Finite Difference Simulation of Nonlinear Convection in Magnetohydrodynamic Flow in the Presence of Viscous and Joule Dissipation over an Oscillating Plate

Abstract

Engineers and researchers are interested in the study of nonlinear convection, viscous dissipation, and Joule heating in various flow configurations due to their various applications in engineering processes. That is why the present study deals with the influence of nonlinear convection, viscous, and Joule dissipation of the temperature and velocity profile of incompressible fluid over a flat plate. In this study, the magnetic field acts perpendicular to the fluid flow and is supposed to be of uniform magnitude. Further, the Newtonian fluid, which is electrically conducting, passes over an infinite vertical flat plate under an oscillatory motion. The term representing the influence of the nonlinear convection phenomenon is integrated into the Navier–Stokes equation. The governing equations of the mentioned study were modeled in the form of non-linear PDEs and modified as non-dimensional equations via appropriate scaling analyses, which resulted in coupled and non-linear PDEs. For the numerical solution of the transformed non-linear PDEs, the finite difference method was applied. Finally, we present the effects of various flow parameters via graphical illustrations.

1. Introduction

Fluid dynamics have immense importance in research and can be observed in engineering and technology. The properties of fluid flow have been explored and are used in modern innovation and sciences. Fluid dynamics applications can be found in heat exchange processes, such as the cooling of electrical apparatuses, lubrication industries, gas turbines, the cooling of atomic reactors, plasma physical science, petroleum industries, geothermal energy, food processing industries, and so on.

Further, the application of viscous dissipation is of great concern. In industries where lubrication processes are dominant, viscous dissipation cannot be disregarded since diffusions generated by liquid particles due to their kinetic energy influence the fluid flow and conduct heat dissipation in the system [1]. One of the best applications of a thermal energy system is a heat exchanger. The irreversible cycle involves work being conducted by the fluid because shear stress is changed into heat energy. This is important for some applications, such as the huge temperature increase seen in polymer industries. Therefore, in engineering and industrial processes and other branches of technological operations that involve copper work, polymer processes, the spinning of metals, and paper production, viscous dissipation plays an important role [2,3], which is why it is always encouraged to innovate the thermal performance of the heat transfer mechanism and processes. Applications of Joule-heating can be seen in various electrical appliances, electronic devices, and industrial processes. As far as electronic devices are concerned, ohmic heating is used in electric stoves, electric and cartridge heaters, soldering irons, etc., along with some food processing equipment, which also makes use of ohmic heating [4]. Work was conducted on Oldroyd-B fluid by Ganesh Kumar et.al [5] on the application of ohmic heating, which influences fluid flow significantly. They noted a rise in the Oldroyd-B nanofluid temperature profile with Joule viscous dissipation. B.K. Swain et al. [6] determined the heat transfer characteristics along with the magnetohydrodynamics effects of a Newtonian fluid, which passes through a stretched sheet that is porous in nature. They stated that the thermal boundary layer thickens with the increasing Eckert number and joule heating parameter, and the porosity parameter works as an assisting force for the velocity profile. The work of Afridi and Qasim [7] cannot be neglected; they determined the combined effects of viscous irreversibility and thermal dissipation, which resulted in the generation of entropy in three-dimensional fluid flow for the Newtonian fluid, which was incompressible in nature. The self-similarity solution under viscous dissipation is a key finding of this research. Extensive research on ohmic heating, along with viscous dissipations, can be seen in recent articles [8,9,10,11,12].

Magnetohydrodynamics deals with magnetic properties and the electrical behavior of conducting fluid. Magnetohydrodynamics applications are very common. There are several research studies on magnetohydrodynamics along with Newtonian/non-Newtonian fluids, including in processed food, dyes, paints, polymer, ketchup, grease, etc. [13,14,15]. In this study, the application of magnetohydrodynamics was also used; it influenced a vertical flat plate that was perpendicularly under oscillatory motion. The concept behind magnetohydrodynamics involves the fluid flow being influenced by the magnetic field; it starts conducting energy that polarizes the fluid and the direction of the magnetic field is reversed [16,17]. When the concept is incorporated into the fluid, it results in the combination of a more complex Navier–Stokes equation. Hence, we used the concept of MHD in this study to see the effect of MHD on the Newtonian fluid. Hence, the influence of magnetohydrodynamics on fluid flow under the phenomenon of heat transfer is significant in many areas, including space physics, astrophysical processes, and engineering [18,19,20].

