# Mixed Convection in a Double Lid-Driven Wavy Shaped Cavity Filled with Nanofluid Subject to Magnetic Field and Internal Heat Source

Authors: Kakali Chowdhury; Md. Abdul Alim
Journal: Journal of Applied Mathematics
 (2023)

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## Abstract

A numerical investigation is carried out to analyze the impacts of internal heat source size, solid concentration of nanoparticles, magnetic field, and Richardson number on flow characteristics in an oppositely directed lid-driven wavy-shaped enclosure. The left and right vertical walls of the enclosure are cooled isothermally and moving with fixed velocity in upward and downward directions, respectively. The bottom wall is wavy shaped and isothermally cooled as the vertical walls while the top wall is kept adiabatic. A rectangular heater is placed horizontally in the center of the cavity. The physical problems are characterized by 2D governing partial differential equations accompanying proper boundary conditions and are discretized using Galerkin’s finite element formulation. The study is executed by analyzing different ranges of geometrical and physical parameters, namely, internal heat source length0.2≤CH≤0.6, solid concentration of nanoparticles 0≤φ≤0.09, Hartmann’s number 0≤Ha≤70, and Richardson’s number 0.1≤Ri≤10. The results indicate that the overall heat transfer rate declines with the increasing length of internal heat source. The presence and rising values of solid concentration of nanoparticles cause the augmentation of heat transfer whereas the magnetic field has a negative influence and the Richardson number has a positive influence on heat transfer.


