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

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Introduction
Convective heat transfer in confned enclosures is of great interest to the engineers and researchers as it is frequently encountered in heat transport processes in various engineering felds 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 fow caused by temperature diferences and shear fow caused by additional forces such as moving surfaces, rotating surfaces, or inlet-outlet fow. In addition, heat transfer associated with magnetic felds 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 purifcation, and petroleum industry. In this case, magnetic force interacts with the buoyancy fow and produces Lorentz force which infuences the fuid fow 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. Tey have tried to break down this limitation and developed novel heat transfer fuids by amalgamating nanoparticles in diferent base fuids named nanofuids which have superior properties such as higher thermal conductivity, improved stability, minimal clogging in the fow 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 diferent confgurations flled with diferent nanofuids under the infuence of the magnetic feld or in absence of magnetic feld. 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 fnite volume method and recommended that heat transfer increases with solid volume fraction at a fxed Reynolds number. Tey also noted that it is increased for Ri and Re at a fxed solid volume fraction. Later on, Muthtamilselvan and Doh [2] conducted a similar study considering the magnetic feld efect in the vertical direction and found the fow, heat, and mass transfer characteristics strongly depends on the magnetic feld strength. Ismael et al. [3] examined the efects of partial slip and inclined magnetic feld on mixed convection and showed that the convection due to partial slip is controlled with the strength and orientation of the magnetic feld. Kasaeipoor et al. [4] utilized a fnite volume approach to study mixed convection in a T-shaped cavity under the magnetic feld efect. Tey demonstrated the heat transfer in nanofuid increases while the cavity aspect ratio was increased. Kefayati [5] used Buongiorno's mathematical model to investigate mixed convection in nonNewtonian nanofuid fow 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 flled by nanofuid and noted that enhancement of heat transfer for volume fraction depends on Ha and Ri. Armaghani et al. [7] studied natural convection in a nanofuid-flled bafed L-shaped cavity. In their study, fnite volume method was implemented and found heat transfer increasing with the aspect ratio and bafes length. Karimdoost Yasuri et al. [8] conducted similar analysis in a square cavity with a bafe and an external magnetic feld. Tey recorded that heat transfer increases for increasing Ra and bafes length but decreases for Ha. After that, Hussein et al. [9] numerically investigated natural convection in inclined cavity having a cold bafe flled by Al 2 O 3 /water nanofuid 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 nanofuid model to analyze natural convection in a bafed U-shaped cavity. Tey 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. Tey recommended the local and average Nusselt numbers are increasing function of ϕ, Re, and Ra. Tey 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 fns and showed the highest length and the widest gap of the fns cause superior heat transfer situation within the cavity. Shah et al. [13] analyzed the magnetized viscous fuid fow of a water-based hybrid nanofuid containing single-wall carbon nanotubes through a permeable channel by implementing a nanolayer approach and demonstrated that heat fux 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 fuid. Bilal et al. [14] investigated natural convection fow of power-law fuids in trapezoidal cavity with a nonuniformly U-shaped fn by using the fnite element method and COMSOL Multiphysics software (5.6) and highlighted that heat transportation and momentum profle increase with the increase in the Rayleigh number which decline for increased viscosity of the fuid. After that, Bilal et al. [15] performed a similar investigation for a square cavity with nonuniformly T-shaped fn. Later on, Shah et al. [16] studied natural convection fow of power-law fuids an isosceles triangular cavity by using similar software and found noticeable infuence of governing parameters on local and average heat fux coefcients 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 fuid 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 coefcient. 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 nanofuid fow between two coaxially rotating disks with magnetic feld efects and found that hybrid nanofuids containing Cu-Al 2 O 3 nanoparticles showed signifcantly better results than those of other hybrid nanoparticles. It was also recorded that the thermal conductance of nanofuids is directly correlated with the morphology of any hybrid nanoparticles. Bilal et al. [20] considered the Cattaneo-Christov heat fux model and transverse magnetic feld to analyze heat transfer of Williamson fuid fow over an exponentially stretched the surface by shooting and Runge-Kutta-method and showed that the velocity feld is suppressed by the magnetic feld and Williamson fuid parameter. Later on, Bilal et al. [21] performed fnite diference simulations of viscous fuid fow over a permeable rotating disk to explore the fow in the presence of the transverse magnetic feld 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. Te fow and thermal felds within simple geometrical models with/without bafes or fns were numerically explored in these studies  by using diferent nanofuids in the presence of the magnetic feld or not. It needs to be highlighted that geometrical confguration 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 fuid