Enhancement of mixed convection in a lid driven enclosure based on magnetic field presence with nanofluid

Abstract

Computational analysis is utilized to examine the effects of introducing nanoparticles to a square lid-driven cavity to enhance the hydro-magnet mixed convection. Constant temperatures are imposed along the square container’s vertical edges. Both the top and bottom surfaces are covered with insulation. The lid is thought to move in two ways: increasing or decreasing free convection. In addition, a horizontal magnetic field is applied uniformly. For various Hartmann numbers (Ha) (0:100), Richardson numbers (Ri) (0.001:10), and solid volume fractions (0:0.1), the results are reported. This research is based on a constant Grashof number (Gr) of 104. The effects of parameters, including the Richardson number, Hartmann number, solid volume percentage on the stream, isothermal lines, and local Nusselt numbers (Nu), were investigated numerically. In addition, given various parametric settings, the anticipated results for the average Nusselt (Nu_avg) are shown and discussed. For all tested parameters, increasing the magnetic field makes the orientation of the lid more effective on heat and fluid movement. The magnetic field reduces heat and fluid flow. The heat transmission is aided by increasing the solid volume percentage. The effect of nanoparticles on flow and heat transmission is being studied. At Ri = 0.001, the effect of lid orientation is not significant. The highest reduction in heat transfer occurs when adding flow at Ri = 10,  = 0, and Ha = 100.

Introduction

Mixed convection heat transfer and fluid flow in square cavities with moving lids are essential subjects of research because of their impact on many engineering applications such as cooling of electronic systems, nuclear reactors, chemical processing equipment, lubricating grooves, solar collectors, heat exchangers, crystal growth, food processing, and float glass production. The use of magnetic fields to create semiconductor crystals has been explored for many decades.

The authors of references^1–3 studied magnetohydrodynamics. Pal et al. ¹ investigated the unstable natural convection heat transfer using two-dimensional numerical analysis inside a two-heated circular cylinder enclosure. As the transport mechanism is diffusion, smaller values of the Rayleigh number (Ra) for heat transfer remained unaffected by the applied magnetic field for a given value of dimensionless separation between the cylinders (S), and Nuavg stayed essentially unchanged with Ha. The heat transfer rate is determined by a complex interplay between the magnetic field and the physico-thermal characteristics of increasing Ra. However, Nu declines with Ha at first and then remains almost constant, and with higher Ra levels, the shift between conductive and convective heat transfer occurs. Furthermore, Abdulkadhim et al. ² investigated a difficult engineering geometry: magnetohydrodynamic natural convection of nanofluid (Cu-water) in a wavy-walled enclosure with a circularly heated cylinder within. Increased Ha has a negligible effect on Nu at low Ra, but it drastically reduces Nu by up to 33% at higher Ra due to restricted convection. Authors discovered that the heat coefficient, q, significantly influences Nu at low Ra but that its importance diminishes as Ra increases. Heat absorption generates a heat sink for q 0, increasing Nu by 34%, whereas heat production (q > 0) lowers Nu by 48%. Aside from that, a variation in the heat coefficient does not affect the improvement impact of nanoparticles. Experimentally and statistically, the effects of both time-unvarying and alternating magnetic fields on natural convection heat transfer of Cu-water nanofluid in a cubic enclosure are investigated. The effect of the nanofluid volume fraction on natural convection heat transfer in the presence of a constant magnetic field is explored for various Gr values. Kargarsharifabad ³ studies the influence of an alternating (in time) magnetic field in the shape of a rectangular wave with different frequencies on the heat transmission rate. In the absence of a magnetic field, the results showed that adding Cu nanoparticles to water increases heat transfer performance. As the volume percentage of the nanofluid rises, the Nu expands. However, if the enclosure is subjected to a strong enough magnetic field, the negative effect of the nanoparticles’ greater electrical conductivity overwhelms the beneficial effect of the nanofluid’s better thermal efficiency. As electrical conductivity rises, the magnetohydrodynamic effect, which acts against the buoyancy force, becomes stronger. As a result, raising the volume percentage of nanoparticles influences heat transfer performance if the magnetic field strength is greater than a threshold value. In addition, numerical analysis is done on the investigation of natural convection in a vertical cylindrical annular enclosure filled with Cu-water nanofluid under magnetic fields. A finite volume approach is used to numerically solve the governing equations. Numerous Rayleigh numbers, Hartmann numbers, and nanoparticle volume fractions have been subjected to numerical study in Medebber et al. ⁴