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## Body

## 1. Introduction

The application of heat transfer mechanism in various shaped enclosures by different types of convection is swelling day by day. Besides cooling of electronic devices and heating and cooling buildings, glass industry, etc., it has many applications such as nuclear reactors, thermal storage, food processing, and solar collectors. The less thermal conductivity of conventional fluids such as water, mineral oils, and ethylene glycol is the main barrier to thermal systems enhancing effectiveness. To circumvent this restrain, researchers are continuously trying to develop some fluids that have advanced properties of heat transfer by appending metallic nanoparticles with base fluid.Abu-Nada and Chamkha [1] examined the mixed convective CuO-water nanofluid flow in a moving lid sinusoidal enclosure. They reported that wavy wall effects strongly on heat transfer. They have shown that avg Nu enhances in case of a wavy wall and reduces when the wall is smooth. Ali et al. [2] examined conjugate combined convection of nanofluid in an alternated baffles equipped horizontal channel. Alsabery et al. [3] investigated nanofluid flow and energy dissipation using rotating cylinders. Alsabery et al. [4] studied on nanofluid using a curvy shaped enclosure for investigation of the effect of heat source and amplitude on free convection. Alsabery et al. [5] investigated combined convection nanofluid flow using entropy generation and rotating cylinder. Armaghani et al. [6] used L-shaped enclosure to study MHD combined convection equipped with heat source/sink and hybrid nanofluid. They remarked that at solid volume fraction 0.1, entropy creation for sink heat generation is 20% less than that of source heat generation. They have also found that local Nu for Cu-water is 29% more than that of pure water. Basak and Chamkha [7] studied nanofluid flow of free convection using square enclosure of different temperatures. Their report shows that, at Ra=105, average Nu is higher for all nanofluids and a larger enhancement of heat transfer exhibit for Cu-water and alumina-water nanofluid. Bensouici and Boudebous [8] focused on combined convection for various nanofluids cramped in a moving lid cavity furnished with a heat source. Their study shows that the nanofluids with high thermal conductivity materials transfer more heat than low thermal conductivity materials. Chamkha et al. [9] used a moving lid cavity to analyze MHD combined convection of nanofluid to determine the impacts of entropy generation, heat sink, and source. Choi and Eastman [10] were pioneers to hoist the thermal conductivity of water by appending high thermal conductivity metallic nanoparticles. Ghaneifar et al. [11] used horizontal channel equipped with Al2O3 nanofluid and double heat sources to study combined convection fluid flow. They reported that the Reynolds number has a negative impact on heat transfer. They also showed that heat transfer rate in a channel can be controlled by changing the distance between two heat sources. Ismael et al. [12] used magnetic field and heat flux for examining combined convection of nanofluid within partial slip enclosure. Kakac and Pramuanjaroenkij [13] used nanofluid to examine the enhancement of convective heat transfer. They have summarized the significant published articles which were conducted on nanofluids for the augmentation of forced convection heat transfer. Their literature investigation shows that suspending nanoparticles with pure fluids enhances the heat transfer ability of base fluids. Kalteh et al. [14] analyzed combined convection using a moving lid enclosure equipped with a heat source of triangular shaped. They reported that the mean Nusselt number upsurges with the enhancement of solid concentration of nanoparticles and downfalls with the increasing value of nanoparticle diameter. They also found that the mean Nusselt number is highest for Ag nanoparticles and lowest for TiO2 nanoparticles. Kasaeipoor et al. [15] examined nanofluid using a T-shaped enclosure to show the effect of magnetic field. Keya et al. [16] used a double-pipe heat exchanger in a moving lid enclosure to investigate conjugate combined convection flow. Mahalakshmi et al. [17] focused on MHD mixed convective fluid flow in a moving lid enclosure with a center heater for Ag-water nanofluid. They reported that higher center heater length and solid concentration of nanoparticles boost up the mean Nusselt number. They also found that the augmentation of Ha reduces the average Nu. Nasrin [18] analyzed MHD mixed convective fluid flow in a moving lid sinusoidal wavy walled cavity with a central heat-conducting body. She reported that average Nu raises with the elevation of the Prandtl number and with the degradation of Ha. She also showed that thermal boundary layer increases with the increasing value of the Richardson number. Nasrin et al. [19] investigated the hybrid and mono nanofluid using moving lid and wavy shaped enclosure. Their finding shows that the augmentation of mean Nu is 14.7% more in the wavy enclosure than that of flat enclosure. They also found that amongst various types of hybrid nanofluid, heat transfer rate is higher in water-Cu-CNT hybrid nanofluid. Rahman et al. [20] used a circular cylinder as a heat source within rectangular enclosure to study combined convection. They showed that mean Nu and fluid temperature in the cavity change nonmonotonically with the position of the heat-conducting cylinder in the cavity. They also showed that the maximum heat transfer rate is found for a cavity aspect ratio of 0.5 and the fluid average temperature is the lowest for a cavity aspect ratio of 2.0. Rahman et al. [21] studied nanofluid using a moving lid cavity to analyze the intensification of heat transfer. They reported the volume fraction of nanoparticle as a high-level control parameter because they noticed, under the same Ri, nanofluid transfers more heat than the original fluid. They also reported that mean Nu upsurges significantly by raising the Richardson number and maximum Nu is found in the combined convection regime. Rashad et al. [22] used a moving lid and partly slip enclosure to investigate the effect of heated source and sink on MHD combined convection nanofluid flow. Roy and Basak [23] analyzed natural convection flow in an enclosure with nonuniformly heated walls. They reported that uniformly heated wall transfers more heat than nonuniformly heated wall. Saha et al. [24] used a wavy walled moving lid enclosure to investigate combined convection fluid flow. Taghikhani [25] examined MHD mixed convective nanofluid flow using moving lid enclosure furnished with Joule heating and two heat sources in wavy pattern. They reported that the increasing value of the solid volume fraction and Reynolds number significantly enhances average Nu which boosts up convective cooling. They also showed that augmentation of the Hartmann number reduces the heat transfer rate and augments conduction heat transfer. Tiwari and Das [26] investigated the augmentation of heat transfer of a nanofluid in a double lid-driven nonuniformly heated enclosure. Zahan et al. [27] studied MHD combined convection of hybrid nanofluid subject to Joule heating using wavy walled moving lid enclosure. They reported that with the increasing value of Ha from 0 to 50, heat transfer rate decreases by 12% and it increases by 16.89% at the undulation number three than that of flat surface.By going through the above literature, the authors acknowledge that no investigation on mixed convection visualization with arrow surface approach into an oppositely directed lid-driven wavy walled cavity furnished with nanofluid and a central heat source of different sizes has been published. Thus, this paper is aimed at investigating the impacts of heat source length, solid concentration of nanoparticles, and magnetic field on flow characteristics under the above-mentioned circumstances for 2D laminar flow.