fow and heat transfer characteristics inside confned enclosures where fuid fow needs to be restricted or bifurcated. Tus, confgurations with appropriate obstructions can signifcantly afect the fow and temperature felds 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 confgurations studied. Majdi et al. [23] used the Fluent 6.3 commercial program to analyze combined convection in a triangular cavity with an insulated cylinder. Tey 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 flled with nanofuids containing Al 2 O 3 nanoparticles. Teir 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 fow and heat transfer of nonlinear fuid 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. Te problem of heat transportation of mixed convection in Newtonian fuid 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 fuid, whereas thermal buoyancy forces and Nusselt number increase for increasing Grashof number. Shah et al. [28] examined the heat and mass transportation of double difusive natural convection in Casson non-Newtonian fuid enclosed by a hexagonal enclosure with an isothermally heated cylinder in the presence of an inclined magnetic feld by practicing COMSOL Multiphysics software and found that heat and mass fux coefcients diminish for the magnetic feld efect, whereas heat and mass fux distribution are found upsurging with a Casson fuid parameter. Bansal and Chatterjee [29] performed a numerical study of magneto-convective fow in a lid-driven cavity that included a rotating and heat conducting cylinder. In their study, the fow and thermal felds 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 fuid-porous layer using ANSYS fuent 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 fow and heat transfer in mixed convection through grooved channels, and they noticed the fow and heat transfer mechanisms, due to governing parameters, are more efective 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 fow function and heat transfer enhancement become maximum at increasing Ra. Tus, it is clearly seen that fuid motion and temperature distribution inside a cavity are infuenced 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 fow function and heat transfer enhancement in a closed or open cavity.
Te irregular surfaces, either static or in motion, of a geometrical confguration generate contributory recirculating regions that can enhance fow mixing within the cavity and heat transfer compared to other smooth confgurations. Tus, such complicated confgurations can cause substantial improvement in the heat transfer performance and fow mixing and play signifcant 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 diferent thermal conditions and demonstrated the fow 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 nanofuid 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 specifc circumstances. Tayebi and Chamkha [37] carried out parametric analysis of natural convection in a nanofuid feld enclosure with a wavy cylinder and pointed out that fow and heat transfer characteristics are controlled by the presence of a wavy conductive cylinder. Ismael [38] utilized a fnite diference method to examine mixed convection in an enclosure with an arcshaped moving wall and showed that heat transfer due to rotational speed is irrespective at lower Ra but signifcant 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 afected 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 nanofuid-flled cavity Mathematical Problems in Engineering associated with a magnetic feld and heat fux. Tey 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 feld is of considerable interest for researchers and engineers and is applicable in science and engineering felds. A lot of research has been conducted for mixed convection in different geometries flled with diferent fuids. Tough mixed convection in diferent geometries is considered in the open literature, mixed convective heat transfer in a lid-driven nanofuid-flled partially heated cavity having a rotating cylinder with roughness components in the presence of bafes has never been investigated. But it has a signifcant infuence to improve the heat transfer efciency by enhancing the efective fow circulation due to the presence of roughness components and nanofuid thermal conductivity and is more applicable in engineering felds. Accordingly, authors have been interested to examine the fuid fow and heat transfer behaviours in a nanofuid-flled partially heated cavity equipped by centred rotating cylinder with roughness components and moving top wall and also a pair of bafes attached to the cavity vertical walls under the infuence of transverse magnetic feld. As per literature survey and author's knowledge, no such work was reported yet. Galerkin's fnite element method is utilized to simulate the modifed governing equations in this study. Te numerical results are obtained for diferent physical parameters and explained via streamlines, isotherms, and average Nusselt number bar charts. Tis type of confguration 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. Figure 1 represents the geometrical confguration of a two-dimensional square cavity of length L. A pair of horizontal bafes of 20% cavity length is fxed 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. Te free region between the cavity and the obstacle is flled with alumina water nanofuid that is heated partially from the cavity bottom wall and cooled from its top moving wall at a uniform velocity U 0 . Remaining walls and bafes are maintained no-slip condition and kept adiabatic. Te mixed convection is induced due to the rotating rough cylinder and top moving wall along with temperature diferences of the active thermal condition. A uniform magnetic feld afects the fow feld, and gravitational acceleration is also activated in the downward direction. Te thermophysical properties of nanoparticles and water are available in [41,42].

