Magneto-hydrodynamic behavior of magnetic nanofluids in mini-channel heat sinks for electronics cooling

Abstract

The rapid advancement of high-density electronic devices and data centres has heightened the demand for effective thermal management solutions capable of handling elevated heat fluxes within compact domains. Conventional cooling techniques often fail to meet these requirements efficiently. This study presents a numerical investigation of heat transfer enhancement in a mini-channel heat sink through the combined use of passive vortex generators (ribs) and externally applied magnetic fields. A two-dimensional simulation was conducted for a 40 mm × 4 mm mini-channel employing a 2% Fe₃O₄–water nanofluid, with magnets positioned at X = 15 mm and X = 25 mm to generate non-uniform magnetic fields ranging from 800 to 2000 G. Three rib configurations parallel, staggered, and ribbed were evaluated across a Reynolds number range of 50, 75, 100, 150, 180, and 210. Results indicate that the ribbed configuration provides the highest heat transfer improvement, achieving up to a 65% increase relative to the baseline, while the parallel arrangement attained the highest absolute Nusselt number. The friction factor increased with stronger magnetic fields but decreased with higher Reynolds numbers. The thermal enhancement factor remained consistently above unity, with peak values of 2.06 for ribbed, 1.77 for parallel, and 1.52 for staggered layouts. Overall, this study demonstrates that integrating rib-induced vortex generation with magnetic field effects offers a promising strategy for enhancing the thermal performance of mini-channel heat sinks, addressing the cooling demands of next-generation electronic and data centre applications.

Introduction

Thermal management has become a crucial issue in system design and reliability in the age of highly efficient and compact electronic gadgets. Resistors, capacitors, MOSFETs, and IGBTs are among the increasingly dense arrays of microelectronic components used in modern electronic systems, and each one produces a considerable amount of heat when in use. These parts have tight thermal working limitations, and going over them could result in failure or performance reduction. When exposed to high temperatures, active components like IGBTs have major reliability problems that could result in thermal runaway and total device failure, but other protective devices, like fuses, depend on thermal thresholds for circuit protection.

In modern electronic systems, traditional cooling methods which are mostly reliant on forced air convection present a number of drawbacks. Within small device enclosures, the recirculation of warm air may unintentionally impact nearby components, resulting in thermal interference and localized hot spots. Furthermore, as a result of significantly increased heat flux densities brought about by the continuous trend toward device reductions in size, traditional cooling techniques are becoming less effective¹. This phenomenon of thermal intensification required creative cooling methods that can handle the twin problems of increased thermal loads and limited space. Therefore, it is now crucial to create sophisticated cooling systems to guarantee device dependability, increase operational longevity, and preserve ideal performance characteristics². In closely packed electronic assemblies, good thermal management must not only remove heat but also avoid thermal cross-talk between components. In order to satisfy these exacting standards, this study explores innovative cooling techniques, emphasizing solutions that get around the drawbacks of conventional techniques and support the ongoing downsizing of electronic equipment³. Promvonge and Eiamsa-ard⁴ experimental study investigates heat transfer, friction factor, and enhancement efficiency in a circular tube fitted with conical-ring turbulators and twisted-tape swirl generators under constant wall heat flux conditions. Results show that combining both devices increases the Nusselt number by 4–10% and enhancement efficiency by 4–8% compared to using the conical-ring alone, with a maximum heat transfer improvement of 367% at Y = 3.75. Empirical correlations for Nusselt number, friction factor, and performance evaluation criteria are developed to assess the effectiveness of these enhancement techniques. Rashidi et al.⁵ conducted a three-dimensional numerical study on nanofluid flow in a square duct with transverse twisted-baffles, using the finite volume method to simulate forced convection. The study found that a baffle pitch of c = 360 maximized heat transfer, while c = 540 minimized pressure drop. Increasing nanoparticle volume fraction or adding baffles reduced thermal entropy generation, optimizing heat transfer performance and thermodynamic efficiency. Alvarado et al.