Non-linear convection is important in the literature due to its application in nuclear reactors, atomic plants, solar power technology, combustion chambers, propulsion devices, high-temperature chemical processes, and aircraft systems. Nonlinear convective flow has an extensive range of uses in engineering, specifically in industrial and manufacturing engineering, geophysics, and astrophysics. Further, research shows that nonlinear convective flow (due to its non-linearity) plays a significant role in heat transfer [21]. It is established that nonlinear convection reduces the temperature profile. Mahanthesh et al. [22] determined the effect of nonlinear heat dissipative fluid that flowed over a stretched surface. They used the homotopic approach to report the analytical solution and discovered that the volume fraction and temperature profiles of nanoparticles were stronger in the presence of solar radiation than they were in the absence of solar radiation. Vasu et al. [23], T. Hayat and M. Qasim [24], and Qayyum et al. [25] worked on the Maxwell fluid and Powell–Eyring fluid and observed the mechanism of nanoparticles with viscous dissipations of fluid flow. Qayyum et al. [26] further worked on nonlinear convective fluid flow, which undergoes chemical reactions of a non-Newtonian fluid. They found that the relationship between temperature and the heat generation/absorption parameter is increasing, whereas the rate of heat transfer is decreasing. Further, Adesanya et al. [27] worked on nonlinear convective fluid flow under the application of entropy through a closed channel. Adesanya et al. [27] found that entropy generation dramatically rises with increasing buoyancy parameters. Hayat et al. [28,29,30] worked on various similar models of the nonlinear convective flow to study the influence of the non-linear property of heat transfer of fluid. Due to emerging applications involving non-linear convective flow, various research studies have been carried out on Newtonian and non-Newtonian fluids in [31,32,33,34,35,36]. Therefore, this article deals with non-linear convective fluid flow moves on a vertical flat plate.

The oscillating phenomenon has a significant role in fluid mechanics [37,38,39]. The oscillating boundary layer phenomenon can be observed in chemical engineering, heat conduction, industrial manufacturing processes, and geophysical flows. The literature shows that the analyses of convective flow and nonlinear convective flow have been performed by using Stokes’ problem. Naeema Ishfaq et al. [40] determined the velocity profile of nanofluid under the influence of oscillation. According to their findings, the skin friction coefficient and the density of nanofluids both go up as the volume fraction of nanoparticles goes up. Moslem Uddin and Abdullah Murad [41] determined analytical solutions for the oscillatory Couette flow and Stokes’ problem for double-layer fluid. Recently, Nepal Chandra Roy et al. [42] investigated water hybrid nanofluid with Stokes’ problem along with a heat transfer application with buoyancy force. They obtained the exact solution in two different forms, i.e., in exponential form and the form of error functioning. Further, they claimed that mixed convection has significant effects on the velocity profile. Hussanan et al. [43] determined an analytical solution by using the Laplace transformation for the natural convective flow of fluid, which was kept on a flat plate under oscillatory motion. Khan and Abro [44] determined a solution for nanofluid that was influenced by a porous medium and magnetic field over an oscillating surface. Moreover, Khan et al. [45] worked on Burgers’ fluid to determine the properties of heat transfer where the fluid passed over a plate having oscillatory motion. It was discovered that when the Hartmann number rises, the velocity falls, while the shear stress rises. Further, they developed Stokes’ first problem solutions for the hydrodynamic Burgers’ fluids as part of a limiting case.

Motivated by the above-cited research, the current research deals with the numerical solution of velocity and temperature where the fluid is influenced by Joule and viscous dissipation with additional application of magnetohydrodynamics. The external magnetic field was applied perpendicularly to a non-linear convective flow. The fluid moved over a vertical flat plate that was under oscillatory motion. Further, governing equations were modeled as nonlinear PDEs and transformed into non-dimensional equations via mathematical transformations that were numerically solved by the finite difference method (known as FTCS). To our knowledge, this is the first attempt to examine the non-linear convection in magnetohydrodynamic fluid flow along with viscous and Joule dissipation.

