Discretization of thermo-fluidic behavior of the combined effects of MHD conjugate mixed convection and Joule heating in a lid-driven Z-shaped cavity containing double rotating cylinders

Discretization of thermo-fluidic behavior of the combined effects of MHD conjugate mixed convection and Joule heating in a lid-driven Z-shaped cavity containing double rotating cylinders | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 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Data may be preliminary. 10 July 2025 V1 Latest version Share on Discretization of thermo-fluidic behavior of the combined effects of MHD conjugate mixed convection and Joule heating in a lid-driven Z-shaped cavity containing double rotating cylinders Authors : Mallik Nadim Arman Omi 0009-0003-9960-4786 [email protected] and Md Abir Khan Authors Info & Affiliations https://doi.org/10.22541/au.175213355.57640830/v1 407 views 116 downloads Contents Abstract Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract The present numerical study focuses on investigating the effects of MHD mixed convection and Joule heating in a lid-driven Z-shaped cavity containing double rotating cylinders in a comprehensive manner. An ample range of practical applications, including nuclear reactor cooling, magnetically controlled solar collectors, metallurgical processes, and electronic device cooling, make it a captivating field of research. The current study considers a Z-shaped cavity that has a hot wall at the lower portion of the cavity and a cold wall at the top that is a moving lid, and other walls are kept adiabatic. It comprises two rotating circular cylinders that can rotate in both clockwise and counterclockwise rotation. The Galerkin finite element method-based computational process is utilized to resolve two-dimensional steady continuity, momentum, and energy equations under specified boundary conditions. Both quantitative and qualitative outcomes are assessed to predict the fluid flow and thermal behavior for a limited range of some pertinent parameters : Hartmann number (Ha = 20, 30, and 40), rotational Reynolds number (Re c = -10 and 10), heat generation coefficient (∆ = 0, 2, and 5), Grashof number (10 3 ≤ Gr ≤ 10 6 ), and Reynolds number (100 < Re < 700 ). Results reveal a maximum 40% and 45% reduction in the Nusselt number occurring for enhancing the magnetic effect and heat generation coefficient, respectively. Moreover, clockwise rotation of the cylinders or aiding flow facilitates over 38% elevation in the Nusselt number by improving overall flow circulation in the cavity. As well as increasing fluid Reynolds number leads to a 150% augmentation in the Nusselt number due to intensified turbulent flow. Hence, the comparative results of this meticulous numerical study, blended with novel geometrical contour, can be utilized to improve the thermal efficiency of other intricate systems. 1. Introduction Mixed convection heat transfer within a vented chamber is an exciting field of research studies due to its immense practical applications, including industrial cooling systems, food processing, electronic chip cooling, nuclear energy production, ventilation systems, and so on. Mixed convection combines the effects of natural convection, which occurs due to buoyancy forces, and forced convection, which prevails when the fluid motion is driven by external means, such as a fan or a pump. In the case of conjugate mixed convection, thermal interaction exists between fluid and solid boundaries. So it is a much more complex as well as challenging phenomenon to consider in the designs of these studies. For many years, researchers have been developing their ideas to investigate new methods to fully understand the intricate concepts of conjugate mixed convection [1, 2, 3, 4]. Several literature have explored the thermo-fluidic characteristics of mixed convection heat transfer in different-shaped vented cavities. Saha et al. [5] investigated the effect of transverse mixed convection in a vented enclosure numerically by applying a constant heat flux at the bottom. They concluded that the position of inlet and outlet ports has significant influence on heat transfer enhancement. Rahman et al. [6] computationally explored the heat transfer characteristics in a vented square cavity containing a heat-generating block. They used various geometric sizes, locations, and thermal conductivities of the element and expressed their findings in terms of average Nusselt number and average fluid temperature. They found that heat transfer increased for larger block sizes, and the highest Nusselt number was obtained at a location of the block near the heated wall. A numerical analysis of a square cavity filled with Al₂O₃-water nanofluid was conducted by Sourtiji et al. [7]. The findings of their study demonstrated that the Nusselt number increases with increasing the Reynolds number, Richardson number, and nanoparticle volume fraction. Moreover, some experimental works [8, 9, 10] were done to explore mixed convection heat transfer in a vented cavity and proposed correlations for the Nusselt number with respect to some pertinent parameters such as the Reynolds number, Grashof number, Richardson number, etc. Recently, an experimental study in mixed convection heat transfer was conducted by Harizi et al. [11] in a vented cavity filled with a porous medium. They were able to distinguish three different heat and flow regimes depending on the Richardson number. They showed that, at higher Ri numbers, natural convection dominates near the heated wall, whereas at low Ri numbers, the flow structure resembles pure forced convection. Mixed convection heat transfer can be increased by applying one or more moving boundaries to create sufficient vortex flow in the working domain and eventually increase the average Nusselt number. The moving lid can be applied horizontally (top or bottom boundary) or in vertical walls, sliding in a constant velocity. Such a type of technique is used in many practical applications, including