Numerical Simulations of Hydromagnetic Mixed Convection Flow of Nanofluids inside a Triangular Cavity on the Basis of a Two-Component Nonhomogeneous Mathematical Model

Nanofluids have enjoyed a widespread use in many technological applications due to their peculiar properties. Numerical simulations are presented about the unsteady behavior of mixed convection of Fe3O4-water, Fe3O4kerosene, Fe3O4-ethylene glycol, and Fe3O4-engine oil nanofluids inside a lid-driven triangular cavity. In particular, a two-component non-homogeneous nanofluid model is used. The bottom wall of the enclosure is insulated, whereas the inclined wall is kept a constant (cold) temperature and various temperature laws are assumed for the vertical wall, namely: 1⁄4 1(Case 1), 1⁄4 Yð1 YÞ(Case 2), and 1⁄4 sinð2 YÞ(Case 3). A tilted magnetic field of uniform strength is also present in the fluid domain. From a numerical point of view, the problem is addressed using the Galerkin weighted residual finite element method. The role played by different parameters is assessed, discussed critically and interpreted from a physical standpoint. We find that a higher aspect ratio can produce an increase in the average Nusselt number. Moreover, the Fe3O4-EO and Fe3O4-H2O nanofluids provide the highest and smallest rate of heat transfer, respectively, for all the considered (three variants of) thermal boundary conditions.


Introduction
The overheating limits the lifespan of the usage of electronic pieces of equipment (for example, computer processor) while operating. It is a big challenge for the industries which produce such sophisticated types of equipment. In a recent study, Bayomy et al. [1] reported that the efficiency rate of electronic devices decreases exponentially due to heat generation within them. The traditional fluids (water, mineral oils, and ethylene glycol) most of the time used for industrial cooling applications limit their use as efficient heat transfer agent. For the growing need in modern technology (chemical production, power station, a computer processor, and micro-electronics), researchers developed nanofluid [2], which efficiently transmit heat. Nanofluid exhibits higher thermal conductivity hence enhanced heat transfer compared to the conventional fluids [3][4][5][6][7][8][9][10][11][12][13] even in the presence of a small amount (1%-5% volume fraction) of nanoparticles.
The heat transfer enhancement inside cavities has become a paramount issue in the industrial and energy sectors. Many researchers studied nanofluids experimentally, analytically as well as numerically for heat enhancement in cavities. In this respect, Khanafer et al. [14] studied the heat transfer enhancement in a differentially heated square cavity. They found that the suspended nanoparticles considerably increase the heat transfer rate. Oztop et al. [15] conducted a numerical study considering the natural convection flow inside the partially heated rectangular enclosure filled with nanofluids. They found that the mean Nusselt number increases with the increase of the nanoparticles volume fraction. They further reported that the low aspect ratio of the geometry significantly enhances the heat transfer rate in nanofluids compared to the corresponding heat transfer for a high aspect ratio.
Magnetohydrodynamics (MHD) convective flow has widespread applications in science and engineering such as extraction of geothermal energy, oil recovery from the petroleum reservoirs, thermal insulation, cooling of nuclear reactors, crystal growth, and plasma confinement [16][17][18]. In light of the various applications of MHD and nanofluids, Al Kalbani et al. [19,20] investigated the buoyancy induced heat transfer flow inside a tilted square cavity filled with nanofluids in the presence of an oriented magnetic field. Their results confirm that the nanoparticle volume fraction, shape, and size significantly intensified the heat transfer rate inside a crater. The applied magnetic field and its direction also played a vital role in heat enhancement. Al Balushi et al. [21,22] further investigated the free convection heat transfer flow of nanofluids inside square cavities utilizing nanofluids under the action of an applied inclined magnetic field to the flow domain. They used a nonhomogeneous dynamic model for nanofluid modeling. They found that heat enhancement in nanofluids depends on the nanoparticle loading, magnetic field's direction and strength, and the location of the heater that supplies heat to the flow field.
The thermal discharge in lid-driven enclosures has direct applications in many engineering fields such as in rheology for lubrication mechanisms, cooling of electronic devices, constructing buildings roofs and attics, processing food, and cooling nuclear reactors (see [23]). Flack et al. [24,25] studied experimentally as well as numerically the convective heat transfer in triangular enclosures. Later on, many researchers conducted research and reported results on triangular-cavities [26][27][28][29][30][31]. All of these studies involved heat transfer in regular fluids. Due to the growing need for nanofluid research in triangular cavities, Ghasemi et al. [32] studied numerically; the steady natural convection flow of CuO-water nanofluid inside a fixed-walls right triangular enclosure. They reported that the Brownian motion of nanoparticles takes part in enhancing the thermal performance of nanofluids in a cavity. Ghasemi et al. [33] further studied steady mixed convection in a lid-driven triangular enclosure filled with Al 2 O 3 -water nanofluid. They confirmed that enhancement in heat transfer within the cavity is due to the addition of nanoparticles, and it depends on the direction of the sliding wall motion. Rahman et al. [34] conducted a numerical study on hydromagnetic free convection flow of nanofluids inside an isosceles-triangular cavity. In their simulation, they used the two-component nonhomogeneous mathematical model and different thermal conditions. The results show that the variable thermal boundary conditions have significant effects on the flow and thermal fields. Rahman [35] studied the hydromagnetic natural convection flow and heat transfer within an equilateral triangular enclosure. In his work, he used water-based as well as kerosene-based ferrofluids in the presence of a sloping magnetic field. The results indicate that increased magnetic field strength diminishes the heat transfer rate, whereas it enhances with the increment of the magnetic field inclination angle. Rahman [36] further studied steady heat transfer in Fe 3 O 4 -water nanofluid inside a triangular cavity with fixed walls under a sloping magnetic field. They conclude that a higher degree of heat transfer is accomplished by reducing the dimension of nanoparticles and increasing the strength of the buoyancy force. Azam et al. [37][38][39][40][41] published a series of papers on unsteady heat and mass transfer flow of nanofluids in different geometries with the various flow and thermal conditions proposed by the Buongiorno mathematical model. In a recent study, Uddin et al. [42] explored heat transportation in copper oxide-water nanofluid inside different triangular cavities. Their results show that heat enhancement in nanofluids strongly depends on the shape of the triangular shape cavity and the applied buoyancy force.
Despite significant research studies on various cavities reported in the literature, there is a substantial lack of information regarding the problem of time-dependent hydromagnetic fluid flow and heat transfer enhancement in the lid-driven right triangular-cavity filled with nanofluids. Therefore, the present paper aims to investigate numerically unsteady mixed convection flow and heat transfer in a lid-driven righttriangular cavity filled with different types of nanofluids in the presence of an oriented magnetic field varying aspect ratio of the enclosure taking into account the Buongiorno mathematical model. We used the Galerkin weighted residual-based finite element method for numerical simulation. Finally, we depicted the mean rate of heat transfer in terms of Nusselt number in varying different model parameters. The organization of the remainder of the paper is as follows: In Section 2, we formulate the problem physically as well as mathematically. Section 3 explains the method of solution in detail. The numerical outcomes we discuss from physical and engineering viewpoints in Section 4. In the end, in Section 5, we conclude our study.

