Effect of symmetric and asymmetric non-uniform magnetic field on hydrodynamically developed and ther

2.2. Numerical approach and governing equations

In this study, COMSOL Multiphysics was applied to model the coupled behavior of fluid flow, heat transfer, and magnetic fields. The fluid flow and thermal energy equations are solved using a stationary linear AMG (Algebraic Multigrid) GMRES (Generalized minimal residual) solver with a relative tolerance of 0.001. This solver configuration includes a smoothed-aggregation AMG solver

with SSOR (Symmetric Successive Over-Relaxation) pre-smoother and post-smoother. Furthermore, the magnetic field equations are solved using the conjugate gradient method with a residual tolerance of 0.001. The computational mesh is highly refined, and the total number of degrees of freedom (DOF) used in computational study is 43,100,279. Ferrofluid is considered a single-phase, electrically non-conducting, homogeneous, Newtonian, and incompressible fluid. These assumptions are physically justified by the specific operating conditions considered, mainly small spherical magnetite particle size (d=10 nm), low volume concentration (Ø=0.02), low Reynolds number and low magnetic field intensity regime. At low magnetic field strengths, the magnetic torque acting on the very small and less concentrated suspended nanoparticles is insufficient to significantly overcome thermal agitation (Brownian motion) [41]. So, the formation of long particle chains or large aggregates which are mainly responsible for inducing non-Newtonian shear-thinning behavior and the magnetoviscous effect is minimal. Under these conditions, the rotational viscosity component arising from the misalignment of magnetic moments remains negligible compared to the hydrodynamic shear viscosity. Therefore, we can reasonably assume constant fluid properties. Nanomagnetic particles in ferrofluids become magnetized in the presence of an external magnetic field, i.e., one generated by a permanent magnet. This magnetized ferrofluid experiences a magnetic force in the vicinity of a magnetic field, corresponding to the Kelvin body force. This external force disturbed the hydrodynamic and thermal boundary layers of the ferrofluid flow. The flow physics is basically governed by the continuity, momentum, and energy equations as given below [27,38,42]:

By considering non conduction of ferrofluid and current free of surrounding air, the Maxwell-Ampere’s law for magnetic field (H) and flux density (B) by Gauss law is expressed as [41]: d # H = 0 (4)

Magnetic remanent flux density (Br) is used in the permanent magnet for generating magnetic field (H). Within the magnet, it is expressed as [38]: d. ^ n 0 n rec H + B r h = 0 (6)

Here, µ0 and µrec represent, respectively, the magnetic permeability of free space and the recoil permeability of the magnet. The magnetic field generated by the permanent magnet magnetizes the ferrofluid. To get the relation between the magnetic field and the magnetization of ferrofluid, Langevin’s classical theory is adopted [42]. M 1 = a cotha - a k = L ^ a h (7) 4M d a=

Here, α represents the interaction of magnetic and thermal energies of ferroparticles. For large values of α, L(α) moves towards unity. It means that the magnetization of ferrofluid reaches a saturated condition. It can be clearly understood in Figure 2 between the magnetization of ferrofluid and the applied magnetic field.

Magnetic flux density and magnetic field for the ferrofluid are expressed as [38].

The momentum equation is the consequence of Newton’s second law of motion applied to a fluid control volume:

With the help of the above equations (7) and (8), equation (9) is modified as:

2.3. Magnetic field and kelvin body force

The last term on the right-hand side of the equation (2) represents the Kelvin body force. This Kelvin body force is calculated by solving the following equations:

Here Ø represents the volume concentration of the nanoparticles, 5 d is the diameter of Al2O3 nanoparticles and Md (4.46 × 10 A/m) is the domain magnetization of magnetite nanoparticles. Table 2 summarizes the parameters used to simulate the magnetic field of the NdFeB permanent magnet.