solar energy collectors, furnace cavities, lubrication systems, electronic component cooling inside an enclosure, etc. Many literatures upheld the investigated results regarding fluid flow and mixed convective heat transfer in a vented cavity with a sliding lid [12, 13, 14, 15, 16]. Cheng [12] conducted a numerical study to find out the characteristics of mixed convection heat transfer in a 2-D square cavity with a moving upper lid that created the fluid motion. He observed that as the lid velocity increased, there was a significant enhancement in the heat transfer rate, and the flow regime shifted towards the forced convection state. A similar type of investigation was conducted by Nada and Chamka [13], where they used an inclined square enclosure to extend the previous work. They found that the inclination effect was significant in increasing heat transfer performance in a forced convection-dominated regime, but it was more prominent in the case of a natural convection-dominated regime. They also reported that using nanoparticles in the working fluid facilitated a higher Nusselt number. In another work, Ibrahim and Hirpho [14] applied a magnetic field in a trapezoidal cavity with non-uniform temperature distribution to observe mixed convection flow. They concluded that the magnetic field suppressed the fluid flow in the cavity and resulted in lower heat transfer performance. However, a higher Ri number leads to a higher Nusselt number as well as better heat transfer. Aljabair et al. [15] explored a numerical study in an arc cavity filled with Cu-water nanofluid with non-uniform heating and presented correlation equations for Nusselt number calculations. They concluded that increasing the Reynolds number, Richardson number, and volume percentage of nanoparticles results higher heat transfer rate and a higher Nusselt number. A recent study is performed by Tulu et al. [16] in a hexagonal cavity with fluctuating sinusoidal temperature distribution at the lower side. They showed that lid velocity governed the fluid flow inside the enclosure and temperature intensity rose from 1.85 to 12.22 when the Eckert number increased from 0 to 50. Magnetohydrodynamic (MHD) mixed convection is a complex term in the field of fluid and heat transfer research areas. Generally a magnetic field is applied perpendicular to the experimental domain to induce Lorentz force within an electrically conductive fluid, such as molten metal, plasma, liquid metal, salt water, nanofluid, etc. This Lorentz force reduces the flow turbulence significantly. Therefore, by controlling the parameters involved in the MHD effect, the flow pattern and thermal behavior can be handled efficiently to meet particular system demands. This type of research has great significance in terms of practical applications, including metallurgical processes, energy systems, MHD generators, nuclear reactor cooling systems, thermal cooling systems, etc. [17, 18, 19, 20]. A numerical study on the mixed convection heat transfer was conducted by Chamkha [21] on a lid-driven cavity along with the effect of magnetic field and internal heat generation. His study concluded that the MHD effect and heat-generating element substantially reduced the heat transfer rate within the fluid. Chatterjee and Halder [22] investigated MHD mixed convection in a 2-D square enclosure containing two circular cylinders, and the enclosure was filled with electrically conducting fluid. Larger magnetic force intensity or Hartmann number resulted in ordered isotherm plots, and rotating circular cylinders induced strong force convective flow. Selimefendigil and Chamkha [23] added a surface corrugation effect to compute numerically mixed convectional behavior in a 3-D vented cavity filled with CuO-water nanofluid with an inner rotating circular cylinder. They found that the magnetic effect increases the heat transfer rate for nanofluid, but the surface corrugation of the base reduced the thermal performance of the system. Recently, Jiang et al. [24] scrutinized MHD mixed convective heat transfer in a 3-D porous cubic enclosure containing double rotating cylinders and hybrid nanofluid. They stated that increasing the Hartmann number reduced the thermal performance by reducing streamline density and entropy generation. Joule heating occurs when electrical energy is converted into heat energy because of the resistive elements or fluids. MHD effects may generate electricity in electrically conductive fluids, resulting in Joule heating that may alter heat transfer and fluid flow behavior in an enclosure. Several researchers thoroughly investigated the MHD and Joule heating conditions in different shape geometries [25, 26, 27, 28, 29]. Tasnim et al. [25] numerically investigated MHD mixed convection and Joule heating in a tilted square enclosure filled with TiO₂-water nanofluid along with two heat-generating elements. Pure water performed better than nanofluid by enhancing the Hartmann number and entropy generation. They also reported that inclination angle had a profound effect on increasing fluid flow and heat generation rate. The combined effect of MHD mixed convection and Joule heating on an obstructed lid-driven enclosure was computationally analyzed by Rahman et al. [27]. Applying the Joule effect increased the average temperature of the experimental contour, resulting in a reduction in heat transfer rate. A similar kind of study was conducted by Ray and Chatterjee [28] by adding a circular solid object inside the domain. They found magnetic force had a great impact on velocity and temperature distributions of the fluid, but the effect of resistive heating or Joule heating was negligible. Mixed convection in a cavity along with spinning cylinders is a classic thermo-fluidic research interest, as it combines the forced and free convectional effects as well as the rotating body effect. Several practical applications, including micro-scale fluid mixing, rotary machines, electronic cooling with rotary heat sinks, materials