Numerical Computations of Entropy Generation and MHD Ferrofluid Filled in a Closed Wavy Configuration: Finite Element Based Study

In this article, the thermal flow effects of inclined magnetohydrodynamics ferrofluid filled in a wavy cavity are studied by adopting the finite element method (FEM). The non-dimensional governing equations and model for different parameters are evaluated. The system of non-linear algebraic equations is computed by adopting the Newton method. A space involving quadratic polynomials (${\mathrm{\mathbb{P}}}_2$) has been selected to compute for the velocity profile, while the pressure and temperature profiles are approximated by linear (${\mathrm{\mathbb{P}}}_1$) finite element space of functions. The discrete systems of non-linear algebraic equations are computed by utilizing the Newton method. The vertical walls are considered cold, whereas the bottom wavy surface is considered hot and the top wavy surface is insulated. The effect of the pertinent parameters, like $Ra={10}^5$, volume fraction $\left(0.00\le \varphi \le 0.06\right),$ inclination angle $\left(0^{\circ }\le \gamma \le {90}^{\circ }\right)$, and amplitude of the wavy surface $(0.02\le AR\le 0.08)$ is investigated. Computational results are addressed as streamlines out, isotherms, and proper graphs for substantial amounts of interest. Increasing Hartmann number ($Ha)$ leads to an increase in Bejan number $\left(Be\right)$, while opposite behavior can be observed in the case of viscous, magnetic, and thermal irreversibility, that is, curves are decreased by increasing $Ha$. Under the influence of an inclined magnetic field, the mathematical structuring of the problem is manifested by continuity, momentum, and energy equations. These equations are solved by using the finite element method computation. The graphs of velocity and isotherm are compared to the relevant parameters. To predict the flow characteristics at different locations, cross-sectional lines representing the velocity field in the horizontal and vertical directions are also drawn. Magnetization’s impact on flow control, heat transfer, and various irreversibilities are also discussed. The $N{u}_{avg}$ progressively declined with the enhancement in the amplitude of the wavy surface $AR$.

Introduction

Within the computational fluid dynamics field, lid-driven cavity flow is widely researched. Since the geometry of cavity flow is simple, the algorithm can be coded easily and boundary conditions can be applied. Despite the way that the issue shows up simple in many regards, the flow in a cavity holds all of the fluid mechanics, with counter-turning vortices showing up at the cavity’s corners [1]. Cavities with wavy walls can be found in many engineering applications, such as heat exchangers, solar panels, and buildings. A few specialized applications, like cooling of electronic parts and fixed electrical or electronic boxes, cooling or heating rooms, heat exchanger plans, and sunlight-based energy authority plans, are keen on regular convection heat move from wavy surfaces. When considering all of these factors, natural convection heat transmission becomes quite important [2]. K. S. Mushate [3] has numerically examined natural convection thermal flow within a square cavity with two wavy sides. [4] considered the complex wavy walls and offered important knowledge into imminent answers for further developing convection heat move execution inside encased depressions with wavy wall (complex) surfaces. R. C. Mohapatra [5] tackled the natural convection issue in a cavity with three flat walls and a right upward wall with one undulation and three undulations. H. R. Ashorynejad [6] utilized the lattice Boltzmann method (LBM) to investigate the thermal flow of uniformly magnetized hybrid nanofluid within a wavy cavity. K. Javaherdeh [7] conducted a numerical study to research the thermal flow properties in a wavy enclosure inside the nanomaterials.

[8] conducted research and utilized 3D numerical simulation to determine the impact of a magnetized ferrofluid and thermal flow in a wavy channel. [9] used a unique approach to explore the behavior of ferrofluid inside a permeable district when presented to an electric field. [10] employed computational study of thermal flow peculiarities in a kerosene-based ferrofluid. [11] considered a numerical study of ferrofluid-filled lid-driven cavity for various heater configurations numerically.

