MHD Fluid Flow Bounded by Two Parallel Vertical Plates in A Porous Media with Heat Transfer

MHD Fluid Flow Bounded by Two Parallel Vertical Plates in A Porous Media with Heat Transfer

John King’ori Kariuki, Maurine Maraka Wafula, Lawrence Njau

Kenyatta University

DOI: https://doi.org/10.51584/IJRIAS.2024.90302

Received: 25 January 2024; Revised: 18 February 2024; Accepted: 23 February 2024; Published: 27 March 2024

ABSTRACT

We are investigating Magneto hydrodynamic fluid flow between two parallel vertical plates in a porous media with heat transfer. We will nondimensionalize the governing equations and then transform the resulting equations using the central difference approximation method. We will simulate the resulting equations using MATLAB to establish the effect of non dimensionalized parameters on the fluid velocity and temperature. We will graphically present the effect of the Darcy, Hartmann, Prandtl, and Reynolds numbers on the velocity and temperature profile. We have made the following conclusions from this study. The fluid velocity is inversely proportional to Darcy’s and Hartmann’s numbers. Fluid velocity increases as the Reynolds number is increased. Fluid temperature declines with a rise in the Prandtl number when the distance between the plates is greater than one and increases the fluid temperature when the distance between plates is less than one. Fluid temperature increases as Hartmann and Darcy numbers are increased. No velocity and temperature profile effect exists on varying Prandtl and Reynolds numbers. The findings have practical applications in industrial processes such as cooling systems.

INTRODUCTION

When an MHD fluid flows, the induced current experiences mechanical forces due to the magnetic field; in return, the current induces a magnetic field that affects the original field. MHD study for fluid in porous media contributes immensely to physiological laws and engineering. Darcy’s law explains how a fluid flows in a porous medium and states that the rate of fluid discharge is proportional to the gradient in the hydraulic head and hydraulic conductivity. In engineering fields, MHD is applicable in MHD power generators, petroleum engineering, industrial cooling systems, etc. In the above applications, cooling effects help achieve the desired characteristics. Therefore, continuing with the MHD study is imperative to improve applications in physics, chemistry, engineering, metallurgy, heat exchangers, accelerators, etc.

The flow field characteristics depend on the orientation and magnitude of the magnetic field. For instance, a change in magnetic field strength influences the nature and concentration of the fluid particles and consequently impacts heat transfer [1]. The fluid flow phenomenon past a permeable material is conceptualized as a natural mechanism like absorbing water and minerals in plants and the flow of nutrients in the human body. Industrial processes such as pulp drying, detergent manufacturing, diffusion, capillarity, and gas management in fuel cells rely on porous material phenomena in fluid dynamics.

[2]investigated unsteady two-dimensional MHD flow via a permeable material, considering it in a revolving parallel plate channel. The flow parameters, heat transfer rate, velocity, shear stress, mass transfer rate, and volume flow rate were analyzed non-dimensionally. The researcher concluded that increasing the Hartmann number reduces velocity and enhances permeability parameters (K). Furthermore, increasing the Grashof number (Gr), mass Grashof number (Gc), and slip parameters enhanced the resultant velocities. Notably, an increase in thermal radiation produces a corresponding reduction in heat transfer rate and temperature at any specific point. However, thermal radiation has no significant effect on velocity.

MHD transient fluid flow passing through two vertical walls in the presence of radiation was examined to investigate the effect on mass and heat transfer rates[3]. The researcher adopts a perturbation technique to determine PDEs governing velocity and temperature fields. Furthermore, assumed a fluctuating temperature on one plate and graphically represented the flow parameters. Also, he developed equations governing skin friction on the parallel plates.