2. Mathematical Modeling

Consider a magnetic field $\overrightarrow{B}=\left[0,B_0,0\right]$ that is applied externally to the unsteady non-linear convective flow of an incompressible fluid. The fluid moves in the x-direction, where the y-axis is situated in a perpendicular direction to the fluid flow. Here, the plate and fluid are at rest initially and are at a constant temperature $T_{\infty }$. After time $t=0^{+}$, the plate oscillates vertically on its own axis with a periodic velocity $U_0\cos \left(\omega t\right)$, as shown in Figure 1. Meanwhile, the temperature of the vertical plate is raised to $T_w$, which is subsequently constantly maintained. Furthermore, the term representing non-linear convection is included in the momentum equation. The energy equation included terms that indicate viscous and magnetic dissipation effects.

The PDEs depicting the present flow problem are as follows:

$$\frac{\partial u}{\partial t}=\nu \frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^{2}u}{\rho }+g{\beta }_0(T-T_{\infty })+g{\beta }_1{(T-T_{\infty })}^2$$

(1)

$$\frac{\partial T}{\partial t}=\frac{k}{\rho C_p}\frac{{\partial }^{2}T}{\partial y^2}+\frac{\mu }{\rho C_p}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{\sigma B_0^2{u}^2}{\rho C_p}$$

(2)

The initial conditions and boundary conditions are:

$$u\left(y,0\right)=0,T\left(y,0\right)=T_{\infty },y\ge 0,$$

(3)

$$u\left(0,t\right)=U_0\cos \left(\omega t\right),T\left(0,t\right)=T_w,\mathrm{for}t>0,$$

(4)

$$u\left(y\to \infty ,t\right)\to 0,T\left(y\to \infty ,t\right)\to T_{\infty },$$

(5)

In the above-mentioned equations, u denotes the x-component of the velocity vector, T indicates the fluid temperature, and $t$ represents the dimensional time. Further, U₀ is the velocity amplitude of the plate oscillations, $T_{\infty }$ denotes the ambient temperature, T_w is the wall temperature, C_p shows the specific heat of the fluid at constant pressure, ω represents the frequency of the oscillating plate, ρ is the fluid density, µ is the dynamic viscosity of the Newtonian fluid, υ shows kinematic viscosity, σ indicates the electrical conductivity of the fluid, g shows gravitational acceleration, B₀ shows the uniform magnetic field strength, β₀ and β₁ are thermal expansion coefficients, and k is the thermal conductivity of the working fluid.

For the conversion of governing Equations (1) and (2) along with their corresponding initial and boundary conditions (3)–(5) into non-dimensional form, we used the dimensionless variables, which are:

$$U=\frac{u}{U_0},\tau =\omega t,\eta =\sqrt{\frac{\omega }{\vartheta }}y,\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}.$$

(6)

By substituting the dimensional variables (y, t, u, T) by (η, τ, U,$\theta$) in Equations (1)–(5), the following non-dimensional system is obtained.

$$\frac{\partial U}{\partial \tau }=\frac{{\partial }^{2}U}{\partial {\eta }^2}-MU+Gr\theta \left[1+N_c\theta \right]$$

(7)

$$\frac{\partial \theta }{\partial \tau }=\frac{1}{\mathrm{Pr}}\frac{{\partial }^2\theta }{\partial {\eta }^2}+Ec{\left(\frac{\partial U}{\partial \eta }\right)}^2+MEc{U}^2$$

(8)

$$U\left(0,\eta \right)=0,\theta \left(0,\eta \right)=0,\eta >0$$

(9)

$$U\left(\tau ,0\right)=\mathrm{Cos}\tau ,\theta \left(\tau ,0\right)=1,\tau >0$$

(10)

$$U\left(\tau ,\eta \right)\to 0,\theta \to 0,as\eta \to \infty .$$

(11)

where shows the Grashof number, $M=\frac{\sigma B_0^2}{\rho \omega }$ represents the magnetic parameter, $\mathrm{Pr}=\frac{\mu C_p}{k}$ shows the Prandtl number, $Nc=\frac{{\beta }_1\left(T_w-T_{\infty }\right)}{{\beta }_0}$ shows the non-linear convection parameter, and $Ec=\frac{U_0^2}{C_p\left(T_w-T_{\infty }\right)}$ shows the Eckert number. The skin friction coefficient (Cf) and Nusselt number (Nu) are defined as

$$\left.\begin{array}{l}\frac{Cf\sqrt{\mathrm{Re}}}{2}=\frac{\partial U\left(\eta =0,\tau \right)}{\partial \eta }\\ Nu\sqrt{St}=\frac{\partial \theta \left(\eta =0,\tau \right)}{\partial \eta }\end{array}\right\}.$$