Mathematical Analysis
Te vector form of conservation equations representing the fow model is defned considering the Boussinesq approximation and the described physical model as follows [29,[42][43][44]: Te source terms used in the governing equation (2) are presented in Table 1.
Te used properties of nanofuid are as follows [30,32,41,42,[45][46][47][48]:  Mathematical Problems in Engineering Corresponding boundary conditions based on the physical model are specifed as follows: On the vertical walls and baffles: V � 0 and ∇T. n � 0, On the on the cylinder: Te dimensionless variables in equation (6) are introduced in the governing equations to make its dimensionless form: and which are  [43,44].
Mathematical Problems in Engineering 5 Te modifed form of the source terms of Table 1 and equation (8) are provided in the following Table 2.
With the help of equation (6), the dimensionless boundary conditions (shown in equation (5)) are written as the following: On the top wall: On the vertical walls and baffles: On the on the cylinde r: V * c � (Sr sin θ, Sr cos θ)and ∇θ * .

Evaluation of Average Nusselt Number.
Te local Nusselt number is computed at the heated midsectional wall of the cavity, which is defned in nondimensional form as follows [51,52]: and corresponding average Nusselt number is estimated as follows [51,52]: 3.2. Numerical Procedure. Te numerical simulation of the governing equations (7)- (9) along with the boundary conditions 10a-d has been performed using Galerkin's weighted fnite element method. In this simulation procedure, the governing partial diferential equations are converted into integral equations by implementing the Galerkin weighted residual method [53]. Te obtained integral equations are as follows: Here, A is the element area, N α (α � 1, 2, ..., 6) are the element shape functions or interpolation functions for the velocity components and temperature, and H λ (λ � 1, 2, 3.) are the element shape functions for the pressure. Te Gaussian divergence theorem has been introduced in equations (13)- (16). to generate the boundary integral terms associated with the surface tractions and heat fux in the momentum and energy equations and we obtained the following equations: 6 Mathematical Problems in Engineering where the surface tractions (A x , A y ) along the outfow boundary A 0 and velocity components and fuid temperature or heat fux (q w ) that fows into or out from the domain along the wall boundary A w . Te basic unknown for the above diferential equations is the velocity components (U, V), temperature θ, and the pressure P. For the development of the fnite 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. Tis means that a lower order polynomial is chosen for pressure and which is satisfed through the continuity equation. Te velocity components, temperature profles, and linear interpolation for the pressure distribution according to their highest derivative orders for the diferential equations (7) to (9) are as follows: 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 fnite element equations can be written in the following form:  [43,44,49,50].
Now, we consider the coefcients in the above governing equations as follows: Tese 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 coefcients, the fnite element equations can be written in the following type:

Mathematical Problems in Engineering
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]. Te convergence criterion of the numerical solution along with error estimation has been set to |φ m+1 − φ m | ≤ 10 − 5 , where m is the number of iteration and φ is a function of U, V, and θ. Te details of the computational procedure are also available in the earlier studies [32,56,57], which are well described in [58,59]. Te simple algorithm of this study is exposed through the following fowchart (in Figure 2):

Mesh Generation.
In two-dimensional confgurations, mesh generation is a procedure of subdivision of the geometrical domain considering elements as triangular or quadrilateral called fnite elements. Tese elements are connected to their neighbouring elements using characteristic points known as "nodes". Te required values of the physical quantities of a problem are calculated at every node. Meshing a complicated geometry into a signifcant number of elements is essential to the numerical simulations, which makes the FEM a powerful tool for solving engineering problems encountered in practical applications. Te mesh confguration with triangular elements of this study is presented in Figure 3.

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 diferent mesh systems, and fve of them have been presented in Table 3 and Figure 4 which confrm that further meshing has very small impacts on the computed average Nusselt number. Based on the presented mesh confgurations 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.

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 diferent confgurations: 3.5.1. Validation Case One. Rashad et al. [60] simulated mixed convection in a lid-driven cavity flled with nanofuids by implementing the fnite 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][61][62] by using a fnite element method-based numerical code. Te 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.

Validation Case Two.
Du et al. [63] used the fnite diference method to investigate the infuence of a magnetic feld on open cavity fow and validated their numerical code with numerical results available in [50,64]. Tey 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).
Tus, the above comparisons ensure that the present numerical procedure is suitable to simulate mixed convection in a nanofuid-flled square domain equipped with bafes and a rotating hollow cylinder having roughness components and also an external magnetic feld which leads us to carry on our investigation.

Results and Discussion
In this section, simulated results are obtained for mixed convection of Al 2 O 3 -water nanofuid in a lid-driven cavity with horizontal bafes and a cantered rotating rough cylinder, and the fow and thermal felds 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 fxed 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 efects of the two governing parameters have been exhibited combinedly. Te parametric ranges of the analysis are presented in Table 6.

Efects of Speed Ratio (Sr) and Magnetic Field (Ha) on
Streamlines. Figure 6 illustrates the fow feld by plotting streamlines for diferent values of the speed ratio of the rotating rough cylinder and magnetic feld strength in the presence of mounted horizontal bafes. In Figure 6(a) at motionless state of the rough cylinder and Ha � 0, heatedfuid ascends near the left vertical wall due to thermal induced-buoyancy fow and then descends quickly from top to bottom near the right vertical walls as the top moving cold wall and insulated vertical wall fow circulates in downward direction. As a result, clockwise streamline circulation is developed within the cavity, which is also confrmed by the negative fow strength. Tis convective circulation occupies the whole cavity, and it is found suppressing by the presence of horizontal bafes. Moreover, the highest fow 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 fow circulation is restructured by compressing the clockwise-circulations toward the top wall and forming anticlockwise circulation around the rough cylinder with the infuence of rotating inertia of rough cylinder. It is also found that fow velocity increased rapidly, and the strength as well as intensifcation of fow 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 efect of the moving wall is dominated. As a result, streamline circulation becomes more intensifed 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 fow velocities and these are 0.065 at Sr � 0, 0.060 at Sr     Ha � 25, streamlines are spaced out with lower strength compared to the case at Ha � 10. In addition, streamlines are more afected and spaced out at Ha � 50 and also gets minimum strength. Te physical consequences behind it is the Lorentz's force retards fow velocity which is produced due to the interaction of nanofuid buoyancy fow and shear fow with an imposed magnetic force, and the Lorentz's force escalates with each increment in the magnetic efect. As a result, fow circulation declines with an increase in Ha for all Ra. Tus, one can conclude that the variation in fow characteristics is more noticeable at greater magnetic feld strengths than at lower ones,while compared in the absence of a magnetic feld.

