# Magnetic-Mixed Convection in Nanofluid-Filled Cavity Containing Baffles and Rotating Hollow-Cylinders with Roughness Components

Authors: Mohammad Mokaddes Ali; Rowsanara Akhter; Md. Abdul Alim; Md. Musa Miah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Mixed convective heat transfer in a nanofluid-filled lid-driven square cavity equipped with a rotating cylinder, horizontal baffles, and an external magnetic field is numerically examined in this study. A cylinder with triangular components is set at the centre of the cavity while two horizontal baffles are fixed to its vertical walls. The cavity is under the impact of the external magnetic field. Modified Maxwell’s model is taken into consideration to estimate the thermal conductivity of nanofluids. Galerkin FEM is applied to simulate nondimensional governing equations. The computations are carried out for specific ranges of physical parameters, and the results are illustrated through streamlines, isotherms, and average Nusselt number bar charts. Contours plotting indicate that flow circulation and distribution of temperature are significantly affected by the speed of a rotating rough cylinder. The fluid velocity remarkably increases with an increase in speed ratio and Reynolds number but it declines with Hartmann number, baffle length, and volume fraction. Heat transfer rate is substantially augmented by increasing the rotational speed of the rough cylinder, heights of triangular components, and suspended-nanoparticles which are also optimized for increasing baffle’s length and its horizontal arrangement. The findings of this investigation can be applied to improve the cooling efficiency of engineering equipment such as heat exchangers, energy storage systems, electronic equipment, solar collectors, and nuclear reactor safety devices.


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## Body

## 1. Introduction

Convective heat transfer in confined enclosures is of great interest to the engineers and researchers as it is frequently encountered in heat transport processes in various engineering fields such as heat exchangers, electronic cooling, energy storage systems, solar technologies, and nuclear reactor systems. Among the heat transfer processes, convective heat transfer is a complex phenomenon which is developed by the interaction of thermal buoyancy flow caused by temperature differences and shear flow caused by additional forces such as moving surfaces, rotating surfaces, or inlet-outlet flow. In addition, heat transfer associated with magnetic fields has taken researcher’s attention considering its extensive applications in engineering, for example, crystal growth in liquid, electronic packaging, solar technology, nuclear reactor cooling, molten metal purification, and petroleum industry. In this case, magnetic force interacts with the buoyancy flow and produces Lorentz force which influences the fluid flow and heat transport mechanisms. Recently, researchers found limitations in using convectional liquids such as water, oil, and ethylene glycol as coolants named lower thermal conductivity. They have tried to break down this limitation and developed novel heat transfer fluids by amalgamating nanoparticles in different base fluids named nanofluids which have superior properties such as higher thermal conductivity, improved stability, minimal clogging in the flow domain, and reduced pumping power. In this context, many researchers and engineers were interested in accomplishing their theoretical and experimental studies of natural convection/mixed convection/forced convection in different configurations filled with different nanofluids under the influence of the magnetic field or in absence of magnetic field. Some of the related studies have been presented here.Fereidoon et al. [1] investigated mixed convection in a double lid-driven square cavity by using the finite volume method and recommended that heat transfer increases with solid volume fraction at a fixed Reynolds number. They also noted that it is increased for Ri and Re at a fixed solid volume fraction. Later on, Muthtamilselvan and Doh [2] conducted a similar study considering the magnetic field effect in the vertical direction and found the flow, heat, and mass transfer characteristics strongly depends on the magnetic field strength. Ismael et al. [3] examined the effects of partial slip and inclined magnetic field on mixed convection and showed that the convection due to partial slip is controlled with the strength and orientation of the magnetic field. Kasaeipoor et al. [4] utilized a finite volume approach to study mixed convection in a T-shaped cavity under the magnetic field effect. They demonstrated the heat transfer in nanofluid increases while the cavity aspect ratio was increased. Kefayati [5] used Buongiorno’s mathematical model to investigate mixed convection in nonNewtonian nanofluid flow and observed heat and mass transfer enhance with the buoyancy ratio number, whereas mass transfer ameliorated with thermophoresis and Brownian motion. Öztop et al. [6] investigated mixed convection in a wavy-walled cavity filled by nanofluid and noted that enhancement of heat transfer for volume fraction depends on Ha and Ri. Armaghani et al. [7] studied natural convection in a nanofluid-filled baffled L-shaped cavity. In their study, finite volume method was implemented and found heat transfer increasing with the aspect ratio and baffles length. Karimdoost Yasuri et al. [8] conducted similar analysis in a square cavity with a baffle and an external magnetic field. They recorded that heat transfer increases for increasing Ra and baffles length but decreases for Ha. After that, Hussein et al. [9] numerically investigated natural convection in inclined cavity having a cold baffle filled by Al2O3/water nanofluid and found Nusselt number increases with Ra and ϕ, but it decreases with the cavity aspect ratio and inclination angle. Ziam et al. [10] proposed a single-phase nanofluid model to analyze natural convection in a baffled U-shaped cavity. They considered Brinkman and Wasp models and suggested heat transfer rate increases for increased Ra and ϕ but diminishes for Ha. Aljabair et al. [11] used FORTRAN code to solve mixed convection in a nonuniformly heated arc-shaped cavity. They recommended the local and average Nusselt numbers are increasing function of ϕ, Re, and Ra. They also presented numerical results based on correlation equations for the average Nusselt number. Al-Farhany et al. [12] studied magnetohydrodynamic natural convection in an inclined porous cavity with conducting horizontal fins and showed the highest length and the widest gap of the fins cause superior heat transfer situation within the cavity. Shah et al. [13] analyzed the magnetized viscous fluid flow of a water-based hybrid nanofluid containing single-wall carbon nanotubes through a permeable channel by implementing a nanolayer approach and demonstrated that heat flux increases by the induction of single-wall carbon nanotubes in water, and the nanolayer as well as the particle radius causes an enhancement in the thermal conductance of the base fluid. Bilal et al. [14] investigated natural convection flow of power-law fluids in trapezoidal cavity with a nonuniformly U-shaped fin by using the finite element method and COMSOL Multiphysics software (5.6) and highlighted that heat transportation and momentum profile increase with the increase in the Rayleigh number which decline for increased viscosity of the fluid. After that, Bilal et al. [15] performed a similar investigation for a square cavity with nonuniformly T-shaped fin. Later on, Shah et al. [16] studied natural convection flow of power-law fluids an isosceles triangular cavity by using similar software and found noticeable influence of governing parameters on local and average heat flux coefficients and kinetic energy. It was also inferred that a higher Rayleigh number causes enrichment in local heat transfer and kinetic energy, whereas the reverse phenomenon occurs for the power-law index. Another numerical study [17] pointed out that more heat transfer is achieved with the addition of metallic particles to base fluid compared to nonmetallic particles. Moreover, an elevated Nussult number is found for increasing the magnitude of nanoparticle volume fraction, but it depreciates the skin friction coefficient. Qureshi et al. [18] optimized entropy generation for the induction of hybrid magnetized nanoparticles between two coaxially rotating porous disks and showed that entropic generation is reduced with upsurge in inducted hybridized particles volume fraction and Bejam number is found inversely related with entropy generation. After that, Qureshi et al. [19] examined three-dimensional triadic hybrid nanofluid flow between two coaxially rotating disks with magnetic field effects and found that hybrid nanofluids containing Cu-Al2O3 nanoparticles showed significantly better results than those of other hybrid nanoparticles. It was also recorded that the thermal conductance of nanofluids is directly correlated with the morphology of any hybrid nanoparticles. Bilal et al. [20] considered the Cattaneo–Christov heat flux model and transverse magnetic field to analyze heat transfer of Williamson fluid flow over an exponentially stretched the surface by shooting and Runge-Kutta-method and showed that the velocity field is suppressed by the magnetic field and Williamson fluid parameter. Later on, Bilal et al. [21] performed finite difference simulations of viscous fluid flow over a permeable rotating disk to explore the flow in the presence of the transverse magnetic field and recommended that the tangential and radial velocity components are enhanced with the magnetic strength parameter, whereas the opposite trend occurred in the axially directed velocity. The flow and thermal fields within simple geometrical models with/without baffles or fins were numerically explored in these studies [1–21] by using different nanofluids in the presence of the magnetic field or not. It needs to be highlighted that geometrical configuration with appropriate obstruction was not considered in those investigations.Insertion of detached obstructions of various shapes such as circular, square, tilted-square, and triangular has an impact on controlling fluid flow and heat transfer characteristics inside confined enclosures where fluid flow needs to be restricted or bifurcated. Thus, configurations with appropriate obstructions can significantly affect the flow and temperature fields compared to other simple shapes. A large number of studies have been conducted considering these phenomena. Rahman et al. [22] investigated mixed convection in a rectangular cavity with a circular cylinder and pointed out that the dimensionless temperature and heat transfer were strongly dependent on the dimensionless parameters and configurations studied. Majdi et al. [23] used the Fluent 6.3 commercial program to analyze combined convection in a triangular cavity with an insulated cylinder. They illustrated that heat transfer rate is enhanced with the volume fraction of nanoparticles but it decreases with Ri. Ishak et al. [24] studied mixed convection in a trapezoidal cavity having a solid cylinder and filled with nanofluids containing Al2O3 nanoparticles. Their results showed the heat transfer and Bejan number improved with the location and size of the solid cylinder. Aly et al. [25] used the ISPH method to investigate mixed convection in a lid-driven cavity with circular cylinders in motion and suggested that the SPH tool can be applied easily to solve complicated 2D and 3D problems. Bilal et al. [26] analyzed the flow and heat transfer of nonlinear fluid in a square cavity with a cold cylinder and revealed that convective heat transfer and kinetic energy arise for Ra but decrease for the power-law index. The problem of heat transportation of mixed convection in Newtonian fluid enclosed by square cavity with a placement of adiabatic square cylinder was investigated by Khan et al. [27] and revealed that increased Reynolds number causes decrement in kinetic energy of fluid, whereas thermal buoyancy forces and Nusselt number increase for increasing Grashof number. Shah et al. [28] examined the heat and mass transportation of double diffusive natural convection in Casson non-Newtonian fluid enclosed by a hexagonal enclosure with an isothermally heated cylinder in the presence of an inclined magnetic field by practicing COMSOL Multiphysics software and found that heat and mass flux coefficients diminish for the magnetic field effect, whereas heat and mass flux distribution are found upsurging with a Casson fluid parameter. Bansal and Chatterjee [29] performed a numerical study of magneto-convective flow in a lid-driven cavity that included a rotating and heat conducting cylinder. In their study, the flow and thermal fields were found qualitatively change with mixed convective strength, magnetic vied, and concentration of nanoparticles. Another numerical study [30] pointed out 14.2% more heat transfer in a rotating state compared to a motionless state. Later on, Malla et al. [31] studied a similar problem considering a fluid-porous layer using ANSYS fluent software, and a stable system was found at lower Da and minimum heat transfer at about Ri = 10. Ali et al. [32] numerically analyzed the flow and heat transfer in mixed convection through grooved channels, and they noticed the flow and heat transfer mechanisms, due to governing parameters, are more effective with the presence of a rotating heat source. Al-Kouz et al. [33] investigated irreversibilities and heat transfer of mixed convection in wavy enclosure containing a rotating cylinder and showed the flow function and heat transfer enhancement become maximum at increasing Ra. Thus, it is clearly seen that fluid motion and temperature distribution inside a cavity are influenced by the presence of internal blockage. However, irregular surfaces of internal blockage were not considered in these studies, but they can be used to improve or control the flow function and heat transfer enhancement in a closed or open cavity.The irregular surfaces, either static or in motion, of a geometrical configuration generate contributory recirculating regions that can enhance flow mixing within the cavity and heat transfer compared to other smooth configurations. Thus, such complicated configurations can cause substantial improvement in the heat transfer performance and flow mixing and play significant roles in engineering processes such as heat exchangers, electronic cooling systems, solar technologies, nuclear reactors, lubrication systems, and chemical processing equipment. Based on this idea, many studies have been accomplished by the researchers. Gangawane and Manikandan [34] studied natural convection in a square cavity containing a hexagonal block with different thermal conditions and demonstrated the flow and temperature pattern changes were remarkable for both thermal conditions. It was also found that heat transfer decreases with the power-law index. Sheikholeslami et al. [35] investigated transportation of nanoparticles in a circular porous cavity with complex inner shaped in the presence of magnetic force and illustrated nanofluid motion that decreases with the magnetic force, and the Nusselt number increases for reducing cavity porosity. Later on, Alhashash [36] accomplished a similar study considering a square cavity with a hot corrugated cylinder and recommended that heat transfer from a corrugated cylinder is better than a smooth cylinder under specific circumstances. Tayebi and Chamkha [37] carried out parametric analysis of natural convection in a nanofluid field enclosure with a wavy cylinder and pointed out that flow and heat transfer characteristics are controlled by the presence of a wavy conductive cylinder. Ismael [38] utilized a finite difference method to examine mixed convection in an enclosure with an arc-shaped moving wall and showed that heat transfer due to rotational speed is irrespective at lower Ra but significant at higher Ra. Ali et al. [39] numerically investigated mixed convection in a concentric rotating cylinder and an inner sinusoidal cylinder. In their study, COMSOL 5.2a was used to solve the modeled governing equations and illustrated the formation of stream and temperature lines, as well as how the Nusselt number was affected by the number of corrugatings of the inner cylinder and the governing parameters studied. Hamzal et al. [40] analyzed the mixed convection of a rotating cylinder immersed in a nanofluid-filled cavity associated with a magnetic field and heat flux. They concluded the enhancement of heat transfer processes for NPs volume fraction, which is promoted with increasing Ra.Based on the presented literature review, it is evident that mixed convective heat transfer in the presence of a magnetic field is of considerable interest for researchers and engineers and is applicable in science and engineering fields. A lot of research has been conducted for mixed convection in different geometries filled with different fluids. Though mixed convection in different geometries is considered in the open literature, mixed convective heat transfer in a lid-driven nanofluid-filled partially heated cavity having a rotating cylinder with roughness components in the presence of baffles has never been investigated. But it has a significant influence to improve the heat transfer efficiency by enhancing the effective flow circulation due to the presence of roughness components and nanofluid thermal conductivity and is more applicable in engineering fields. Accordingly, authors have been interested to examine the fluid flow and heat transfer behaviours in a nanofluid-filled partially heated cavity equipped by centred rotating cylinder with roughness components and moving top wall and also a pair of baffles attached to the cavity vertical walls under the influence of transverse magnetic field. As per literature survey and author’s knowledge, no such work was reported yet. Galerkin’s finite element method is utilized to simulate the modified governing equations in this study. The numerical results are obtained for different physical parameters and explained via streamlines, isotherms, and average Nusselt number bar charts. This type of configuration may be set up in engineering equipment such as high-performance heat exchangers, energy storage systems, cooling of electronic equipment, solar collectors, space thermal management, and reactor safety devices.