Physical Modeling
We consider an unsteady, laminar, incompressible two-dimensional mixed convection flow inside a right-angle triangular cavity that is filled with Fe 3 O 4 -water nanofluid as shown in Fig. 1, where x and y are the Cartesian coordinates. Here, L is the bottom wall length, and H is the height of the vertical wall. We assumed that the vertical wall temperature is T h while the inclined wall (hypotenuse) is T c (where T h > T c ). The bottom-wall is insulated; thus, no heat can escape along the transverse direction of it. Initially, we considered that nanofluid concentration is C C , but for t > 0, it is assumed as C h in the entire domain so that C h > C C . The vertical wall is allowed to move with constant speed V 0 in its plane while the remaining walls have no speed. Here the gravity acts in the vertical direction, along the y-axis. We included the thermophoresis and Brownian diffusion effects in the mathematical model in the absence of any chemical reaction and thermal radiation. The base fluid and the nanoparticles are in thermal equilibrium, and hence no slip occurs between them. Surfactant or surface charge technology disperses the nanoparticles within the nanofluid. The Boussinesq approximation tackled the density variation in the buoyancy force. The cavity is permeated by a uniform magnetic field B ¼ B x i þ B y j of constant magnitude , where i and j are the unit vectors along the coordinate axes. Also, the direction of the magnetic field makes an angle c with the positive x-axis. We may use this type of cavity filled with nanofluid to model a solar thermal collector.

Mathematical Modeling
Within the framework of the above-noted assumptions, the governing conservation equations for this model are expressed in dimensional form as follows [9,[34][35][36]: where r 2 ¼ @ 2 @x 2 þ @ 2 @y 2 and the descriptions of the physical variables are mentioned in the nomenclature.

