Analysis of Magnetohydrodynamics Flow of Incompressible Fluids over Oscillating Bottom Surface with Heat and Mass Transfer

Analysis of magnetohydrodynamics flow of incompressible fluids over an oscillating bottom surface with heat and mass transfer is discussed. The flow is free convection in nature. Momentum, energy, and concentration equations are obtained for computation of their respective profiles. The unsteady flow two-dimensional governing equations are solved numerically by the explicit finite difference method of the Forward Time Backward Space scheme. The numerical results show that the applied parameters have significant effects on the fluid flow and heat transfer and have been discussed with the help of graphical illustrations.


Introduction
e problems of hydromagnetic free convection flow of incompressible fluids over corrugated vibrating surface have drawn considerable attention from several medical researchers and engineers, resulting in the enhanced heat transfer performance by increasing the area over which heat transfer takes place. is technology is applied in the design of processing equipment that complies with cheap, environmental friendly, and energy-saving with high efficiency of mass and heat transfers. Temperature control is important in corrugated structures manufacturing because it helps to ensure a strong bond between the layers of the corrugated surface and indicates moisture content [1].
Fluid flow and heat transfer on corrugated channel laminar in nature were first studied by [2] for transitional and low Reynolds number turbulent flow. Magnetohydrodynamic free convection flow past an infinite vertical plate oscillating in its plane was studied in the case of an isothermal plate [3]. Rizwan et al. [4] investigated MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium with a complete structure of corrugated surface for heat transfer effects in the presence of the uniform magnetic field. e unsteady flow of second-order thermoviscous incompressible forced oscillations of a fluid bounded by rigid bottom was studied by [5]. Also, Schlichting [6] observed experimentally and numerically that corrugated channels do not have significant effects on heat transfer enhancement if operated in a steady regime. Garg and Maji [7] numerically studied the heat transfer of sinusoidal wavy channels at zero degrees phase shift. A numerical analysis of laminar forced convection in corrugated-plate channels with a sinusoidal, ellipse, and roundedvee wall shapes were studied [8]. Furthermore, Gbadeyan et al. [9] investigated Soret and Dufour effects on heat and mass transfer in chemically reacting MHD flow through a wavy channel using amplitude as the perturbation parameter.
is work attempts to study the effects of velocities, concentration, and temperature fields on the unsteady flow of incompressible fluid over the heated oscillating bottom for the various material parameters.