## 2. Physical Model

Figure1 represents the geometrical configuration of a two-dimensional square cavity of length L. A pair of horizontal baffles of 20% cavity length is fixed to the vertical walls of the cavity, and a circular obstacle of radius R (=l0% L) with triangular roughness components is positioned at its centre which is rotated at angular velocity ω. The free region between the cavity and the obstacle is filled with alumina water nanofluid that is heated partially from the cavity bottom wall and cooled from its top moving wall at a uniform velocity U0. Remaining walls and baffles are maintained no-slip condition and kept adiabatic. The mixed convection is induced due to the rotating rough cylinder and top moving wall along with temperature differences of the active thermal condition. A uniform magnetic field affects the flow field, and gravitational acceleration is also activated in the downward direction. The thermophysical properties of nanoparticles and water are available in [41, 42].Figure 1
Physical model of the present study.


## 3. Mathematical Analysis

The vector form of conservation equations representing the flow model is defined considering the Boussinesq approximation and the described physical model as follows [29, 42–44]:(1)∇.V=0,(2)V.∇V=−1ρnf∇p+νnf∇2V+F,(3)V.∇T=anf∇2T.The source terms used in the governing equation (2) are presented in Table 1.Table 1
Elaboration of source terms [43, 44].
F1/ρnfI×B⇀−g⇀βnfT−Tc: Iσnf−∇φ+V×B and ∇2.φ=0The used properties of nanofluid are as follows [30, 32, 41, 42, 45–48]:(4)ρnf=1−ϕρbf+ϕρnp,ρcpnf=1−ϕρcpbf+ϕρcpnp,βnf=1−ϕβbf+ϕβnp,σnf=σbf1+3σnp/σbf−1ϕσnp/σbf+2−σnp/σbf−1ϕ,μnf=μbf1+39.11ϕ+533.9ϕ2,knf=knp+2kbf−2ϕkbf−knpknp+2kbf+ϕkbf−knpkf+ϕρnpcp,np2KBTref3πdnpμbf.Corresponding boundary conditions based on the physical model are specified as follows:(5)Onthetopwall:V=U0,0,T=Tc,Onthebottomwall:V=0,∇T.n=0whenx<L3andx>2L3T=ThwhenL3≤x≤2L3,Ontheverticalwallsandbaffles:V=0and∇T.n=0,Ontheonthecylinder:Vc=VpU0sinθ,VpU0cosθand∇T.n=0,whereVp=ωRand0≤θ≤2π.The dimensionless variables in equation (6) are introduced in the governing equations to make its dimensionless form:(6)x∗,y∗=x,yL,V=V∗U0,p=p∗ρnfU02andθ∗=T−TcTh−Tc,and which are(7)∇.V∗=0,(8)V∗.∇V∗=−∇p∗+νnfνbf1Re∇2V∗+F∗,(9)V∗.∇θ∗=αnfαbf1RePr∇2θ∗.The modified form of the source terms of Table1 and equation (8) are provided in the following Table 2.Table 2
Source terms are [43, 44, 49, 50].
Fx∗0Fy∗βnf/βbfGr/Re2θ∗−ρbf/ρnfσnf/σbfHa2/ReV∗With the help of equation (6), the dimensionless boundary conditions (shown in equation (5)) are written as the following:(10)Onthetopwall:V∗=1,0,θ∗=0,Onthebottomwall:V∗=0,∇θ∗.n=0whenx∗<L3andx∗>2L3θ∗=1,whenL3≤x∗≤2L3,Ontheverticalwallsandbaffles:V∗=0,∇θ∗.n=0,Ontheonthecylinder:Vc∗=Srsinθ,Srcosθand∇θ∗.n=0whereSr=VpU0.