References^5–12 study the Al₂O₃/water nano-fluid effect in the following ways. Aghakhani et al. ⁵ investigates the free convection of Al₂O₃/water nano-fluid in a slanted container with a curved form fin in the lower section under a magnetic field. The results revealed that raising Ra and AR enhanced the Heat Transfer Rate (HTR) and maximum entropy production, whereas increasing Ha lowered the HTR and maximum entropy generation. Furthermore, when the Nu_avg and entropy generation is ignored, the Nu_avg and entropy generation rise at a given concentration. Liu et al. ⁶ investigated the entropy production and spontaneous convection of an Al₂O₃/water nanofluid in an inclined cavity made up of two linked inclined triangular enclosures in the presence of a horizontal magnetic field. Their findings show that by increasing the Raleigh number, the rate of heat transmission increases by 12% and the entropy generation increases by 13%. The heat transmission and entropy formation rate drop by 6.5% and 8%, respectively, as the Ha rises. A greater Ra and a lower Ha reduce the Bejan number. The average Nu computed on the right wall constantly decreases as the cavity angle increases. Entropy production lowers for a more inclined angle, and the Bejan number rises.

Similarly, Pordanjani et al. ⁷ quantitatively evaluated the influence of a magnetic field on the natural convection of Al₂O₃/water nanofluid inside a square enclosure with isothermal barriers and sinusoidal wall temperature distribution. The governing equations were converted to algebraic form using the finite volume approach, and the SIMPLE algorithm was used to solve them all at once. The coefficient of heat conductivity was calculated using Vajjha’s suggested model, which considered Brownian particle motion. The results showed that raising the Ha reduced the fluid velocity and the Nu at all volumetric percentages of nanoparticles. The Nu rose as the volumetric percentage of nanoparticles grew; thus, at a 6% nanoparticle concentration, the mean Nu increased by 9.04% above the base fluid. Furthermore, raising the magnetic field angle and Rayleigh number raised the Nu whilst increasing the aspect ratio lowered it.

Besides that, Mourad et al. ⁸ analyzed the constant laminar natural convection flow and heat transfer between a cold, wavy porous enclosure and a heated elliptic cylinder using a three-dimensional numerical simulation. Alumina nanoparticles are disseminated in the water to improve the heat exchange process. The nanofluid flow is assumed to be laminar and incompressible, and the Darcy–Forchheimer model accounts for the advection inertia effect in the porous layer. The dimensionless version of the governing equations is used to elucidate the issue, which is then solved using the finite element technique. ⁸ The results show that increasing Ra improves heat transmission. The increase in permeability resulted in a 12.73% increase in heat transfer rate. Additionally, when Ha is changed from 0 to 100, the Nu values are reduced by up to 22.22%.

Furthermore, for the convective process, the ideal inclination angle is 45°. In an inclined enclosure impacted by a magnetic field, the formation of entropy and free convective heat transport of Al₂O₃/water nanofluid were investigated by Zheng et al., ⁹ as well as radiation impacts. In the lower half of the enclosure’s left wall, there is a circular quadrant at hot temperature Th. The enclosure’s right wall is kept at a constant cold temperature of Tc. The remainder of the walls are insulated. The SIMPLE method is used to solve the governing equations for fluid flow. The data show that increasing the Ra and reducing the Ha increases the Nu by 160% and 40%, respectively. By increasing Ra and decreasing Ha, Be decreases. By increasing the Ra and decreasing the Ha, the maximal creation of entropy increases by 288% and 39%, respectively. At a 30° inclination angle, the maximum average Nu and total creation of entropy occur. ⁹

Moreover, in Berrahil et al., ¹⁰ a numerical study of the natural convection of Al₂O₃/water nanofluid in a differentially heated vertical annulus under a uniform magnetic field is carried out. Farid Berrahil and others created an in-house Fortran algorithm to solve the equations regulating the magneto-hydrodynamic flow. The theoretical findings show that when the nanofluid volume percentage grows, the average Nu on the inner cylinder wall falls for nanoparticles with a diameter of dp = 47 nm. Nu decreases until it reaches a volume fraction of 0.05, then rises again. It is proven that raising the Ra number leads Nu to grow while increasing the Ha number and increasing the magnetic field causes Nu to drop across the whole range of volumetric fractions.