Mathematical Formulation
Consider a two-dimensional unsteady flow of a viscous incompressible fluid that is electrically conducting flows upwards on an oscillating bottom surface. e x-axis is taken along the infinite surface and the y-axis normal to it. e corrugated surface and the fluid are initially put at the same temperature. e fluid is initially at rest, responds to the fluctuations of the bottom and the periods of oscillation of the fluid response, and the temperature distribution are assumed to be oscillatory with the same frequency. At time t > 0, the plate starts vibrating with a frequency of oscillation ω and reference velocity U R forming boundary oscillating velocity u � U R sin ωt approximated [10]. A magnetic field B 0 of uniform strength is applied perpendicular to the plate along the positive z-axis and the magnetic Reynolds number is assumed to be small since the electric intensity E is zero at the plate; therefore, it is assumed to be zero everywhere within the flow [11][12][13]. e bottom surface is subjected to perturbations through forced transverse oscillations.
Linear sinusoidal displacement of the fluid along the vibrating surface, along the x-axis, is generated. Under the Navier-Stokes equation approximation, the viscous stress tensor is zero. e restoring force that produces the oscillation is the buoyancy force, and the waves are associated with these vibrations. e boundary contains the effect of confining the wave energy to a region of finite extent. Corrugation causes periodic variation in the force component and the oscillations due to vortex shedding [14]. Fluid dumping is generated as a result of relative fluid movement to the vibrating structure [15]. It means that the cross-flow vibration is caused by the lift force while the inline flow vibration is caused by the drag force, which in all cases are vortex-induced vibrations represented by periodic corrugation [2]. From Lorentz force, a moving particle with velocity, carrying a charge, contains a force acting on this particle. Also, the magnetic field vector B is perpendicular to the vibrating surface.
A current is induced in a conducting loop when magnetic flux linking the loop changes [16,17]. e electric field intensity in a region of time with varying magnetic flux density is present. When the magnetic Reynolds number is small [3,18] the induced magnetic field is negligible in comparison with the applied magnetic field, therefore becoming constant. Since the corrugated bottom surface is nonconducting, therefore the heat flux is zero at the surface and hence zero everywhere in the flow.
ere is a variation in the temperature and the density, but density is neglected everywhere [19,20] apart from the buoyancy terms which varies linearly with the local temperature and mass fraction.
Under the above-stated conditions and assuming variation of density in the body force term under Boussinesq's approximation [7], the problem is governed by the following momentum equations describing velocity profiles: e heat due to viscous dissipation is taken into an account and thermal radiation is assumed to be present in the form of a unidirectional flux in the z-direction denoted by q r .
By using the Rosseland approximation [19], the radiative heat flux q r is given by Since temperature differences within the flow are sufficiently small, then (5) can be linearized by expanding T 4 in Taylor series about T ∞ , which after neglecting higher-order terms is Substituting the partial derivative with respect to T of (5) in (4), the rate of change of radiative heat flux becomes 2 International Journal of Mathematics and Mathematical Sciences With initial and boundary conditions, Introducing nondimensional numbers, Equations (1)-(3) and (6) are nondimensionalised as follows.
Momentum equations in x− axis and y− axis are given as zv * zt * + u * zv * zx * + v * zv * zy * � 1 Re e energy equation is given as e concentration equation is given as where International Journal of Mathematics and Mathematical Sciences With the following initial and boundary conditions, e Skin friction, Nusselt number, and Sherwood number at the corrugated surfaces are estimated as follows: 3. Numerical Technique e partial differential equations (10)- (14) show the solutions to highly nonlinear coupled governing equations of velocity, concentration, and temperature with the various physical parameters, and the associated boundary conditions in (15) are solved numerically using the explicit finite difference method of the Forward Time Backward Space (FTBS) scheme since this method is stable and is validated using computer software. is is carried out by discretizing the computational domain with nonuniform grids of sinusoidal elements. e flow is in two dimensions and therefore flow domain is confined by the x, y, and t axes. e approximate values of u * , v * , C * , and T * are found at every nodal point for particular i at (k + 1)th time level. A necessary condition for time stability, the Courant-Friedrichs-Lewy (CFL) condition, which depends on time and space discretization, is used. e FTBS finite difference method is applied to replace continuous derivatives with difference formulas that involve only the discrete values associated with positions on the mesh. e basic unknowns for the above differential equations are the velocity components (U m+1 k,j , V m+1 k,j ), the temperature (T m+1 k,j ), and the concentration (C m+1 k,j ). Momentum equations expressed in finite differences is given as e concentration equation expressed in finite differences is given as 4 International Journal of Mathematics and Mathematical Sciences e energy equation expressed in finite differences is given as To get the analytical results of velocities in (17) and (18), concentration in (19), and temperature in (20) for various values of physical parameters, the code of the algorithm has been executed in MATLAB running on a PC. is is done by applying numerical calculations, and the mesh size is fixed at Δx � 0.2, Δy � 0.3, and Δt � 0.0001, where a sinusoidal shape is formed by the x -axis and y-axis containing 15 × 20 meshes. As shown in Figure 1, the magnitude of the Prandtl number, Pr, is varied in Table 1, that determines whether the thermal boundary layer is larger for Pr ≤ 1, where buoyancy forces are in balance with the thin viscous boundary layer, or smaller for Pr ≥ 1, where inertial and buoyancy forces are in balance with the momentum boundary layer; this is shown at the oscillating bottom surface with the wave-like motion. A smaller value of Pr is an indication that heat diffuses faster than velocity; therefore, it is clear that fluids with small Prandtl numbers are free-flowing liquids with high thermal conductivity and are therefore a good choice for heatconducting liquids, as shown in Figure 1. Figure 2 shows the effect of Reynolds number on secondary velocity profiles. Since Re is associated with the smoothness of fluid flow, at lower velocities the flow is laminar and this is pictured as a series of parallel layers moving at different velocities. In the presence of oscillations, the fluid flows vigorously and reaches a velocity at which the velocity changes from laminar to turbulence. When a small Re is used, it applies that the viscous force is predominant thus imposing drug in the fluid and reducing the fluid flow. e effect of local mass Grashof number Gr m on velocity is shown in Figure 3. In this case, mass transfer natural convection is as a result of concentration gradients rather than temperature gradients. It is clear that when the value of Gr m increases, the velocity rises as it reaches the greatest value near the surface due to the enhancement in the buoyancy force (Tables 1-3). e inline vibration of a structure is caused by the oscillating drag force with different ranges in the reduced velocities. A similar effect is experienced when thermal Grashof number Gr θ is used, as shown in Figure 4. By varying the values of Gr θ , the effects of free convection currents on the flow are indicated and the fluid's velocity increases since fluid flow is aided by the free convection currents. Figure 4 shows that when the values of Gr θ causes a rise in velocity profiles on a cooled surface due to the varying nature of boundary conditions, an indication that the thermal radiation parameter produces significant increases in the thermal conditions of the fluid temperature which consequently induces more fluid in the boundary layer through buoyancy effect to the viscous force, therefore enhancing fluid velocity. Variation in Gr θ and Gr m , as shown in Figure 5, has an increasing effect on velocity near the International Journal of Mathematics and Mathematical Sciences center as a result of thermal and mass buoyancy forces due to cooling of the surface, by making the bond between the fluids to become weaker, strengthening the internal friction to reduce, and the gravity becoming stronger enough. Due to oscillation, thermal and flow patterns adjacent to the boundary are mainly affected.