[4]studied a non-Newtonian MHD flow in a magnetic field through a permeable material with varying wall temperatures. The governing equations were solved using regular perturbation methods and reduced to boundary value problems. The solutions on flow parameters indicate that velocity increases with increasing hall and permeability parameters. Furthermore, higher values of the viscoelastic parameter lead to a corresponding increase in velocity. On the other hand, low values of Hartmann’s number correspond to a velocity decrease, while temperature and concentration decrease correspond to a decrease in the Prandtl and Schmidt numbers, respectively.

Flow Model

Figure 1 Flow Model

We are making the following assumptions for this study:

1. The fluid is in compressible
2. No slip condition.
3. The flow is steady and in the x-direction.
4. The magnetic field is in the Y-direction.
5. The plates are maintained at a constant temperature of T. Also, the temperature T near the plate is higher than the temperature T_f.

MATHEMATICAL FORMULATION

Governing Equations

Continuity Equation

We are simplifying the continuity equation since the flow is steady, in compressible, and in the x-direction to:-

Momentum Equation

where ρ- fluid density, u -velocity vector, t- time, p – pressure, μ viscosity of the fluid in the porous medium, g – acceleration due to gravity, J – current density vector, B- magnetic field vector, α – thermal expansion coefficient, T – temperature of the fluid, T_f is the temperature of the porous medium, κ is the Darcy-Forchheimer coefficient.

Introducing the assumptions stated above to the momentum equation,we obtain

Energy Equation

We are modifying the energy equation to account for the energy balance between the convective heat transfer, conductive heat transfer, and Joule heating.

Further, simplifying the energy equation since the flow is steady, in the x-direction, and incompressible, we have that

The convective term u∂T/∂x implies that the fluid flow is only in the x-direction and through the porous medium. The thermal conductivity term k ((∂^2 T)/(∂x^2 )) represents the heat conduction in the x-direction, while the term (δu² B_y²)/μ gives energy dissipation per unit of viscosity.

Boundary Conditions

The flow is steady and in the x-direction.

Non dimensionalization

Using the following non-dimensional variables,

U, H, P^*, and ∇T are characteristic velocity, length, pressure, and temperature difference scales.

Momentum Equation

We are substituting the non-dimensional variables into the momentum equation (4) to obtain

Multiply each term by -H/(ρU²)

Re- Reynolds Number, Ha- Hartmann Number, Gr- Grashof number, Da- Darcy Number

Energy Equation

We are substituting the non-dimensional variables in the energy equation (6)to obtain

Dividing each term by ρC_p we obtain

Multiply both sides by μ/(U² ρ∆T)

Re- Reynolds number, Pr- Prandtl Number, Pr_m – Prandtl Magnetic Number

The nondimensionalized boundary conditions are

We are going to apply central difference approximations to the nondimensionalized momentum equation and energy equation to obtain equation (16) and (17)

RESULTS AND DISCUSSION

Figure 2: Effect of Hartmann number on velocity profile

In Figure 2, the velocity profile declines as the value of Hartmann is increased due to retarding effect of Lorentz force. As Ha increases, the magnetic forces become more dominant compared to viscous forces. The higher Ha implies stronger magnetic effects, which can suppress fluid motion. Magnetic suppression can lead to a decline in velocity because the fluid is resisted by the magnetic forces, thus dampening its kinetic energy. In physical terms, high Hartmann numbers generate a counter-force to fluid flow [5]; thus, fluid velocity will be high at low Hartmann numbers. High Hartmann number values tend to decrease fluid movement [6].

Figure 3: Effect of Hartmann Number on the temperature profile

Figure 3 shows thermal temperature increases as the Hartmann number increases since high Ha values result in low viscosity. The thermaltemperature increases when the Hartmann number increases because when the magnetic field induces suppression of fluid motion, it hinders convective heat transfer. The finding agrees with [7], who noted that an increase in Ha value leads to a rise in temperature. Furthermore, [8] observes that the Lorentz force produces heat energy, raising the fluid temperature.