Furthermore, when the magnetic field is directed radially, the heat transfer enhancement ratio increases as the Ha number grows. Finally, new Nu versus Ra, Ha, and correlations are determined for the axial and radial magnetic field situations. For the dissipating Al₂O₃-Cu nanoparticles in Slimani et al., ¹¹ the governing equations subject to the physical boundary conditions are numerically solved using the Galerkin finite element technique. The average Nusselt Number responds inversely to magnetic fields but directly to dynamic fields flowing through the porous material. These variables play a crucial role in controlling the fluid flow and heat transfer rate.

Using a computational model, Yan et al. ¹² investigated the effect of nanopowder shape on natural convection and irreversibilities in water/alumina nanofluids. A square enclosure with consistent cold (Tc) and hot (Th) temperatures on the right and left walls was employed to achieve this. The container was subjected to a magnetic field, and the horizontal surfaces were segregated. The left wall had two fins that were the same temperature as the wall. The algebraic form of the equations was derived using the control volume approach, and the simulations were performed using a SIMPLE program developed in Fortran. ¹² According to the data, this model forecasts the maximum heat transfer rate and entropy generation. The most important conclusion of this research is that adding nanopowder to nanofluids does not always boost free heat convection. In certain situations, the Nu number and irreversibilities can be reduced due to the significant rise in nanofluid viscosity. It was also discovered that the Nu increases with Ra and decreases with magnetic field strength. The angled and straight fins generally produced the highest and lowest heat transfer coefficients, respectively.

The hybrid Fe₃O₄/MWCNT-water nanofluid is numerically simulated using the finite element method (FEM). The simulation is run for various parameter values, such as the Hartmann and Darcy numbers. ¹³ Moreover, the authors in Al-Kouz et al. ¹⁴ described a multi-walled carbon nanotube-iron oxide (MWCNT-Fe3O4) hybrid nanofluid MHD nano liquid convective flow in an unusually shaped cavity. The interior and external boundaries of the hollow are isothermally maintained at high and low temperatures. Mehryan et al. ¹⁵ examined the effects of a periodic magnetic field on natural convection and entropy production of Fe₃O₄/water nanofluid flowing in a square enclosure. Streamlines, isotherms, and entropy generation contours, as well as the local and Nu_avg, are used to display the findings. Nu_avg and total entropy generation St for the periodic magnetic field were more significant than the uniform magnetic field, regardless of Ha and. Dogonchi et al. ¹⁶ also studied natural convection heat transfer in a square enclosure with a wavy circular heater in the presence of a magnetic field and nanoparticles. The governing equations, written in dimensionless form, are solved using a verified FORTRAN code and the control volume finite element method. The results show that in the presence of waves on the annulus’ inner wall, the HTR is an ascending function of Ra, nanoparticle volume percent, and a less noticeable relation of their shape factor, but a descending function of the Ha.

Similarly, the natural convection of a porous enclosure subjected to a nonuniform magnetic field was numerically studied using the LTNE model. The buoyancy, Lorentz, and magnetization forces are all applied to the hybrid nanofluid under these circumstances. Izadi et al. ¹⁷ discretized and solved the set of governing equations relevant to the current situation using the finite element technique. The results show that as γ_r increases, the Nu of the two phases of porous material converges. However, these two thermal indices fluctuate when γ_r decreases. When the Ha and Lorentz forces operating on the nanofluid grow, it is reasonable to use the local thermal equilibrium. Lu et al. ¹⁸ investigated the influence of a magnetic field on double-diffusive natural convection in an enclosure with a heat-conducting partition. The left and right walls are kept at constant temperatures and concentrations, while the horizontal walls are insulated. The dimensionless governing equations are solved using the lattice Boltzmann technique. The findings show that the Ha impacts heat and mass transport processes. The magnetic field angle has a considerable impact on the flow pattern. Furthermore, there is an ideal magnetic field angle of 90° at which the heat and mass transfer rates are constant and maximal. The relationships between the Nu_avg and Sherwood numbers also fit.