processing, etc., depend on this intricate system. Researchers investigated over many years to fully understand the flow pattern and thermal behavior of such systems through computational and experimental observations [30, 31, 32, 33]. Sadr et al. [34] simulated mixed convection heat transfer in a square cavity with a rotating circular cylinder, and they used water and nano-enhanced phase change materials as a working fluid. They reported that, by adding nano-encapsulated phase change materials, the convectional heat transfer rate increased by 13%. The effect of MHD mixed convection and power law non-Newtonian fluid in a cavity containing a circular cylinder was conducted numerically by Abderrahmane et al. [35]. Counterclockwise rotation of the cylinder resulted in the best performance as the cylinder approached near the hot wall. Ali and Alomer [36] assessed a numerical observation on a square cavity containing two aligned circular cylinders to obtain the mixed convective flow pattern. Their results revealed that cylinder spacing size is a decisive parameter for thermal performance, and the optimum spacing size was S = 0.375, which led to the highest Nusselt number. Recently, in another similar study, Al-Omar and Ali [37] utilized one hot and one cold cylinder to explore the mixed convective flow behavior in a square enclosure and found the highest increment in the Nusselt number by 36.8% at spacing size S = 0.375 and opening size O = 0.375. To the authors’ best knowledge, there is a remarkable research gap on exploring MHD mixed convection and Joule heating in a novel lid-driven Z-shaped cavity containing double rotating cylinders. This research work aims to investigate the flow pattern and thermal performance of such a system by exhibiting both streamline and isothermal plots for a wide range of pertinent parameters such as Reynolds number, Grashof number, Richardson number, Hartmann number, and rotational Reynolds number. Moreover, quantitative analysis is performed to indicate the thermal performance increment in terms of the Nusselt number, average fluid temperature, and drag coefficient. The outcomes of this research would be significant for many practical applications, including solar energy collectors, nuclear power plants, heat sinks with rotating objects, material processing, controlling chemical reactions, etc., to manage excess heat and increase the efficiency of the aforementioned systems. 2. Geometric model description : Fig. 1 portrays the physical model considered for the present study. It encompasses a movable upper lid and an immovable lower wall with an equal length, L. The lower wall is kept at a higher temperature, T h , while the upper lid is taken as a cold condition of temperature T c . The top lid can slide from left to right or in the positive X-direction at a constant velocity of u. There are two enlarged sections in the geometric domain with a length of W = 0.2L. The remaining walls, other than the upper and lower walls, are considered adiabatic. The two circular cylinders with diameter D are positioned at (x1, y1) = (0.5L, 0.75L) and (x2, y2) = (0.5L, 0.25L). These cylinders can rotate in both clockwise and counterclockwise directions with a rotational velocity ω and a circumferential velocity of u c = ωD/2. A magnetic field of strength B 0 is applied in a horizontal direction at the left wall of the enclosure. Gravitational acceleration operates in a vertical downward direction inside the cavity. The remaining portion is filled with electrically conductive salt water that has a Prandtl number, Pr = 8.5, density, ρ = 1023 kg/m 3 , thermal expansion coefficient, β = 2×10 -4 K -1 and thermal conductivity, k = 0.63 (properties are taken at 3.5% NaCl concentration). A volumetric heat generation rate, Q (W/m³), is applied in the enclosure to mimic internal heating. Table 1 shows the necessary boundary conditions that are being considered in this paper where N designates the non-dimensional normal distance measured from the wall. [insert Table 1 here] [insert Fig. 1 here] 3. Mathmatical formulations : The governing equations for the present numerical simulations comprises with continuity of mass, momentum and energy equations. The following assumptions are considered : • The flow is laminar, two dimensional and steady. • The fluid is Newtonian and incompressible and has constant thermophysical properties. • The effects of viscous dissipation and radiation are neglected. • The Boussinesq approximation is applied to account for the density change with temperature. The governing equations in dimensional form are given below [38, 39, 40] : \(\frac{\partial u}{\partial x}\) + \(\frac{\partial v}{\partial y}\ \)= 0 (1) \(\rho\left(\text{\ u\ }\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\ \right)=\ -\frac{\partial p}{\partial x}+\ \mu\left(\frac{\partial^{2}u}{{\partial x}^{2}}+\ \frac{\partial^{2}u}{\partial y^{2}}\right)\)(2) \(\rho\left(\text{\ u\ }\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\ \right)=\ -\frac{\partial p}{\partial y}+\ \mu\left(\frac{\partial^{2}v}{{\partial x}^{2}}+\ \frac{\partial^{2}v}{\partial y^{2}}\right)+\rho g\beta\left(T-T_{c}\right)-\text{\ σ}B_{0}^{2}v\)(3) \(\rho C_{p}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\ \right)+Q+\sigma B_{0}^{2}v\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \)(4) Where, u and v denotes the dimensional velocity of the fluid in the x and y direction respectively in Cartesian coordinate system.