Within the sight of an isothermally heated corner, a numerical investigation of MHD natural convection in a wavy open permeable tall cavity loaded up with a Cu–water nanofluid was researched by [12]. [13] used local thermal non-equilibrium conditions to investigate magnetize hybrid nanomaterial in a wavy porous media. [14] implemented a finite volume approach to analyze the magnetohydrodynamic non-Newtonian ferrofluid to better understand the properties of heat transfer in a duct. [15] studied the entropy-based characteristics of thermal flow rate in a cavity having a rotating obstacle. Furthermore, in everyday life, thermal conductivity has a basic impact on re-scaling the thermal flow characteristics of fluids. In this regard, [16–23] mention some new modifications, addressing major physical aspects and computational techniques. Selimefendigil and Öztop [24] have investigated the characteristics of thermal enhancement through MHD nanofluid in a cavity. Furthermore, they have explored the magnetic field is imposed and its strength is steadily raised, it reduces heat transmission and convective fluid motion. [25] analyzed the effects of heat transfer and a phase change material (PCM) inside the cavity in the presence of nanofluid via finite element computation. In addition, they have considered a lower PCM height and a better thermal conductivity ratio result in enhanced local and average temperature distribution. [26] investigated the influence of entropy generation and MHD nanofluid inside the permeable cavity. The fluctuation of the local Nusselt number for the corrugated wall is affected by the triangle wave’s imposed frequency. The corrugation frequency near the heated wall has a big impact on velocity and temperature implementations.

The objective of the present article is to elucidate the entropy generation process by including the magnetic field and natural convection process. A wide range of irreversibility types have been discussed including the irreversibilities of heat transfer, viscous effects, and magnetic field.

Problem Description

The wavy surface is considered here in this problem and the physical diagram of the problem is portrayed in Figure 1. In a cavity, $T_h$ represents temperature of the hot wall, while $T_c$ denotes the temperature of the cold wall, and the top wavy wall is insulated.

FIGURE 1

FIGURE 1. Schematic diagram of the problem.

The non-dimensional mathematical description of ferrofluid flow governed by the conservation law of mass, momentum, and energy are given as [14]

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,(1)$

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\frac{\partial p}{\partial x}= \frac{{\mu }_{ff}}{{\rho }_{ff }}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+\frac{{\rho }_f}{{\rho }_{ff }}\frac{{\sigma }_{ff}}{{\sigma }_f}P{r}_{f }H{a}^2\left(vsin\gamma cos\gamma -usi{n}^2\gamma \right) ,(2)$

$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+\frac{\partial p}{\partial y}= \frac{{\mu }_{ff}}{{\rho }_{ff }}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+\frac{{\left(\rho \beta \right)}_{ff}}{{\rho }_{ff {\beta }_f}}R{\alpha }_{f}P{r}_{f }T+\frac{{\rho }_f}{{\rho }_{ff }}\frac{{\sigma }_{ff}}{{\sigma }_f} P{r}_{f }H{a}^2\left(usin\gamma cos\gamma -vco{s}^2\gamma \right),(3)$

$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}= \frac{{\alpha }_{ff}}{{\alpha }_f}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right),(4)$

where symbols have been established in the nomenclature portion. The boundary conditions are given as follows:

i) $T_h$ = 1, at the lower wavy surface.

ii) $(u,v)$ = 0, at all sides of the cavity.

iii) $\frac{\partial T}{\partial n}$ = 0, at the upper wavy surface.

iv) $T_c=0$, at the vertical walls.

Thermophysical Features of Nanofluid

The effective equations for the thermophysical characteristics of the nanoparticle employed in this investigation. Table 1 also takes into account the ferrofluid’s thermophysical characteristics [14].

TABLE 1

TABLE 1. Description of the base fluid and nanoparticles properties.

Entropy Generation

The measurement of local entropy production derived by adding conjugated fluxes and produced forces is referred to as irreversibility analysis. The non-dimensional local entropy production in a convective process and under the effect of a magnetic field is given by [14]:

$Si=\frac{k_{ff}}{k_f}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]+\psi \left[\begin{array}{c}2{\left(\frac{\partial u}{\partial x}\right)}^2\\ +\end{array}+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right]+\frac{{\sigma }_{ff}}{{\sigma }_f} H{a}^2\psi {\left(usin\gamma -vcos\gamma \right)}^2 ,(5)$

$Si=S_{Ther}+S_{Vis}+S_{Mag},(6)$

where

$S_{Ther}=\frac{k_{ff}}{k_f}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right],(7)$

$S_{Vis}=\psi \left[\begin{array}{c}2{\left(\frac{\partial u}{\partial x}\right)}^2\\ +\end{array}+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right],(8)$

$S_{Mag}=\frac{{\sigma }_{ff}}{{\sigma }_f} H{a}^2\psi {\left(usin\gamma -vcos\gamma \right)}^2.(9)$

Here, $Si,$ $S_{Ther},S_{Vis}$, and $S_{Mag}$ represent the local entropy generation, entropy generation due to heat transfer $(S_{Ther}$), irreversibility of viscous $\left(S_{Vis}\right)$, and the irreversibility of magnetic $\left(S_{Mag}\right)$ field. In Eq. 5, the formula for distribution irreversibility ratio ($\psi$) is given in Eq. 10.