Discussion of the Results
A reverse effect in the case of heating of the surface, where Gr m < 0 and Gr θ < 0, is shown in Figure 6. Reducing the effect on velocity near the center as a result of thermal and mass buoyancy forces due to the heating of the surface is carried out by making the bond between the fluids to become stronger thus strengthening the internal friction to increase and the gravity becoming weaker enough.
It is observed from Figure 7 that while all other participating parameters are held constant, the values of Sc from  International Journal of Mathematics and Mathematical Sciences hydrogen to atmosphere pressure reduce the velocity due to oscillation. Since Sc is the ratio velocity boundary layer to the concentration boundary layer which is comparable to Prandtl number in heat transfer, therefore Sc is applied to characterize flows when there are simultaneous momentum and mass transfer. Figure 8 depicts the effects of velocity on the magnetic parameter M. Velocity rises as values of M improves because the frictional or drag force (Lorentz force) in the magnetic field is responsible, which affects the velocity field that opposes the fluid motion, causing the velocity to decrease.
An increase in magnetism significantly reduces the thickness of the boundary layer, thereby reducing the velocity components. A reversal in the direction of the secondary velocity profiles is achieved by using large values of M. Here, the effective conductivity of the fluid rises with a rise in M as a result of damping force due to oscillation. e Soret effect causes the main-flow shear stress to rise and the cross-flow shear stress to fall, as shown in Figure 9. By decreasing the values of Sr effect leads to a rise in the main flow and cross-flow velocities, as an indication that the velocity boundary layer thickness decreases with an increase in Sr as a result of mass buoyancy force. is brings about the thermal diffusion effect. Figure 10 shows that when the values of the Dufour number increases, velocity rises as an indication that the velocity boundary layer thickness increases due to mass diffusion effect. International Journal of Mathematics and Mathematical Sciences and Pr � 7.56 for water at 20 ∘ C and one-atmosphere pressure. It is clear from Figure 11 that an increase in Pr causes a fall in concentration due to the Brownian motion of the fluid as a result of an increase in migration from the high concentration regions to the regions with low concentration. e Reynolds number used is assumed to be small so that the induced magnetic field is neglected within the fluid particles as the fluid moves due to vibration, as shown in Figure 12. It is clear that the Reynolds number is varied in Table 2, to help predict flow patterns in different fluid flow situations.

Concentration
As the values of Schmidt number rises, i.e., 0.22 (hydrogen), 0.62 (water vapour), and 0.78 (ammonia), from Figure 13, the concentration profile rises because the concentration profile and the boundary layer thickness decreases, corresponding to a thinner concentration boundary layer relative to the momentum boundary layer. e effect of oscillation on velocity is overcome by freestream velocity, leading to the observed crossover of concentration profiles. When the values of Dufour number increases, the fluid concentration field reduces the boundary layer thickness due to oscillation, as shown in Figure 14.

Temperature
Profiles. An increase in Prandtl number results in a fall in temperature, as shown in Figure 15, because the thermal boundary layer thickness decreases with increasing Pr. is is because the fluid viscosity becomes larger and reduces the thickness of the thermal boundary layer. In cases where Pr is high in liquids, the instability is hydrothermal and the related mechanism involves communication between free-surface temperature perturbations and bulk-liquid temperature. By eliminating the free-surface temperature, oscillations caused by hydrothermal wave coupling could be broken and they would cease. A smaller value of Pr is an indication that heat diffuses quickly compared to the velocity.
Reynolds number incorporates the physical properties of liquid density and dynamic viscosity which are directly related to temperature. is means that dynamic viscosity decreases in response to falling density. From Figure 16, as the temperature rises, the change in viscosity decreases due to the presence of the inertia force. Reynolds number is directly proportional to the temperature. e Eckert number influences the self-heating of a fluid due to dissipation as a result of internal friction of the fluid. If dissipation is neglected at Ec ≤ 1 as shown in Table 3. Using Figure 17, it is shown that for higher values of the Eckert number Ec, the rate of heat transfer decreases. All the terms in the energy equation describing the effects viscous dissipation and body forces on the energy balance can be neglected, and the equation reduces to a balance between conduction and convection. e effect of viscous dissipation on the flow field is to increase  International Journal of Mathematics and Mathematical Sciences the energy, resulting in greater fluid temperature and as a consequence greater buoyancy force. e increase in the buoyancy force due to an increase in the dissipation parameter enhances the temperature. Figure 18 shows that the Dufour number is directly proportional to the fluid temperature when other parameters are kept constant as a result of thermal boundary layer thickness. e values of skin friction, Nusselt number, and Sherwood number are computed from random values generated from MATLAB, as shown in Table 4.    Table 4, as time increases, the Nusselt number, Sherwood number, and Skin friction decrease, as it physically implies that shear stresses decrease with an increase in time. A higher value of radiation parameter leads to an increase in magnitudes of skin frictions and Nusselt number as a result of an increase in the rate of species concentration. e effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid, but it decreases in Sherwood number.

Conclusion
A numerical study has been conducted on free convective heat and mass transfer of an incompressible electrically