\(\rho,\ \mu,\ k,\ C_{p}\), β imply the fluid’s density, dynamic viscosity, thermal conductivity, specific heat and volume expansion coefficient, respectively. Several reference scales are used to convert the dimensional equations into non-dimensional forms : \(X=\ \frac{x}{L},\ Y=\ \frac{y}{L},\ U=\ \frac{u}{u_{0}},\ V=\ \frac{v}{v_{0}},\ P=\ \frac{p}{\rho u^{2}},\ \theta=\ \frac{T-T_{c}}{T_{h}-T_{c}},\ \mathrm{\Delta}\ =\ \frac{QL^{2}}{k(T_{h}-T_{c})}\text{\ \ }\) (5) The transformed non-dimensional equtions are written below: \(\frac{\partial U}{\partial X}\) + \(\frac{\partial V}{\partial Y}\ \)= 0 (6) \(\left(\text{U\ }\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\ \right)=\ -\frac{\partial P}{\partial X}+\ \frac{1}{\text{Re}}\left(\frac{\partial^{2}U}{{\partial X}^{2}}+\ \frac{\partial^{2}U}{\partial Y^{2}}\right)\)(7)\(\left(\text{\ U\ }\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\ \right)=\ -\frac{\partial P}{\partial Y}+\ \frac{1}{\text{Re}}\left(\frac{\partial^{2}V}{{\partial X}^{2}}+\ \frac{\partial^{2}V}{\partial Y^{2}}\right)+Ri\theta-\ \frac{\text{Ha}^{2}}{\text{Re}}V\)(8)\(\left(U\frac{\partial\theta}{\partial X}+V\frac{\partial T}{\partial Y}\right)=\frac{1}{\text{RePr}}\left(\frac{\partial^{2}\theta}{\partial X^{2}}+\frac{\partial^{2}\theta}{\partial Y^{2}}\ \right)+JV^{2}+\ \frac{\mathrm{\Delta}}{\text{RePr}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \)(9) The corresponding characterizing equations and range of values of the non-dimensional parameters associated with the equations (6)-(9) are tabuled in the Table 2. [insert Table 2 here] To assess the qualitative outcomes of the present study, Nusselt number (Nu) at the hot bottom wall, drag coefficient (C d ) of the moving lid and non-dimensional average temperature (Ɵ) of the fluid are measured by utilizing the following equations : Nu = -\(\int_{0}^{1}{\frac{\partial\theta}{\partial Y}\text{dX}}\), C d =\(\frac{2}{\text{Re}}\int_{0}^{1}{\frac{\partial U}{\partial Y}dX\,\ \ }\)Ɵ av =\(\frac{1}{\text{πD}}\int_{0}^{\text{πD}}\text{θdS}\) (10) 4. Computational details 4.1. Numerical procedure The Galerkin finite element method-based specialized software “COMSOL Multiphysics” is utilized to solve the non-linear and non-dimensional governing equations (6)-(9) that govern the mathematical formulations in the corresponding computational domain. Necessary boundary conditions are employed in this method, which are tabulated in Table 1. At first, this method splits the computational domain into a large number of triangular and quadrilateral mesh elements. Some edge mesh elements are also generated in border regions to avoid the formation of any singular points and enhance precision. The finite element method transforms the non-linear equations into integral forms by applying a weighted residual approach. Then the Gauss Quadrature technique is employed to perform the integral operations. Lastly, the Newton-Raphson method is used to resolve the algebraic equations found from previous operations. The Newton-Raphson method is an iteration-based technique that continuously updates the solutions until proper convergence is acquired. In the present study, the convergence criterion is set at 10 -3 . Thus, the discretization method promotes quick convergence as well as ensures the finite element method’s accuracy. Fig. 2 contains a flowchart showing sysmetic procedures of finite elemnet method used in the present study. [insert Fig. 2 here] 4.2. Mesh refinement test As the finite element method discretizes the whole computational domain into smaller triangular and quadrilateral regions, there is a possibility of variation in the output depending on element and node numbers. Hence, a mesh independence test is performed to ensure that mesh size and numbers don’t influence the result of the simulations. In the present study, a mesh refinement test is assessed by using a physics-controlled mesh generation tool in COMSOL Multiphysics software at Pr = 8.5, Re = 700, Gr = 1\(\times\)10 4 , Ri = 0.1, Ha = 20, and Re c = -10. Table 3 shows the obtained results of this test, where the Nusselt number is calculated for each mesh arrangement to make comparisons between different mesh types. Results reveal that, after a particular number of elements or mesh type “Extra fine,” the average Nusselt number doesn’t depend much on further refinement. Therefore, an “Extra fine” meshing type is selected for all other simulations to obtain precise results with minimal computational cost. Table 4 illustrates the detailed properties of the optimum mesh type. The formula that is used to define the percentage change in the average Nusselt number in the mesh sensitivity test is given below : \begin{equation} \bigtriangleup Nu\ =\ \frac{\left|\text{Nu}_{i+1}\ -\ \text{Nu}_{i}\right|}{\text{Nu}_{i+1}}\ \times\ 100\%\nonumber \\ \end{equation} Where Nu i and Nu i+1 refer to the Nusselt number measured at a particular mesh type and subsequent better mesh arrangement, respectively. Fig. 3 shows the optimum mesh geometry for the current study. [insert Table 3 here] [insert Table 4 here] [insert Fig. 3 here] 4.3. Geometric model validation : The present numerical model needs to be verified before conducting any observational simulation to prove its correctness and alignment with other research works. Firstly, our current model is compared with the numerical work conducted by Khanafer and Aithal [41]. They performed their simulations at Re = 100, Pr = 0.7 (for air), ∆ = 0, d/L = 0.4, Re c = ωL/u₀ = +10 and -10 for rotating double circular cylinders inside a lid-driven square cavity. Fig. 4 depicts the relationships between these two models by comparing Nusselt number calculation with respect to the variation of Richardson number for a range of 0 to 10. There is a close proximity between these two observations with a maximum error of 4.68%. Another verification assessment is conducted to compare Nusselt number variation with respect to Hartmann number with the work published by Rahman and Alim [42]. They conducted their experiment at Re = 1000, Gr = 100, Pr = 0.71, and ∆ = 0 for a rotating circular cylinder in a lid-driven square enclosure with MHD mixed convectional effect. Table 5 shows the comparative results between the present model and the model proposed by Rahman and Alim. These two models exhibit a good similarity in Nusselt number calculation with a maximum error of 4.889%. Therefore, our present model can be utilized to predict the thermo-fluidic characteristics of a circular cylinder in a cavity, combining the MHD mixed