## 2. Physical Model

The treated problem is investigated inside an enclosure with heightH and width L where H/L=1 filled with Cu-water nanofluid. The schematic diagram together with the rectangular Cartesian coordinate system is shown in Figure 1. The top wall is adiabatic whereas the vertical side walls and the curvy bottom wall of the cavity are maintained with same cold temperature Tc. A rectangular heat source with temperature Th>Tc is positioned horizontally in the middle of the cavity. The left and right vertical walls are assumed to move in the upward and downward directions with velocity V0 and −V0, respectively. In standard temperature, the thermophysical properties of copper and water are given in Table 1. B0 is the applied magnetic field along the negative x-axis normal to the vertical wall, and acceleration of gravity g is acting in the downward direction. The nanofluid inside the cavity is considered as laminar, incompressible, and Newtonian. The equilibrium temperature is maintained for water and copper nanoparticles.Figure 1
Schematic diagram with coordinate system.Table 1
Thermophysical properties of water and copper [9, 15].
Physical propertiesUnitsWaterCopper (Cu)CpJ/(kgK)4179385ρkg/m3997.18933μkg/(ms)9.09×10−4—kW/(mK)0.613401βK-121×10−51.67×10−5σSm-10.055.96×10−7


## 3. Mathematical Formulation

Based on the above assumptions, the conservation equations for steady two-dimensional MHD mixed convection flow are given in a dimensional form as follows [2]:
(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=−1ρnf∂p∂x+μnfρnf∂2u∂x2+∂2u∂y2,(3)u∂v∂x+v∂v∂y=−1ρnf∂p∂y+μnfρnf∂2v∂x2+∂2v∂y2+βnfgT−Tc−σnfρnfB02v,(4)u∂T∂x+v∂T∂y=knfρCpnf∂2T∂x2+∂2T∂y2.Energy equation for rectangular center heater:(5)∂2Ts∂x2+∂2Ts∂y2=0.The associated conditions of heat source and boundaries are as follows:(6)u=0,v=V0,T=Tcatx=0and0≤y≤L,u=0,v=−V0,T=Tcatx=Land0≤y≤L,u=v=0,∂T∂y=0aty=Land0≤x≤L,u=v=0,T=Tcatthewavybottomsurface,u=0,v=0,T=Thonthecenterheatersurface.Density of nanofluid at standard temperature is as Ali et al. [2]:
(7)ρnf=1−ϕρf+ϕρs.The other properties of nanofluid are given as follows:Heat capacity as Alsabery et al. [5]:
(8)ρCpnf=1−ϕρCpf+ϕρCps.Coefficient of thermal expansion as Alsabery et al. [5]:
(9)ρβnf=1−ϕρβf+ϕρβs.Thermal diffusivity as Ali et al. [2]:
(10)αnf=knfρCpnf.Dynamic viscosity initiated by Brinkman as Rashad et al. [22]:
(11)μnf=μf1−φ2.5.Thermal conductivity knf introduced by Maxwell as Nasrin [18]:
(12)knfkf=ks+2kf−2∅kf−ksks+2kf+∅kf−ks.To convert the governing equations (1) to (5) into nondimensional form, the following mentioned nondimensional parameters are used:
(13)X=xL,Y=yL,U=uV0,,V=vV0,P=pL2ρnfV02,θ=T−TcTh−Tc,Gr=gβfΔTL3υf2,Re=V0Lυf,Ri=GrRe2,Pr=υfαf,Ha=B0Lσfρfυf,where V0 is the velocity of the vertical walls of the cavity and the nondimensional parameters Gr, Pr, Re, Ri, and Ha are the Grashof, Prandtl, Reynolds, Richardson, and Hartmann numbers, respectively.Nondimensional forms of equations (1) to (5) are as follows:
(14)∂U∂X+∂V∂Y=0,U∂U∂X+V∂U∂Y=−∂P∂X+μnfρnfνf1Re∂2U∂X2+∂2U∂Y2,U∂V∂X+V∂V∂Y=−∂P∂Y+μnfρnfνf1Re∂2V∂X2+∂2V∂Y2+βnfβfGrRe2θ−Ha2ReV,U∂θ∂X+V∂θ∂Y=knfρCpnf1RePr∂2θ∂X2+∂2θ∂Y2.Energy equation for rectangular center heater:(15)∂2θs∂X2+∂2θs∂Y2=0.The nondimensional forms of the associated conditions of heat source and boundaries (6) are given as follows:
(16)U=0,V=1,θ=0atX=0and0≤Y≤L,U=0,V=−1,θ=0atX=Land0≤Y≤L,U=V=0,∂θ∂Y=0atY=Land0≤X≤L,U=V=0,θ=0atthewavybottomsurface,U=V=0,θ=1onthecenterheatersurface.LocalNu along vertical side walls is given as Mahalakshmi et al. [17]:
(17)NuY=−knfkf∂θ∂XX=0,L.After integration (equation (17)), the average Nu is obtained as Mahalakshmi et al. [17]:
(18)Nu¯Y=12∫0LNuYdYX=0+∫0LNuYdYX=L.For the wavy bottom surface,(19)NuX=−knfkf∂θ∂NY=0,where N is the spaces along y-axis or x-axis normal to the wavy wall. The nondimensional temperature gradient is given as
(20)∂θ∂N=1L∂θ∂X2+∂θ∂Y2,Nu¯X=−knfkf1S∫0S∂θ∂NdS,where S is the length of wavy wall.