### 3.1. Evaluation of Average Nusselt Number

The local Nusselt number is computed at the heated midsectional wall of the cavity, which is defined in nondimensional form as follows [51, 52]:(11)Nu=−khnfkbf∂θ∗∂y∗,and corresponding average Nusselt number is estimated as follows [51, 52]:(12)N¯u=1LT∫L/32L/3Nudx∗whereLT=2L3−L3,


### 3.2. Numerical Procedure

The numerical simulation of the governing equations (7)–(9) along with the boundary conditions 10a-d has been performed using Galerkin’s weighted finite element method. In this simulation procedure, the governing partial differential equations are converted into integral equations by implementing the Galerkin weighted residual method [53]. The obtained integral equations are as follows:(13)∫ANα∂U∂X+∂V∂YdA=0,(14)∫ANαU∂U∂X+V∂U∂YdA=−∫AHλ∂P∂XdA+νnfνbf1Re∫ANα∂2U∂X2+∂2U∂Y2dA,(15)∫ANαU∂V∂X+V∂V∂YdA=−∫AHλ∂P∂YdA+νnfνbf1Re∫ANα∂2V∂X2+∂2V∂Y2dA+βnfβbfGrRe2∫ANαθdA−ρbfρnfσnfσbfHa2Re∫ANαVdA,(16)∫ANαU∂θ∂X+V∂θ∂YdA=αnfαbf1PrRe∫ANα∂2θ∂X2+∂2θ∂Y2dA.Here,A is the element area, Nαα=1,2,...,6 are the element shape functions or interpolation functions for the velocity components and temperature, and Hλλ=1,2,3. are the element shape functions for the pressure. The Gaussian divergence theorem has been introduced in equations (13)–(16). to generate the boundary integral terms associated with the surface tractions and heat flux in the momentum and energy equations and we obtained the following equations:(17)∫ANα∂U∂X+∂V∂YdA=0,(18)∫ANαU∂U∂X+V∂U∂YdA+∫AHλ∂P∂XdA−νnfνbf1Re∫A∂Nα∂X∂U∂X+∂Nα∂Y∂U∂YdA=∫A0NαAxdA0(19)∫ANαU∂V∂X+V∂V∂YdA+∫AHλ∂P∂YdA−νnfνbf1Re∫A∂Nα∂X∂V∂X+∂Nα∂Y∂V∂YdA−βnfβbfGrRe2∫ANαθdA+ρbfρnfσnfσbfHa2Re∫ANαVdA=∫A0NαAydA0,(20)∫ANαU∂θ∂X+V∂θ∂YdA−αnfαbf1PrRe∫A∂Nα∂X∂θ∂X+∂Nα∂Y∂θ∂YdA=∫AwNαqwdAw,where the surface tractions (Ax,Ay) along the outflow boundary A0 and velocity components and fluid temperature or heat flux (qw) that flows into or out from the domain along the wall boundary Aw. The basic unknown for the above differential equations is the velocity components (U, V), temperature θ, and the pressure P. For the development of the finite element equations, the six-node triangular element is used in this work. All six nodes are associated with velocities as well as temperature, only the three corner nodes are linked with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through the continuity equation. The velocity components, temperature profiles, and linear interpolation for the pressure distribution according to their highest derivative orders for the differential equations (7) to (9) are as follows:(21)UX,Y=NβUβ,(22)VX,Y=NβVβ,(23)θX,Y=Nβθβ,(24)PX,Y=HλPλ,where β=1, 2, … …, 6 and λ=1, 2, 3. Substituting the element velocity component distributions, temperature distributions, particles distributions, and the pressure distribution from equations (21)–(24) into the equations (17)–(20), the finite element equations can be written in the following form:(25)Uβ∫ANαNβ,xdA+Vβ∫ANαNβ,ydA=0,(26)UβUγ∫ANαNβNγ,xdA+VβUγ∫ANαNβNγ,ydA+Pμ∫AHλHμ,xdA−νnfνbf1ReUβ∫ANα,xNβ,x+Nα,yNβ,ydA=∫A0NαAxdA0,(27)UβVγ∫ANαNβNγ,xdA+VβVγ∫ANαNβNγ,ydA+Pμ∫AHλHμ,ydA−νnfνbf1ReVβ∫ANα,xNβ,x+Nα,yNβ,ydA−βnfβbfGrRe2θβ∫ANαNβdA+ρbfρnfσnfσbfHa2ReVβ∫ANαNβdA=∫A0NαAydA0,(28)Uβθγ∫ANαNβNγ,xdA+Vβθγ∫ANαNβNγ,ydA−αnfαbf1PrReθβ∫ANα,xNβ,x+Nα,yNβ,ydA=∫AwNαqwdAw.Now, we consider the coefficients in the above governing equations as follows:(29)Kαβx=∫ANαNβ,xdA,Kαβy=∫ANαNβ,ydA,Kαβγx=∫ANαNβNγ,xdA,Kαβγy=∫ANαNβNγ,ydA,Kαβ=∫ANαNβdA,Sαβxx=∫ANα,xNβ,xdA,Sαβyy=∫ANα,yNβ,xdA,Mλμx=∫AHλHμ,xdA,Mλμy=∫AHλHμ,ydA,Sαβxγx=∫ANαNβ,yNγ,xdA,Sαβyγy=∫ANαNβ,yNγ,ydA,Qαu=∫A0NαAxdA0,Qαv=∫A0NαAydA0,Qαθ=∫AwNαqwdAw.These element matrices are evaluated in closed form for numerical simulation. Details of the derivation for these element matrices are omitted for brevity.With the help of the above coefficients, the finite element equations can be written in the following type:(30)KαβxUβ+KαβyVβ=0,(31)KαβγxUβUγ+KαβγyVβUγ+MλμxPμ−νnfνbf1ReUβSαβxx+Sαβyy=Qαu,(32)KαβγxUβVγ+KαβγyVβVγ+MλμyPμ−νnfνbf1ReVβSαβxx+Sαβyy−βnfβbfGrRe2Kαβθβ+ρbfρnfσnfσbfHa2ReKαβVβ=QαV,(33)KαβγxUβθγ+KαβγyVβθγ−αnfαbf1PrReθβSαβxx+Sαβyy=Qαθ.Using Newton–Raphson method explained by Reddy [54], the obtained nonlinear equations (29) to (32) are converted into linear algebraic equations. Finally, these linear equations are solved by employing the triangular factorization method and reduced integration method expressed by Zeinkiewicz et al. [55]. The convergence criterion of the numerical solution along with error estimation has been set to φm+1−φm≤10−5, where m is the number of iteration and φ is a function of U, V, and θ. The details of the computational procedure are also available in the earlier studies [32, 56, 57], which are well described in [58, 59]. The simple algorithm of this study is exposed through the following flowchart (in Figure 2):Figure 2
Flowchart of the FEM.


### 3.3. Mesh Generation

In two-dimensional configurations, mesh generation is a procedure of subdivision of the geometrical domain considering elements as triangular or quadrilateral called finite elements. These elements are connected to their neighbouring elements using characteristic points known as “nodes”. The required values of the physical quantities of a problem are calculated at every node. Meshing a complicated geometry into a significant number of elements is essential to the numerical simulations, which makes the FEM a powerful tool for solving engineering problems encountered in practical applications. The mesh configuration with triangular elements of this study is presented in Figure3.Figure 3
Mesh generation of the square cavity.


### 3.4. Grid Sensitivity Test

A grid sensitivity test is performed to acquire a grid-irrespective solution. In this regard, the average Nusselt number has been estimated at different mesh systems, and five of them have been presented in Table3 and Figure 4 which confirm that further meshing has very small impacts on the computed average Nusselt number. Based on the presented mesh configurations with an average Nusselt number, it can be decided that meshing with 24444 nodes and 47648 elements is appropriate for an accurate solution of this study.Table 3
Average Nusselt number at different mesh systems.
Nodes (elements)8527 (16526)13125 (25482)17342 (33712)24444 (47648)32702 (63999)Nuav9.4399.5219.5819.6079.618Figure 4
Grid refinement test.