(12)

where $\mathrm{Re}=\frac{U_0^2}{\nu \omega }$ (Reynold number) and $St=\frac{\omega }{\nu }{L}^2$ (Strouhal number).

Without the thermal analysis and in the absence of the magnetic field, the present problem reduces to Stokes’ second problem, given below

$$\frac{\partial U}{\partial \tau }=\frac{{\partial }^{2}U}{\partial {\eta }^2},$$

(13)

$$\left.\begin{array}{l}U\left(0,\eta \right)=0,\eta >0\\ U\left(\tau ,0\right)=\mathrm{Cos}\tau ,\\ U\left(\tau ,\eta \to \infty \right)\to 0,\end{array}\right\}.$$

(14)

The exact solution of Equation (13), along with the initial and boundary conditions, are obtained in the form of an exponential–trigonometric function, given below

$$U\left(\eta ,\tau \right)=e^{-\frac{\eta }{\sqrt{2}}}\cos \left(t-\frac{\eta }{\sqrt{2}}\right).$$

(15)

The skin friction coefficient takes the form of

$$\frac{Cf\sqrt{\mathrm{Re}}}{2}=\frac{1}{\sqrt{2}}\left(\sin (\tau )-\cos \left(\tau \right)\right).$$

(16)

3. Numerical Methodology (FTCS)

The coupled, dimensionless, and non-linear PDEs are numerically solved by the finite difference method, which is forward time centered space, i.e., FTCS. The FTCS method is an explicit technique; a direct calculation of dependent variables is conducted in terms of known quantities. Here, there is a simple updating procedure that does not depend on other values at the current level, only previous values are used to compute the current values. The continuous function is discretized into the discrete function by replacing derivatives in the governing equations via difference quotients. Since the working variables depend on both time and length, temporal and spatial discretization is done by forward and central differences, respectively.

Spatial discretization of velocity U by using the central difference method

$$\frac{\partial U}{\partial {\eta }_i}=\frac{U_{i+1}-U_{i-1}}{2\Delta \eta },$$

(17)

$$\frac{{\partial }^{2}U}{\partial {\eta }_i{}^2}=\frac{1}{4{(\Delta \eta )}^2}\left[U_{i-2}-2U_i+U_{i+2}\right].$$

(18)

Equation (18) can be transformed into Equation (19)

$$\frac{{\partial }^{2}U}{\partial {\eta }_i{}^2}=\frac{U_{i-1}-2U_i+U_{i+1}}{{(\Delta \eta )}^2}.$$

(19)

Similarly, for temperature θ

$$\frac{{\partial }^2\theta }{\partial {\eta }_i{}^2}=\frac{{\theta }_{i-1}-2{\theta }_i+{\theta }_{i+1}}{{(\Delta \eta )}^2}.$$

(20)

Here, $\Delta \eta ={\eta }_i-{\eta }_{i-1}$ represents the space step size, which is constant throughout the simulation.

The temporal discretization of velocity U and temperature θ by using forward differences are

$$\frac{\partial U}{\partial \tau }=\frac{U_i{}^{j+1}-U_i{}^j}{\Delta \tau },$$

(21)

$$\frac{\partial \theta }{\partial \tau }=\frac{{\theta }_i{}^{j+1}-{\theta }_i{}^j}{\Delta \tau }.$$

(22)

Here, i and j are indexing parameters used for spatial and temporal indexing, where $i=1,2,3,\dots N$, $j=1,2,3,\dots M$, and $\Delta \tau ={\tau }_j-{\tau }_{j-1}$ represent the time step size, which is constant throughout the simulation. The obtained numerical results are accurate up to the order ~ ${10}^{-7}$. To obtain this accuracy, the steps sizes $\Delta \eta =0.025$ and $\Delta \tau =0.00001$ are chosen. Further, the FTCS scheme is stable with considered time and space steps as discussed in the book by Hoffman and Chiang [46]