⁶ investigated the heat transfer performance of liquid-cooled heat sinks with conventional and novel microchannel flow field configurations using CFD simulations in ANSYS FLUENT under laminar flow conditions. Results highlight improved temperature uniformity, flow distribution, and reduced pumping power in novel configurations compared to conventional ones. The study concludes that novel flow fields offer significant advantages for heat sink applications in electronics, fuel cells, and solar cells. Khaleduzzaman et al.⁷ experimentally analyzed the energy and exergy performance of a rectangular minichannel heat sink using Al₂O₃-water nanofluid as a coolant. Results showed a maximum energy efficiency of 94.68% at 0.25 vol.% nanofluid with a 0.375 l/min flow rate, while the highest exergy improvement (60.86%) occurred at 1.0 l/min. Exergy efficiency increased with nanoparticle concentration, while friction factor decreased with higher flow rates but increased with nanoparticle volume fraction. Bahiraei and Hangib⁸ reviewed recent advancements in magnetic nanofluids (MNFs), which consist of magnetic nanoparticles suspended in a non-magnetic base fluid. Their study highlights MNFs’ unique thermophysical properties, controllable heat transfer under magnetic fields, and applications in natural and forced convection, boiling, and other thermal processes. The review also discusses existing challenges and potential future research directions in MNF technology. Muhammad Ali et al.⁹ experimentally optimized heat transfer in electronic circuits using pin–fin heat sinks filled with phase change materials (PCMs). Various pin–fin configurations (rectangular, round, and triangular) were tested with six PCMs to evaluate thermal performance, operational time, and storage efficiency. Results indicate that triangular pin-fins provide the highest heat transfer efficiency, both with and without PCM integration. Kumar et al.¹⁰ conducted a thermofluidic analysis of an Al₂O₃-water nanofluid-cooled branched wavy heat sink microchannel (BWHS MC) using ANSYS Fluent and validated it against experiments with a straight channel heat sink (SCHS MC). The BWHS MC enhanced heat transfer through secondary flow, vortex formation, and boundary layer reinitialization, albeit with a higher pressure drop. Results showed a 154% increase in the heat transfer coefficient at 2% nanofluid concentration and Re = 300 compared to SCHS MC. Adnan et al.¹¹ developed a hybrid nanofluid model using graphene and Fe₃O₄ nanoparticles to study fluid dynamics in a channel with expanding/contracting walls, incorporating effects like Joule heating and magnetic fields. Numerical results show that increasing nanoparticle concentration enhances density and thermal conductivity, with hybrid nanofluids exhibiting superior heat transfer performance. The study highlights the influence of wall motion, energy dissipation, and thermal radiation on fluid behavior, demonstrating the potential of hybrid nanofluids for advanced thermal applications. Barbar et al.¹² investigated the thermal performance of liquid-cooled straight channel heat sinks for high-power electronics, analyzing the effects of a hydrophobic coating under varying heating powers, flow rates, and orientations. Results showed that while higher Reynolds numbers improved heat transfer, the hydrophobic coating reduced the Nusselt number by up to 20.85% due to bubble retention, though it slightly decreased pressure drop. The study highlights the broader applicability of heat sinks in industries like automotive, aerospace, and renewable energy systems. Bahiraei and Heshmatian¹² reviewed the application of nanofluids in electronics cooling, highlighting their potential to enhance heat dissipation and enable further miniaturization. The study examines factors like liquid block type, nanoparticle material, energy consumption, and second-law efficiency, identifying key benefits and challenges. Findings suggest that nanofluids can significantly improve cooling performance in liquid blocks and heat pipes, paving the way for advanced thermal management in electronics. Bezaatpour and Goharkhah¹³ proposed an innovative method to enhance convective heat transfer in mini heat exchangers using an external magnetic field to induce swirling flow in magnetic nanofluids. Numerical simulations showed up to 320% heat transfer enhancement with minimal pressure drop due to improved mixing and boundary layer disruption. The study concludes that optimal performance is achieved at low Reynolds numbers, high magnetic field intensities, and high nanofluid concentrations. Everts et al.