### 3.5. Validation of Computational Procedure

As code validation is required to evaluate the accuracy of the computational procedure of a problem, we have accomplished comparisons considering mixed convection in different configurations:


#### 3.5.1. Validation Case One

Rashad et al. [60] simulated mixed convection in a lid-driven cavity filled with nanofluids by implementing the finite volume method and compared their results with earlier studies reported by Khanafer and Chamkha [61] and Iwatsu et al. [62]. We have simulated the similar studies of those [60–62] by using a finite element method-based numerical code. The obtained results are compared with their results and presented in Table 4. In addition, comparative results of isotherm plotting are also presented in Figure 5(a). In both cases, numerically and graphically, rational agreements are found.Table 4
Comparison of calculated Nuav at Pr = 0.71 and Gr = 102.
ReIwatsu et al. [62]Khanafer and Chamkha [61]Rashad et al. [60]Present studyDev (%)[61]1001.942.011.932.020.494003.843.913.913.940.7710006.336.336.316.390.95Figure 5
(a) Comparison of temperature contours for the present study against the earlier studies [59, 61]. (b) Comparison of streamline and isotherm plotting.
(a)(b)


#### 3.5.2. Validation Case Two

Du et al. [63] used the finite difference method to investigate the influence of a magnetic field on open cavity flow and validated their numerical code with numerical results available in [50, 64]. They also validated their numerical code by comparing stream function and temperature contours against the stream function and temperature contours reported by Ghasemi et al. at [50]. In this study, we have implemented our numerical procedure to solve the problems relating to [50, 63, 64] in special cases, and the obtained results are validated by performing comparisons as shown in Table 5 and Figure 5(b).Table 5
Comparison of average Nusselt number.
Ha=0Ha=30ϕ=0ϕ=0.02ϕ=0ϕ=0.02Ghasemi et al. [50]4.7384.8203.1503.138Pordanjani et al. [64]4.6544.8133.1473.143Du et al. [63]4.7464.8373.1693.238Present study4.7364.8193.1493.221Dev (%)[63]0.21110.4980.6350.527Thus, the above comparisons ensure that the present numerical procedure is suitable to simulate mixed convection in a nanofluid-filled square domain equipped with baffles and a rotating hollow cylinder having roughness components and also an external magnetic field which leads us to carry on our investigation.


## 3.1. Evaluation of Average Nusselt Number

The local Nusselt number is computed at the heated midsectional wall of the cavity, which is defined in nondimensional form as follows [51, 52]:(11)Nu=−khnfkbf∂θ∗∂y∗,and corresponding average Nusselt number is estimated as follows [51, 52]:(12)N¯u=1LT∫L/32L/3Nudx∗whereLT=2L3−L3,


## 3.2. Numerical Procedure

The numerical simulation of the governing equations (7)–(9) along with the boundary conditions 10a-d has been performed using Galerkin’s weighted finite element method. In this simulation procedure, the governing partial differential equations are converted into integral equations by implementing the Galerkin weighted residual method [53]. The obtained integral equations are as follows:(13)∫ANα∂U∂X+∂V∂YdA=0,(14)∫ANαU∂U∂X+V∂U∂YdA=−∫AHλ∂P∂XdA+νnfνbf1Re∫ANα∂2U∂X2+∂2U∂Y2dA,(15)∫ANαU∂V∂X+V∂V∂YdA=−∫AHλ∂P∂YdA+νnfνbf1Re∫ANα∂2V∂X2+∂2V∂Y2dA+βnfβbfGrRe2∫ANαθdA−ρbfρnfσnfσbfHa2Re∫ANαVdA,(16)∫ANαU∂θ∂X+V∂θ∂YdA=αnfαbf1PrRe∫ANα∂2θ∂X2+∂2θ∂Y2dA.Here,A is the element area, Nαα=1,2,...,6 are the element shape functions or interpolation functions for the velocity components and temperature, and Hλλ=1,2,3. are the element shape functions for the pressure. The Gaussian divergence theorem has been introduced in equations (13)–(16). to generate the boundary integral terms associated with the surface tractions and heat flux in the momentum and energy equations and we obtained the following equations:(17)∫ANα∂U∂X+∂V∂YdA=0,(18)∫ANαU∂U∂X+V∂U∂YdA+∫AHλ∂P∂XdA−νnfνbf1Re∫A∂Nα∂X∂U∂X+∂Nα∂Y∂U∂YdA=∫A0NαAxdA0(19)∫ANαU∂V∂X+V∂V∂YdA+∫AHλ∂P∂YdA−νnfνbf1Re∫A∂Nα∂X∂V∂X+∂Nα∂Y∂V∂YdA−βnfβbfGrRe2∫ANαθdA+ρbfρnfσnfσbfHa2Re∫ANαVdA=∫A0NαAydA0,(20)∫ANαU∂θ∂X+V∂θ∂YdA−αnfαbf1PrRe∫A∂Nα∂X∂θ∂X+∂Nα∂Y∂θ∂YdA=∫AwNαqwdAw,where the surface tractions (Ax,Ay) along the outflow boundary A0 and velocity components and fluid temperature or heat flux (qw) that flows into or out from the domain along the wall boundary Aw. The basic unknown for the above differential equations is the velocity components (U, V), temperature θ, and the pressure P. For the development of the finite element equations, the six-node triangular element is used in this work. All six nodes are associated with velocities as well as temperature, only the three corner nodes are linked with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through the continuity equation. The velocity components, temperature profiles, and linear interpolation for the pressure distribution according to their highest derivative orders for the differential equations (7) to (9) are as follows:(21)UX,Y=NβUβ,(22)VX,Y=NβVβ,(23)θX,Y=Nβθβ,(24)PX,Y=HλPλ,where β=1, 2, … …, 6 and λ=1, 2, 3. Substituting the element velocity component distributions, temperature distributions, particles distributions, and the pressure distribution from equations (21)–(24) into the equations (17)–(20), the finite element equations can be written in the following form:(25)Uβ∫ANαNβ,xdA+Vβ∫ANαNβ,ydA=0,(26)UβUγ∫ANαNβNγ,xdA+VβUγ∫ANαNβNγ,ydA+Pμ∫AHλHμ,xdA−νnfνbf1ReUβ∫ANα,xNβ,x+Nα,yNβ,ydA=∫A0NαAxdA0,(27)UβVγ∫ANαNβNγ,xdA+VβVγ∫ANαNβNγ,ydA+Pμ∫AHλHμ,ydA−νnfνbf1ReVβ∫ANα,xNβ,x+Nα,yNβ,ydA−βnfβbfGrRe2θβ∫ANαNβdA+ρbfρnfσnfσbfHa2ReVβ∫ANαNβdA=∫A0NαAydA0,(28)Uβθγ∫ANαNβNγ,xdA+Vβθγ∫ANαNβNγ,ydA−αnfαbf1PrReθβ∫ANα,xNβ,x+Nα,yNβ,ydA=∫AwNαqwdAw.Now, we consider the coefficients in the above governing equations as follows:(29)Kαβx=∫ANαNβ,xdA,Kαβy=∫ANαNβ,ydA,Kαβγx=∫ANαNβNγ,xdA,Kαβγy=∫ANαNβNγ,ydA,Kαβ=∫ANαNβdA,Sαβxx=∫ANα,xNβ,xdA,Sαβyy=∫ANα,yNβ,xdA,Mλμx=∫AHλHμ,xdA,Mλμy=∫AHλHμ,ydA,Sαβxγx=∫ANαNβ,yNγ,xdA,Sαβyγy=∫ANαNβ,yNγ,ydA,Qαu=∫A0NαAxdA0,Qαv=∫A0NαAydA0,Qαθ=∫AwNαqwdAw.These element matrices are evaluated in closed form for numerical simulation. Details of the derivation for these element matrices are omitted for brevity.With the help of the above coefficients, the finite element equations can be written in the following type:(30)KαβxUβ+KαβyVβ=0,(31)KαβγxUβUγ+KαβγyVβUγ+MλμxPμ−νnfνbf1ReUβSαβxx+Sαβyy=Qαu,(32)KαβγxUβVγ+KαβγyVβVγ+MλμyPμ−νnfνbf1ReVβSαβxx+Sαβyy−βnfβbfGrRe2Kαβθβ+ρbfρnfσnfσbfHa2ReKαβVβ=QαV,(33)KαβγxUβθγ+KαβγyVβθγ−αnfαbf1PrReθβSαβxx+Sαβyy=Qαθ.Using Newton–Raphson method explained by Reddy [54], the obtained nonlinear equations (29) to (32) are converted into linear algebraic equations. Finally, these linear equations are solved by employing the triangular factorization method and reduced integration method expressed by Zeinkiewicz et al. [55]. The convergence criterion of the numerical solution along with error estimation has been set to φm+1−φm≤10−5, where m is the number of iteration and φ is a function of U, V, and θ. The details of the computational procedure are also available in the earlier studies [32, 56, 57], which are well described in [58, 59]. The simple algorithm of this study is exposed through the following flowchart (in Figure 2):Figure 2
Flowchart of the FEM.


## 3.3. Mesh Generation

In two-dimensional configurations, mesh generation is a procedure of subdivision of the geometrical domain considering elements as triangular or quadrilateral called finite elements. These elements are connected to their neighbouring elements using characteristic points known as “nodes”. The required values of the physical quantities of a problem are calculated at every node. Meshing a complicated geometry into a significant number of elements is essential to the numerical simulations, which makes the FEM a powerful tool for solving engineering problems encountered in practical applications. The mesh configuration with triangular elements of this study is presented in Figure3.Figure 3
Mesh generation of the square cavity.