A magneto-free convection of a hybrid nanofluid with a heated side wall and an interior adiabatically spinning cylinder was analyzed. ¹⁹ For the permeable domain, the Darcy-Forchheimer model has been applied. We visually explore the effects of critical factors on velocity, stream functions, isotherms, and mean Nusselt Number. While the purpose of the study in Koulali et al. ²⁰ is to ascertain how the temperature gradient direction influences the heat transfer between two superposed fluid layers separated by zero wall thickness. The use of continuous heat flow together with a no-slip condition at the interface allowed for the coupling of the two layers.

Shuvo et al. ²¹ investigated the steady entropy generation and mixed convective flow of nanofluid within an inclined lid-driven trapezoidal enclosure have been investigated numerically. The Reynolds, Grashof, and Richardson numbers, the tilt angle of the base wall, and the nanoparticle volume fraction are the primary parameters of interest in the present study. Nath and Murugesan ²² showed the results of numerical experiments are reported for thermo-solutal buoyancy induced mixed convective heat and mass transfer with an influence of inclined magnetic field in a backward facing step channel filled with hybrid nanofluid for part heating load condition.

The constant, laminar, hydromagnetic, double-diffusive forced convection flow inside a square enclosure filled with nanofluid in the presence of heat generation or absorption was not explained in the previous review. Because this scenario is of primary importance and has numerous potential applications, such as crystal formation, geothermal reservoirs, nuclear fuel debris removal, and metal alloy solidification, it is of particular interest to consider it in the current work. The effects of the Richardson number, Hartmann number, and solid volume percentage on the stream will be investigated numerically in the presence of nanoparticles and magnetic fields for square lid-driven cavities.

Mathematical model

The physical model of the system under discussion is depicted in Figure 1. A two-dimensional square cavity enclosure of side length L, Th and Tc temperatures are evenly applied to the vertical walls and are kept constant and uniform, resulting in a double-diffusive convection flow field. It is assumed that the top and bottom surfaces are adiabatic. The source of heat is the left wall. A magnetic field of uniform strength Bo is applied in the horizontal direction. The induced magnetic field is ignored since the magnetic Reynolds number is considered minimal. In addition, the enclosure is filled with a Cu spherical nanoparticle-containing water-based nanofluid (Pr = 6.2). The base fluid and nanoparticles are assumed to be thermally balanced, the nanofluid Newtonian and incompressible, and the flow laminar. The thermophysical properties of the base fluid and nanoparticles are listed in Table 1. The fluid in this container is assumed to be incompressible and Newtonian. The Boussinesq approximation equation (1) with opposing and compositional buoyancy forces is used for the body force elements in the momentum equations. The balance laws of mass, linear momentum, and thermal energy in a steady state in two dimensions serve as the foundation for the governing equations for the issue at hand. The continuity, momentum, and energy in two-dimensional equations can be expressed as follows in light of the assumptions mentioned above ²³ :

$$\rho ={\rho }_o[1-\beta (T-T_C)]$$

(1)

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(2)

$$(\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{u}}{\partial \mathrm{y}})=-\frac{1}{{\mathrm{\rho }}_{\mathrm{nf}}}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}+{\upsilon }_{\mathrm{nf}}(\frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{y}}^2})$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{(\mathrm{u}\frac{\partial \mathrm{v}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{v}}{\partial \mathrm{y}})=-\frac{1}{{\mathrm{\rho }}_{\mathrm{nf}}}\frac{\partial \mathrm{p}}{\partial \mathrm{y}}+{\upsilon }_{\mathrm{nf}}(\frac{{\partial }^2\mathrm{v}}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathrm{v}}{\partial {\mathrm{y}}^2})}\\ +\mathrm{g}{\left(\mathrm{\rho }{\beta }_{\mathrm{T}}\right)}_{\mathrm{nf}}\left[(\mathrm{T}-\mathrm{Tc})\right]+\frac{{\sigma }_{\mathrm{nf}}{\mathrm{B}}^2}{{\mathrm{\rho }}_{\mathrm{nf}}}v\hfill \end{array}$$

(4)

$$(\mathrm{u}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}+\mathrm{v}\frac{\partial \mathrm{T}}{\partial \mathrm{y}})={\alpha }_{\mathrm{nf}}(\frac{{\partial }^{2}T}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{y}}^2})$$

(5)

Figure 1.

Physical model and coordinates system.

Table 1.

Grid independence study for Ri = 1, Ha = 50, φ = 10%, and U = 1.