¹⁴ studied mixed convective flow in vertical tubes at low laminar Reynolds numbers, analyzing heat transfer characteristics for upward and downward flows under forced convection. Experiments and simulations showed that as Reynolds numbers dropped below 250 (upward) and 600 (downward), free convection effects became significant, affecting Nusselt numbers. Correlations were developed to predict Nusselt numbers for assisting and opposing laminar flows. Goharkhah and Ashjaee¹⁵ numerically investigated forced convective heat transfer of Fe₃O₄-water nanofluid in a 2D channel under an alternating non-uniform magnetic field. Results showed a maximum heat transfer enhancement of 13.9% at Re = 2000 and f = 20 Hz, with an optimal frequency for different Reynolds numbers. Although heat transfer improved with increasing magnetic field intensity, a minor pressure drop penalty of up to 6% was observed. Bhattacharyya et al.¹⁶ numerically analyzed heat transfer in a wavy minichannel using Fe₃O₄-water nanofluid under an external magnetic field at low Reynolds numbers. Results showed up to 103.54% heat transfer enhancement at 3000 G, with downstream magnet placement improving heat transfer but increasing pressure drop. The wavy minichannel demonstrated better performance than a plain channel, especially at low magnetic fields. Bezaatpour and Goharkhah¹⁷ proposed an active vortex generator using a uniform magnetic field to enhance heat transfer in ferrofluid-cooled heat sinks. Numerical simulations showed up to 37.8% heat transfer enhancement with a 29.18% pressure drop reduction due to improved flow mixing and reduced surface contact. Increasing magnetic field intensity and adding a second vortex generator further optimized performance. Bhattacharyya et al.¹⁸ numerically analyzed the effect of an external magnetic field on Fe₃O₄-water nanofluid flow in an inclined channel. Results showed up to 19.27% heat transfer enhancement with a 2000G field, accompanied by an 89.23% increase in friction factor. Magnetic fields reduced pressure drop for positive inclinations but increased it for negative ones. The thermal enhancement factor improved by up to 12.50%, indicating better performance across various inclinations. Bhattacharyya et al.¹⁹ experimentally investigated heat transfer, pressure drop, and thermal performance in a solar air heater tube with hybrid tapes under turbulent flow. Results showed that increasing width ratio and decreasing pitch ratio enhanced both Nusselt number and friction factor, with respective increases of 91% and 39%. The thermal performance factor remained above unity for all configurations. This highlights the effectiveness of hybrid tapes in improving heat transfer efficiency. Bhattacharyya et al.²⁰ numerically analyzed the thermal and flow performance of Fe₃O₄-water nanofluid in a 2D channel under a magnetic field at low Reynolds numbers. The magnetic field acted as a vortex generator, enhancing heat transfer by up to 47.64% at x = 25 mm but also inducing pressure drop variations. In some cases, frictional pressure drop reduction led to a net decrease in overall pressure drop. The study highlights the trade-off between heat transfer enhancement and pressure drop due to vortex formation. Vishwakarma et al.²¹ conducted forced convection experiments in a circular duct with spring tape inserts to investigate heat transfer and pressure drop across various Reynolds numbers. The study found that decreasing the spring ratio led to an earlier onset of transition and an increased transition length. The transition Reynolds number range varied with different spring ratios under constant heat flux conditions. The study also provides predictive Nusselt number and friction factor correlations for various flow regimes. Kumar and Sarkar²² analyzed heat transfer and pressure drop characteristics in a minichannel heat sink using Al₂O₃–TiO₂ hybrid nanofluids. The study showed that the two-phase mixture model provided better agreement with experimental data than the single-phase model. The convective heat transfer coefficient was enhanced by 12.8% experimentally and 8.5% numerically with Al₂O₃ (10:0) hybrid nanofluid. Pressure drop and friction factor increased with nanoparticle volume fraction and decreased temperature, but no synergistic effect was observed with the hybrid nanofluid. Xuan et al.²³ used the lattice-Boltzmann method to develop mesoscopic models for simulating flow and thermal processes of ferrofluid in microchannels. The models accounted for various forces and potentials acting on the ferrofluid system, including heat exchange between magnetic nanoparticles and the surrounding liquid. Numerical examples demonstrated how adjusting the magnetic field gradient’s orientation and magnitude could either enhance or suppress heat transfer in the ferrofluid. This study provides insights into optimizing ferrofluid flow and heat transfer using external magnetic fields. Li et al.²⁴ conducted a combined experimental and numerical study on liquid-cooled aluminum foam (AF) heat sinks for high-power electronics cooling. The study revealed that AF heat sinks with higher pore densities (20 PPI) exhibited improved thermal performance compared to lower pore densities (10 PPI), with the Nusselt number up to 1.76 times greater than that of an empty channel. The study used the Brinkman-Forchheimer model for momentum and both LTNE and LTE models for heat transfer analysis. The findings showed that the LTNE model provided more accurate predictions of temperature distributions, while non-equilibrium effects were less significant at higher flow velocities. Zhou et al.