## 3.4. Grid Sensitivity Test

A grid sensitivity test is performed to acquire a grid-irrespective solution. In this regard, the average Nusselt number has been estimated at different mesh systems, and five of them have been presented in Table3 and Figure 4 which confirm that further meshing has very small impacts on the computed average Nusselt number. Based on the presented mesh configurations with an average Nusselt number, it can be decided that meshing with 24444 nodes and 47648 elements is appropriate for an accurate solution of this study.Table 3
Average Nusselt number at different mesh systems.
Nodes (elements)8527 (16526)13125 (25482)17342 (33712)24444 (47648)32702 (63999)Nuav9.4399.5219.5819.6079.618Figure 4
Grid refinement test.


## 3.5. Validation of Computational Procedure

As code validation is required to evaluate the accuracy of the computational procedure of a problem, we have accomplished comparisons considering mixed convection in different configurations:


### 3.5.1. Validation Case One

Rashad et al. [60] simulated mixed convection in a lid-driven cavity filled with nanofluids by implementing the finite volume method and compared their results with earlier studies reported by Khanafer and Chamkha [61] and Iwatsu et al. [62]. We have simulated the similar studies of those [60–62] by using a finite element method-based numerical code. The obtained results are compared with their results and presented in Table 4. In addition, comparative results of isotherm plotting are also presented in Figure 5(a). In both cases, numerically and graphically, rational agreements are found.Table 4
Comparison of calculated Nuav at Pr = 0.71 and Gr = 102.
ReIwatsu et al. [62]Khanafer and Chamkha [61]Rashad et al. [60]Present studyDev (%)[61]1001.942.011.932.020.494003.843.913.913.940.7710006.336.336.316.390.95Figure 5
(a) Comparison of temperature contours for the present study against the earlier studies [59, 61]. (b) Comparison of streamline and isotherm plotting.
(a)(b)


### 3.5.2. Validation Case Two

Du et al. [63] used the finite difference method to investigate the influence of a magnetic field on open cavity flow and validated their numerical code with numerical results available in [50, 64]. They also validated their numerical code by comparing stream function and temperature contours against the stream function and temperature contours reported by Ghasemi et al. at [50]. In this study, we have implemented our numerical procedure to solve the problems relating to [50, 63, 64] in special cases, and the obtained results are validated by performing comparisons as shown in Table 5 and Figure 5(b).Table 5
Comparison of average Nusselt number.
Ha=0Ha=30ϕ=0ϕ=0.02ϕ=0ϕ=0.02Ghasemi et al. [50]4.7384.8203.1503.138Pordanjani et al. [64]4.6544.8133.1473.143Du et al. [63]4.7464.8373.1693.238Present study4.7364.8193.1493.221Dev (%)[63]0.21110.4980.6350.527Thus, the above comparisons ensure that the present numerical procedure is suitable to simulate mixed convection in a nanofluid-filled square domain equipped with baffles and a rotating hollow cylinder having roughness components and also an external magnetic field which leads us to carry on our investigation.


## 3.5.1. Validation Case One

Rashad et al. [60] simulated mixed convection in a lid-driven cavity filled with nanofluids by implementing the finite volume method and compared their results with earlier studies reported by Khanafer and Chamkha [61] and Iwatsu et al. [62]. We have simulated the similar studies of those [60–62] by using a finite element method-based numerical code. The obtained results are compared with their results and presented in Table 4. In addition, comparative results of isotherm plotting are also presented in Figure 5(a). In both cases, numerically and graphically, rational agreements are found.Table 4
Comparison of calculated Nuav at Pr = 0.71 and Gr = 102.
ReIwatsu et al. [62]Khanafer and Chamkha [61]Rashad et al. [60]Present studyDev (%)[61]1001.942.011.932.020.494003.843.913.913.940.7710006.336.336.316.390.95Figure 5
(a) Comparison of temperature contours for the present study against the earlier studies [59, 61]. (b) Comparison of streamline and isotherm plotting.
(a)(b)


## 3.5.2. Validation Case Two

Du et al. [63] used the finite difference method to investigate the influence of a magnetic field on open cavity flow and validated their numerical code with numerical results available in [50, 64]. They also validated their numerical code by comparing stream function and temperature contours against the stream function and temperature contours reported by Ghasemi et al. at [50]. In this study, we have implemented our numerical procedure to solve the problems relating to [50, 63, 64] in special cases, and the obtained results are validated by performing comparisons as shown in Table 5 and Figure 5(b).Table 5
Comparison of average Nusselt number.
Ha=0Ha=30ϕ=0ϕ=0.02ϕ=0ϕ=0.02Ghasemi et al. [50]4.7384.8203.1503.138Pordanjani et al. [64]4.6544.8133.1473.143Du et al. [63]4.7464.8373.1693.238Present study4.7364.8193.1493.221Dev (%)[63]0.21110.4980.6350.527Thus, the above comparisons ensure that the present numerical procedure is suitable to simulate mixed convection in a nanofluid-filled square domain equipped with baffles and a rotating hollow cylinder having roughness components and also an external magnetic field which leads us to carry on our investigation.


## 4. Results and Discussion

In this section, simulated results are obtained for mixed convection of Al2O3-water nanofluid in a lid-driven cavity with horizontal baffles and a cantered rotating rough cylinder, and the flow and thermal fields are demonstrated using physical quantities streamlines, temperature contours, and average Nusselt number bar charts consistently. During simulation and graphic presentation, variation in two parameters is considered simultaneously while the values of others are kept fixed as Pr = 6.2, Gr = 100, Sr = 10, Ha = 10, Re = 100, ϕ = 1%, Bh = 0.20 L, and Δ = 0.0275 L. As a result, the sole effects of the two governing parameters have been exhibited combinedly. The parametric ranges of the analysis are presented in Table 6.Table 6
Parametric ranges.
Sr: Ha0–20 : 0–50Re: Ha10–200 : 0–50ϕ: Ha0%–5% : 0–50Δ: Ha0.0000 L–0.0375 L : 0–50Bl: Ha0.10 L–0.25 L : 0–50Baffles orientation: HaBaffles positions: 0–50


### 4.1. Effects of Speed Ratio (Sr) and Magnetic Field (Ha) on Streamlines

Figure6 illustrates the flow field by plotting streamlines for different values of the speed ratio of the rotating rough cylinder and magnetic field strength in the presence of mounted horizontal baffles. In Figure 6(a) at motionless state of the rough cylinder and Ha = 0, heated-fluid ascends near the left vertical wall due to thermal induced-buoyancy flow and then descends quickly from top to bottom near the right vertical walls as the top moving cold wall and insulated vertical wall flow circulates in downward direction. As a result, clockwise streamline circulation is developed within the cavity, which is also confirmed by the negative flow strength. This convective circulation occupies the whole cavity, and it is found suppressing by the presence of horizontal baffles. Moreover, the highest flow strength is noticed at -0.065 within the core circulation near the top moving wall. After that, rough cylinder gets rotating at speed Sr = 5, the flow circulation is restructured by compressing the clockwise-circulations toward the top wall and forming anticlockwise circulation around the rough cylinder with the influence of rotating inertia of rough cylinder. It is also found that flow velocity increased rapidly, and the strength as well as intensification of flow circulations around the hollow cylinder is much stronger than those close to the top wall. Later on, when speed ratio increased to Sr = 10, more rotating inertia is produced within the cavity, and hence, the effect of the moving wall is dominated. As a result, streamline circulation becomes more intensified with higher strength around the rotating obstacle. Further increase in rotating speed to Sr = 20, the anticlockwise circulation gets maximum size and strength, whereas the clockwise circulation becomes small and close to the top wall. In order to visualize the impact of the speed of the rotating rough cylinder, we have recorded the numerical flow velocities and these are 0.065 at Sr = 0, 0.060 at Sr = 5, 1.10 at Sr = 10, and 2.40 at Sr = 20. In all diagrams, streamline circulations are found squeezing from its both sides due to the presence of adiabatic baffles, as a result flow circulation is retarded by the baffles and gets lower strength. On the other hand, mixed convection circulations are remained almost similar for varying rotating speed while magnetic field is imposed at strength Ha = 10 compared to the earlier case (Ha = 0), but its strength decreases remarkably. Further increase in Ha = 25, streamlines are spaced out with lower strength compared to the case at Ha = 10. In addition, streamlines are more affected and spaced out at Ha = 50 and also gets minimum strength. The physical consequences behind it is the Lorentz’s force retards flow velocity which is produced due to the interaction of nanofluid buoyancy flow and shear flow with an imposed magnetic force, and the Lorentz’s force escalates with each increment in the magnetic effect. As a result, flow circulation declines with an increase in Ha for all Ra. Thus, one can conclude that the variation in flow characteristics is more noticeable at greater magnetic field strengths than at lower ones,while compared in the absence of a magnetic field.Figure 6
Variation in streamline plotting for different Sr and Ha.
(a)(b)(c)(d)