Efects of Speed Ratio (Sr) and Magnetic Field (Ha) on
Isotherms. Figure 7 represents the thermal feld via temperature contour plotting as well as its topology due to combined variation in rotational speed of rough cylinder and magnetic feld strength at fxed 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 fow 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 reshufed by forming a spinning shape around the rough cylinder that occupies a fuid 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 bafes where the lower isotherm pack seems to be connected to the lower end of the left adiabatic bafe and another isotherm pack seems to initiate from the top end of the right adiabatic bafe 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 signifcantly afected, 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 feld is imposed at strength Ha � 10, small changes are noticed in temperature contour distribution for diferent 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 feld strength at Ha � 50, signifcant changes are observed in isotherm distributions within the closed domain which refects the magnetic feld has a signifcant impact on the distribution of isotherms as well as temperature zones inside the cavity. Moreover, temperature plotting is found diferent for both increasing speed of rotation and strength of magnetic felds. Consequently, convective heat transferring reduces with Ha. In addition, diferent colours of temperature zones in the fuid domain indicate the topologies of the thermal feld for varying amounts of Sr and Ha, respectively.

Efects of Speed Ratio (Sr) and Bafes Length (Bl) on
Streamlines. Te streamline plotting in nanofuid fow circulation at diferent speed ratios of rotating rough cylinder and bafes length is presented in Figure 8. In all diagrams, streamline circulations are found squeezing by the bafes and it increases with the increase in length of both bafes. As a result, fowing nanofuid encounters more resistive force for longer bafes, and hence the strength of fow circulation is reducednoticeably. Moreover, the streamline pattern of core circulations is unchanged except in the regions where the bafes are horizontally located. In addition, secondary circulations over the core circulation are also more afected by the increased length of bafes. Tus, longer bafes cause more changes in the fow velocity as well as circulations compared to shorter bafes. Moreover, impact of bafes on the fow feld is more efective while rough cylinder is rotated at highest speed ratio than lower as 44.44% reduction is found in fow strength at Sr � 20 that becomes 17.24% at Sr � 0.

Efects of Speed Ratio (Sr) and Bafes Length (Bl) on
Isotherms. Besides this, Figure 9 displays isotherms of distribution when speed ratio and bafe length are increased simultaneously. Te distribution of temperature contours is found and changes gradually especially in the core region, with the increase in length of bafes and denseness of isotherms, are also found getting closer to the bottom heated wall which results in heat transferring increase with longer bafes. It is required to note that the impact of bafe length on the thermal feld decreases while the speed ratio gets lower strength and it seems to be insignifcant in the motionless state of the rough cylinder.

Efects of Reynolds Number (Re) and Magnetic Field (Ha) on Streamlines.
Te plotting of streamlines due to combined variation in Reynolds and Hartmann numbers is demonstrated in Figure 10. In Figures 10(a)-10(d), strong convective counter-clockwise fow circulations are observed around the rough cylinder due to the infuence of cylinder rotating-inertia and the imposed thermal condition. In addition, two symmetric clockwise fow circulations are also visualized over the centred anticlockwise circulation with the assistance of (a) (b) (c) (d) Figure 9: Variation in temperature contour plotting for diferent Sr and Bl.
lid-wall inertia. However, the strength of convective fow circulations is observed increasing with the increase in Reynolds numbers and streamlines are also intensifed toward the cylinder because increase in Re causes increment in fuid 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 fow velocity gets lower strength, and distribution of streamlines is spaced out toward the boundaries with an increasing Ha for all Re. Tese variations are more noticeable in higher magnetic strength than comparative lower ones.

Efects of Reynolds Number (Re) and Magnetic Field (Ha) on Isotherms.
On the other hand, isotherm distributions at diferent Re and Ha are depicted in Figure 11. It is observed that isotherms are generated from the heated wall and distorted around the rough cylinder covering the fuid domain due to the rotating rough cylinder and moving wall in the presence of bafes. As lower Re causes lower fuid 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 fuid 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. Ten 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 feld is imposed at Ha � 10, isotherms distribution slightly changed for all Re. Further increase in Ha, circulation, and distortion of isotherms remarkably get afected, and the efect 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 confrms the substantial efect of the magnetic feld on isotherm distribution for all Re.