## 4. Numerical Procedure

The governing equations have been simulated using finite element technique of Galerkin’s method [28, 29]. Finite element method is useful to solve the problems with complicated geometries where analytical solutions cannot be obtained. It divides up a very complicated problem into small elements that can be solved in relation to each other. This method has become an integral part of the design and development of numerous engineering systems. In this simulation process, the whole domain has been subdivided into definite number of triangular meshes. After that, the conservation equations (14) and (15) accompanying boundary conditions (16) are converted into a system of integral equations with the help of Galerkin’s method. By using the Newton-Raphson approach, these equations in nonlinear form are converted into linear form, and then, triangular factorization process is used for solving these linear algebraic equations to find the values of velocity components U,V, pressure P, and temperature θ. The convergence technique for the solution benchmark is demarcated as Ψm+1−Ψm≤10−6 with m as the iteration number and Ψ as ΨU,V,θ. Figure 2 expresses the generated mesh of the computational domain for simulation. It consists of 25,657 domain elements and 829 boundary elements.Figure 2
Mesh configuration for the domain (25,657 domain elements and 829 boundary elements).


## 5. Code Verification

To verify the present solution procedure, it has been compared with the study of mixed convection using moving lid enclosure equipped with center heater and nanofluid of Mahalakshmi et al. [17]. The comparison has been concurred between current study and those of Mahalakshmi et al. demonstrated for center heater length CH=0.75 and various Hartmann numbers at Ri=1 in Figures 3(a) and 3(b). These figures are almost concurrent. The comparison gives a very close propinquity with other published works. The verification with other research works boosts up the credence for graphical and numerical results of the current work.Figure 3
(a) Comparison of the present study against Mahalakshmi et al. [17] through streamlines at CH=0.75,Ri=1, and various Ha. (b) Comparison of the present study against Mahalakshmi et al. [17] through isotherms at CH=0.75,Ri=1, and various Ha.
(a)(b)


## 6. Results and Discussion

MHD mixed convection in an oppositely directed lid-driven cavity filled with Cu-water nanofluid and furnished with wavy bottom wall and internal heat source is considered in this study. The nondimensional parameters characterizing the flow behavior and thermal state within the cavity are as follows: the Reynolds numberRe is considered constant as 100, Grashof number 103≤Gr≤105, Richardson number 0.1≤Ri≤10, and Hartmann number 0≤Ha≤70. The length of internal heat source is considered as 0.2≤CH≤0.6 and is positioned horizontally. The whole investigation is conducted taking Cu-water nanofluid as functional fluid within the enclosure with the Prandtl number Pr=7.56 and volume fraction of nanoparticle 0≤∅≤0.06. The flow characteristics and thermal distribution inside the cavity are investigated through streamlines and isothermal lines whereas heat transfer rate is analyzed through the average Nusselt number in three different regions, namely, forced, mixed, and free convection regions.