Parameter used for the magnetic field simulation Remanent Magnetic flux density (Br)

To obtain a second-order solution of the energy equation, a thermally developing flow with a uniform fluid temperature of 300 K was maintained at the channel inlet, and diffusive heat conduction at the outlet was set to zero. A constant heat flux is applied to the channel wall. W Tin = 300 K, n. ^ -k ff dT h = 0, q " = n. ^ -k ff dT h = 10000 2 m (14) To understand the heat transfer phenomenon of ferrofluid in the presence of magnet, we use the local Nusselt number and space average Nusselt number from the obtained velocity and temperature solution as per equations (15) and (17). Local Nusselt Number: q'' # D h Nu z = k (T - T ) ff

Figure 2. Magnetization of ferrofluid Magnetic force or Kelvin body force on the magnetized ferrofluid is defined as [19]: Fmag = n 0 ^ M.d h H

Simplifying the Kelvin body force with help of the above equation (7), we get: Fmag = n 0 4M d L ^ a h dH

The governing equations mentioned above have been solved using the finite element method. The discretization scheme is based on a second-order formulation for the velocity field and a linear formulation for the pressure and temperature fields. The Navier-Stokes equations are solved to obtain the velocity field of the ferrofluid. The energy transport equation is solved using the previously obtained velocity field to obtain the temperature field. To solve the above-mentioned equations, different models use separate boundary conditions. For the present simulation study, different boundary conditions were used to solve the above-mentioned equations. To get the magnetic field for the entire domain, the magnetic insulation (n.B=0) boundary condition is considered at the wall boundary. The continuity and momentum equations are solved by assuming a hydrodynamically developed velocity profile, an average inlet velocity, and atmospheric pressure at the channel outlet. The no-slip condition is applied at the channel surface.

Here, Tff,z represents the bulk fluid temperature at a particular section in the fluid flow domain. Average Nusselt Number ## V $ TdA Tff,z = (16) ## VdA Nu avg =

2.4. Material properties

In the present study, a ferrofluid solution (Fe3O4+ water) prepared by mixing 2% (v/v) Fe3O4 nanoparticles with water was used. The bulk density, specific heat, thermal conductivity, and dynamic viscosity of the ferrofluid are estimated by the commonly quoted and accepted correlation [43,44] given below. t ff = 4t np + ^ 1 - 4 h t w C pff =

RS VW k np SS WW 3c k - 1 m4 S WW w k ff = k np SS1 + k W k np np SS c m c m W S k w + 2 - k w - 1 4 WW T X

The thermophysical properties of the ferrofluid are summarized in Table 3.

Fe3O4 nano particles Distilled water Ferrofluid (water + 2 % v/v, Fe3O4)

2.5. Mesh independency and validation

A fine tetrahedral non-uniform grid has been used in the computational domain, as can be seen from Figure 3 (i). Fine boundary layers have been provided near the walls of the fluid domain to accurately capture the temperature and velocity gradients. To ensure that the results are independent of grid size, a grid independence test was performed. Figure 3 (ii) shows the grid independence test where the variation of the local Nusselt number along the flow direction for various grid sizes is shown. Changing the mesh from 3 to 4 resulted in negligible changes. Hence, we chose mesh 4 to improve the accuracy of the results. Details of the mesh configuration are given in Table 4.

1.302 (z *) -1/3 - 1 for z * # 0.00005 Nu z,H = 1.302 (z *) -1/3 - 0.5 for 0.00005 # z * # 0.0015 4.364 + 8.68 (1000z *) -0.506 e -41z for z * $ 0.0015 *

In the second aspect, the magnetic field result is validated with Asfer et al. [46]. Figure 4 (ii) shows the variation in the magnetic flux density along the centerline in the flow direction. Magnetic flux density decreases as we move away from the magnet. The flux density is higher at the magnet’s edge. A very good quantitative and qualitative agreement is seen for the obtained result with that of Asfer et al. [46]. After successfully validating both aspects separately, move forward and couple together these aspects, and validate the result with Shah et al. [19]. Figure 4 (iii) shows the comparison of the coupled physics (i.e., hydrothermal and magnetic) in terms of lower wall temperature variation along the flow direction with the earlier published result of Shah et al.[19]. Negligible variation was observed between them. Now with the confidence gained from this certification, we can proceed with our studies.