### 4.2. Effects of Speed Ratio (Sr) and Magnetic Field (Ha) on Isotherms

Figure7 represents the thermal field via temperature contour plotting as well as its topology due to combined variation in rotational speed of rough cylinder and magnetic field strength at fixed values of other parameters such as Pr = 6.2, Re = 100, Gr = 100, and ϕ = 1%. In Figure 7(a), at Sr = 0 and Ha = 0, isotherms are densely visible over the heated wall and then shifted in an upward direction near the left vertical wall resulting in an imposed thermal condition and clockwise flow circulation due to the top moving wall. In the case of a rotating rough cylinder at Sr = 5 (in a counter-clockwise direction), isotherms are reshuffled by forming a spinning shape around the rough cylinder that occupies a fluid domain inside the cavity. Moreover, isotherms close to the cold moving wall are found twisting in the direction of lid-wall motion. In all diagrams, especially while the rough cylinder is in motion, isotherm distribution is found reforming with the presence of baffles where the lower isotherm pack seems to be connected to the lower end of the left adiabatic baffle and another isotherm pack seems to initiate from the top end of the right adiabatic baffle and then complete isotherm circulation in the core region. It is worth noting that cylinder rotating -inertia dominates the lid force from the core region to the bottom region, and hence isotherms tend to spin in the direction of the rough cylinder. With each subsequent increment in rotation speed, isothermal distribution within the cavity is significantly affected, core circulation becomes much stronger, and the isothermal pack near the bottom region is also more condensed toward the heated wall. As a result, more convective heat is released from the heated wall for each increment in the rotational speed of the rough cylinder. When the magnetic field is imposed at strength Ha = 10, small changes are noticed in temperature contour distribution for different speed ratios, but for Ha = 25 remarkable changes are recorded in the isotherms distribution for all Sr, especially in the core distribution of temperature contours. Further increment in magnetic field strength at Ha = 50, significant changes are observed in isotherm distributions within the closed domain which reflects the magnetic field has a significant impact on the distribution of isotherms as well as temperature zones inside the cavity. Moreover, temperature plotting is found different for both increasing speed of rotation and strength of magnetic fields. Consequently, convective heat transferring reduces with Ha. In addition, different colours of temperature zones in the fluid domain indicate the topologies of the thermal field for varying amounts of Sr and Ha, respectively.Figure 7
Variation in temperature contour plotting for different Sr and Ha.
(a)(b)(c)(d)


### 4.3. Effects of Speed Ratio (Sr) and Baffles Length (Bl) on Streamlines

The streamline plotting in nanofluid flow circulation at different speed ratios of rotating rough cylinder and baffles length is presented in Figure8. In all diagrams, streamline circulations are found squeezing by the baffles and it increases with the increase in length of both baffles. As a result, flowing nanofluid encounters more resistive force for longer baffles, and hence the strength of flow circulation is reducednoticeably. Moreover, the streamline pattern of core circulations is unchanged except in the regions where the baffles are horizontally located. In addition, secondary circulations over the core circulation are also more affected by the increased length of baffles. Thus, longer baffles cause more changes in the flow velocity as well as circulations compared to shorter baffles. Moreover, impact of baffles on the flow field is more effective while rough cylinder is rotated at highest speed ratio than lower as 44.44% reduction is found in flow strength at Sr = 20 that becomes 17.24% at Sr = 0.Figure 8
Variation in streamline plotting for different Sr and Bl.
(a)(b)(c)(d)


### 4.4. Effects of Speed Ratio (Sr) and Baffles Length (Bl) on Isotherms

Besides this, Figure9 displays isotherms of distribution when speed ratio and baffle length are increased simultaneously. The distribution of temperature contours is found and changes gradually especially in the core region, with the increase in length of baffles and denseness of isotherms, are also found getting closer to the bottom heated wall which results in heat transferring increase with longer baffles. It is required to note that the impact of baffle length on the thermal field decreases while the speed ratio gets lower strength and it seems to be insignificant in the motionless state of the rough cylinder.Figure 9
Variation in temperature contour plotting for different Sr and Bl.
(a)(b)(c)(d)


### 4.5. Effects of Reynolds Number (Re) and Magnetic Field (Ha) on Streamlines

The plotting of streamlines due to combined variation in Reynolds and Hartmann numbers is demonstrated in Figure10. In Figures 10(a)–10(d), strong convective counter-clockwise flow circulations are observed around the rough cylinder due to the influence of cylinder rotating-inertia and the imposed thermal condition. In addition, two symmetric clockwise flow circulations are also visualized over the centred anticlockwise circulation with the assistance of lid-wall inertia. However, the strength of convective flow circulations is observed increasing with the increase in Reynolds numbers and streamlines are also intensified toward the cylinder because increase in Re causes increment in fluid inertia which results rotating inertia and lid inertia are both increased, and hence strength as well as concentration of streamlines increased but the pattern of streamlines remained similar for varying Re. Besides this, it is noticed that the flow velocity gets lower strength, and distribution of streamlines is spaced out toward the boundaries with an increasing Ha for all Re. These variations are more noticeable in higher magnetic strength than comparative lower ones.Figure 10
Variation in streamline plotting for different Re and Ha.
(a)(b)(c)(d)


### 4.6. Effects of Reynolds Number (Re) and Magnetic Field (Ha) on Isotherms

On the other hand, isotherm distributions at different Re and Ha are depicted in Figure11. It is observed that isotherms are generated from the heated wall and distorted around the rough cylinder covering the fluid domain due to the rotating rough cylinder and moving wall in the presence of baffles. As lower Re causes lower fluid inertia, rotating inertia and lid inertia lose their inertia at Re = 10, and hence a weak isotherm circulation is visible around the cylinder and isotherms become wavy with lower gradient near the top wall (as seen in Figure 11(a) at Ha = 0). As a result, lower heat transfer occurs at Re = 10. After that, increasing Re increase fluid inertia consequently, both rotating inertia and lid inertia become prominent, and hence isotherms circulation around the cylinder become stronger and isotherms are more twisted near the top moving wall up to Re = 100. Then at Re = 200, circulation and distortion of isotherms become more stronger having higher thermal gradient which results more convective heat released at Re = 200. When magnetic field is imposed at Ha = 10, isotherms distribution slightly changed for all Re. Further increase in Ha, circulation, and distortion of isotherms remarkably get affected, and the effect of Ha is found prominent at lower Re than at higher Re. Moreover, dense circulation rounding the rough cylinder tends to disappear, and isotherms are spaced out with a lower curvature at Ha = 50, which confirms the substantial effect of the magnetic field on isotherm distribution for all Re.Figure 11
Variation in temperature contour plotting for different Re and Ha.
(a)(b)(c)(d)


### 4.7. Effects of Volume Fraction (ϕ) and Magnetic Field (Ha) on Streamlines

Figure12 delineates streamlines plotting for different volume fractions of nanoparticles in a lid-driven cavity having a rotating cylinder with triangular components in the presence of a magnetic field or not. Due to induced thermal and velocity boundary conditions, a primary convective strong flow circulations along with two secondary small circulation are visible in the base fluid at fixed parametric values as Pr = 6.2, Gr = 100, Sr = 10, Re = 100, Bh = 0.20 L, and Δ = 0.0275 L, which remain almost similar in nanofluids at different concentrations (1%, 3%, and 5%), but flow strength is found declining remarkably for the additional nanoparticles at vol. of 1%, 3%, and 5% in the base fluid. These results are expected since the amalgamation of nanoparticles increases the density of the working fluid and decelerates flow velocity within the cavity. In order to understand the flow magnitude at different amounts of nanoparticles more accurately, one can find the maximum velocity at the core flow circulations which are 1.20, 1.10, 0.90, and 0.80, respectively, at vol. of 1%, 3%, and 5%. It is also found that flow velocity decreases for imposing and increasing magnetic field strength transverse to the flow circulation. Moreover, denseness of streamlines reduces with each increment in Ha and spaced out toward the sidewalls. In addition, maximum declination in flow magnitude id occurred at simultaneous changes in the volume fraction and magnetic field which are recorded as 1.2 (ϕ = 0%, Ha = 0), 1.0 (ϕ = 1%, Ha = 10), 0.70 (ϕ = 3%, Ha = 25), and 0.45 (ϕ = 5%, Ha = 50).Figure 12
Variation in streamline plotting for differentϕ and Ha.
(a)(b)(c)(d)


### 4.8. Effect of Volume Fraction (ϕ) and Magnetic Field (Ha) on Isotherms

Beside these, distributions of temperature contours in base fluid and nanofluid with 1% volume fraction are found (in Figure13(a)) qualitatively similar but minute changes are noticed for vol. of 3% and 5%. The physical reason behind is that the presence of nanoparticles in base fluid decreases convection mode and hence increases the domination of conduction mode heat transfer. It is necessary to note that isotherms distributions substantially change in both base fluid and nanofluid at highest magnetic field strength compared to at lowest.Figure 13
Variation in temperature contour plotting for differentϕ and Ha.
(a)(b)(c)(d)


