# Flow with Buoyancy Ratio in MHD under the Influence of Radiation Absorption and Diffusion-Thermo Effects in a Slip Flow Regime

Sridhar W, Dharmaiah G, Rao DS and Kumar RD

**Published on:** 2023-11-17

#### Abstract

This study focuses on the analysis of magneto hydrodynamic (MHD) buoyancy ratio flow in the presence of radiation absorption and diffusion-thermo effects within a slip flow regime. The governing partial differential equations are solved using a perturbation method. Analytical solutions are obtained for velocity, temperature, and concentration, considering various fluid flow parameters such as Schmidt number (Sc), chemical reaction parameter (Γ), Dufour number (Du), magnetic field parameter (M), radiation absorption parameter (Q1), porous permeability parameter (Φ1), porosity parameter (K), Solutal Grashof number (Gr), and buoyancy ratio parameter (N), which are depicted in figures. The study also presents results for skin friction, Sherwood number, and Nusselt number values, all of which demonstrate good agreement with existing research.

#### Keywords

Buoyancy ratio, MHD, Grashof number, Porous media, Radiation#### Introduction

Magneto hydrodynamics (MHD) is a branch of continuum mechanics that deals with the behavior of electrically conducting fluids in the presence of electric and magnetic fields. Many natural phenomena and engineering challenges warrant investigation within the framework of MHD. Magneto hydrodynamic conditions are essentially electromagnetic and hydrodynamic conditions adapted to examine the interaction between fluid motion and electromagnetic fields. The mathematical formulation of the electromagnetic theory in this context is encapsulated in Maxwell's equations. Gravity's influence is a constant factor in forced convection heat transfer due to buoyancy forces resulting from temperature variations. Typically, these forces are of a small magnitude, allowing us to disregard external forces. However, there has been growing interest in the effects of body forces on forced convection phenomena, especially in specific engineering problems where they cannot be ignored.

It's worth noting that heat transfer in mixed convection can differ significantly from both pure natural convection and pure forced convection. The study of forced and free convection flow and heat transfer in electrically conducting fluids past an infinitely long porous plate influenced by a magnetic field has attracted the attention of many researchers due to its relevance in various engineering applications, including geophysics, astronomy, boundary layer control in aerodynamics, and more. Engineers apply MHD principles in the design of heat transfer pumps and flow meters, space propulsion, thermal protection, braking systems, power generation systems, and more. In all these applications, a deep understanding of the behavior of MHD free and forced convective flow and the various influencing parameters is crucial for designers seeking to harness and control this flow.

For instance, it plays a role in processes such as metal fusion in an electric furnace using a magnetic field and cooling the primary wall inside a nuclear reactor containment vessel, where hot plasma is separated from the wall using a magnetic field. While many researchers have explored this problem in recent years, none of them have comprehensively considered all the relevant aspects that influence flow behaviour.

