Hydrothermal and Entropy Investigation of Nanofluid Natural Convection in a Lid-Driven Cavity Concentric with an Elliptical Cavity with a Wavy Boundary Heated from Below

This work investigates mixed convection in a lid-driven cavity. This cavity is filled with nanofluid and subjected to a magnetic field. The concentric ovoid cavity orientation (γ), 0–90°, and undulation number (N), 1–4, are considered. The Richardson number (Ri) varies between 1 and 100. The nanofluid volume fraction (φ) ranges between 0 and 0.08%. The effect of the parameters on flow, thermal transport, and entropy generation is illustrated by the stream function, isotherms, and isentropic contours. Heat transfer is augmented and the Nusselt number rises with higher Ri, γ, N, and φ. The simulations show that the heat transfer is responsible for entropy generation, while frictional and magnetic effects are marginal.


Introduction
Combined convective flowing and temperature distribution into a wavy frame enclosure imposed by nanofluids have an immeasurable real-life enrollment in diverse industrial systems, engineering, and domestic processes such as heat exchangers, nuclear reactor technologies, solar collectors, electric equipment, refrigeration units, and others. Choi [1] reviewed nanofluid technology for current and future research. Individual sets of geometries, designs, and types of fluid, scrutinized in the recent year, incorporated wavy [2][3][4][5], square [6,7], triangular [8,9], rectangular [10,11], and hexagonal shapes [12], and so on. Mixed convection passing inside a wavy triangular box, loaded by nanofluid exerting viscosity criteria, was explored by Nasrin et al. [13]. Fares et al. [14] considered magneto-free convective inside a non-Darcy porous enclosure, using hybrid nanofluid enclosing an adiabatic rotating cylinder. Srinivas Acharya and Kumar [15] investigated mixed convection across an oblique wavy surface in a non-Darcy porous media saturated with nanofluids in the presence of radiation. Fares et al. [16] investigated the effect of a magnetic field

Physical Model
Depicted in Figure 1 is a lid-driven square cavity (LDSC) that is isothermal (T H ), heated from below, with the top surface maintained at a uniform cold (Tc) temperature. In contrast, the lateral surfaces and the concentric oval-shaped cavity are adiabatic. The LDSC contains a nanofluid Al 2 O 3 , dispersed uniformly. The wavy boundary is sketched according to Equation (1), where A = 0.04, and the undulation, N, varies from 1 to 4.

The Governing Equations
The governing equations (Equations (2)-(4)) are the continuity, momentum, and energy equations [50] for laminar mixed convection and are as follows:
The dimensionless boundary conditions regarding Equations (12)- (15) are: On the bottom wavy heated surface: On the top moving cold surface: On the left and the right surfaces: At the concentric surface, The local Nusselt number evaluated at the heated bottom surface is defined by: The average Nusselt number evaluated at the heated part is: The entropy generation relation is given by [29]: In dimensionless form, local entropy generation can be expressed as: where is the irreversibility distribution ratio and S GEN = S gen The terms of Equation (19) can be separated into the following form: where S θ , S ψ, and S B are the entropy generation due to heat transfer irreversibility (HTI), fluid friction irreversibility (FFI), and entropy generation due to magnetic field effect, respectively: It is appropriate to mention the Bejan number to determine the dominant, heat transfer, or fluid friction irreversibility. The Bejan number is defined as:

Numerical Method and Validation
The dimensionless controlling Equations (12)- (15) are augmented by the boundary conditions. The numerical solutions to Equations (18)- (22) are obtained using the Galerkin weighted residual finite element technique [53][54][55]. The issuing domain is discretized using non-uniform triangular components. The finite element equations are constructed using triangular elements with six nodes. The Galerkin weighted residual approach converts the non-linear partial differential equations to a system of integral equations. Each term is then solved using Gauss's quadrature method. The objective is to obtain a set of non-linear algebraic equations that satisfy the boundary conditions. Grid-independent solutions are discovered by experimenting with a few grids. To conduct all simulations, a grid of 695,244 elements was used. To guarantee the numerical approach converted approved code is accurate, the velocity profile within the cavity with obstacles is compared with Iwatsu et al.'s work [56]. In Figure 2, a perfect agreement between the two results is obtained.   [56] Reproduced with permission from [56]. Elsevier, 1993.

Results and Discussion
To understand the salient features of this problem, the fluid motion streamlines, the temperature field isotherm contours, and entropy distribution depicted by the isentropic contours are utilized. The controlling geometric parameters are the orientation of the concentric ovoid cavity and the undulation number on the heated boundary. Heating from below gives rise to a buoyancy force in conjunction with the sliding top lid, which assists the fluid motion. Here, mixed convection and free convection regimes are investigated.