### 4.9. Effect of Ha along with Sr, Re,ϕ, Δ, Bl, and Baffles Orientations on Average Nusselt Number (Nuav)

Figure14 illustrates heat transferring in the fluid flow domain due to physical parameters via average Nusselt number bar charts. Figure 14(a) confirms the heat transfer rate augments rapidly with increasing rotational speed of rough cylinder because increased rotational inertia due to Sr increases flow circulation significantly. It is also visualized in Figure 14(a) that higher magnetic strength decreases the heat transfer rate at each Sr as the interaction of magnetic force with convective flow circulation generates Lorentz’s force, which reduces the flow velocity and produces more temperature within the fluid domain. From Figure 14(b), it is seen that the heat transfer rate augments monotonically by the increasing Re where maximum heat transfer is recorded in forced convection dominated regime and minimum for mixed convection regime. The physics behinds it is that higher Re increases fluid inertia which accelerates cylinder rotating inertia and also lid inertia that causes maximum flow circulation and heat transfer within the cavity. In Figure 14(c), the heat transfer rate is found increasing the function of the amount of nanoparticles. This phenomenon is expected as higher amounts of nanoparticles improves nanofluid thermal conductivity, and hence capability of energy transportation in the flow domain. A similar trend is also observed in Figure 14(d) for the presence and increase in height of triangular components on the rotating cylinder, as higher heights of triangular components increase rotating inertia and flow velocity as well. In order to understand the impact of roughness components on heat transferring precisely, it can be recorded that 13.20% (at Ha = 0) more heat transfer takes place for rotating rough cylinders compared to rotating smooth cylinders, and it becomes 10.14%, while the magnetic field is activated at strength Ha = 50. On the other hand, enhancement in heat transfer is found for enlarging baffles length (seen in Figure 14(e)) as the temperature contours were found getting closer to the heated wall with longer baffles and also concentrated in its distribution around the rough cylinder, which leads to an increase in heat transferring inside the cavity (shown in Figure 9). Moreover, heat transfer rate increases by 42.42% while baffle length changes from 0.10 L to 0.25 L in the absence of a magnetic field but reduces to 34.53% whilethe magnetic field is activated at a strength of Ha = 50. Results in Figure 14(f) indicate that the average Nusselt number strongly depends on the orientation of baffles, and maximum heat transfer occurs while baffles are horizontally fixed at the cavity’s vertical walls. In all bar charts of Nusselt number, it is observed that the heat transfer rate decelerates monotonically with the increase in magnetic field strength resulting in active Lorentz’s force due to the magnetic field effect.Figure 14
Variation in average Nusselt for different Sr, Ha, Re,ϕ, Δ, Bl, and orientation of baffles.
(a)(b)(c)(d)(e)(f)


## 4.1. Effects of Speed Ratio (Sr) and Magnetic Field (Ha) on Streamlines

Figure6 illustrates the flow field by plotting streamlines for different values of the speed ratio of the rotating rough cylinder and magnetic field strength in the presence of mounted horizontal baffles. In Figure 6(a) at motionless state of the rough cylinder and Ha = 0, heated-fluid ascends near the left vertical wall due to thermal induced-buoyancy flow and then descends quickly from top to bottom near the right vertical walls as the top moving cold wall and insulated vertical wall flow circulates in downward direction. As a result, clockwise streamline circulation is developed within the cavity, which is also confirmed by the negative flow strength. This convective circulation occupies the whole cavity, and it is found suppressing by the presence of horizontal baffles. Moreover, the highest flow strength is noticed at -0.065 within the core circulation near the top moving wall. After that, rough cylinder gets rotating at speed Sr = 5, the flow circulation is restructured by compressing the clockwise-circulations toward the top wall and forming anticlockwise circulation around the rough cylinder with the influence of rotating inertia of rough cylinder. It is also found that flow velocity increased rapidly, and the strength as well as intensification of flow circulations around the hollow cylinder is much stronger than those close to the top wall. Later on, when speed ratio increased to Sr = 10, more rotating inertia is produced within the cavity, and hence, the effect of the moving wall is dominated. As a result, streamline circulation becomes more intensified with higher strength around the rotating obstacle. Further increase in rotating speed to Sr = 20, the anticlockwise circulation gets maximum size and strength, whereas the clockwise circulation becomes small and close to the top wall. In order to visualize the impact of the speed of the rotating rough cylinder, we have recorded the numerical flow velocities and these are 0.065 at Sr = 0, 0.060 at Sr = 5, 1.10 at Sr = 10, and 2.40 at Sr = 20. In all diagrams, streamline circulations are found squeezing from its both sides due to the presence of adiabatic baffles, as a result flow circulation is retarded by the baffles and gets lower strength. On the other hand, mixed convection circulations are remained almost similar for varying rotating speed while magnetic field is imposed at strength Ha = 10 compared to the earlier case (Ha = 0), but its strength decreases remarkably. Further increase in Ha = 25, streamlines are spaced out with lower strength compared to the case at Ha = 10. In addition, streamlines are more affected and spaced out at Ha = 50 and also gets minimum strength. The physical consequences behind it is the Lorentz’s force retards flow velocity which is produced due to the interaction of nanofluid buoyancy flow and shear flow with an imposed magnetic force, and the Lorentz’s force escalates with each increment in the magnetic effect. As a result, flow circulation declines with an increase in Ha for all Ra. Thus, one can conclude that the variation in flow characteristics is more noticeable at greater magnetic field strengths than at lower ones,while compared in the absence of a magnetic field.Figure 6
Variation in streamline plotting for different Sr and Ha.
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## 4.2. Effects of Speed Ratio (Sr) and Magnetic Field (Ha) on Isotherms

Figure7 represents the thermal field via temperature contour plotting as well as its topology due to combined variation in rotational speed of rough cylinder and magnetic field strength at fixed values of other parameters such as Pr = 6.2, Re = 100, Gr = 100, and ϕ = 1%. In Figure 7(a), at Sr = 0 and Ha = 0, isotherms are densely visible over the heated wall and then shifted in an upward direction near the left vertical wall resulting in an imposed thermal condition and clockwise flow circulation due to the top moving wall. In the case of a rotating rough cylinder at Sr = 5 (in a counter-clockwise direction), isotherms are reshuffled by forming a spinning shape around the rough cylinder that occupies a fluid domain inside the cavity. Moreover, isotherms close to the cold moving wall are found twisting in the direction of lid-wall motion. In all diagrams, especially while the rough cylinder is in motion, isotherm distribution is found reforming with the presence of baffles where the lower isotherm pack seems to be connected to the lower end of the left adiabatic baffle and another isotherm pack seems to initiate from the top end of the right adiabatic baffle and then complete isotherm circulation in the core region. It is worth noting that cylinder rotating -inertia dominates the lid force from the core region to the bottom region, and hence isotherms tend to spin in the direction of the rough cylinder. With each subsequent increment in rotation speed, isothermal distribution within the cavity is significantly affected, core circulation becomes much stronger, and the isothermal pack near the bottom region is also more condensed toward the heated wall. As a result, more convective heat is released from the heated wall for each increment in the rotational speed of the rough cylinder. When the magnetic field is imposed at strength Ha = 10, small changes are noticed in temperature contour distribution for different speed ratios, but for Ha = 25 remarkable changes are recorded in the isotherms distribution for all Sr, especially in the core distribution of temperature contours. Further increment in magnetic field strength at Ha = 50, significant changes are observed in isotherm distributions within the closed domain which reflects the magnetic field has a significant impact on the distribution of isotherms as well as temperature zones inside the cavity. Moreover, temperature plotting is found different for both increasing speed of rotation and strength of magnetic fields. Consequently, convective heat transferring reduces with Ha. In addition, different colours of temperature zones in the fluid domain indicate the topologies of the thermal field for varying amounts of Sr and Ha, respectively.Figure 7
Variation in temperature contour plotting for different Sr and Ha.
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## 4.3. Effects of Speed Ratio (Sr) and Baffles Length (Bl) on Streamlines

The streamline plotting in nanofluid flow circulation at different speed ratios of rotating rough cylinder and baffles length is presented in Figure8. In all diagrams, streamline circulations are found squeezing by the baffles and it increases with the increase in length of both baffles. As a result, flowing nanofluid encounters more resistive force for longer baffles, and hence the strength of flow circulation is reducednoticeably. Moreover, the streamline pattern of core circulations is unchanged except in the regions where the baffles are horizontally located. In addition, secondary circulations over the core circulation are also more affected by the increased length of baffles. Thus, longer baffles cause more changes in the flow velocity as well as circulations compared to shorter baffles. Moreover, impact of baffles on the flow field is more effective while rough cylinder is rotated at highest speed ratio than lower as 44.44% reduction is found in flow strength at Sr = 20 that becomes 17.24% at Sr = 0.Figure 8
Variation in streamline plotting for different Sr and Bl.
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## 4.4. Effects of Speed Ratio (Sr) and Baffles Length (Bl) on Isotherms

Besides this, Figure9 displays isotherms of distribution when speed ratio and baffle length are increased simultaneously. The distribution of temperature contours is found and changes gradually especially in the core region, with the increase in length of baffles and denseness of isotherms, are also found getting closer to the bottom heated wall which results in heat transferring increase with longer baffles. It is required to note that the impact of baffle length on the thermal field decreases while the speed ratio gets lower strength and it seems to be insignificant in the motionless state of the rough cylinder.Figure 9
Variation in temperature contour plotting for different Sr and Bl.
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## 4.5. Effects of Reynolds Number (Re) and Magnetic Field (Ha) on Streamlines