Grid size	Nuavg
42 × 42	2.4759
62 × 62	2.5449
82 × 82	2.5659
102 × 102	2.5726
122 × 122	2.5734
132 × 132	2.5726

The boundary conditions for the problem could be written as

$$\mathrm{At}x=0,u=\nu =0.0,T=T_h$$

(6)

$$\mathrm{At}x=L,u=\nu =0.0,T=T_c$$

(7)

$$\mathrm{At}y=0,u=v=\frac{\partial T}{\partial y}=0.0$$

(8)

$$\mathrm{At}y=L;u=\pm u_0,v=\frac{\partial T}{\partial y}=0.0$$

(9)

$$\mathrm{X}=\frac{x}{L},\mathrm{Y}=\frac{y}{L},\mathrm{U}=\frac{u}{u_o},\mathrm{V}=\frac{v}{u_o},\mathrm{P}=\frac{p}{\mathrm{\rho }{u}_o^2},\theta =\frac{T-\mathrm{Tc}}{T_h-T_c}$$

(10)

In the governing equations, the dimensionless governing equations become

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$$

(11)

$$\mathrm{U}\frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\mathrm{V}\frac{\partial \mathrm{U}}{\partial \mathrm{Y}}=-\frac{\partial \mathrm{P}}{\partial \mathrm{X}}+\frac{{\mathrm{\mu }}_{\mathrm{nf}}}{{\mathrm{\mu }}_{\mathrm{f}}}\frac{{\mathrm{\rho }}_{\mathrm{f}}}{{\mathrm{\rho }}_{\mathrm{nf}}}\frac{1}{\mathrm{Re}}(\frac{{\partial }^2\mathrm{U}}{\partial {\mathrm{X}}^2}+\frac{{\partial }^2\mathrm{U}}{\partial {\mathrm{Y}}^2})$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\frac{1}{\mathrm{Re}}(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2})}\\ +\frac{\mathrm{Gr}}{{\mathrm{Re}}^2}\frac{{\beta }_{\mathrm{nf}}}{{\beta }_f}\theta +\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}{\frac{\mathrm{Ha}}{\mathrm{Re}}}^{2}V\hfill \end{array}$$

(13)

$$\mathrm{U}\frac{\partial \theta }{\partial \mathrm{X}}+\mathrm{V}\frac{\partial \mathrm{\theta }}{\partial \mathrm{Y}}=\frac{1}{\mathrm{Re}\mathrm{Pr}}\frac{{\mathrm{\alpha }}_{\mathrm{nf}}}{{\mathrm{\alpha }}_{\mathrm{f}}}(\frac{{\partial }^2\mathrm{\theta }}{\partial {\mathrm{X}}^2}+\frac{{\partial }^2\mathrm{\theta }}{\partial {\mathrm{Y}}^2})$$

(14)

The density, the heat capacity, and the thermal expansion coefficient, of the nanofluid are obtained from the following respective equations^24,25 and shown in Table 2:

$${\rho }_{\mathrm{nf}}=(1-\varphi ){\rho }_f+\varphi {\rho }_p$$

(15)

$$(\rho C_p{)}_{\mathrm{nf}}=(1-\varphi )(\rho C_p{)}_f+\varphi (\rho C_p{)}_p$$

(16)

Table 2.

Thermo physical properties of water and copper. ²⁶

Property	Fluid (water)	Solid (Cu)
Cp (J/kg k)	4179	383
ρ (kg/m3)	997.1	8954
β (K−1)	2.1 × 10−4	1.67 × 10−5
k (W/m K)	0.613	400
σ (Ω m)−1	0.05	5.96 × 107

The effective thermal conductivity of the nanofluid is approximated by the Maxwell–Garnetts model:

$$\frac{k_{\mathrm{nf}}}{k_f}=\frac{(k_s+2k_f)-2\varphi (k_f-k_s)}{(k_s+2k_f)+\varphi (k_f-k_s)}$$

(17)

$$\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=1+\frac{3(\gamma -1)\varphi }{(\gamma +2)-\varphi (\gamma -1)}$$

(18)

Where

$$\gamma =\frac{{\sigma }_s}{{\sigma }_f}$$

(19)

Nusselt number calculation

Equating the heat transfer by convection to the heat transfer by conduction at hot wall;

$$h\Delta T=-k_{nf}{\left(\frac{\partial T}{\partial x}\right)}_{x=0}$$

(20)