Results and Discussion
To understand the salient features of this problem, the fluid motion streamlines, the temperature field isotherm contours, and entropy distribution depicted by the isentropic contours are utilized. The controlling geometric parameters are the orientation of the concentric ovoid cavity and the undulation number on the heated boundary. Heating from below gives rise to a buoyancy force in conjunction with the sliding top lid, which assists the fluid motion. Here, mixed convection and free convection regimes are investigated. The characterizing parameter is the Richardson number. The Hartmann number characterizes the intensity of the applied magnetic field. The thermo-physical controlling parameter is the nanoparticle volume fraction. The numerical experiments are conducted for Richardson number values (Ri: 1, 10, 50, 100), Hartman number values (Ha: 1, 25, 50, 100), orientation (γ: 0 • , 30 • , 50 • , 90 • ), undulation (N: 1, 2, 3, 4), and nanoparticle volume fraction (ϕ: 0, 0.02, 0.04, 0.08). Figure 3a shows the effect of the Richardson number.
To understand the salient features of this problem, the fluid motion streamlines, the temperature field isotherm contours, and entropy distribution depicted by the isentropic contours are utilized. The controlling geometric parameters are the orientation of the concentric ovoid cavity and the undulation number on the heated boundary. Heating from below gives rise to a buoyancy force in conjunction with the sliding top lid, which assists the fluid motion. Here, mixed convection and free convection regimes are investigated. The characterizing parameter is the Richardson number. The Hartmann number characterizes the intensity of the applied magnetic field. The thermo-physical controlling parameter is the nanoparticle volume fraction. The numerical experiments are conducted for Richardson number values (Ri: 1, 10, 50, 100), Hartman number values (Ha: 1, 25, 50, 100), orientation (γ: 0°, 30°, 50°, 90°), undulation (N: 1, 2, 3, 4), and nanoparticle volume fraction ( : 0, 0.02, 0.04, 0.08). Figure 3a shows   It is interesting to consider ∆ , the difference between the maximum and minimum value of the stream function, as Ri is varied. We note that ∆ increases monotonically with Ri. There are two vortices at low Ri: a local vortex near the top moving lid and a global vortex with its sudo-center the oval-shaped cavity. At higher Ri, the local vortex is stretched vertically, and, eventually, the local vortex is merged into the global vortex. The influence of natural convection driven by buoyancy forces dominates the inertial forces of forced convection. Considering Figure 3b, more significant distortion of the isotherms is observed at a higher Ri number, meaning greater homogenization of the temperature field.
Furthermore, at higher Ri, the thermal boundary layer becomes thinner. The isotherms become condensed near the heated wavy boundary, implying enhanced heat transfer dominated by natural convection in contrast to the sparse thermal lines at low Ri, indicating a thick thermal boundary layer and less efficient thermal transport dominated by mixed convection. The merging and increased strength of the re-circulating fluid are reflected in the increase in frictional irreversibility, as depicted in the entropy generation plots shown in Figure 3c, near the wavy boundary and the moving boundary. It is interesting to consider ∆ψ, as the difference between the maximum and minimum value of the stream function, as Ri is varied. We note that ∆ψ increases monotonically with Ri. There are two vortices at low Ri: a local vortex near the top moving lid and a global vortex with its sudo-center the oval-shaped cavity. At higher Ri, the local vortex is stretched vertically, and, eventually, the local vortex is merged into the global vortex. The influence of natural convection driven by buoyancy forces dominates the inertial forces of forced convection. Considering Figure 3b, more significant distortion of the isotherms is observed at a higher Ri number, meaning greater homogenization of the temperature field.
Furthermore, at higher Ri, the thermal boundary layer becomes thinner. The isotherms become condensed near the heated wavy boundary, implying enhanced heat transfer dominated by natural convection in contrast to the sparse thermal lines at low Ri, indicating a thick thermal boundary layer and less efficient thermal transport dominated by mixed  The tip of the major axis squeezes the effective area of the local vortex with an increasing ellipse major axes angle, shown in Figure 4a. The streamlining of the obstruction with the gravity axes invigorates the mixed convective heat transfer, as evident by the distortion in the isotherm plots, shown in Figure 4b, which leads to an increase in entropy generation, as shown in Figure 4c. At low Ri, a horizontally oriented cavity impedes the inertially driven flow from mixing. This is confirmed by Figure 5a which shows the average Nusselt number versus Richardson number at various angles. The tip of the major axis squeezes the effective area of the local vortex with an increasing ellipse major axes angle, shown in Figure 4a. The streamlining of the obstruction with the gravity axes invigorates the mixed convective heat transfer, as evident by the distortion in the isotherm plots, shown in Figure 4b, which leads to an increase in entropy generation, as shown in Figure 4c. At low Ri, a horizontally oriented cavity impedes the inertially driven flow from mixing. This is confirmed by Figure 5a which shows the average Nusselt number versus Richardson number at various angles. At low Ri number, we note a monotonic and uniform increase in the average dimensionless heat transfer coefficient associated with incrementing the angle of the ovoid cavity from 0° to 90°. However, a marginal increase in the heat transfer coefficient is found at higher Ri numbers. Bejan number which represents the ratio of heat transfer to fluid At low Ri number, we note a monotonic and uniform increase in the average dimensionless heat transfer coefficient associated with incrementing the angle of the ovoid cavity from 0 • to 90 • . However, a marginal increase in the heat transfer coefficient is found at higher Ri numbers. Bejan number which represents the ratio of heat transfer to fluid friction Nanomaterials 2022, 12, 1392 9 of 17 irreversibility is illustrated in Figure 5b. Generally, there is a small increase in entropy as the angle of the ovoid cavity is increased which can be attributed to the distortion of the local vortex and nudge in the ratio entropy generated due to frictional losses. In all the investigated cases the generated entropy is clearly dominated by heat transfer irreversibility and attained a value of [0.94, 0.97]. Furthermore, increasing the Ri results in increasing the total entropy however, dominated by heat transfer, the increase is about 2%. The entropy generated peaks out between Ri 50 and 100, as can be seen at higher Ri number a slight decrease in Bejan number is noted, which is indicative of an increase in frictional entropy generation relative to entropy generated by heat transfer.
Shown in Figure 6 is the effect of undulations, N, 1-4 at a fixed ovoid angle of 0 • and selected Ri of 1.
Nanomaterials 2022, 12, x FOR PEER REVIEW 10 of 18 friction irreversibility is illustrated in Figure 5b. Generally, there is a small increase in entropy as the angle of the ovoid cavity is increased which can be attributed to the distortion of the local vortex and nudge in the ratio entropy generated due to frictional losses. In all the investigated cases the generated entropy is clearly dominated by heat transfer irreversibility and attained a value of [0.94, 0.97]. Furthermore, increasing the Ri results in increasing the total entropy however, dominated by heat transfer, the increase is about 2%. The entropy generated peaks out between Ri 50 and 100, as can be seen at higher Ri number a slight decrease in Bejan number is noted, which is indicative of an increase in frictional entropy generation relative to entropy generated by heat transfer. Shown in Figure 6 is the effect of undulations, N, 1-4 at a fixed ovoid angle of 0° and selected Ri of 1.  It appears that the undulation squeezes the space available for the global vortex Figure 6a, which enhances the convective heat transfer as shown by the increased perturbation of the isotherm plots in Figure 6b, the latter leads to the generation of a greater amount of entropy and the thermally driven irreversibility as seen in Figure 6c. Figure 7a is the average heat transfer coefficient, which shows that increasing the undulation is directly proportional to increase Nusselt number. perturbation of the isotherm plots in Figure 6b, the latter leads to the generation of a greater amount of entropy and the thermally driven irreversibility as seen in Figure 6c. Shown in Figure 7a is the average heat transfer coefficient, which shows that increasing the undulation is directly proportional to increase Nusselt number.  At a low Ri number or in the mixed convection regime, the enhancement in heat transfer per undulation is nearly 9%, whilst in the free convection regime, at a higher Ri, it is approximately 10%. The entropy generation is dominated by heat transfer, as explained in Figure 7b. We also notice that the Bejan number peaks at between Ri 50 and 100. The effect of the applied magnetic field on the motion of the fluid and the associated convection is illustrated in Figure 8, having a Ri of 1 and ovoid cavity angle of 0 • . At a low Ri number or in the mixed convection regime, the enhancement in heat transfer per undulation is nearly 9%, whilst in the free convection regime, at a higher Ri, it is approximately 10%. The entropy generation is dominated by heat transfer, as explained in Figure 7b. We also notice that the Bejan number peaks at between Ri 50 and 100. The effect of the applied magnetic field on the motion of the fluid and the associated convection is illustrated in Figure 8 As the Hartman number is increased from 0 to 100, the Lorentz force suppresses the global cavity fluid motion and reduces the effect of the local cavity near the moving lid.