By putting Equations (17)–(22) into dimensionless Equations (7) and (8), the resulting stencil equations are:

$$\frac{U_i{}^{j+1}-U_i{}^j}{\Delta \tau }=\frac{U_{i-1}^j-2U_i^j+U_{i+1}^j}{{(\Delta \eta )}^2}-M{U}_i^j+Gr{\theta }_i^j\left[1+Nc{\theta }_i^j\right],$$

(23)

$$\frac{{\theta }_i{}^{j+1}-{\theta }_i{}^j}{\Delta \tau }=\frac{1}{Pr}\left[\frac{{\theta }_{i-1}^j-{\theta }_i^j+{\theta }_{i+1}^j}{{(\Delta \eta )}^2}\right]+Ec\left[\frac{U_{i+1}^j-U_{i-1}^j}{2\Delta \eta }\right]+MEc{\left(U_i^j\right)}^2.$$

(24)

Rearranging Equations (23) and (24), we have

$$U_i^{j+1}=\frac{\Delta \tau }{{(\Delta \eta )}^2}\left[U_{i-1}^j-2U_i^j+U_{i+1}^j\right]-\Delta \tau M{U}_i^j+\Delta \tau Gr{\theta }_i^j\left[1+Nc{\theta }_i^j\right]+U_i^j,$$

(25)

$$\begin{array}{l}{\theta }_i^{j+1}=\frac{\Delta \tau }{Pr{(\Delta \eta )}^2}\left[{\theta }_{i-1}^j-2{\theta }_i^j+{\theta }_{i+1}^j\right]+\frac{\Delta \tau }{4{(\Delta \eta )}^2}Ec{\left[U_{i+1}^j-U_{i-1}^j\right]}^2+\Delta \tau MEc{U}_i^{j2}\\ +{\theta }_i^j\end{array}$$

(26)

The discretized forms of the corresponding initial and boundary conditions are as follows:

$$\left.\begin{array}{l}U\left({\tau }_j,{\eta }_i\right)=0,\theta \left({\tau }_j,{\eta }_i\right)=0,\mathrm{for}\forall i\mathrm{and}j=1,\\ U\left({\tau }_j,{\eta }_i\right)=\cos \left({\tau }_j\right),\theta \left({\tau }_j,{\eta }_i\right)=1,\mathrm{for}\forall j>1\mathrm{and}i=1,\\ U\left({\tau }_j,{\eta }_i\right)=0,\theta \left({\tau }_j,{\eta }_i\right)=0,\mathrm{for}\forall j\mathrm{and}i=N,\end{array}\right\}.$$

(27)

4. Results and Discussion

For the numerical solution, the FTCS scheme is applied to dimensionless PDEs. Numerical solutions are obtained for distinct parameters, such as Eckert Number $\left(Ec\right)$, magnetic parameter $\left(M\right)$, Grashof number $\left(Gr\right)$, non-linear convection parameter $\left(Nc\right)$, Prandtl number $\left(\mathrm{Pr}\right)$, and dimensionless time $\left(\tau \right)$.

Table 1. Comparison of the numerical values of Cf in the absence of the magnetic field and energy equation.

τ	Cf
	Exact Solution	FDM
0.1	−0.6329813065	−0.6329813064
0.2	−0.5525312920	−0.5525312920
0.3	−0.4665605675	−0.4665605676
0.4	−0.3759281240	−0.3759281240
0.5	−0.2815395311	−0.2815395311
0.6	−0.1843378880	−0.1843378882
0.7	−0.08529440190	−0.08529440189
0.8	0.01460131775	0.01460131773
0.9	0.1143511457	0.1143511455
1.0	0.2129584152	0.2129584151

shows the comparison of the numerical and exact values of the skin friction coefficient in the absence of the magnetic field and without considering the heat transfer analysis.

Table 2. Comparison of the numerical values of $Cf$ and $Nu$ with the published literature for the special case, i.e., when $\mathrm{Pr}=7.0$, $Gr=0.5$ and $Gc=Ec=M=0.0$.