Efects of Volume Fraction (ϕ) and Magnetic Field (Ha) on
Streamlines. Figure 12 delineates streamlines plotting for diferent volume fractions of nanoparticles in a lid-driven cavity having a rotating cylinder with triangular components in the presence of a magnetic feld or not. Due to induced thermal and velocity boundary conditions, a primary convective strong fow circulations along with two secondary small circulation are visible in the base fuid at fxed 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 nanofuids at diferent concentrations (1%, 3%, and 5%), but fow strength is found declining remarkably for the additional nanoparticles at vol. of 1%, 3%, and 5% in the base fuid. Tese results are expected since the amalgamation of nanoparticles increases the density of the working fuid and decelerates fow velocity within the cavity. In order to understand the fow magnitude at diferent amounts of nanoparticles more accurately, one can fnd the maximum velocity at the core fow 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 fow velocity decreases for imposing and increasing magnetic feld strength transverse to the fow circulation. Moreover, denseness of streamlines reduces with each increment in Ha and spaced out toward the sidewalls. In addition, maximum declination in fow magnitude id occurred at simultaneous changes in the volume fraction and magnetic feld 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).

Efect of Volume Fraction (ϕ) and Magnetic Field (Ha) on
Isotherms. Beside these, distributions of temperature contours in base fuid and nanofuid with 1% volume fraction are found (in Figure 13(a)) qualitatively similar but minute   Figure 14 illustrates heat transferring in the fuid fow domain due to physical parameters via average Nusselt number bar charts. Figure 14(a) confrms the heat transfer rate augments rapidly with increasing rotational speed of rough cylinder because increased rotational inertia due to Sr increases fow circulation signifcantly. 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 fow circulation generates Lorentz's force, which reduces the fow velocity and produces more temperature within the fuid 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. Te physics behinds it is that higher Re increases fuid inertia which accelerates cylinder rotating inertia and also lid inertia that causes maximum fow 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. Tis phenomenon is expected as higher amounts of nanoparticles improves nanofuid thermal conductivity, and hence capability of energy transportation in the fow 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 fow 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 feld is activated at strength Ha � 50. On the other hand, enhancement in heat transfer is found for enlarging bafes length (seen in Figure 14(e)) as the temperature contours were found getting closer to the heated wall with longer bafes 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 bafe length changes from 0.10 L to 0.25 L in the absence of a magnetic feld but reduces to 34.53% whilethe magnetic feld 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 bafes, and maximum heat transfer occurs while bafes are horizontally fxed 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 feld strength resulting in active Lorentz's force due to the magnetic feld efect.

Conclusions
In this study, the impacts of the magnetic feld, rotating rough cylinder, and amount of nanoparticles on the fully developed fow and temperature felds in a nanofuid flled, partially heated lid driven cavity with horizontal bafes are numerically investigated. Te Galerkin fnite 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. Te major fndings based on the obtained results are as follows: (i) Intense streamline circulation and fuid fow velocity are occurred at higher speed ratios and Reynolds number, while the reverse phenomenon is occurred at higher magnetic feld strengths, nanoparticle concentrations, and length of the bafes. (ii) Te temperature contour plotting changes signifcantly with the change in speed of rotating rough cylinder, Reynolds and Hartmann numbers, and bafes length, while rough cylinder is in motion but it minutely changes with the increase in concentration of nanoparticles in the base fuid. (iii) Heat transfer rate is augmented substantially at higher speed ratio, height of the triangular components, length of bafes, and Reynolds number, which is respectively maximized and minimized at each increment in nanoparticle volume fraction and magnetic feld strength. (iv) heat transfer rate is optimum, while bafes are horizontally fxed 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.