### 6.1. Effect of Center Heater Length

The effect ofCH length on fluid flow at various Richardson numbers for different CH length is outlined in Figure 4 through streamlines. The flow characteristics and thermal state within the cavity are governed by forced convection in the region Ri<1, by combined convection in the region Ri=1, and by free convection in the region Ri>1. The criterions like solid concentration of nanoparticles and Ha are kept constant as φ=0.06 and Ha=0 in pursuance of observing the unique power of heater size on fluid flow and temperature field. As seen from the figure, at Ri=0.1, the flow within the enclosure is largely generated by forced convection which is created by the shear force emanated from the upward motion of the left wall and downward motion of the right wall. Since the shear stresses induced by the left and right walls of the enclosure are equal and opposite, streamline contours are permeated almost symmetrically by a primary clockwise rotating unicellular vortex around the center heater which is clearly seen in the first column (at Ri=0.1) of Figure 4. With the increasing length of CH, vortex pattern remains almost similar except the center of the vortices is enlarged as the CH takes more space in the center of the enclosure and the streamlines are shrunk near the vertical walls. Few tiny vortices are also created in the center of the vortex. With raising the Richardson number, to Ri=1, where both the shear stress and buoyancy force are equally dominant, streamlines have started to change its pattern irrespective of CH length. And further increase of Ri, at Ri=10, the shear effect is overwhelmed by the buoyancy effect. In this region, due to the variation of temperature between heated center heater and cold side walls of the enclosure, the fluid near the center heater gets warmer and denser which flows upward and the cold fluid of higher density goes downward. Furthermore, the adiabatic wall resists the air entering into the cavity which creates the circulation of flow, and this continuous process boosts up the fluid flow and rises the number of vortices. The figure evidently expresses that the natural upward flow creates two clockwise vortices in the vertical sides of the cavity and an anticlockwise recirculation cell in the cavity center. By augmentation of CH length, the vortex of the middle part becomes horizontal from vertical due to the reduction of flow area. Figure 5 exhibits the arrow surface of fluid flow at different Ri with ϕ=0.06,Ha=0, and CH=0.4. The clockwise and anticlockwise directions of the vortices are clearly seen from these figures.Figure 4
Streamlines at differentRi with ϕ=0.06, Ha=0, and (i) CH=0.2, (ii) CH=0.4, and (iii) CH=0.6.Figure 5
Arrow surface at differentRi with ϕ=0.06, Ha=0, and CH=0.4.The corresponding temperature distribution which is highly influenced by convection is depicted in Figure6. As seen from the figures, the isotherms are denser at the cavity center as the CH exists there. In forced convection regime Ri=0.1 for all values of CH length, the isotherms are distributed around the CH which is the manifestation of sole conduction heat transfer. Similar trend continues in combined convective region Ri=1 with little distortion in isothermal lines, but in the natural convective region Ri=10, a significant variation is discerned. In this region, the isothermal lines are raising in the upward direction creating a plume shape illustrating that more heat is being transferred due to convection. Figure 7 depicts the consequence of CH length on convection through the average Nusselt number in three different regimes with φ=0.06 and Ha=0. As seen from the figure, with the increasing length of CH, average Nu decreases and vice versa at any constant value of Ri. Figure 6 clearly demonstrates that by increasing the CH length, the isotherms occupy larger area around the CH transferring more heat by conduction. Enhancing conduction reduces convective heat transfer accordingly. The analyzed data of Figure 7 depicts that the mean Nusselt number is highest at CH=0.2, and by increasing the length to CH=0.4, it decreases by 30% and further increase of CH length to 0.6 average Nu decreases by 4% only. To envisage the consequence of Ri on convection, keeping the CH length constant, data analysis reports that heat transfer rate upsurges by 1% and 11% in the mixed and natural convection regimes, respectively, compared to that of forced convection regime at CH=0.2, Ha=0, and ϕ=0.06. Overall heat transfer rate is maximum in the natural convection regime for CH=0.2 and minimum in the forced convection regime for CH=0.6.Figure 6
Isotherms at differentRi with ϕ=0.06, Ha=0, and (i) CH=0.2, (ii) CH=0.4, and (iii) CH=0.6.Figure 7
Variation of averageNu with Ri for different CH length at ϕ=0.06 and Ha=0.Figure8 depicts the discrepancy of average thermal state with the Richardson number for different CH length at ϕ=0.06 and Ha=0. From the figure, it is clear that CH length has a positive relation with temperature increase. It is very logical because with the increasing value of CH perimeter, more heat will be disbursed and fluid temperature will increase accordingly.Figure 8
Variation of average temperature withRi for different CH length at ϕ=0.06 and Ha=0.