$\psi =\frac{{\mu }_{ff}}{k_f}{T}_o{\left(\frac{{\alpha }_f}{H\mathrm{\Delta }T}\right)}^2.(10)$

Here, $T_o$ represents the average temperature, that is, $T_o=\frac{T_h+T_c}{2}$ .

In addition, the non-dimensional Bejan number $\left(Be\right)$ is described as the ration of thermal entropy $S_{Ther}$ and local entropy $Si$, which is found as $Be=\frac{S_{Ther}}{Si}$, respectively.

Numerical Procedure

Because of the non-linearity and coupling of the governing Eqs 1–4, the continuity, momentum, and energy equations cannot be solved with analytical solution techniques, so we executed the solution using numerical approach, namely, FEM. The computational grid on a coarse level is shown in Figure 2 where a hybrid meshing is performed to capture the flow dynamics accurately near the boundaries. A space involving the quadratic polynomials (${\mathrm{\mathbb{P}}}_2$) has been selected to compute for the velocity profile, while the pressure and temperature profiles are approximated by linear (${\mathrm{\mathbb{P}}}_1$) finite element space of functions. The discrete systems of non-linear algebraic equations are computed by utilizing the Newton method and the linearized inner system with a direct solver PARDISO. There are many benefits of using the PARDISO solver (see [27–34] for further information). The PARDISO solver uses LU factorization and reduces the number of cycles required for the desired level of convergence.

FIGURE 2

FIGURE 2. Computational grid of the domain.

For various grid levels, the kinetic energy is determined. As the refinement level is increased, the percentage of an error lowers. The eighth refinement level is employed during findings because the difference between levels 8 and 9 is the smallest (Table 2).

TABLE 2

TABLE 2. Grid convergence test for kinetic energy.

Results and Discussion

In this section, we present the flow features pertaining to different ranges of the involved physical parameters. Figure 3 shows the impact of $Ha$ on streamline (left) and isotherm contour (right) with Rayleigh number $Ra=1E5$ and $\varphi$ = 0.04 in a wavy cavity. A couple of counter-rotating cells is shaped in the left and right portions of the enclosure. This fig shows a configuration of flow defined by two thermally convective cells that are symmetrical around the mid-plane $x =0.5$. Compared to the left cell, the right one is squeezed and smaller as a result of the present undulation; however, it has considerably less impact on the left cell. The isotherms for higher $Ra=1E5$ concentrate near the bottom wall because the advection mode of the heat transfer dominates over conduction. At the point when $Ha$ is expanded, the compression of isotherms relaxes, this, in turn, reduces the amount of heat transmitted through hot vertical walls.

FIGURE 3

FIGURE 3. Impacts of $Ha$ on streamline (left) and isotherm contour (right) with $Ra=1E5$ and.$\varphi =\mathrm{0.04.}$

Figure 4 shows the variation of the $S_{Ther}$ production with $Ha$ for three different parameters, that is, volume fraction ( $\varphi$ ), angle of inclination ($\gamma ),$ and amplitude of wavy surface (AR). It is noteworthy that thermal entropy generation in case of $\varphi$ and $\gamma$ has the highest numerical value when $Ha$ is low and afterward diminishes as $Ha$ is expanded and the curve is symmetrical in case of $AR$.

FIGURE 4

FIGURE 4. Impacts of $Ha$ on $S_{Ther}$ for various parameters.

Figure 5 depicted viscous irreversibility, which is related to $Ha$ for different parameters. When we increase $Ha$, we can easily see that the generation of $S_{Vis}$ decreases. This decrease is caused by a decrease in natural convection as a result of the Lorentz force’s latent impact. It can also be examined that in the case of $\gamma$ and $AR$ the curves are decreasing until $Ha=40$ and after that the values of $\gamma$ and $AR$ are equal for a large value of $Ha$.

FIGURE 5

FIGURE 5. Impacts of $Ha$ on $S_{Vis}$ for various parameters.