The plotting of streamlines due to combined variation in Reynolds and Hartmann numbers is demonstrated in Figure10. In Figures 10(a)–10(d), strong convective counter-clockwise flow circulations are observed around the rough cylinder due to the influence of cylinder rotating-inertia and the imposed thermal condition. In addition, two symmetric clockwise flow circulations are also visualized over the centred anticlockwise circulation with the assistance of lid-wall inertia. However, the strength of convective flow circulations is observed increasing with the increase in Reynolds numbers and streamlines are also intensified toward the cylinder because increase in Re causes increment in fluid inertia which results rotating inertia and lid inertia are both increased, and hence strength as well as concentration of streamlines increased but the pattern of streamlines remained similar for varying Re. Besides this, it is noticed that the flow velocity gets lower strength, and distribution of streamlines is spaced out toward the boundaries with an increasing Ha for all Re. These variations are more noticeable in higher magnetic strength than comparative lower ones.Figure 10
Variation in streamline plotting for different Re and Ha.
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## 4.6. Effects of Reynolds Number (Re) and Magnetic Field (Ha) on Isotherms

On the other hand, isotherm distributions at different Re and Ha are depicted in Figure11. It is observed that isotherms are generated from the heated wall and distorted around the rough cylinder covering the fluid domain due to the rotating rough cylinder and moving wall in the presence of baffles. As lower Re causes lower fluid inertia, rotating inertia and lid inertia lose their inertia at Re = 10, and hence a weak isotherm circulation is visible around the cylinder and isotherms become wavy with lower gradient near the top wall (as seen in Figure 11(a) at Ha = 0). As a result, lower heat transfer occurs at Re = 10. After that, increasing Re increase fluid inertia consequently, both rotating inertia and lid inertia become prominent, and hence isotherms circulation around the cylinder become stronger and isotherms are more twisted near the top moving wall up to Re = 100. Then at Re = 200, circulation and distortion of isotherms become more stronger having higher thermal gradient which results more convective heat released at Re = 200. When magnetic field is imposed at Ha = 10, isotherms distribution slightly changed for all Re. Further increase in Ha, circulation, and distortion of isotherms remarkably get affected, and the effect of Ha is found prominent at lower Re than at higher Re. Moreover, dense circulation rounding the rough cylinder tends to disappear, and isotherms are spaced out with a lower curvature at Ha = 50, which confirms the substantial effect of the magnetic field on isotherm distribution for all Re.Figure 11
Variation in temperature contour plotting for different Re and Ha.
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## 4.7. Effects of Volume Fraction (ϕ) and Magnetic Field (Ha) on Streamlines

Figure12 delineates streamlines plotting for different volume fractions of nanoparticles in a lid-driven cavity having a rotating cylinder with triangular components in the presence of a magnetic field or not. Due to induced thermal and velocity boundary conditions, a primary convective strong flow circulations along with two secondary small circulation are visible in the base fluid at fixed parametric values as Pr = 6.2, Gr = 100, Sr = 10, Re = 100, Bh = 0.20 L, and Δ = 0.0275 L, which remain almost similar in nanofluids at different concentrations (1%, 3%, and 5%), but flow strength is found declining remarkably for the additional nanoparticles at vol. of 1%, 3%, and 5% in the base fluid. These results are expected since the amalgamation of nanoparticles increases the density of the working fluid and decelerates flow velocity within the cavity. In order to understand the flow magnitude at different amounts of nanoparticles more accurately, one can find the maximum velocity at the core flow circulations which are 1.20, 1.10, 0.90, and 0.80, respectively, at vol. of 1%, 3%, and 5%. It is also found that flow velocity decreases for imposing and increasing magnetic field strength transverse to the flow circulation. Moreover, denseness of streamlines reduces with each increment in Ha and spaced out toward the sidewalls. In addition, maximum declination in flow magnitude id occurred at simultaneous changes in the volume fraction and magnetic field which are recorded as 1.2 (ϕ = 0%, Ha = 0), 1.0 (ϕ = 1%, Ha = 10), 0.70 (ϕ = 3%, Ha = 25), and 0.45 (ϕ = 5%, Ha = 50).Figure 12
Variation in streamline plotting for differentϕ and Ha.
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## 4.8. Effect of Volume Fraction (ϕ) and Magnetic Field (Ha) on Isotherms

Beside these, distributions of temperature contours in base fluid and nanofluid with 1% volume fraction are found (in Figure13(a)) qualitatively similar but minute changes are noticed for vol. of 3% and 5%. The physical reason behind is that the presence of nanoparticles in base fluid decreases convection mode and hence increases the domination of conduction mode heat transfer. It is necessary to note that isotherms distributions substantially change in both base fluid and nanofluid at highest magnetic field strength compared to at lowest.Figure 13
Variation in temperature contour plotting for differentϕ and Ha.
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## 4.9. Effect of Ha along with Sr, Re,ϕ, Δ, Bl, and Baffles Orientations on Average Nusselt Number (Nuav)

Figure14 illustrates heat transferring in the fluid flow domain due to physical parameters via average Nusselt number bar charts. Figure 14(a) confirms the heat transfer rate augments rapidly with increasing rotational speed of rough cylinder because increased rotational inertia due to Sr increases flow circulation significantly. It is also visualized in Figure 14(a) that higher magnetic strength decreases the heat transfer rate at each Sr as the interaction of magnetic force with convective flow circulation generates Lorentz’s force, which reduces the flow velocity and produces more temperature within the fluid domain. From Figure 14(b), it is seen that the heat transfer rate augments monotonically by the increasing Re where maximum heat transfer is recorded in forced convection dominated regime and minimum for mixed convection regime. The physics behinds it is that higher Re increases fluid inertia which accelerates cylinder rotating inertia and also lid inertia that causes maximum flow circulation and heat transfer within the cavity. In Figure 14(c), the heat transfer rate is found increasing the function of the amount of nanoparticles. This phenomenon is expected as higher amounts of nanoparticles improves nanofluid thermal conductivity, and hence capability of energy transportation in the flow domain. A similar trend is also observed in Figure 14(d) for the presence and increase in height of triangular components on the rotating cylinder, as higher heights of triangular components increase rotating inertia and flow velocity as well. In order to understand the impact of roughness components on heat transferring precisely, it can be recorded that 13.20% (at Ha = 0) more heat transfer takes place for rotating rough cylinders compared to rotating smooth cylinders, and it becomes 10.14%, while the magnetic field is activated at strength Ha = 50. On the other hand, enhancement in heat transfer is found for enlarging baffles length (seen in Figure 14(e)) as the temperature contours were found getting closer to the heated wall with longer baffles and also concentrated in its distribution around the rough cylinder, which leads to an increase in heat transferring inside the cavity (shown in Figure 9). Moreover, heat transfer rate increases by 42.42% while baffle length changes from 0.10 L to 0.25 L in the absence of a magnetic field but reduces to 34.53% whilethe magnetic field is activated at a strength of Ha = 50. Results in Figure 14(f) indicate that the average Nusselt number strongly depends on the orientation of baffles, and maximum heat transfer occurs while baffles are horizontally fixed at the cavity’s vertical walls. In all bar charts of Nusselt number, it is observed that the heat transfer rate decelerates monotonically with the increase in magnetic field strength resulting in active Lorentz’s force due to the magnetic field effect.Figure 14
Variation in average Nusselt for different Sr, Ha, Re,ϕ, Δ, Bl, and orientation of baffles.
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## 5. Conclusions

In this study, the impacts of the magnetic field, rotating rough cylinder, and amount of nanoparticles on the fully developed flow and temperature fields in a nanofluid filled, partially heated lid driven cavity with horizontal baffles are numerically investigated. The Galerkin finite element method is implemented to simulate the governing equations, and the obtained results are validated against existing results available in the literature. Detailed parametric discussion has been performed based on the physical point of view. The major findings based on the obtained results are as follows:(i)
Intense streamline circulation and fluid flow velocity are occurred at higher speed ratios and Reynolds number, while the reverse phenomenon is occurred at higher magnetic field strengths, nanoparticle concentrations, and length of the baffles.(ii)
The temperature contour plotting changes significantly with the change in speed of rotating rough cylinder, Reynolds and Hartmann numbers, and baffles length, while rough cylinder is in motion but it minutely changes with the increase in concentration of nanoparticles in the base fluid.(iii)
Heat transfer rate is augmented substantially at higher speed ratio, height of the triangular components, length of baffles, and Reynolds number, which is respectively maximized and minimized at each increment in nanoparticle volume fraction and magnetic field strength.(iv)
heat transfer rate is optimum, while baffles are horizontally fixed at the cavity walls than other cases.(v)
Maximum heat transfer occurred while triangular components are attached to the rotating cylinder rather than a smooth rotating cylinder.(vi)
Optimization of heat transfer is correlated with the direction of rotating the rough cylinder and lid wall.



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*Source: 3044930-2022-12-14.xml*