convectional effect with greater accuracy. [insert Fig. 4 here] [insert Table 5 here] 5. Results and discussions : The prime objective of this study is to evaluate the airflow pattern and thermal behavior of fluid in a Z-shaped heat-generating lid-driven cavity containing double rotating cylinder when both the MHD mixed convection and Joule heating effects are applied. In order to obtain the results, four types of numerical simulations have been conducted to find the effects of magnetic field strength, direction of rotation of the cylinders, heat generation coefficient, and Reynolds number of the fluids. Quantitative analysis has been conducted by calculating the Nusselt number, the average temperature of the fluid, and the average drag coefficient of the moving lid. Moreover, qualitative analysis is assessed by exhibiting streamline and isothermal contour to explicitly evaluate the flow pattern and thermal condition in the designated cavity. 5.1. Effects of Hartmann number : To find the impacts of the magnetic field effect in the lid-driven cavity, the Nusselt number is calculated by keeping some other parameters at a constant value, such as Pr = 8.5(Prandtl number of salt water at 300 K ), heat generation coefficient, ∆ = 2, Richardson number, Ri = 0.1, rotational Reynolds number, Re c = -10 (clockwise rotation), and Grashof number, Gr = 10 4 . Simulations have been carried out by varying the Reynolds number ranges from 100 to 700. Fig. 5(a) illustrates the effect of magnetic field on Nusselt number by considering three different values of Hartmann number (Ha = 20, 30, and 40). When a magnetic field is applied in a cavity, a Lorentz force is originated, which works as a damping force for convective currents generated in the cavity from temperature difference. As large as the effect of the Lorentz force, this weakens the strength of the convective currents, leading to an increasing influence of conductive heat transfer. As convective heat transfer declines, the average Nusselt number also decreases. From Fig. 5(a), it shows an almost 20% reduction in Nusselt number when Hartmann number increases from 20 to 30 and a 40% reduction in Nusselt number when Hartmann number rises from 20 to 40 at a Reynolds number of 700. This effect is much more prominent when the Reynolds number is lower than 300, as large Reynolds numbers facilitate turbulence in the fluid flow in the cavity. Fig. 5(b) delineates a kind of similar type of comparison in terms of average fluid temperature. Increasing the Hartmann number suppresses the strong convective current generated from the moving lid and rotating cylinder. This phenomenon declines the ability of the system to circulate heat effectively in the cavity, leading to an overall increase in the temperature level. At a Reynolds number of 450, the maximum temperature difference is found, which is almost 45% between the Ha number of 20 and 40. Introducing a magnetic field imposes a resistive force on the moving lid, causing the change in velocity gradient near the surface of the lid wall. The moving lid needs additional energy to overcome the effect of the Lorentz force and preserve a constant velocity. Hence, a stronger magnetic field develops a large wall shear stress in the lid wall, resulting in an increase in the drag coefficient. Fig. 5(c) shows a 10% increment in the drag force when the Ha number rises from 20 to 40 at a Reynolds number of 100. But at a higher Reynolds number, the increasing effect of the magnetic field is much lower. [insert Fig. 5 here] The effect of Hartmann number on the flow pattern is depicted in Fig. 6 by showing the streamlines on the cavity for Reynolds numbers 100 and 700. For Ha = 20 and Re = 100, two equal and symmetrical vortices are created around the cylinders, implying greater influence of the rotational motion over convective motion. But when the Reynolds number increases to 700, keeping the Hartmann number constant, more denser streamlines are visualized around the cylinders and the lower portion of the cavity. This indicates stronger effects of the moving lid and momentum transport over the convective flow. In contrast, at Ha = 30 and Re = 100, a noticeable effect of the Lorentz force can be seen, leaving only a few streamlines in the middle portion of the cavity. A small vortex is generated around the upper cylinder as the overall circulation strength diminishes. However, denser streamlines are observed around the upper cylinder when Re = 700 and Ha = 30. Because the combined effect of the moving lid and turbulent flow surpasses the effect of the magnetic field, though the scenario in the lower portion of the cavity remains unchanged. Notable reduction in the streamlines is characterized at Ha = 40 and Re = 100. Streamlines are mostly pronounced only near the cylinder walls, specially at the upper cylinder, because the magnetic effect dampens the convective effect at the middle portion where the rotational effect of the cylinders is also lower. Nevertheless, the rotational effect is dominant when the Reynolds number increases to 700, generating denser streamlines, specially at the lower right corner of the cavity. [insert Fig. 6 here] Isothermal contours of the designed cavity are depicted in Fig. 7 to visualize the temperature distribution using a non-dimensional temperature unit ranging from 0 to 1. In all of the isothermal plots, the higher temperature isothermal lines are accumulated around the lower cylinder as it is situated near the hot cavity wall. In the case of the upper cylinder, the moving lid facilitates better convection and good circulation of the temperature. Hence, comparatively lower temperature isothermal lines are generated in the upper portion of the cavity. Another important observation is that, by increasing Reynolds number with a constant Hartmann number, better convective currents generate and abate magnetic effects, creating lower temperature isothermal lines in the isothermal plots. [insert Fig. 7 here] 5.2. Effects of rotational Reynolds number : To illustrate the effects of the direction of rotation of circular cylinders, simulations have been carried out for rotational Reynolds number, Rec = -10 (clockwise rotation) and Rec = 10 (counter-clockwise rotation). Some other parameters, such as Pr = 8.5, ∆ = 2, Ha = 20, Ri = 0.1, and Gr = 104, are kept constant. Fig. 8 (a-c) presents the effects of the rotational Reynolds number on the Nusselt number, average fluid temperature, and drag coefficient of the moving lid. As the lid moves from the left to the right section of the cavity, it creates a shear-driven vortex in the fluid. When the cylinders rotate in the clockwise direction, they improve the overall circulation. Hence, better mixed convection prevails in the cavity, generating a higher Nusselt number. Moreover, this act reduces the average fluid temperature significantly. Fig. 8(a) shows a maximum 38% increment in the Nusselt number at a Reynolds number of 450 for clockwise rotation. Fig.8 (b) indicates a 22% reduction in the Nusselt number for clockwise rotation at Re = 700. But the rotation of cylinders has a negligible effect on the drag coefficient of the lid, as the lid velocity is kept constant and the Lorentz force dampens the circular rotation. This effect can be observed in Fig. 8(c), where the plot lines are almost identical. [insert Fig. 8 here] Fig. 9 shows the effect of rotational Reynolds number on the flow pattern in a picturesque manner by plotting streamlines for Reynolds numbers 100 and 700. Rotational effect prevails over magnetic and convectional effects when Re c =-10 and Re=700, creating two symmetrical vortices around the cylinders. But when the Reynolds number increases to 700 with clockwise rotation of the cylinder, two additional vortices generate surrounded with high-density streamlines, indicating initiation of turbulence in the fluid. Clockwise rotation of the cylinders generally opposes the effect of shear force generated in the moving lid. This matter can be perfectly visualized at Reynolds number 700, when the number of the streamlines is declining with an additional vortex established near the lid. Fig. 10 presents the isotherm plots from the simulations to show the effects of the rotational Reynolds number on the temperature distribution. These plots clearly reveal that clockwise rotation of the cylinders or aiding flow can provide better temperature distribution inside the cavity with a higher Reynolds number. [insert Fig. 9 here] [insert Fig. 10 here] 5.3. Effects of heat generation coefficient : Fig. 11(a,b) explores the effects of the heat generation coefficient on the Nusselt number and average fluid temperature of the rotating cylinders. These simulations have been done at Pr = 8.5, Ha = 20, Ri = 0.1, Gr = 104, Rec = -10, heat generation coefficient ∆ = 0, 2, 5, and Reynolds number from 100 to 700. As the heat generation coefficient increases, it adds additional thermal energy in the fluid domain. As a result, the average temperature in the cavity rises, and the temperature gradient between the hot and cold walls decreases. These effects lead to a reduction in the Nusselt number as well as convective flow strength. Fig. 11(a) expresses that the maximum reduction in the Nusselt number is 45% when the heat generation coefficient enhances from 0 to 5 for a fixed Reynolds number of 100. Fig. 11(b) indicates an almost 20% increment in average fluid temperature for maximum rise in heat generation coefficient. [insert Fig. 11 here] Fig. 12 shows the streamline plots developed to understand the effect of the heat generation coefficient in the flow field. However, there is no significant change in these plots. When heat is applied in a cavity, it increases the temperature of the fluid and facilitates buoyant force. But, as the magnetic effect is applied and flow circulation is primarily driven by rotation of the cylinders and the moving lid, temperature rises are suppressed. Therefore, streamline plots are similar for a constant Reynolds number. Fig. 13 draws the isotherm plots to predict the effect of the heat generation coefficient. The isothermal lines are mostly accumulated and more intense near the hot boundary walls and less so near the upper cylinder. As heat generation increases, the temperature gradient reduces between hot and cold walls, and a uniform temperature distribution can be found in the core. [insert Fig. 12 here] [insert Fig. 13 here] 5.4. Effects of Reynolds number : The effects of fluid Reynolds number are examined by comparing three values of Reynolds number of 100, 400, and 700 with respect to Grashof number ranges from 103 to 104 by keeping other parameters such as Pr = 8.5, heat transfer coefficient = 2, Ha = 20, Ri = 0.1, and Re c = -10 as constant. As the Reynolds number of the fluid increases, it creates strong forced convection and better mixing of the flow in the cavity, resulting in an increase in the Nusselt number. Fig. 14(a) indicates over a 150% increment in the Nusselt number when the Reynolds number increases from 100 to 700. But the Reynolds number doesn’t change with respect to increasing the Grashof number. This is because, though natural convection rises with increasing Grashof number, the magnetic field suppresses the strength of the natural convection inside the cavity, and only forced convection dominates. However, overall fluid temperature reduces as expected, and it is about 37%, according to Fig. 14(b). [insert Fig. 14 here] Fig. 15 shows the streamline characteristics for this simulation result. It is noted that a secondary vortex creates above the upper cylinder and continuously approaches from the left to the right section of the cavity with increasing Reynolds number, as higher Reynolds numbers provide faster motion of the lid with counterclockwise rotation of the cylinders. Fig. 16 illustrates the isothermal properties for the effects of Reynolds number, showing a general trend of reducing the high-temperature isothermal lines in the plots with increasing Reynolds number, as higher Reynolds number