Table 4. Different mesh sizes for fluid flow region and total number of mesh element Min element size (mm)

Before gaining insight into the flow physics of the ferrofluid, we validated our computational models against established literature to increase confidence in the results. The current problem has two aspects, a heat transfer study and a magnetic field study. Therefore, we have validated both aspects separately. As an initial part of the validation process, the models were first validated with the Shah and London equations [45] of Nusselt number variation along the flow for hydrodynamically developed and thermally developing flow with constant wall heat flux condition. Figure 4 (i) shows that the results obtained in the present work are in very good agreement with the value obtained from the correlations given in equation (22). Shah and London equations for the hydrodynamically developed and thermally developing flow:

Figure 3. (i) Meshing of computational domain (ii) Mesh independency test by using different mesh size

Figure 5. (i) Magnetic flux density vectors (a) case 2 (b) case 3 (c) o case 4 (θ=45 ) (ii) Magnetic flux density variation along the centre line of the flow channel

Figure 4. (i) Validation graph of present work with Shah and London (ii) Magnetic field variation along the centerline of flow channel validation with Asfer et al.[46] (iii) Coupled physics (Flow + Magnet) validation with shah et al. [19]

3.1. Variation of magnetic field

Figure 5 (i) shows how the magnetic flux vectors vary in the study domain. It is clear from the vectors that, in cases 2 and 3, symmetric, non-uniform magnetic flux has been applied in the flow domain on both sides of the magnet, whereas in case 4, an asymmetric, non-uniform magnetic field has been applied.

Figure 5 (ii) shows how the magnitudes of flux density components vary along the centerline of fluid flow. For cases 2 and 3, away from the magnet, Bx at the centerline is negative because the vector direction changes from upward to downward. On coming closer, the direction of the vector changes upward, causing the maximum value of Bx (the maximum values for case 2 and case 3 are 0.042 Tesla and 0.0765 Tesla, respectively) to be obtained just below the magnet with a positive value. Bx varies similarly on both sides of the magnet along the centerline of the fluid channel. For case 2, the z-component of magnetic flux density, Bz, at the centerline also shows the same pattern on both sides of the magnet but of opposite nature. Its value becomes zero in case 3 because the two magnets are oppositely oriented. In case 4 (θ=45°), because the magnet is angled, the value of Bx on one side decreases on moving away from the magnet and becomes negative, whereas on the opposite side it remains positive. The z component in case 4 (θ=45°) also varies as in case 2, but the magnitude of Bz on both sides of the magnet is different (+ 0.019 Tesla on the left side of the magnet and -0.03 Tesla on the right side of magnet) because of its angular position.

3.2. Effect on hydrothermal characteristics of ferrofluid flow

It is quite clear from Figure 6 (i) that in the left part from the vertical center of the magnet, the horizontal component of the magnetic 5 3 force (at centerline in the range of Fz=0.42 x 10 N/m ) will be supporting the inertia force. On the other hand, in the right-hand part, it will oppose the inertial force with the same magnitude. Thus, the flow field in the domain results from the resultant of the inertial and magnetic forces. From the velocity vector, it is observable that in upstream locations of the magnet, w’ (velocity component in flow direction) velocity increases in the nearby region where magnetic force supports the inertia, while in the downstream locations, a larger magnitude of ‘w’ is obtained away from the magnet. Fluid on 5 both sides of the magnet feels attraction toward it with a 1.7 x 10 N/ 3 m maximum magnitude. Due to this attractive force, the velocity vector of the fluid on the left side of the magnet changes direction toward the magnet and increases in magnitude (maximum velocity w = 0.061 m/s). This velocity distribution, the result of the interaction between inertia and magnetic forces, causes disturbances in the boundary layer. This magnet creates a counterclockwise recirculation region, 4.8 mm long and 0.52 mm wide, whose tail end is oriented opposite to the flow direction. Since the velocity vector is changing direction towards the magnet with increasing magnitude, the tail part of this recirculation region in the left part of the magnet acts as a hot region, as the conduction process from wall to fluid is continuous but the advection process is restricted. Consequently, the wall temperature in the tail of the recirculation region increases; as the flow moves downstream, convection intensifies and the wall temperature subsequently decreases. This phenomenon is clearly observed in all cases except case 1 in Figure 6 (ii). This phenomenon is mostly pronounced in case 3. Because the two magnets are face-to-face, the magnetic force only pulls the ferrofluid towards itself, in the direction perpendicular to the magnets’ faces within the magnetic field zone. Due to this, recirculation regions (5 mm long and 0.43 mm wide) form near the channel wall, and the advection process becomes very slow. Therefore, the temperature of the wall increases. However, due to the formation of this narrow region beneath the magnet, the velocity of the convective flow increases. Due to this increased velocity, convection intensifies and the wall temperature begins to decrease. In case 4 (θ=135°), the asymmetric magnetic field also causes the magnetic force to appear uneven. Deep recirculation zones (6 mm long and 1.5 mm wide on the left, and 3.5 mm long and 1 mm wide on the right) are formed on both sides of the magnet, thereby disturbing both the thermal and hydrodynamic boundary layers.