### 6.2. Effect of Magnetic Field and Solid Concentration of Nanoparticle

Figure9 presents the consequence of the Hartmann number on fluid flow at various volume fraction of nanoparticles through streamlines. In order to perceive the solitary effect of magnetic field and volume fraction of nanoparticles on fluid flow and temperature field, other parameters like the CH length and Richardson number are kept constant as CH=0.4 and Ri=1. It is evident from the figure that, at Ha=10, the consequence of magnetic field is very insignificant and the streamlines are portrayed by a primary clockwise rotating unicellular vortex within the enclosure which is mostly produced by the mutual consequence of shear stress and buoyancy effect. The core of the recirculating vortex becomes anemic with swelling the value of Ha from 10 to 30 then to 50 gradually. Since the Lorentz force generated by magnetic field resists the buoyancy force, the magnetic field and fluid velocity take action in opposite direction and the augmentation of Ha slows down the locomotion of fluid in the enclosure which declines the strength of the circulation. Furthermore, no major change is noticed in the pattern of streamlines by escalating the value of solid concentration of nanoparticles. Figure 10 exhibits the arrow surface of streamlines at different Ha with Ri=1, CH=0.4, and ϕ=0.03. The clockwise and anticlockwise directions of the vortices are clearly seen from these figures.Figure 9
Streamlines at differentHa with Ri=1, CH=0.4, and (i) ϕ=0, (ii) ϕ=0.03, and (iii) ϕ=0.06.Figure 10
Arrow surface at differentHa with Ri=1, CH=0.4, and ϕ=0.03.The corresponding impact of the Hartmann number and solid concentration of nanoparticles on the thermal state is demonstrated in Figure11. Some heat is generated within fluid of cavity by the internal friction which is created by the magnetic field. At Ha=10, the isotherms are symmetrically spread around the center heater occupying the whole enclosure, but with the increasing value of Ha, the isotherms are more scattered.Figure 11
Isotherms at differentHa with Ri=1, CH=0.4, and (i) ϕ=0, (ii) ϕ=0.03, and (iii) ϕ=0.06.In order to present the heat transfer rate as a function ofHa, the mean Nusselt number at the hot surface for the above-mentioned diverse solid concentration of nanoparticles is depicted in Figure 12. Here, the center heater length and Richardson number are considered constant as CH=0.4 and Ri=1. As shown in the figure, the heat transfer rate upsurges gradually by the escalating value of solid concentration of nanoparticles. The figure also expresses that the augmentation of the Hartmann number reduces the mean Nusselt number for all values of solid concentration of nanoparticles. The reason is that, by addition of Cu nanoparticles with base fluid ϕ=0 water, thermal conductivity of fluid enhances and the increasing value of Cu nanoparticles ϕ=0.03,0.06 accelerate the elevation of this thermal conductivity which boosts up the thermal transport of nanofluid inside the enclosure. The analyzed data of Figure 12 reveals that in the combined convection regime, the addition of 3%, 6%, and 9% of solid nanoparticles escalates the heat transfer rate by 8%, 16%, and 24%, respectively, at a constant magnetic field Ha=30. Heat transfer rate is maximum at Ha=10 with ϕ=0.09 and minimum at Ha=70 with ϕ=0.Figure 12
Variation of averageNu with Ha for various solid concentration of nanoparticles at Ri=1 and CH=0.4.