facilitates proper allotment of temperature and reduces the temperature gradient between hot and cold walls. [insert Fig. 15 here] [insert Fig. 16 here] 6. Conclusion : The current numerical investigations intensified with both quantitative and qualitative assessments to interpret the results of the effects of MHD mixed convection and Joule heating in a specialized lid-driven Z-shaped cavity containing double rotatin cylinders. Important outcomes of the present study are listed below: • The damping effect of the Lorentz force suppresses the convective flow inside the cavity. Hence, increasing the Hartmann number leads to a reduction in the Nusselt number as well as an increment in the fluid average temperature. Moreover, this effect increases the drag coefficient of the moving lid due to the resistive force applied by the magnetic effect. • Clockwise rotation of the cylinders or aiding flow develops the overall circulation of the flow. That’s why clockwise rotation increases the Nusselt number and decreases the average temperature of the fluid. Though, this condition has a negligible effect on the drag coefficient of the lid because of the magnetic effect. • An increase in the heat generation coefficient adds more energy to the simulation contour, leading to a decrease in the Nusselt number and convective flow strength. • Higher Reynolds number generally implies more turbulent flow conditions inside the cavity, resulting in a better mixing of the fluid and strong forced convection. Therefore, a higher Reynolds number increases the Nusselt number and reduces the average fluid temperature. Future studies should focus on more rigorous simulations by incorporating nanofluid and considering turbulent flow in the cavity. Moreover, investigations with complex geometry, such as curved surfaces, inclination of magnetic effect, and oscillating motion of the cylinders, would be helpful to predict the fluid flow and thermal conditions in intricate geometrical applications. Declaration of competing interest : This research did not receive any specific grants from funding agencies in the public, commercial or non-profit sectors. Data Availability Statement : The data that support the findings of this study are available on request from the corresponding authors. The data are not publicly available due to privacy or ethical restrictions. Authorship contribution statement Mallik Nadim Arman Omi : Idea generation, Data curation, Preliminary analysis, Figure generation, Methodology, Visualization, Simulation, Writing manuscript. MD Abir Khan : Conceptualization, Equation formation, Methodology, Writing manuscript. Nomenclature x,y Cartesian coordinates (m) X,Y Non-dimensional cartesian coordinates u Velocity component along x-axis (m/s) U Dimensionless velocity component along x-axis v Velocity component along y-axis (m/s) V Dimensionless velocity component along y-axis Q Volumetric heat generation rate (W/m 3 ) p Pressure (Pa) P Dimensionless pressure T Temperature ( K) B 0 Magnetic field strength (T) C p Specific heat capacity ( J/kgK) D Diameter of the cylinder (m) L Length of the enclosure (m) g Gravitional acceleration (m 2 /s) k Thermal conductivity (W/mK) Ha Hartmann nubmer Gr Grashof number Ri Richardson number Pr Prandtl number Re Reynolds number Re c Circular Reynolds number Nu Nusselt number J Joule heating parameter i Index number N Non-dimensional normal distance measured from wall Greek symbols μ Dynamic viscosity (Pa s) β Volumetric coefficient of thermal expansion (K -1 ) ∆ Heat generation coefficient Ω Non-dimensional velocity of the cylinder Ɵ Non-dimensional temperature ρ Density ( kg/m 3 ) σ Electrical conductivity ( S/m) Abbreviations MHD Magnetohydrodynamics CW Clockwise CCW Counter clockwise References 1. 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Hasan, Nahid, and Sumon Saha. ”MHD conjugate mixed convection along with internal heat generation and Joule heating in a closed/open cavity with rotating solid cylinder.” International Journal of Numerical Methods for Heat & Fluid Flow 34, no. 9 (2024): 3438-3461. 41. Khanafer, Khalil, S. M. Aithal, and K. Vafai. ”Mixed convection heat transfer in a differentially heated cavity with two rotating cylinders.” International Journal of Thermal Sciences 135 (2019): 117-132. 42. Rahman, M. M. M., and M. A. A. Alim. ”MHD mixed convection flow in a vertical lid-driven square enclosure including a heat conducting horizontal circular cylinder with Joule heating.” Nonlinear Analysis: Modelling and Control 15, no. 2 (2010): 199-211. Table captions : Table 1. Boundary and cylinder interface conditions in non-dimensional form considered for the current problem. Table 2. Characterizing equations and range of values of the non-dimensional parameters used in this study. Table 3. Mesh refinement test in terms of average Nusselt number . Table 4 : Detailed properties of the optimum mesh type ( extra fine ) Table 5 : Comparison of the results between the present model and Rahman & Alim [42] by calculating Nusselt number with respect to the variation of Hartmann number Top wall u = 1, v = 0 Ɵ = 0 Bottom wall u = v = 0 Ɵ = 1 Left surfaces u = v = 0 \(\frac{dƟ}{\text{dx}}=0\) Right surfaces u = v = 0 \(\frac{dƟ}{\text{dx}}=0\) Upper Cylinder surface u = 2(Y-0.75)(Re c /Re)/(D/L) v = 2(X-0.5)(Re c /Re)/(D/L) k (\(\frac{dƟ}{\text{dN}})\ \)= 0 Upper Cylinder surface u = 2(Y-0.25)(Re c /Re)/(D/L) v = 2(X-0.5)(Re c /Re)/(D/L) k (\(\frac{dƟ}{\text{dN}})\ \)= 0 Table 1 : Boundary and cylinder interface conditions in non-dimensional form considered for the current problem. Reynolds number (Re) \(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\text{ρuL}}{\mu}\) 100-700 Prandtl number (Pr) \(\frac{\mu C_{p}}{k}\) 6.2 Grashof number (Gr) \(\frac{\text{gβ}\rho^{2}(T_{h}-T_{c})L^{3}}{\mu^{2}}\) 10 3 -10 6 Richardson number (Ri) \(\frac{\text{Gr}}{\text{Re}^{2}}\) 0.1 Hartmann number (Ha) \(B_{0}L\sqrt{\frac{\sigma}{\mu}}\) 20-40 Joule heating parameter (J) \(\frac{\text{gβL}\text{Ha}^{2}\text{Re}}{\text{Gr}C_{p}}\) 