Figure 6. For Re=60 (i) Variation of velocity vector and temperature colour contour on zx plane (A) case1 (B) Case 2 (C) Case 3 (D) O Case 4 (θ=135 ) (ii) Variation of wall temperature (iii) Variation of Local Nusselt Number along the direction of flow

3.3. Effect on Nusselt number

The augmentation of the convective heat transfer of ferrofluid under the presence of an external magnetic field mainly depends on the ratio of magnetic to inertial forces acting on the ferrofluid, and on the formation of a recirculation zone. Magnetic force arises from the magnetic field gradient, and the inertial force varies with the flow velocity of the ferrofluid. Figure 6 (iii) shows the fluctuation of the local Nusselt number with respect to the dimensional axial distance (z/Dh) for all examined cases (1 to 4) at Re=60. The graphical representations for all cases show that away from the magnet the Nusselt number decreases to a limiting value owing to the formation of the hot recirculation region; thereafter, when advection becomes

dominant, the Nusselt number increases, reaches a peak, and then declines. In case 4, the progression of the Nusselt number is found to be higher than the increase observed in case 2 for nearly the same magnetic field intensity. This phenomenon can be attributed to the formation of deep recirculation regions, which arise from the asymmetric magnetic field and significantly affect the thermal and fluid dynamics within the system. Case 3, under the influence of a symmetric non-uniform magnetic field, showed a comparatively greater increase in the local Nusselt number (Nuz=10.86) than the other cases subjected to magnetic fields. This leads to the formation of stagnation recirculation zones in certain regions on the channel wall, which subsequently hinder the efficient transfer of thermal energy and significantly affect the complex interplay of thermal and fluid dynamics within the system. Comparative analysis of the local Nusselt-number increments across cases, relative to case 1, shows that case 4 exhibits the largest increase (73.16%), whereas case 2 shows the smallest increase (67.31%).

Table 5. Space average Nusselt number enhancement by considering base is case1 Re no.

3.4. Effect of Reynolds number in symmetric non-uniform

magnetic field condition Extensive numerical investigations covering relevant cases have been carefully conducted at low Reynolds numbers, particularly in the range from 20 to 100, where significant variations in the local Nusselt number have been systematically observed. By carefully analyzing from panels (i) and (ii) of Figure 7, one can observe the complex way the local Nusselt number varies along the specified direction of fluid flow, providing important insights into the thermal performance of the ferrofluid. In a localized context, it has been observed that local enhancement can reach a maximum of 122.68% in case 2 and 177.87% in case 3, representing a significant increase achieved by the application of magnetic fields. Examination of the graphs reveals a significant increase in the local Nusselt number, particularly at low Reynolds numbers. This phenomenon can be attributed to the predominance of magnetic forces under such conditions. Therefore, with increasing Reynolds number, the observed enhancement of the local Nusselt number systematically decreases relative to the initial case. If one is to analyze extensive changes within the flow field, it becomes imperative to employ equation (17), which provides the mathematical framework for determining the space-averaged Nusselt number and offers a quantitative basis for understanding heat-transfer characteristics in fluid dynamics. Findings from this analytical method indicate that maximum heat transfer predominantly occurs at low Reynolds numbers, demonstrating a significant relationship between heat transfer and Reynolds numbers. Furthermore, as illustrated in Table 5, scenarios characterized by symmetric non-uniform magnetic fields show a clear enhancement in thermal performance, especially at low Reynolds numbers, with Case 2 showing a 21.72% increase and Case 3 showing a 32.01% increase.