## 6.1. Effect of Center Heater Length

The effect ofCH length on fluid flow at various Richardson numbers for different CH length is outlined in Figure 4 through streamlines. The flow characteristics and thermal state within the cavity are governed by forced convection in the region Ri<1, by combined convection in the region Ri=1, and by free convection in the region Ri>1. The criterions like solid concentration of nanoparticles and Ha are kept constant as φ=0.06 and Ha=0 in pursuance of observing the unique power of heater size on fluid flow and temperature field. As seen from the figure, at Ri=0.1, the flow within the enclosure is largely generated by forced convection which is created by the shear force emanated from the upward motion of the left wall and downward motion of the right wall. Since the shear stresses induced by the left and right walls of the enclosure are equal and opposite, streamline contours are permeated almost symmetrically by a primary clockwise rotating unicellular vortex around the center heater which is clearly seen in the first column (at Ri=0.1) of Figure 4. With the increasing length of CH, vortex pattern remains almost similar except the center of the vortices is enlarged as the CH takes more space in the center of the enclosure and the streamlines are shrunk near the vertical walls. Few tiny vortices are also created in the center of the vortex. With raising the Richardson number, to Ri=1, where both the shear stress and buoyancy force are equally dominant, streamlines have started to change its pattern irrespective of CH length. And further increase of Ri, at Ri=10, the shear effect is overwhelmed by the buoyancy effect. In this region, due to the variation of temperature between heated center heater and cold side walls of the enclosure, the fluid near the center heater gets warmer and denser which flows upward and the cold fluid of higher density goes downward. Furthermore, the adiabatic wall resists the air entering into the cavity which creates the circulation of flow, and this continuous process boosts up the fluid flow and rises the number of vortices. The figure evidently expresses that the natural upward flow creates two clockwise vortices in the vertical sides of the cavity and an anticlockwise recirculation cell in the cavity center. By augmentation of CH length, the vortex of the middle part becomes horizontal from vertical due to the reduction of flow area. Figure 5 exhibits the arrow surface of fluid flow at different Ri with ϕ=0.06,Ha=0, and CH=0.4. The clockwise and anticlockwise directions of the vortices are clearly seen from these figures.Figure 4
Streamlines at differentRi with ϕ=0.06, Ha=0, and (i) CH=0.2, (ii) CH=0.4, and (iii) CH=0.6.Figure 5
Arrow surface at differentRi with ϕ=0.06, Ha=0, and CH=0.4.The corresponding temperature distribution which is highly influenced by convection is depicted in Figure6. As seen from the figures, the isotherms are denser at the cavity center as the CH exists there. In forced convection regime Ri=0.1 for all values of CH length, the isotherms are distributed around the CH which is the manifestation of sole conduction heat transfer. Similar trend continues in combined convective region Ri=1 with little distortion in isothermal lines, but in the natural convective region Ri=10, a significant variation is discerned. In this region, the isothermal lines are raising in the upward direction creating a plume shape illustrating that more heat is being transferred due to convection. Figure 7 depicts the consequence of CH length on convection through the average Nusselt number in three different regimes with φ=0.06 and Ha=0. As seen from the figure, with the increasing length of CH, average Nu decreases and vice versa at any constant value of Ri. Figure 6 clearly demonstrates that by increasing the CH length, the isotherms occupy larger area around the CH transferring more heat by conduction. Enhancing conduction reduces convective heat transfer accordingly. The analyzed data of Figure 7 depicts that the mean Nusselt number is highest at CH=0.2, and by increasing the length to CH=0.4, it decreases by 30% and further increase of CH length to 0.6 average Nu decreases by 4% only. To envisage the consequence of Ri on convection, keeping the CH length constant, data analysis reports that heat transfer rate upsurges by 1% and 11% in the mixed and natural convection regimes, respectively, compared to that of forced convection regime at CH=0.2, Ha=0, and ϕ=0.06. Overall heat transfer rate is maximum in the natural convection regime for CH=0.2 and minimum in the forced convection regime for CH=0.6.Figure 6
Isotherms at differentRi with ϕ=0.06, Ha=0, and (i) CH=0.2, (ii) CH=0.4, and (iii) CH=0.6.Figure 7
Variation of averageNu with Ri for different CH length at ϕ=0.06 and Ha=0.Figure8 depicts the discrepancy of average thermal state with the Richardson number for different CH length at ϕ=0.06 and Ha=0. From the figure, it is clear that CH length has a positive relation with temperature increase. It is very logical because with the increasing value of CH perimeter, more heat will be disbursed and fluid temperature will increase accordingly.Figure 8
Variation of average temperature withRi for different CH length at ϕ=0.06 and Ha=0.