Introducing the dimensionless variables, defined in equation (10), into equation (20), gives:

$$N{u}_l=-\frac{{k_n}_f}{k_f}{\left(\frac{\partial \theta }{\partial X}\right)}_{X=0}$$

(21)

The average Nusselt Number is obtained by integrating the above local Nusselt Number over the vertical hot wall:

$$\mathrm{Nu}=\underset{0}{\overset{1}{\int }}N{u}_l\mathrm{dY}$$

(22)

Numerical method

Patankar ²⁴ proposed a finite-volume approach for solving governing equations. This method is based on leveraging the significant difference in space to discretize the governing equations. The Number of nodes utilized was examined first. Five grid sizing are checked (42 × 42), (62 × 62), (82 × 82), (102 × 102), (122 × 122), and (132 × 132) for Ri = 1.0, Ha = 50, ϕ = 0.01 and U = 1 are shown in Table 1. The deviations between the results for the average Nusselt Number on the left vertical wall obtained for the domain (22 × 22) and (102 × 102) were less than 3.9%, as shown in Figure 2. The control-volume-based finite volume approach is used to reduce the partial differential equations driving double-diffusive forced convection in the presence of a heat source and magnetic fields to a system of simultaneous algebraic equations. For cell face values, the Power-law approach described by Patankar ²⁴ is used to model the discretization of the combined convective and diffusive fluxes over the control surfaces. The control volumes center contains the pointers. Patankar’s SIMPLER technique is used to address pressure velocity coupling. Relaxation techniques are used to obtain steady-state solutions. Because of the linked structure of the system equations, under-relaxation hindered convergence yet was required to generate solutions.

Figure 2.

Comparison of Stream and isothermal lines with Abu-Nada and Chamkha ²⁵ for Gr = 100, Ha = 0 at φ = 0 (__) and ϕ = 0.1 (----): (a) Ri =0.001 and (b) Ri = 1.

As a result, the number of nodes (102 102) was employed throughout the research. The tiny boundary layer at the vertical walls was resolved with 102 grid points in the X-direction. The GausseSeidel technique was used to solve the discretization equations. This program’s iteration approach is a line-by-line procedure that combines the direct method with the resultant Tri Diagonal Matrix Algorithm (TDMA). The Nu_avg and other dependent variables must vary less than 1.2% from their starting value after one hundred iterations to indicate that the iteration has reached its conclusion.

Program validation and comparison with previous work

The code is validated using two different studies to ensure that the numerical technique used to solve this research problem is accurate. The initial step was to use Abu-Nada and Chamkha ²⁵ to validate the code. Simulating mixed convection flow in a vertical square container with temperatures Th and Tc uniformly imposed along the bottom and top walls, respectively. The vertical walls are deemed adiabatic and impermeable, while the top surface moves right. Figure 2 shows the plotting of streams and isotherms. By using the current code, the results were satisfactory.

Results and Discussion

In a lid-driven square enclosure, the influence of Richardson and Hartmann numbers and the volume concentration of nanoparticles on mixed convection is investigated. The Grashof number, Gr, is held constant in this investigation at Gr = 104. In mixed convection, the controlling parameter is the Richardson number, R_i = Gr/Re = 2, which describes the relative significance of natural and forced convection. When the top plate moves to the right, natural convection from the heated to the cold side is aided. However, when the plate shifts to the left, it resists natural convection from the heated to the cold side. The Reynolds number is modified to attain the desired Richardson number by altering the lid velocity u_o. For the top moving plate, this research includes Richardson numbers ranging from 0.001 to 10, Hartmann numbers ranging from 0 to 100, and volume concentrations ranging from 0 to 10% in both directions (right and left).

Effect of Richardson number

The effect of varying the Richardson number on both stream and isothermal lines is illustrated in Figure 3 for adding and opposing flow in the absence of magnetic effect and without addition for solid particles, ϕ = 0%.

Figure 3.

Effect of Richardson number on the Stream and isothermal lines for Gr = 104, Ha = 0 at φ = 0 for assisting and opposing flow.