²⁵ developed a hybrid oscillating heat pipe (OHP) for electric vehicle (EV) battery cooling using CNT nanofluids in ethanol–water mixtures. The experimental results showed that CNT nanofluids, particularly at a 0.2 wt% concentration, enhanced heat transfer performance and reduced evaporator temperature and thermal resistance compared to ethanol–water mixtures. The OHP significantly improved the cooling efficiency, keeping the battery pack temperature below 45 °C with a minimal temperature difference of 1 °C. This approach offers a promising solution for efficient cooling during rapid charging and discharging processes in EVs. In this work, fluid flow and heat transfer properties in mini-channels with three different configurations—parallel, staggered, and ribbed geometries—are thoroughly and methodically investigated. Rahaman et al.²⁶numerically analyzed mixed convective heat transfer in a grooved channel cavity using CuO-water nanofluid under an inclined magnetic field. Results showed that optimal heat transfer occurred when the heater was positioned at the right corner, enhancing heat transfer by up to 168.53% compared to bottom heating. The magnetic field inclination significantly influenced thermal performance, initially increasing and then declining with angle. This study highlights the importance of heater positioning and magnetic field orientation for improving heat transfer in electronic cooling and heat exchange applications.Manna et al.²⁷ numerically investigated the transition from unsteady to steady flow in a magneto-nanofluidic thermal system with a recto-triangular shape using CuO-water nanofluid. Results showed the evolution of multi-vortical structures, transitioning from four-cell to one-cell and finally to two-cell configurations as Rayleigh number (Ra) varied from 10³ to 10⁵. The inclination of the magnetic field significantly influenced flow dynamics, with the system rapidly stabilizing except for specific Ra and Hartmann number (Ha) values. This study provides insights into the impact of magnetic fields on multi-cellular convection structures in thermal systems. Pandit et al.²⁸ analyzed thermal-fluid behavior in a tilted porous enclosure filled with Cu-Al/water hybrid nanofluid under segmented magnetic fields, wavy cooling, and distributed heat sources. Using FVM and the SIMPLE algorithm, the study showed that wavy walls and segmented heating enhanced heat transfer by up to 38%, while strategic magnetic field orientation improved it by 26%. The results highlighted the role of surface area increase, boundary layer disruption, and localized convection in thermal performance enhancement. This study offers insights for optimizing heat transfer in electronics cooling, solar collectors, and nuclear reactors. Datta et al.²⁹ numerically investigated buoyancy-driven free convection in a solar air heating system using an ‘H’-shaped cavity filled with a porous medium. The study analyzed fluid flow and heat transfer using air and Cu-water nanofluid across various Rayleigh and Darcy numbers, porosity levels, nanoparticle concentrations, and heater aspect ratios. Results showed that nanofluids enhanced heat transfer compared to air, with optimal aspect ratios improving thermal performance at higher Rayleigh numbers. This research provides insights into optimizing SAH systems for efficient solar energy utilization.Halder et al.³⁰ explored thermal management in a semi-circular vented cavity with multi-segmental bottom heating, hybrid nanofluids, and a magnetizing field. The study analyzed heat transfer performance using various control parameters, including Reynolds, Rayleigh, and Hartmann numbers. Results showed a 68% enhancement in heat transfer with segmented heating, with the optimal configuration achieved using two heating segments. This research provides insights for improving thermal performance in electronics cooling, solar power, and industrial heat exchangers.Biswas et al.³¹ investigated mixed convection heat transfer enhancement in a grooved channel using flow injection under an assisting flow configuration. The study analyzed the effects of injection position, size, and flow rate for different Reynolds and Richardson numbers. Results showed heat transfer improvement ranging from 50 to 218%, demonstrating the effectiveness of injection in optimizing thermal performance .Li et al.³² investigated the oscillating flow of Jeffrey fluid in a rough circular microchannel with slip boundary conditions using the perturbation method. Their findings reveal that velocity and volumetric flow rate are influenced by slip length, wall roughness, and wave numbers across different angular Reynolds numbers. The study highlights the significance of these parameters in biomedical applications, particularly in modeling physiological fluid flow. Akbar et al.