations (5) and (6) in equation (2), the energy equation becomes as follows:

On introducing the following nondimensional quantities and parameters

into equations (1), (3), (4), and (7), the following dimensionless governing coupled partial differential equations of the flow are obtained.

The corresponding initial and boundary conditions in a dimensionless form become as follows:

3. Numerical Procedure

The constructing coupled nonlinear partial differential equations of the model equations (9)–(11) subject to the initial and boundary conditions (12) have been solved numerically via DuFort–Frankel’s explicit finite difference method given in the study by Jain et al. [40], which is unconditionally stable. The corresponding finite difference equations to the equations (9)–(11) are derived as follows:

The initial and boundary conditions (13) become as follows:

Here, corresponds to infinite. The suffix corresponds to space and the suffix corresponds to time Also, and . The region of integration of the problem is considered as a rectangle with sides and where corresponds to ; this has been done after some numerical experimentation. A value greater than 4 does not result in any significant change in the numerical results. Numerical results of the velocity , temperature , and concentration are obtained from equations (13)–(15) under the initial and boundary conditions (16). The grid size along direction and along direction was selected after some numerical experimentation with grid seizes and ; no significant change in the results was noticed. To verify the accuracy of the present numerical scheme, a comparison of the computed Nusselt number in the present work was made with that of the Nusselt number obtained by the Laplace transform technique in the earlier published work of Pattnaik et al. [3] shown in Table 1; a perfect agreement was achieved. This verifies the accuracy of the present numerical scheme.

4. Results and Discussion

To evaluate the physical significance of the problem, numerical calculations were made for different values of thermophysical parameters controlling the flow, such as magnetic parameter permeability parameter inclination of angle plate acceleration radiation parameter heat source parameter thermodiffusion parameter chemical reaction rate and time which are displayed through the graphs and tables to analyze the variations of velocity, temperature, concentration profiles, skin friction, and Nusselt and Sherwood numbers. We set default values of critical parameters as , , , , , , , , , , , , and in all figures and tables unless specified.

Figure 2 illustrates the effect of magnetic parameter and porosity parameter on velocity profiles against . It is seen from Figure 2(a) that an increase in the magnetic field intensity leads to a decrease in the fluid motion. Physically, increasing values of generates resistive forces strong enough so that they can oppose the fluid motion, and as a result, fluid velocity decreases. Also, it is observed that there is a sudden rise in the fluid velocity near the plate in the absence of a magnetic field. In Figure 2(b), the reverse impact on the fluid velocity to that of the magnetic field is seen. This is due to the fact that uprising values of reduce the drag force; as a result, they increase the fluid momentum. Figure 3 depicts the variation of the velocity profiles against for various values of angle of inclination and acceleration parameter It is observed from Figure 3(a) that an increasing angle of inclination reduces the fluid velocity due to forcing forces being drained due to the factor It is noticed from Figure 3(b) that the fluid momentum improves with increasing acceleration parameter. Also, it is clear that when the case of constant plate velocity and satisfies the boundary condition implies that a thinner boundary layer. The variation of velocity and temperature profiles under the effects of the thermal radiation parameter against is shown in Figure 4. It is perceived that both fluid velocity and temperature deflate with the expansion of the radiation parameter. Physically, when radiation is presented in the energy equation, the thermal boundary layer is always starting to thicken; as a result, fluid temperature diminishes and the speed of velocity is lower. The impact of the heat source/sink parameter on the velocity and temperature profiles against is displayed in Figure 5. It is found that fluid velocity, as well as temperature, amplifies with growing heat source parameter, whilst the overturn is noted in the case of heat sink Physically, more heat is generated from the surface to the fluid when increasing values of heat source parameter, as a result, enhanced fluid temperature consequently boost up fluid velocity whilst when heat sink present in the boundary layer, engross energy, which causes to reduce the fluid temperature as a consequence slowdown fluid flow. Figure 6 illustrates the effects of the thermodiffusion parameter on the velocity and concentration profiles against It can be seen that an increasing behavior in both fluid velocity and concentration with rising values of . Physically, increasing the thermodiffusion effect tends to raise mass buoyancy force, which results in an increase of species concentration as outcome growth flow transports in the boundary layer. The variation of velocity and concentration profiles for various values of chemical reaction parameter against is depicted in Figure 7. This indicates that the destructive chemical reaction rate leads to descending both fluid velocity and concentration, which is a physical reality. Since an increase in weakens the buoyancy effects in the boundary layer due to concentration gradients, both velocity and concentration decrease. Figure 8 demonstrates the variation of velocity, temperature, and concentration profiles under the variation of time against It is observed from Figure 8(a) that an increasing effect on the fluid velocity with time progression due to increasing buoyancy effect causes to enhance momentum boundary layer thickness. It is noticed from Figures 8(b) and 8(c) that both fluid temperature and concentration enhance in the boundary layer on escalating values of . Further initially, the plate temperature and concentrations are equal to the time, and for larger values of , both tend to zero with growing values of time which is a clear validation of the thermal and concentration boundary conditions.

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(c)

From Table 1, it is perceived that the Nusselt number at the plate surface increases with an increase in Prandtl number, radiation parameter, and time, whereas it decreases with an increase in heat source parameter. The presence of a heat sink clearly supports the appreciation of the Nusselt number. From Tables 2 and 3, it is seen that the skin friction at the plate surface decreases with an enhancement in permeability parameter, heat source parameter, thermodiffusion parameter, time, thermal, and mass buoyancy forces, whereas it increases with an increase in magnetic parameter, inclination of angle, Prandtl number, radiation parameter, acceleration parameter, Schmidt number, and chemical reaction parameter. Also, it is clear that heat sink helps to improve the skin friction. It is observed from Table 3 that an increase in Schmidt number, chemical reaction rate, and time increases the Sherwood number at the plate surface, whereas it decreases with an increase in the thermodiffusion parameter.

5. Conclusions

The governing coupled nonlinear partial differential equations of the problem are solved numerically by the finite difference method to analyze the effects of thermodiffusion and chemical reaction on magnetohydrodynamic unsteady-free convective flow past an exponentially accelerated inclined permeable plate embedded in a porous medium in the presence of radiation and heat source/sink. We found that magnetic field force and angle of inclination decelerate fluid motion, whereas a reverse trend is noted with enhancement in porosity parameter, plate acceleration, and time. The fluid velocity and temperature increase with the growing heat source parameter, whereas radiation and heat sink have overturned fluid velocity and temperature results. The thermodiffusion effect enhances fluid velocity and concentration, whereas the chemical reaction rate has a reverse impact. Thermal and mass buoyancy forces, heat source, the permeability of the medium, and time tend to lessen the skin friction, whereas magnetic field, angle of inclination, radiation, chemical reaction rate, and plate acceleration tend to raise the skin friction. The Nusselt number increases with increasing radiation parameters, heat sink, and time at the plate surface, whereas the heat source has the opposite effect. An increase in the Schmidt number and chemical reaction rate helps to raise the Sherwood number, whereas the thermodiffusion effect has the opposite tendency.

Data Availability

The authors confirm that the data supporting the findings of this research are available within the article.

Disclosure

Preprint Source: https://www.authorea.com/doi/full/10.22541/au.164865092.20993641.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are thankful to the management of the University of Dodoma, Tanzania, for their constant encouragement.

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Copyright © 2023 B. Prabhakar Reddy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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