Shown in
The magnetic field reduces convection and the conduction mode dominates. This is approved by the temperature contour distortions that are smoothed out with the increasing Ha number; this results in a stratified temperature field and an aligned temperature gradient between the two isothermal surfaces, as shown in Figure 8b. Naturally, this led to a decrease in the entropy generation attributed to heat transfer, as seen in Figure 8c, As the Hartman number is increased from 0 to 100, the Lorentz force suppresses the global cavity fluid motion and reduces the effect of the local cavity near the moving lid.
The magnetic field reduces convection and the conduction mode dominates. This is approved by the temperature contour distortions that are smoothed out with the increasing Ha number; this results in a stratified temperature field and an aligned temperature gradient between the two isothermal surfaces, as shown in Figure 8b. Naturally, this led to a decrease in the entropy generation attributed to heat transfer, as seen in Figure 8c, and entropy generation increased due to the magnetic field effect. Figure 9a shows the Nusselt number as the magnetic field is activated. The suppression of the convective effect results in the diminution of the heat transfer process and, hence, an extreme reduction in the Nusselt number. and entropy generation increased due to the magnetic field effect. Figure 9a shows the Nusselt number as the magnetic field is activated. The suppression of the convective effect results in the diminution of the heat transfer process and, hence, an extreme reduction in the Nusselt number.  Figure 9b shows the Bejan number increase with the increasing Hartman number, which can be attributed to irreversibility associated with magnetic field effects.
Generally, introducing nanoparticles in the cavity fluid causes it to flow slower, resulting in a more substantial temperature gradient and an increase in the heat transfer rate. Figure 10a shows the Nusselt number versus Ri for an increasing nanoparticles volume fraction. Increasing the Hartman number from 25 to 100 has a marginal effect on the further decrease in the Nusselt number. Figure 9b shows the Bejan number increase with the increasing Hartman number, which can be attributed to irreversibility associated with magnetic field effects.
Generally, introducing nanoparticles in the cavity fluid causes it to flow slower, resulting in a more substantial temperature gradient and an increase in the heat transfer rate. Figure 10a shows the Nusselt number versus Ri for an increasing nanoparticles volume fraction. Doubling the nanoparticle fraction increases the Nusselt number by approximately 10%. Figure 10b shows that increasing the volume fraction also increases the entropy generation, which can be attributed to an increase in the apparent viscosity of the fluid, the agent for fluid friction, and the source of irreversibility.