Watanabe [1] presented a laminar forced and free mixed convection flow on a flat plate with uniform suction or injection that was theoretically investigated. Non-similar partial differential equations are transformed into non-similar ordinary ones by means of the difference-differential method. Also, Ahmed and Liu [2] analyzed the effects of mixed convection and mass transfer in the three-dimensional oscillatory flow of a viscous incompressible fluid past an infinite vertical porous plate in the presence of transverse sinusoidal suction velocity oscillating with time and a constant free stream velocity. Hussain [3] considered the problem of natural convection boundary layer flow, induced by the combined buoyancy forces from mass and thermal diffusion from a permeable vertical flat surface with non-uniform surface temperature and concentration but a uniform rate of suction of fluid through the permeable surface. Alom [4] investigated the steady MHD heat and mass transfer by mixed convection flow from a moving vertical porous plate with induced magnetic, thermal diffusion, constant heat, and mass fluxes, and the non-linear coupled equations were solved by the shooting iteration technique. Orhan and Kaya [5] investigated the mixed convection heat transfer about a permeable vertical plate in the presence of magneto and thermal radiation effects using the Keller box scheme, an efficient and accurate finite-difference scheme. Ghosh [6] considered an exact solution for the hydro magnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects, and the transformed ordinary differential equations are solved exactly. Dharmaiah [7] analysed MHD-free convection flow through a porous medium along an aertical wall. Dharmaiah [8] expalined the effects of radiation, chemical reactions, and sores on unsteady mhd-free convective flow over a vertical porous plate. Baby Rani [9] examined synthetic responses and radiation impacts. Balamurugan [10] studied MHD-free convective flow past a semi-infinite vertical permeable moving plate with heat absorption. Takhar [11] discussed radiation effects on the magneto hydrodynamic (MHD) free convection flow of a radiating gas past a semi-infinite vertical plate. Muthukumaraswamy [12] studied the effects of diffusion and first-order chemical reactions on an impulsively started infinite vertical plate with variable temperature. Ibrahim [13] considered the effect of chemical reactions and radiation absorption on the unsteady MHD-free convection flow past a semi-infinite vertical permeable moving plate with a heat source and suction. Patil [14] discussed the effect of chemical reactions on the free convection flow of a polar fluid through a porous medium in the presence of internal heat generation. Kandaswamy [15] studied the chemical reaction effect on heat and mass transfer flow along a wedge in the presence of suction or injection. MHD-free convection flows occur in nature frequently. Fluid flows through porous mediums have been attracting the attention of many researchers in recent days based on their wide applications in many areas of science and technological fields, namely the study of groundwater resources in agricultural engineering and petroleum technology to study the flow of ordinary gas, oil, and water through oil reservoirs. Ibrahim et al. [16] investigated the effect of the chemical reaction and radiation absorption on unsteady MHD-free convection flow past a semi-infinite vertical permeable moving plate with a heat source and sink. Prasad et al. [17] studied the radiation effect on MHD unsteady-free convection flow with mass transfer past a vertical plate with variable surface temperature and concentration. Cookey [18] analysed the effect of unsteady MHD-free convection and mass transfer flow past an infinitely heated porous vertical plate with time-dependent suction. Gireesh Kumar [19] studied the effect of mass transfer on MHD unsteady-free convective Walters’s memory flow with constant suction and heat sink.

Fluid flows through porous mediums are seriously attracted by engineers and scientists. Now a days, due to their applications in the emerging trends in science and technology, namely in the field of agricultural engineering, especially while studying water resources in the ground, to study the moment of natural gas, oil, and water through the reservoirs in petroleum technology. Chen [20] discussed the effect of free convection of non-Newtonian fluid along a vertical plate embedded in a porous medium. Chamkha [21] studied the effect of heat and mass transfer on a non-Newtonian fluid flow along a surface embedded in a porous medium on wall heat and mass fluxes and heat generation or absorption. Panda [22] considered the effect of unsteady free convection flow and mass transfer past a vertical porous plate. Mahapatra [23] analysed the effect of chemical reactions on free convection flow through a porous medium surrounded by a vertical surface. Mishra [24] investigated the effect of mass and heat transfer on the MHD flow of a visco-elastic fluid through a porous medium with variable suction and heat sources. Reddy [25] considered unsteady-free convection MHD non-Newtonian flow through a porous medium bounded by an infinitely inclined porous plate. Raju [26] investigated the effect of radiation and mass transfer effects on a free convection flow past a porous medium bounded by a vertical surface. Seddek [27] studied the effects of chemical reaction and variable viscosity on hydro-magnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation. Ravikumar [28] discussed the combined effect of heat absorption and MHD on convective Rivlin-Erichsen flow past a semi-infinite vertical porous plate with variable temperature and suction. Ibrahim [29] analyzed the effect of the chemical reaction and radiation absorption on the unsteady MHD-free convective flow past a semi-infinite vertical permeable moving plate with a heat source and suction. Chamkha [30] analyzed the unsteady MHD-free convection flow past an exponentially accelerated vertical plate with mass transfer, chemical reactions, and thermal radiation. Umamaheswar [31] discussed the combined radiation and ohmic heating effects on MHD-free convective visco-elastic fluid past a porous plate with viscous dissipation. Rao [32] focused their efforts on the study of unsteady MHD-free convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate with radiation, heat absorption, chemical reactions, and Soret effects. Ravikumar [33] studied the magnetic field effect on transient free convection flow through a porous medium past an impulsively started vertical plate with fluctuating temperature and mass diffusion. Rao [34] considered the unsteady MHD-free convective double diffusive and dissipative visco-elastic fluid flow in a porous medium with suction.