Initial and Boundary Conditions
The appropriate initial and boundary conditions for the above-stated model are as follows: 1) For t 0: 2) For t > 0: ðaÞ On the vertical wall ðx ¼ 0; 0 y HÞ: ðbÞ On the bottom wall ðo x L; y ¼ 0Þ:

Introduction of Non-Dimensional Variables
The governing differential Eqs. (1)-(5) representing conservation laws are rarely solved using dimensional variables. The common practice is to write these dimensional equations in a non-dimensional form using dimensionless quantities obtained through proper characteristics scales. Writing the conservation equations in non-dimensional forms results in dimensionless numbers that are very useful for performing parametric studies of engineering problems. Again, the use of non-dimensional variables has several advantages. It allows reducing the number of appropriate parameters for the problem considered, revealing the relative magnitude of the various terms in the conservation equation that are less important. This process simplifies the equation to be solved and leaves only the terms of a similar order of magnitude, which results in better numerical accuracy. Besides, the generated solution will apply to all dynamically similar-problems. A dimensional variable is transformed into a non-dimensional one by dividing the variable by a quantity (composed of one or more physical properties) having the same dimension as the original variable. Thus the non-dimensional forms of the governing conservation Eqs.
(1)-(5) together with the initial and boundary conditions (6)-(9) are obtained by employing the following dimensionless parameters: Substituting (10) into (1)-(5), we obtain the dimensionless equations as follows: The non-dimensional boundary conditions become 1. For s 0: 2. For s > 0: ðaÞ On the vertical wall ðX ¼ 0; 0 Y ARÞ: ðbÞ On the bottom wall ð0 X 1; Y ¼ 0Þ: The parameters appeared in (11)- (19) are defined by is the Lewis number, The dimensionless Eqs. (11)-(15) determine the physical parameters that affect the solutions. The role of these parameters on the flow and thermal fields are discussed in the results and discussion section.

Average Nusselt Number
The significant physical quantity in this model is the calculation of the average Nusselt number Nu ave along the left heated wall. The Nusselt number Nu is the ratio of convective to conductive heat transfer across the boundary, and the local Nusselt number is defined by where DT ¼ T h À T c , H is the height of the triangle (the vertical heated wall), j f is the thermal conductivity of the base fluid. The convective heat transfer coefficient of the nanofluid flow h is defined by Using the dimensionless variables defined in Eq. (10), the heat transfer coefficient of nanofluid at the left heated wall turns into Hence, the local Nusselt number for nanofluid at the left heated wall can be expressed as The average Nusselt number is expressed as follows: 3 Numerical Procedure We applied the Galerkin weighted residual-based finite element method (FEM) to solve the governing dimensionless Eqs. (11)- (15) and boundary conditions (17)- (19). A finite element method is a numerical tool that approximates the solution of boundary value problems of partial differential equations. The finite element method exhibits high accuracy of calculation and easily handles complex geometries in engineering problems. In FEM, we construct approximation functions using the weighted-integral technique to find a solution of differential equations. We accomplished this by dividing the whole domain into a set of small sub-domains called finite elements. These elements can be of different types. In 2D problems, we usually use either triangular or quadrilateral shape elements. Besides, in 3D, the most commonly used elements' shape is tetrahedral or hexahedral. Here, we used six node triangular shape elements for developing the finite element equations. All six nodes are connected with velocities, temperature, and concentration fields, while only the corner nodes are associated with pressure. In the finite element method, the approximate solutions are expressed in terms of the shape (or interpolation) functions, which can be linear or quadratic depending on the number of nodes per element. Also, in 2D problems, the x, y-coordinates (global coordinate) are mapped into n, g coordinates (or local coordinates), and the shape functions are defined as functions of n and g. Such local coordinates n; g ð Þ are useful in the numerical evaluation of the integration. Now in terms of local coordinates, the quadratic shape functions for the velocities, temperature, and concentration are as follows: where Q i ðn; gÞ ¼ 1 at node i 0 at every other node & and X 6 i¼1 Q i ðn; gÞ ¼ 1 for all n; η in À 1 n 1 and À 1 η 1 Also, the linear shape functions for the pressure are as follows: with the property and X 3 i¼1 L i ðn; gÞ ¼ 1 for all n; g in À 1 n 1 and À 1 g 1 Again, for the triangular shape element, the coordinates x, y can be represented in terms of nodal coordinates using the same shape functions and this is known as isoparametric representation. Thus, for isoparametric representation, the transformation between x; y ð Þ and n; g ð Þ is accomplished by a coordinate transformation of the form In the 2D problem discussed here, each node is permitted to displace along with the two directions, x and y: Thus, each node has two degrees of freedom. As a result, the number of unknown variables for velocities, temperature, concentration, and pressure is 27 per element, and hence there are 27 degrees of freedom. Thus, in terms of the above-defined shape functions, the approximate solutions of U ; V ; h; f and P can be expressed as follows: where U i ; V i ; h i ; f i and P i are the corresponding nodal values of the unknown functions.
In the Galerkin weighted residual-based finite element method, the weight functions that we choose are the same as the shape functions that have been used in the approximate solutions (32). Thus employing the Galerkin weighted residual approach on Eqs. (11)-(15) and also using the Gauss's divergence theorem on the second derivative terms that contain in Eqs. (12)-(15), we get finally, the following finite element equations: Here, e is the typical triangular element area, À e is the boundary of the element e , ds is the arc length of an infinitesimal line element along the boundary À e , n ¼ n x ; n y À Á is the unit outward normal vector on the boundary À e , S x and S y are the outflows from the boundary along the x and y-directions respectively. q w ¼ rh:n denotes the heat flux normal to the boundary of the element and q 2w ¼ rh:n þ rf:n is the sum of heat and mass fluxes which are normal to the boundary of the element e .
We used a three-point Gaussian quadrature formula to evaluate the integrals in the residual Eqs. (33)- (37). Using the Newton-Raphson method, non-linear residual Eqs. (33)-(37) are solved to determine the coefficients of the expansions in Eq. (32). The details of this technique are well documented in the textbook by Reddy et al. [43]. The readers can also consult the work of Uddin et al. [44]. We set À rþ1 À À r 10 À5 , where À is the general dependent variable ðU; V ; h; fÞ, and r is the number of iteration in order to calculate the error and to determine the convergence of the solution. We tabulated the thermophysical properties of the base fluids and nanoparticles in Tab. 1.