Figure 7. Local Nusselt number change in the flow direction (i) case 2 (ii) case 3

3.5. Effect of Reynolds number in asymmetric non uniform

magnetic field condition Figure 8 (i) clearly shows that each magnet configuration at Re=20 produces a distinct enhancement of the local Nusselt number, which can be attributed to the differing regions of the magnetic field arising from the magnet orientations. The figure shows how the local Nusselt number changes with the angular position of the magnet relative to the reference axis (i.e., the z-axis). The graph clearly shows a different behavior where the peak of the local Nusselt number shifts toward the left as the angular position of the magnet increases. Furthermore, the magnitude of this peak increases as

the magnet’s angular position increases. The maximum local heat transfer is achieved at the θ = 135° magnet position, yielding a peak value of Nuz = 10.44. This corresponds to a substantial 129.8% local enhancement compared to the base case. By contrast, the peaks for θ = 45° and θ = 90° occur at 10.11 and 10.17, respectively, whereas the θ = 0° and θ = 180° configurations yield a lower peak at 8.68. These results demonstrate that the magnet’s position significantly influences heat transfer performance. This variation occurs because the changing position creates an asymmetric, non-uniform magnetic field, exerting distinct magnetic forces that accelerate or decelerate the ferrofluid flow in specific regions. o

Having identified that the θ =135 magnet position yields the maximum increment in heat transfer, we continue to investigate how this enhancement behaves across the selected Reynolds number range (20-100). Figure 8 (ii) shows that the largest enhancement occurs at lower Reynolds numbers; as flow velocity increases, the peak local Nusselt number declines. Specifically, we observed a maximum enhancement of 129.8% at Re = 20. This value gradually diminishes to 85.68% at Re = 40, 74.07% at Re = 60, and 54.62% at Re = 80. Finally, at the highest tested velocity of Re = 100, the enhancement reduces to 38.68%. Figure 8 (iii) depicts the variation of the space-averaged Nusselt number for case 4 across the given Reynolds number range and for all angular positions of the magnet. A remarkably similar enhancement trend was consistently observed across all angular positions assessed in this study. A thorough analysis of the graphical representation indicates that the peak enhancement occurs at the angular o position θ = 135 . Notably, an impressive enhancement of 23.54 % is recorded at a Reynolds number of 20, occurring at the mago net’s angular position of θ=135 . For the asymmetric nonuniform magnetic field case, a generalized empirical correlation is derived from reported data for the spatially averaged Nusselt number as a function of Reynolds number and magnet inclination angle θ (in degrees). The proposed relation, valid for 20 ≤ Re ≤ 100 and 45 ≤ θ ≤ 180, is expressed as: Nu = 4.06 # Re avg

61 - ^ 3.70 # 10 h i + ^ 3.71 # 10 h i - ^ 1.10 # 10 h i @ -3

This correlation demonstrates excellent agreement with the numerical data, with a mean absolute percentage error of approximately 1.1%, effectively capturing the optimal thermal performance obo served near θ =135 .

Figure 8. (i) Variation of Local Nusselt Number along the flow direction for case 4 (Re=20) (ii) Variation of Local Nusselt number o along the direction of flow, case 4 (θ =135 ) (iii) Space average Nusselt Number variation for case 4

21.72. % is documented in Case 2 and of 32.01% in Case 3.

• The magnet’s angular position plays a critical role in thermal performance because it generates asymmetric, o non-uniform magnetic fields. Magnet position at θ=135 identified as the optimal position, showing the maximum local Nusselt number enhancement of 129.8% and a space-average enhancement of 23.54% compared to the base case. • An inverse relationship is observed between the flow velocity and the effectiveness of the magnetic field. The heat transfer augmentation is most pronounced in the low-Reynolds-number regime where magnetic forces dominate inertial forces. Specifically, at the optimal ano gular position (θ=135 ), the local enhancement peaks low Reynolds number but progressively diminishes as inertial forces increase. The current 3D numerical investigation is restricted to laminar flow under steady-state magnetic fields, in which increasing inertia is observed to suppress thermal enhancement, thereby leaving the impact in turbulent regimes unexplored. Future research should therefore extend this analysis to high Reynolds numbers and examine time-varying or oscillating magnetic fields, which may reduce stagnation zones and optimize performance in dynamic flow environments.