τ	Cf		Nu
	* GGDQM [37]	** FDM	GGDQM [37]	FDM
0.1	−1.77812702	−1.77812702	4.72034872	4.72034871
0.2	−1.21946930	−1.21946929	3.33779059	3.33779061
0.3	−0.93773608	−0.93773608	2.72529460	2.72529460
0.4	−0.73915669	−0.73915670	2.36017436	2.36017438

* Gear generalized differential quadrature method (GGDQM). ** Finite difference method (FDM).

and

Table 3. Comparison of the numerical values of $Cf$ and $Nu$ with the published literature for the special case, i.e., when $\mathrm{Pr}=1.0$, $Gr=0.5$ and $Gc=Ec=M=0.0$.

τ	Cf		Nu
	GGDQM [37]	FDM	GGDQM [37]	FDM
0.1	−1.67115676	−1.67115677	1.78412411	1.78412412
0.2	−1.06843318	−1.06843319	1.26156626	1.26156625
0.3	−0.75321389	−0.75321390	1.03006454	1.03006456
0.4	−0.52680054	−0.52680051	0.89206205	0.89206205

, respectively, show the comparison of the FDM and GGDQM for $\mathrm{Pr}=7.0$ and $\mathrm{Pr}=1.0$ in the absence of nonlinear convection, magnetic field, and viscous dissipation. All three tables validate the accuracy of our computational framework. The obtained numerical results are plotted against various parameters; in this section, their impacts on velocity and temperature are presented via graphical illustrations.

The Eckert number’s influence on temperature along with velocity can be observed, respectively, in Figure 2a,b. Fluid accelerates with the enhancing Eckert number. True to form, the thermal boundary layer shows increments with $Ec$. This is on the grounds that by increasing the Eckert number, the fluid contact between the contiguous layers increments and the friction between the layers of fluid increases, thus transforming kinetic into heat energy (in this manner, increasing the temperature of the fluid). Figure 3a,b shows the impact of the Grashof number $Gr$ on temperature and velocity profiles. Velocity as well as temperature increase with growing $Gr$. This is because as the Grashof number increases, the buoyancy force also increases, which accelerates the fluid, subsequently increasing its temperature. Figure 4a,b illustrate how the magnetic parameter $M$ influences temperature and velocity. Here, the temperature rises as the magnetic parameter increases whereas the velocity profile of fluid diminishes. From Figure 4b, it is presumed that the Lorentz force produced opposes the fluid flow. Because of the resulting deceleration, the impact of viscosity increases, which produces frictional heating, thus increasing the temperature. In Figure 5a,b, the impact of the non-linear convection parameter $Nc$ is shown on the temperature and velocity of the fluid. It is obvious (by graphical representation) that both the temperature and velocity profiles show increments with the increase in $Nc$. This is valid because by increasing $Nc$, the heat dissipation of the fluid surface enhances due to more diffusion, therefore showing increments in the temperature and velocity of the fluid. Figure 6a,b represents the impact of the Prandtl number $\mathrm{Pr}$ on temperature and velocity. Here, the temperature as well as the velocity decline with the increment in $\mathrm{Pr}$. By increasing the Prandtl number, there is a decrement in thermal conductivity, which slows down the heat transfer rate and, consequently, reduces the temperature. The impact of time $\tau$ on the temperature and velocity is outlined in Figure 7a,b. It was observed that there is an increase in the thermal boundary layer with the preceding time, thus resulting in an increased temperature.

5. Closing Remarks

This study revolves around a nonlinear convective, time-dependent, electrically conducting, and incompressible dissipative fluid flow that was influenced by the magnetic field. The fluid was present on an infinite vertical flat plate that was under oscillation. This was mathematically modeled in terms of non-linear PDEs that were modified into non-dimensional forms and later solved by the application of the FTCS scheme.

Influences of various fluid flow parameters were determined. The conclusions of this article are summarized as follows:

• The velocity field diminishes with the increase in the magnetic parameter and Prandtl number.
• Reverse impacts are observed with the non-convective flow parameter; the Eckert number and Grashof number show increments, i.e., when these parameters increase, the velocity also increases.
• With preceding time, as the values of Eckert number, magnetic parameter, nonlinear convective flow parameter, and Grashof number increase, the temperature profile shows increasing behavior.
• The temperature field decrements in value as the Prandtl number increases.