1.8×10 -7 – 7.25×10 -7 Table 2. Characterizing equations and range of values of the non-dimensional parameters used in this study. Extreamely coarse 258 270 5.1727 - Extra coarse 622 432 4.8136 6.9422 Coarser 704 608 4.5701 5.0585 Coarse 1422 1092 4.4063 3.5842 Normal 1894 1398 4.2881 2.6825 Fine 2848 1945 4.1832 2.4463 Finer 6812 4497 4.1073 1.8144 Extra fine 19956 12079 4.0584 1.1905 Extreamely fine 22952 13577 4.0562 0.0542 Table 3. Mesh refinement test in terms of average Nusselt number . Mesh vertices 12079 Triangles 19956 Quads 1680 Edge elements 844 Vertex elements 16 Number of elements 21636 Minimum element quality 0.2217 Average element quality 0.8017 Element area ratio 0.001496 Mesh area 0.6172 m 2 Table 4 : Detailed properties of the optimum mesh type ( extra fine ) 0 2.1085 2.2069 4.458 10 2.0836 2.1132 1.410 20 1.7316 1.8206 4.889 50 1.1592 1.186 2.259 Table 5 : Comparison of the results between the present model and Rahman & Alim [42] by calculating Nusselt number with respect to the variation of Hartmann number Figure captions : Fig. 1. Schematic diagram of a Z-shaped cavity used in this study with a sliding top lid and double rotating cylinders Fig. 2 : Flowchart of the finite element method adopted in this study Fig. 3. Optimum mesh geometry selected after mesh refinement test in the present study Fig. 4 depicts the relationships between these two models by comparing Nusselt number calculation with respect to the variation of Richardson number Fig. 5. Variation of the (a) Nusselt number, (b) average fluid temperature and (c) drag coefficient as a function of Reynolds number for different values of Hartmann numbers. Fig. 6. Visualization of streamlines for several values of Hartmann number and Reynolds number Fig. 7. Visualization of isothermal plots for different values of Hartmann number and Reynolds number Fig. 8. Variation of the (a) Nusselt number, (b) average fluid temperature and (c) drag coefficient as a function of Reynolds number for different values of rotational Reynolds number. Fig. 9. Visualization of streamlines for several values of rotational Reynolds number and Reynolds number Fig. 10. Visualization of isothermal plots for different values of rotational Reynolds number and Reynolds number Fig. 11. Variation of the (a) Nusselt number and (b) average fluid temperature coefficient as a function of Reynolds number for different values of heat generation coefficient Fig. 12. Visualization of streamlines for several values of heat generation coefficient and Reynolds number Fig. 13. Visualization of isothermal plots for different values of heat generation coefficient and Reynolds number Fig. 14. Variation of the (a) Nusselt number and (b) average fluid temperature coefficient as a function of Grashof number for different values of Reynolds number Fig. 15. Visualization of streamlines for several values of Grashof number and Reynolds number Fig. 16. Visualization of isothermal plots for different values of Grashof number and Reynolds number Fig. 1. Schematic diagram of a Z-shaped cavity used in this study with a sliding top lid and double rotating cylinders Fig. 2 : Flowchart of the finite element method adopted in this study Fig. 3. Optimum mesh geometry selected after mesh refinement test in the present study Fig. 4 depicts the relationships between these two models by comparing Nusselt number calculation with respect to the variation of Richardson number (a) (b) (c) Fig. 5. Variation of the (a) Nusselt number, (b) average fluid temperature and (c) drag coefficient as a function of Reynolds number for different values of Hartmann numbers. Re = 100 Re = 700 Fig. 6. Visualization of streamlines for several values of Hartmann number and Reynolds number Re = 100 Re = 700 Fig. 7. Visualization of isothermal plots for different values of Hartmann number and Reynolds number (a) (b) (c) Fig. 8. Variation of the (a) Nusselt number, (b) average fluid temperature and (c) drag coefficient as a function of Reynolds number for different values of rotational Reynolds number. Re = 100 Re = 700 Fig. 9. Visualization of streamlines for several values of rotational Reynolds number and Reynolds number Re = 100 Re = 700 Fig. 10. Visualization of isothermal plots for different values of rotational Reynolds number and Reynolds number (a) (b) Fig. 11. Variation of the (a) Nusselt number and (b) average fluid temperature coefficient as a function of Reynolds number for different values of heat generation coefficient Re = 100 Re = 700 Fig. 12. Visualization of streamlines for several values of heat generation coefficient and Reynolds number Re = 100 Re =700 Fig. 13. Visualization of isothermal plots for different values of heat generation coefficient and Reynolds number (a) (b) Fig. 14. Variation of the (a) Nusselt number and (b) average fluid temperature coefficient as a function of Grashof number for different values of Reynolds number Gr = 10 3 Gr = 10 6 Fig. 15. Visualization of streamlines for several values of Grashof number and Reynolds number Gr = 10 3 Gr = 10 6 Fig. 16. Visualization of isothermal plots for different values of Grashof number and Reynolds number Information & Authors Information Version history V1 Version 1 10 July 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Collection Heat Transfer Keywords convection fluid dynamics heat transfer Authors Affiliations Mallik Nadim Arman Omi 0009-0003-9960-4786 [email protected] Bangladesh University of Engineering and Technology View all articles by this author Md Abir Khan Bangladesh University of Engineering and Technology View all articles by this author Metrics & Citations Metrics Article Usage 407 views 116 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Mallik Nadim Arman Omi, Md Abir Khan. Discretization of thermo-fluidic behavior of the combined effects of MHD conjugate mixed convection and Joule heating in a lid-driven Z-shaped cavity containing double rotating cylinders. Authorea . 10 July 2025. 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