Moving to right (adding flow)

At Ri = 0.001, the flow patterns form a strong clockwise vortex cell at the center of the cavity as the forced convection is dominated. Due to forced convection generating thinner boundary layers, the isothermal lines are congested at the vertical walls, causing severe temperature gradients in the horizontal direction near the left wall, enhancing heat transfer. The vortex travels higher, approaching the influence of forced convection at Ri = 0.1. At the vertical walls, the isothermal lines are less dense. As Ri is increased to 10, the vortex is weakened as the natural convection is dominated. The density of the isothermal lines near the left wall is less as the effect of the natural convection is dominated.

Moving to left (opposing flow)

Due to the impact of forced convection caused by the surface sliding to the left at Ri = 0.001, flow patterns are defined by a counterclockwise vortex at the cavity’s center. In the case of the top lid moving right, the stream and isothermal lines are mirror images. The flow is split into two vortices when Ri = 1. Forced convection causes the top vortex, whereas natural convection causes the lower clockwise vortex. Because of the impact of natural convection, the isothermal lines are dense near the moving lid and tend horizontally as you get away from it. At Ri = 10, the bottom vortex expands even farther, occupying 75% of the cavity. The isothermal lines are dense near the lid at 25% of the cavity, while the rest is horizontal as the natural convection effect.

Hartmann number Effect

The effect of Hartman on the stream and isothermal lines is illustrated in Figures 4–6 at Ri = 0.001, 1, 10 to simulate the forced dominated, mixed and natural dominated at ϕ = 0% for both lids moving right and left.

Figure 4.

Effect of Hartmann number on the Stream and isothermal lines for Gr = 104, Ri = 0.001 at ϕ = 0 for assisting and opposing flow.

Figure 5.

Effect of Hartmann number on the Stream and isothermal lines for Gr = 104, Ri = 1 at φ = 0 for assisting and opposing flow.

Figure 6.

Effect of Hartmann number on the Stream and isothermal lines for Gr = 104, Ri = 10 at φ = 0 for assisting and opposing flow.

Moving to right (adding flow)

At Ri = 0.001, where the forced convection is dominated, at Ha = 10, a strong clockwise vortex at the center of the cavity with Ψ_max = −0.038, as the effect of forced convection is more vital than the magnetic field. The isothermal lines are denser at the vertical walls. As Hartmann increases, Ha = 50, the flow strength is reduced and concentrates at the upper half of the cavity. The isothermal lines at the upper half of cavity is dense at the vertical walls while it spreads over the other half of cavity as the effect of magnetic field. At Ha = 100, the effect of magnetic field increases, the vortex is reduced to 25% of the cavity near the upper wall with Ψmax = −0.01. The isothermal lines are less dense near the wall. The isothermal lines are less dense at the left wall near the upper wall while the rest of cavity the isothermal is parallel to the wall as the conduction regime is highly dominant. At Ri = 1, where the forced and natural convection are equal and at the same direction. At Ha = 10, clockwise vortex near the upper wall with Ψmax = −0.055, the isothermal lines spread at the cavity. As increasing the magnetic field, the streamlines are concentrated near the upper cavity as the effect of forced convection while the rest of cavity is stagnated. The isothermal lines are dense at the vertical wall near the lid while the rest of cavity under conduction regime. At Ri = 10, where the natural convection is dominated. At Ha = 10, clockwise vortex near the upper wall with Ψ_max = −0.065, the isothermal lines spread at the cavity. As increasing the magnetic field, the streamlines are concentrated near the upper cavity as the effect of forced convection while the rest of cavity is stagnated. The isothermal lines are dense at the vertical wall near the lid while the rest of cavity under conduction regime.

Moving to left (opposing flow)

At Ri = 0.001, the streamlines and isothermal mirror the case of lid moving rights. At Ri = 1, the forced and the convection are equal and in opposite directions. At Ha = 10, a counterclockwise with Ψ_max =  0.065 fills the upper half of the cavity. The isothermal lines are denser at the vertical walls, while the lower half of the isothermal lines are spread horizontally. As the Hartman Number increased, the stream lines concentrated more near the moving lid while the rest of the cavity stagnated. The isothermal lines near the cavity are spread horizontally, while the rest are under a conduction regime.

At Ri = 10, for Ha = 10, two vortices are formed in the cavity. An upper counterclockwise vortex a with Ψ_max = 0.034 due to forced convection while the lower clockwise vortex with Ψ_max = −0.02. As the magnetic field increases, the upper vortex concentrates near the moving lid while the lower vortex strength decreases.