Conclusions
Numerical simulations of mixed convection within a concentric cavity with wavy boundaries are carried out. The inner cavity is an adiabatic oval, and the outer is a liddriven square cavity. The space in the cavity is filled with a nanofluid. The convective heat transport is set up by the moving lid and heating from below. A uniform magnetic field is imposed on the working fluid. The dimensionless numbers of interest are Richardson, Hartman, and Bejan. The undulation of the heated wall and orientation of the inner cavity is varied. The results are displayed in in-stream function, isothermal, and isentropic contour plots.
Additionally, the Nusselt number and Bejan number are calculated. It is determined that a single incrementation of undulation increases the Nusselt number by an average of 9.5% within the investigated range (1 N 4 . Furthermore, doubling the Doubling the nanoparticle fraction increases the Nusselt number by approximately 10%. Figure 10b shows that increasing the volume fraction also increases the entropy generation, which can be attributed to an increase in the apparent viscosity of the fluid, the agent for fluid friction, and the source of irreversibility.

Conclusions
Numerical simulations of mixed convection within a concentric cavity with wavy boundaries are carried out. The inner cavity is an adiabatic oval, and the outer is a liddriven square cavity. The space in the cavity is filled with a nanofluid. The convective heat transport is set up by the moving lid and heating from below. A uniform magnetic field is imposed on the working fluid. The dimensionless numbers of interest are Richardson, Hartman, and Bejan. The undulation of the heated wall and orientation of the inner cavity is varied. The results are displayed in in-stream function, isothermal, and isentropic contour plots.
Additionally, the Nusselt number and Bejan number are calculated. It is determined that a single incrementation of undulation increases the Nusselt number by an average of 9.5% within the investigated range (1 ≤ N ≤ 4). Furthermore, doubling the nanoparticle volume fraction increased the Nusselt number by nearly 8%. Meanwhile, incrementing the major ovoid axes from the horizontal plane by 10 • raises the Nusselt number by 2.8% and 0.81% in the mixed convection regime (low Ri) and free convection regime (high Ri), respectively. The Nusselt number increases with increasing the Richardson number for any of the factors mentioned above, as the natural convection heat transfer mode dominates the mixed convection mode. An increase in entropy generation is associated with efficient thermal transport due to heat transfer irreversibility. Imposing a magnetic field shuts down the convection mode, rendering conduction heat transfer the sole mechanism of thermal transport.

Conflicts of Interest:
The authors declare no conflict of interest.