Test for Grid Independence
The scrutiny of grid sensitivity on a converged solution is essential for the correct usage of the finite element method. Here, we examined five non-uniform grids named coarse, normal, fine, finer, and extrafine. Each of them has 297, 668, 1075, 1643, and 7435 number of elements within the resolution field. We have calculated the average Nusselt number (Nu ave ) for the afore-said mesh elements and tabulated in Tab. 2 for understanding grid fineness. From Tab. 2, we notice that the values of the average Nusselt number for 1643 and 7435 mesh elements remain almost the same. It indicates that either 1643 or 7435 mesh elements are sufficient to obtain a grid-independent solution. To save run time and memory, we used 1643 mesh elements for numerical computation.

Code Validation
We tallied our simulated results with the work of Ghasemi et al. [33]. They studied a mixed convection flow in a lid-driven right-angled triangular cavity in the absence of mass transfer. They have considered the insulated horizontal wall, hot inclined wall, and uniformly moving cold vertical wall. In Tab. 3, we compared our calculated average Nusselt number with Ghasemi et al. [33] varying Richardson number, Ri. The comparisons show an excellent agreement among the data and inspire us to use the current code.

Numerical Results and Discussion
In this section, we mainly presented average Nusselt numbers computed for different values of the model parameters. Due to the Brownian diffusion and thermophoresis, it is expected that there is a minimal concentration difference say, DC ¼ 0:01 within the flow field. For Fe 3 O 4 nanoparticle with diameter d p ¼ 50 nm, and assuming reference temperature T c ¼ 300 K, temperature difference DT ¼ 10K, the Brownian diffusion and thermophoresis coefficients are calculated as D B ¼ 8:7591 Â 10 À12 and D T ¼ 3:9597 Â 10 À12 respectively (see [44,45]). The corresponding values for the other physical parameters are Pr ¼ 6:8377, Nb ¼ 1:24 Â 10 À5 , Nt ¼ 9:58 Â 10 À7 , and Le ¼ 16795. It is good mention that in nanofluid research using Buongiorno model, the values of the Brownian diffusion parameter Nb, thermophoresis parameter, and Lewis number are very poorly determined in a significant number of studies in the open literature (see [46]). In our simulation, we have used the realistic values of the aforesaid-parameters, which make this study unique. For numerical computation, we considered Ri ¼ 10 4 , Pr ¼ 6:838, Ha ¼ 25, c ¼ 15 , Nb ¼ 1:24 Â 10 À5 , Nt ¼ 9:58 Â 10 À7 , Nr ¼ 0:001, Le ¼ 16795, and AR ¼ 1 as default unless otherwise specified.
To explore the time progression of numerical solutions, we have calculated the streamlines of Fe 3 O 4water nanofluid for different values of the dimensionless time s keeping other model parameter values fixed. We have taken the snapshot of the unsteady solution at s ¼ 0:01, 0.1, 1, and 1.5 and depicted in Fig. 2. This series of figures show the time progression of solutions from the transient state to a steady state. We can see that when s ! s s ¼ 1, there are no changes in the structure of streamlines, which means that the solution reached a steady-state. As dimensionless time increases, the fluid flow intensity increases and approaches a steady-state. Fig. 3 shows the dimensionless time s s needed to reach the solution in a steady-state for different Richardson number Ri. We detected that for increasing Richardson's number the solution requires more time to be in a steady-state. It is because increased Ri weakens inertia force over the buoyancy force as a result of the external driving force, the nanofluid motion diminishes. Hence, the system required more time to be in a steady-state.  The average Nusselt number decreases with time and approaches a steady-state after a certain-time s ! s s . Also, it can be seen that as the Richardson number increases, the average Nusselt number also increases due to the natural convection. From these figures, we observe that higher values of the Hartmann number, as well as the magnetic field inclination angle, forced the solution to reach a steadystate earlier compared to the absence of the magnetic field within the flow domain. We also found that the heat transfer rate decreases with the increase of the values of s, whereas when the Hartmann number Ha, as well as the magnetic field inclination angle, increases, the rate of heat transfer decreases.   When the Hartmann number increases, the Lorentz force becomes energetic and dominates over the buoyancy force that causes a reduction the heat transfer for all considered values of Ri. It means that a stronger magnetic field may delay the onset of convection. Thus, the rate of heat transfer can be controlled by controlling the strength of the applied magnetic field.