Figure 4: Effect of varying Reynolds number on the velocity profile

Figure 4 shows that fluid velocity increases as Reynolds values are increased since inertial forces dominate over viscous forces. An increase in Reynolds number means that inertial forces within the fluid are becoming more dominant relative to viscous forces, and thus fluid velocity increases. The results agree with [9], who made similar observations.

Figure 5: Effect of Reynolds number on the temperature profile.

Figure 5 shows that a variation in the Reynolds number does not affect the temperature profile. It implies that heat transfer in the system is by conduction rather than convection. Raising the Reynolds value indicates that inertia forces are more than viscous forces, which leads to forced convection.

Figure 6: Effect of Prandt l Number on the velocity profile

Figure 6 shows no change in the velocity profile on varying Prandtl numbers. Pr numbers show the relationship between momentum and thermal boundary layers near the walls and thus have no direct effect on fluid velocity. According to [4],the Pr number weakly influences fluid velocity, especially in a porous medium.

Figure 7: Effect of Prandtl Number on Temperature Profile

Figure 7 shows that when the distance between the plates is greater than 1, the fluid temperature drops with a rise in the Prandtl number since it lowers thermal boundary layer thickness [10]. However, temperature increases when the distance between the plates is less than 1. According to [11], increasing the Prandtl values at fixed values of Re enhances the heat transfer rate. The finding indicates that when the distance between the plates is increased, the velocity of the fluid changes, implying that between a distance of 1 and 1.5, the fluid temperature is nearly equal to the bulk fluid velocity since the thermal boundary layer is minimal.

Figure 8: Effect of Darcy Number on Velocity Profile

Figure 8 shows fluid velocity decreases as Darcy value is increased.An increased Darcy number indicates that viscous forces become more dominant than inertia forces. This means there is a greater resistance to flow due to increased viscous dissipation within the porous structure. Thus, increasing Darcy’s number changesthe fluid flow rate through the permeable medium. Raising Darcy number leads to a corresponding increase in kinematic viscosity, hindering fluid free flow [12]. Furthermore, increasing Da leads to a change in fluid permeability and, thus, in velocity distribution. Consequently, the fluid velocity decreases [5].

Figure 9: Effect of Darcy Number on Temperature Profile

Figure 9 shows fluid temperature increases as the Darcy number is raised since enhancing medium permeability boosts convective heat transfer. Increasing the Darcy number improves the permeability rate of the fluid in the medium through the pores and thus enhances heat flow through convection.

CONCLUSION

We investigated Magneto hydrodynamic fluid flow between two parallel vertical plates in a porous media with heat transfer. We simulated the resulting nondimensionalized equation using MATLAB and made the following conclusions. Fluid velocity is inversely proportional to Darcy number and Hartmann number. Fluid velocity is proportional to the Reynolds number. Fluid temperature declines with a rise in the Prandtl number when the distance between the plates is greater than one and increases the fluid temperature when the distance between plates is less than one. Fluid temperature increases as Hartmann and Darcy numbers are increased.No velocity and temperature profile effect exists on varying Prandtl and Reynolds numbers.

ACKNOWLEDGMENTS

Glory to God for helping us compile this paper.

The paper is extracted from an MSc thesis by John King’ori Kariuki, supervised by Maurine Maraka Wafula, PhD, and Lawrence Njau, PhD at Kenyatta University, Nairobi, Kenya.

Declaration of interest statement

The authors declare no conflict of interest.

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APPENDICES

Figure 1 Flow Model 4

Figure 2 Effect of Hartmann number on velocity profile. 9

Figure 3 Effect of Hartmann Number on the temperature profile. 10

Figure 4 Effect of varying Reynolds number on the velocity profile. 11

Figure 5 Effect of Reynolds number on the temperature profile. 12

Figure 6 Effect of Prandtl Number on the velocity profile. 13

Figure 7 Effect of Prandtl Number on Temperature Profile. 14

Figure 8 Effect of Darcy Number on Velocity Profile. 15

Figure 9 Effect of Darcy Number on Temperature Profile. 16