## 6.2. Effect of Magnetic Field and Solid Concentration of Nanoparticle

Figure9 presents the consequence of the Hartmann number on fluid flow at various volume fraction of nanoparticles through streamlines. In order to perceive the solitary effect of magnetic field and volume fraction of nanoparticles on fluid flow and temperature field, other parameters like the CH length and Richardson number are kept constant as CH=0.4 and Ri=1. It is evident from the figure that, at Ha=10, the consequence of magnetic field is very insignificant and the streamlines are portrayed by a primary clockwise rotating unicellular vortex within the enclosure which is mostly produced by the mutual consequence of shear stress and buoyancy effect. The core of the recirculating vortex becomes anemic with swelling the value of Ha from 10 to 30 then to 50 gradually. Since the Lorentz force generated by magnetic field resists the buoyancy force, the magnetic field and fluid velocity take action in opposite direction and the augmentation of Ha slows down the locomotion of fluid in the enclosure which declines the strength of the circulation. Furthermore, no major change is noticed in the pattern of streamlines by escalating the value of solid concentration of nanoparticles. Figure 10 exhibits the arrow surface of streamlines at different Ha with Ri=1, CH=0.4, and ϕ=0.03. The clockwise and anticlockwise directions of the vortices are clearly seen from these figures.Figure 9
Streamlines at differentHa with Ri=1, CH=0.4, and (i) ϕ=0, (ii) ϕ=0.03, and (iii) ϕ=0.06.Figure 10
Arrow surface at differentHa with Ri=1, CH=0.4, and ϕ=0.03.The corresponding impact of the Hartmann number and solid concentration of nanoparticles on the thermal state is demonstrated in Figure11. Some heat is generated within fluid of cavity by the internal friction which is created by the magnetic field. At Ha=10, the isotherms are symmetrically spread around the center heater occupying the whole enclosure, but with the increasing value of Ha, the isotherms are more scattered.Figure 11
Isotherms at differentHa with Ri=1, CH=0.4, and (i) ϕ=0, (ii) ϕ=0.03, and (iii) ϕ=0.06.In order to present the heat transfer rate as a function ofHa, the mean Nusselt number at the hot surface for the above-mentioned diverse solid concentration of nanoparticles is depicted in Figure 12. Here, the center heater length and Richardson number are considered constant as CH=0.4 and Ri=1. As shown in the figure, the heat transfer rate upsurges gradually by the escalating value of solid concentration of nanoparticles. The figure also expresses that the augmentation of the Hartmann number reduces the mean Nusselt number for all values of solid concentration of nanoparticles. The reason is that, by addition of Cu nanoparticles with base fluid ϕ=0 water, thermal conductivity of fluid enhances and the increasing value of Cu nanoparticles ϕ=0.03,0.06 accelerate the elevation of this thermal conductivity which boosts up the thermal transport of nanofluid inside the enclosure. The analyzed data of Figure 12 reveals that in the combined convection regime, the addition of 3%, 6%, and 9% of solid nanoparticles escalates the heat transfer rate by 8%, 16%, and 24%, respectively, at a constant magnetic field Ha=30. Heat transfer rate is maximum at Ha=10 with ϕ=0.09 and minimum at Ha=70 with ϕ=0.Figure 12
Variation of averageNu with Ha for various solid concentration of nanoparticles at Ri=1 and CH=0.4.


## 7. Conclusion

MHD mixed convective Cu-water nanofluid flow has been studied in a wavy walled lid-driven enclosure. Based on the above discussion, the summary of the findings for varying several pertinent parameters is as follows:(i)
The minutest center heaterCH=0.2 gives the maximum heat transfer rate in the natural convection regime(ii)
Magnetic field controls heat transfer rate robustly, and it is maximum atHa=10. For base fluid, heat transfer rate is decreased by 23%, and for nanofluid, it is decreased by 25% with the increasing value of Ha from 10 to 70 at φ=6%(iii)
Solid concentration of nanoparticles influences strongly the flow behavior. In the mixed convection regime, the dissolved nanoparticles boost up the heat transport system and the nanofluid achieves higher averageNu in comparison with pure base fluid. Average Nu is upsurged by 8%, 16%, and 24% for nanofluid with solid volume concentrations 3%, 6%, and 9%, respectively, in comparison with that of pure base fluid under the same magnetic field Ha=30(iv)
Heat transfer rate muscularly depends on the Richardson number. It is minimum at low Richardson numberRi=0.1. It enhances slightly in the combined convection region but vigorously in the free convection region for all center heater length, and it is maximum at Ri=10(v)
Center heater length has a strong influence on thermal state of fluid inside the cavity. The increasing length of center heater enhances average temperature in all the regions from forced to natural convection



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*Source: 7117186-2023-03-14.xml*