The effects of the orientation of the magnetic field on the average Nusselt number are displayed in Fig. 6. From this figure, we see that Nu ave decreases with the increase of c when c < 20, but a further escalation in c enhances the rate of heat transfer. Thus, we can say that the magnetic field inclination angle, as well as the Hartmann number, significantly controls the heat transfer rate. One of the most critical characteristics of the problem is the change of the aspect ratio (AR) between the height (the vertical-wall) and length (the horizontal wall) of the triangular enclosure. The diagrams in Fig. 7 show the effect of change in the aspect ratio for different Richardson numbers on the average Nusselt number and resulting heat transfer. As the aspect ratio increases, the average Nusselt number decreases for AR < 1. For AR ! 1, the average Nusselt number increases with the increment of AR for all Ri.
We can see from Fig. 8 that changing the aspect ratio has a significant effect on the time required for the solution to reach a steady-state with increasing Richardson number. It is observed from this figure that at the lower value of aspect ratio and for increasing Richardson's number, the solution needs less time to reach a steady-state. Thus, we may conclude that changing the aspect ratio of the triangular enclosure helps the solution reach a steady-state faster with increasing Ri:  In our physical model, we have considered that the vertical wall of the triangular enclosure is uniformly heated (h ¼ 1). But in reality, this configuration may change depending on the specific applications. Thus, it is essential to study the present model taking into the effects of varying the thermal boundary conditions to a non-uniformly heated wall such as h ¼ Y ð1 À Y Þ and h ¼ sinð2pY Þ. Consideration of these conditions   The average Nusselt number for different aspect ratios with various types of base fluids (H 2 O, Ke, EG, and EO) is displayed in Fig. 11. As stated before, the average Nusselt number decreases with the rise of the aspect ratio for all four types of base fluids. Besides, as we observed in Fig. 9, Fe 3 O 4 -EO nanofluid gives the highest rate of heat transfer, whereas Fe 3 O 4 -H 2 O has the lowest heat transfer. So changing the aspect ratio leads to the same trend of heat transfer for different types of base fluids.

Conclusions
In this work, we numerically studied the problem of unsteady natural convection flow and heat transfer in a lid-driven right-angle triangular shaped enclosure filled with Fe 3 O 4 nanoparticles in four different types of base fluids such as water, kerosene, ethylene glycol, and engine oil in the presence of an inclined magnetic field. The model used for the binary nanofluid incorporates the effects of Brownian motion and thermophoresis. The influence of changing the thermal boundary conditions of the enclosure was also  The actual value of the average Nusselt number for EO is divided by 20 to fit within the diagram investigated, taken into account different aspect ratios of the cavity. Furthermore, we analyzed the time evolution of the solution from unsteady to the steady-state. In the physical model, the effects of the different model parameters such as Richardson number ðRiÞ; Hartmann number ðHaÞ; the inclination angle (c) of the magnetic field, aspect ratio (AR), and various thermal boundary conditions on the average Nusselt number were investigated in details and discussed their physical significance. From the numerical simulations, we found that a strong magnetic field may suppress the convection mechanisms in nanofluids, as a consequence rate of heat transfer decreases. The magnetic field orientation significantly controls the rate of heat transfer in nanofluids. For the higher value of Ri, the heat transfer rate decreases through lower-values of c; but increases through the large-value of c. The higher Ri confirms better heat transfer through convection than conduction.