While in case of inclination angle ($0\le \gamma \le {90}^{\circ }$) $Be$ attains the maximum value at $\gamma$ = $0^{\circ }$ and minimum value at $\gamma ={90}^{\circ }$. Figure 6 shows the impact on magnetic irreversibility against $Ha$ for different parameters. It is analyzed that in case $\varphi$ and $\gamma$ all curves at first increase then steadily decline as $Ha$ increases due to the Lorentz force. The curves show a similar trend in the case of AR. Figure 7 shows the impact on $Be$ against $Ha$ for different parameters. It can be noted that in the case of volume fraction $\left(0\le \varphi \le 0.06\right)$ and amplitude of the wavy surface $\left(0.02\le AR\le 0.08\right)$ the curves are initially decreasing up to certain values of $Ha=10$ and after that the curves increase by increasing $Ha$. Figure 8 illustrates the variation of $N{u}_{avg}$ with $Ha$ for different parameters. It can be noted that the curves continuously decrease as the value of $Ha$ increase. The curves are symmetrical in the case of $AR.$ Figure 9 shows the impact of velocity components on Ha and $\varphi$. The velocity profile has the peak values at the middle and least values near the wavy surface.

FIGURE 6

FIGURE 6. Impacts on $S_{Mag}$ against $Ha$ for different parameters.

FIGURE 7

FIGURE 7. Impacts on $Be$ against $Ha$ for different parameters.

FIGURE 8

FIGURE 8. Impacts on $N{u}_{avg}$ against $Ha$ for different parameters.

FIGURE 9

FIGURE 9. Impacts of velocity components on Hartman number (Ha) and volume fraction.$\varphi .$

Conclusion

In this study, the thermal flow effects of inclined MHD ferrofluid filled in a wavy cavity are studied by adopting the FEM. The study has been considered by varying parameters, like $Ra=1E5$, volume fraction $\left(0.00\le \varphi \le 0.06\right),$ inclination angle $\left(0^{\circ }\le \gamma \le {90}^{\circ }\right)$, and amplitude of the wavy surface $(0.02\le AR\le 0.08)$ by maintaining the Prandtl number ($Pr= 6.2$) fixed. The results have been illustrated using graphs to show the physical repercussions. Magnetic, viscous, and thermal entropies are the three types of entropy variations that are evaluated. The primary goal of our current effort is to reveal variation in entropy generation in enclosures and to make predictions about factors that influence entropy measurements. The main findings of the aforementioned study are summarized as follow:

• $N{u}_{avg}$ is decreased with an increase in $Ha$, while it is pronounced with an increase in $\gamma$.

• $Be$ displayed an increasing trend with $Ha$.

• Increasing $Ha$ leads to a decrease in viscous, thermal, and magnetic irreversibilities due to the Lorentz force.

• $N{u}_{avg}$ progressively declined with the enhancement in the amplitude of the wavy surface $AR$.

• A forward flow is observed via the horizontal component of the velocity in the upper half of the cavity while a backward flow in the lower half, with a strength directly proportional to the volume fraction $\varphi$.

• The vertical component of the velocity is enhanced in the central region of the cavity with an increase in the volume fraction $\varphi$

• For all cases of $\gamma$ and $R$, viscous irreversibility is dominated by the thermal irreversibility at low $Ha$.The influence of corrugation amplitude, various corrugation types, different nanofluid types, and transient flow effects could all be addressed as extensions of this work.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

YK has carried out modeling and computing data. RM has supervised us. AM and NR have written a complete manuscript. AA and NF have performed revisions and re-wrote the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research, University of Hafr Al Batin for funding this work through the research group project no. (0033-1443-S).

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Nomanclature

Abbreviations

$C_p$ Specific heat

$Ha$ Hartmann number

$B_o$ Magnetic field

$H$ Cavity height

$Pr$ Prandtl number

$Ra$ Rayleigh number

$p$ Pressure

$S_{Ther}$ Thermal entropy

$S_{Mag}$ Magnetic entropy

$S_{Vis}$ Viscous entropy

$T$ Temperature

$u,v$ Non-dimensional velocities

$x,y$ Non-dimensional coordinates

$\varphi$ Volume fraction

$T_h$ Temperature of the hot wall

$T_c$ Temperature of the cold wall

$\rho$ Density

$\text{Δ}T$ Temperature gradient

$\mu$ Dynamic viscosity

$Be$ Bejan number

$\alpha$ Thermal diffusivity

$\beta$ Expansion coefficient

$\psi$ Distribution irreversibility ratio

$k$ Thermal conductivity

$AR$ Amplitude of wavy surface

$N{u}_{avg}$ Average Nusselt number