In recent times, chemical reactions and radiation absorption have influenced fluid flow and attracted the attention of engineers and scientists. This type of fluid flow plays an important role in food processing, flow in desert coolers, generating electrical power, groves of fruit trees, etc. The present study is motivated in this direction. The main objective of the present investigation will, therefore, be to study the effects of chemical reactions, radiation absorption, and the dufour effect over an infinite vertical porous plate in the presence of a transverse magnetic field by means of analytical solutions. These analytically approximated solutions under the perturbation technique have a wider applicability in understanding the basic physics and chemistry of the problem, which are particularly important in industrial and technological fields. In this article, the effects of chemical reactions as well as magnetic fields on the heat and mass transfer of Newtonian fluids over an infinite vertical oscillating permeable plate with variable mass diffusion are considered. The magnetic field is imposed transversely on the plate. The temperature and concentration of the plate are oscillating with time to a constant nonzero mean value. The dimensionless governing equations involved in the present analysis are solved using a closed analytical method and discussed.

#### Formulation Of The Problem

Consider a flow scenario in which we have a two-dimensional, unsteady flow of an incompressible, viscous, electrically conducting, and heat-absorbing fluid. This fluid is flowing past a semi-infinite vertical permeable plate that's embedded in a uniform porous medium. Additionally, this flow is subjected to a uniform transverse magnetic field and is influenced by thermal and concentration buoyancy effects. Here are the key assumptions and conditions for this scenario:

The applied magnetic field is relatively weak, allowing us to neglect Hall and ion slip effects.

We consider Dufour effects by describing them as a second-order concentration derivative concerning the transverse coordinate in the energy equation.

No applied voltage is present, implying the absence of an electric field.

The semi-infinite plate is maintained at a constant temperature, Tw, and a constant concentration, C_{w}, both of which are higher than the ambient temperature, T∞, and concentration, C∞, respectively.

Chemical reactions occur within the flow, and we assume that all thermophysical properties remain constant.

Due to our semi-infinite plane surface assumption, the flow variables depend solely on the dimensionless vertical coordinate y* and time t*.

Given these conditions and under typical boundary layer approximations, the governing equations and the initial and boundary conditions for velocity distribution, including slip flow, temperature, and concentration distributions, are as follows:

The boundary conditions are

From Eq. (1) it is clear that the suction velocity at the plate surface is either constant or a function of time only. Hence, it is assumed that

Where V_{0} is the mean suction velocity and εA << 1. The negative sign indicated that the suction velocity is directed towards the plate.

In the free stream Eq. (2) gives

Non-dimensional quantities are given by

Considering the above dimensionless variables, the basic field of Eqs. (2) Through (4) can be expressed in a dimensionless form as:

#### Method Of Solution

The Eqs. (10) - (12) are coupled non-linear partial differential equations whose solutions in closed-form are difficult to obtain. To solve these coupled non-linear partial differential equations, we assume that the unsteady flow is superimposed on the mean steady flow, so that in the neighborhood of the plate, we have

By substituting the set of Eqs. (15) - (17) into Eqs.(10) - (12) and equating the harmonic and non-harmonic terms, and neglecting the higher order terms in ε, we obtain

Where the prime denotes the differentiation with respect to y.

The corresponding boundary conditions are:

By solving the set of Eqs. (18) - (23) using boundary conditions (24) - (25), the boundary layer flow solutions velocity, temperature, concentration, the coefficient of Skin-friction, rate of heat transfer in terms of Nusselt number and rate of mass transfer are:

#### Results And Discussion

The analytical solutions are performed for velocity, temperature and concentration for various values of fluid flow parameters such as Schmidt number Sc, chemical reaction parameter Γ, Dufour number Du, magnetic field parameter M, radiation absorption parameter Q1, porous permeability parameter Φ1, porocity parameter K, Solutal Grashof number Gr, and buoyancy ratio parameter N, which are presented in figures 1-18. Throughout the calculations, the parametric values are chosen as Pr = 0.71, A = 0.5, ε = 0.02, n = 0.1, and t = 1. Various values of the thermal buoyancy ratio parameter N are plotted in Fig. 1. As seen from the figure, as buoyancy ratio values are increasing, the velocity is decreasing from zero to one and enhancing from 1 to 6. Figure 2 displays the effects of the radiation absorption parameter on the velocity field. It is obvious from the figure that an increase in the absorption radiation parameter results in a decrease in the velocity profiles from 0 to 1 and an increase in the velocity profiles from 1 to 6. Figure 3 illustrates the variation of velocity distribution across the boundary layer for various values of the porous permeability parameter Φ_{1}. It is observed that velocity increases near the source and reaches the free-stream condition. Permeability Φ_{1} is directly proportional to the square root of the actual permeability K. Hence, an increase will decrease the resistance of the porous medium, which will tend to accelerate the flow and increase the velocity. The velocity profiles for different values of magnetic parameter M are depicted in Fig. 4. From this figure, it is clear that as the magnetic field parameter increases, the Lorentz force, which opposes the fluid flow, also increases and leads to an enhanced deceleration of the flow. This result qualitatively agrees with expectations since the magnetic field exerts a retarding force on the free convection flow. Figure 5 represents the influence of the porosity parameter K on velocity. It is clear that velocity increases significantly with increasing values of K. Different values of the thermal buoyancy force parameter Gr are plotted in Fig. 6. As seen from the figure, the thermal buoyancy force increases, fluid velocity increases, and the boundary layer thickness increases. Fig. 7 displays the velocity profiles for various values of the Dufour number. The velocity decreases from 0 to 1 and enhances from 1 to 6 with an increase in the Dufour number. Fig. 8 displays the velocity profiles for various perturbation parameters. The velocity increases as the perturbation parameter increases. Figure 9 illustrates the effect of the chemical reaction parameter Γ. It is clear that the velocity decreases uniformly with increasing values of Γ. It is interesting to observe that the peak values of the velocity profiles are attained near the porous boundary surface. The influence of Schmidt number Sc on the velocity profiles is shown in Fig. 10. As Sc values increase, velocity decreases. Further, it is observed that the momentum boundary layer decreases with an increase in the value of Sc. Figure 11 displays the effects of the radiation absorption parameter on the temperature field. It is obvious from the figures that an increase in the absorption radiation parameter results in an increase in the temperature profiles within the boundary layer as well as an increase in the momentum and thermal thickness, because the large values correspond to the increased dominance of conduction over absorption radiation, thereby increasing the buoyancy force and thickness of the thermal and momentum boundary layers. Fig. 12 displays the temperature profiles for various values of perturbation parameters. As the perturbation parameter is increased, the temperature rises. The temperature profiles for different values of the Dufour number are shown in Fig. 13. It can be seen that the fluid temperature decreases with the Dufour number. Physically, the Dufour term that appears in the temperature equation measures the contribution of the concentration gradient to thermal energy flux in the flow domain. Figure 14 illustrates the effect of the chemical reaction parameter Γ. It is clear that the temperature profiles decrease uniformly with increasing values of Γ. The effect of the Schmidt number Sc on the temperature is shown in Fig. 15. As the Schmidt number increases, the temperature decreases. The concentration profiles for different values of perturbation parameters are presented in Fig. 16. From this figure, it is seen that the concentration increases with an increase in perturbation parameter ε. The concentration profiles for different values of chemical reaction parameters are presented in Fig. 17. From this figure, it is seen that the concentration decreases with an increase in the chemical reaction parameter. The effect of Schmidt number Sc on the species concentration profiles is shown in Fig. 18. It is clear that the concentration decreases exponentially and reaches the free-stream condition. Also, it is noticed that the concentration boundary layer thickness decreases with Sc.

**Skin friction:** The absolute value of Skin friction is depicted in Table 1, which illustrates the effect of the parameters Gr, Φ_{1}, N, K, M, Du, Q_{1}, Γ and Sc on Skin friction at plate. It is noticed that absolute value of Skin friction at plate decreases with the increase of Φ_{1} and Q_{1}. Whereas increases with the increase of Gr, N, K, M, Du, Γ and Sc.

**Nusselt Number:** The absolute value of Nusselt number is depicted in Table 2.which illustrates the effect of the parameters Q1, Du, Γ, Sc and ε. It is noticed that absolute value of Skin friction at plate decreases with the increase of ε and Q_{1}. Whereas increases with the increase of Du, Γ and Sc.

**Sherwood Number:** Table 3 shows the effect of the parameters Sc, Γ and ε on absolute value of Sherwood number at plate, it is observed that absolute value of Sherwood number at the plate increases with the increase of Sc, Γ and ε.

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