Figure 7 show the effect of the Hartmaan number on the local Nusselt Number for adding and opposing flow at Ri = 1.0 and ϕ =  0%. It observed that the local Nusselt Number is highest near the upper wall for both adding and opposing flow due to the effect of forced convection. As the Hartmaan Number increases, the local Nusselt Number near the upper wall is highest, while for the rest of the wall’s length, the magnetic strength effect is more significant on the local Number.

Figure 7.

Effect of Hartmann number on the local Nusselt Number for Gr = 104, Ri = 1 at φ = 0 for assisting and opposing flow.

Figure 8 shows the effect of the Hartmann number on the average Nusselt Number for ϕ = 0% at different Ri for adding and opposing flow. As the effect of magnetic increases, the average Nusselt Number decreases for adding and opposing flow. For low magnetic strength, the average Nusselt Number for adding flow is higher than opposing. At Ri = 0.1, the point of inflexion, the average Nusselt Number for opposing flow is higher than adding after this point at Ha = 11. As increasing Ri, the value of this point decreases to 6.4 for Ri = 5.0, and the effect of natural convection is significant. From equation (13), the term of Ri is opposite to (Ha²/Re), which causes the average Nusselt Number for opposing is higher than for adding flow.

Figure 8.

Effect of Hartmann number on the average Nusselt Number at φ = 0 for assisting and opposing flow at different Ri.

Effect of nanoparticle volume concentration

Figure 9 represents the effect of solid volume fraction on the stream and isothermal lines for both adding and opposing flow at Ri = 1, Ha = 50. It observed that increasing the solid volume fraction increases the vortex’s strength in the cavity’s upper part slightly, as the effect of forced convection is significant. In contrast, the effect is more significant in the rest of the cavity. The addition of nanoparticles increases the convection that augments the heat transfer for both adding and opposing flow. The effect of increasing the solid volume fraction on the local Nusselt Number for both adding and opposing flow is shown in Figure 10. For both adding and opposing flow, 0.9≤Y≤1, the effect of adding nanoparticles is not significant as the forced convection effect is strong. As the length decreases, the effect of nanoparticles is significant.

Figure 9.

Effect of Hartmann number on the Stream and isothermal lines for Ri = 1 and Ha = 50 for assisting and opposing flow.

Figure 10.

Effect of Hartmann number on the local Nusselt Number for Gr = 104, Ri = 1 at φ = 0 for assisting and opposing flow.

The average Nusselt Number for adding and opposing flow

The effect of increasing the solid volume fraction on the average Nusselt Number for both adding and opposing flow at different Hartmann numbers and Ri is shown in Figure 11. The effect of increasing nanoparticles on the average Nusselt Number is linearly for the range of Ri, 0.001 ≤ Ri ≤10. The maximum enhancement for the average Nusselt Number is 25.15% at Ri = 10, Ha = 0, and ϕ = 0.01 for adding flow. The minimum increase is 11.73% for Ri = 10, Ha = 100, and ϕ = 0.01 for opposing flow.

Figure 11.

Effect of nanofluid volume concentration on the average Nusselt Number at different Ri and Hartmann Number for assisting and opposing flow.

Conclusion

In the present study, the effect of a magnetic field on a steady lid-driven cavity on the flow and heat transfer for a wide range of Richardson, 0.001 ≤ Ri ≤10, is studied numerically for adding and opposing flow.

• The addition of nanoparticles is investigated on the flow and heat transfer enhancement. The lid direction effect is insignificant at Ri = 0.001, as the forced convection is dominated in the absence or under magnetic effect.

• As the Richardson number increases, the average Nusselt Number for adding flow is higher till the inflection point. At Ha > Ha_cr, the heat transfer due to opposing is higher than adding flow. The magnetic field reduces the flow and heat transfer for both adding and opposing flow. The maximum decrease in heat transfer is 59% for adding flow at Ri = 10, ϕ = 0, and Ha = 100.

• Adding Cu nanoparticles at different volume fractions to the base fluid water increases the flow and heat transfer. The maximum increase for adding flow is at Ha = 0, Ri = 10, and ϕ = 0.1, while for opposing flow is 25.07% at Ha = 10, Ri = 0.001, and ϕ = 0.1.

• The different types of nanoparticles and the cavity shape will be considered in the future work.