³³ developed a numerical solver using a two-layer backpropagation Levenberg–Marquardt artificial neural network (BLMS-ANN) to analyze MHD effects on thermal radiation in nanofluid flow between rotating plates. The MHD-TRTM model was transformed into ODEs and validated using Homotopy Analysis Method (HAM) datasets for training and testing. Their approach demonstrated high accuracy (10^-10 to 10⁻¹²) in predicting solutions across various physical scenarios. Shoaib et al.³⁴ utilized the Levenberg–Marquardt backpropagation neural network (TLMB-NN) to analyze heat transfer in Maxwell nanofluid flow with MHD over a vertical moving surface. The study incorporated thermal energy effects, Brownian motion, and radiation, transforming governing equations into nonlinear ODEs using similarity transformation. The TLMB-NN model, validated through regression analysis and error metrics, achieved high accuracy (10^-9 to 10^-10) across various parameter variations. Ullah et al.³⁵ developed a Levenberg–Marquardt algorithm-based artificial neural network (LMA-BANN) model to obtain an accurate series solution for micropolar flow in a porous channel with mass injection. The model was trained, tested, and validated using data from the optimal homotopy asymptotic (OHA) method, with performance evaluated through mean square error and absolute error metrics. The LMA-BANN model demonstrated high accuracy (E-05 to E-08) and was further assessed using error histogram and regression plots. Zeng et al.³⁶ proposed a cavitation detection method for vortex pumps using dual-tree complex wavelet transform (DT-CWT) and variational mode decomposition (VMD) to analyze current signals. A Bayesian-optimized locally weighted k-nearest neighbor (LW-KNN) algorithm was employed for accurate identification, achieving an overall recognition accuracy of 94.22%. This approach enhances reliability and fault diagnosis in fluid mechanical systems, improving operational efficiency and maintenance strategies. Wang et al.³⁷ developed a permanent magnet-based flow velocity meter to address high output drift in traditional marine electromagnetic sensors. By measuring electrode output current instead of voltage, the proposed design significantly reduces drift by 92% and achieves high measurement accuracy (R² = 0.998) within a velocity range of 0–0.875 m/s. This innovation enhances underwater flow sensing for robotic and marine applications. Ullah et al.³⁸ analyzed the effects of electric and magnetic fields on micropolar nanofluid flow between rotating parallel plates under Hall current influence (EMMN-PPRH) using an artificial neural network with Levenberg–Marquardt backpropagation (ANN-SLMB). The model, trained on homotopy analysis method (HAM) data, was validated through regression analysis and error metrics, achieving high accuracy (10⁻⁹ to 10⁻¹¹). This approach enhances predictive modeling for complex fluid dynamics.Ullah et al.³⁹ employed the Levenberg–Marquardt backpropagation neural network (LMBT-NN) to analyze heat and mass transfer in MHD nanofluid flow over a vertical cone under convective boundary conditions. By transforming PDEs into ODEs and utilizing numerical techniques, the model was validated through regression analysis and error metrics, achieving high accuracy (E-9 to E-10). This study enhances predictive capabilities for complex thermal-fluid systems. Akbar et al.⁴⁰ utilized the Levenberg–Marquardt backpropagation neural network (LMB-NNS) to model MHD nanofluid flow over a rotating disk with partial slip effects based on the Buongiorno model. By transforming PDEs into ODEs and applying numerical methods, the model was validated through regression analysis and error metrics, achieving high accuracy (10⁻⁹ to 10⁻¹²). This study enhances predictive modeling for complex fluid flow applications. Sheikholeslami et al.⁴¹ investigated the thermal management of lithium-ion battery packs using four advanced mini-channel designs—Smooth, Grooved, Tooth, and Pin Fin combined with a hybrid Fe₃O₄-SWCNT nanofluid. Their conduction-based simulations showed that the Pin Fin configuration significantly improved heat transfer, achieving a Nusselt number over five times higher than the Smooth channel. The study emphasizes the critical role of channel geometry and nanofluid properties in optimizing battery cooling performance and safety. Aliabadi et al.⁴² numerically investigated heat sinks with various convergent and divergent minichannel geometries cooled by turbulent supercritical CO₂ flow, using 3D finite volume simulations. Their results showed that converging channels significantly enhance heat transfer coefficients while managing pressure drops efficiently due to CO₂’s thermophysical variations near the critical point. This study highlights the potential of supercritical CO₂ as an advanced coolant for high-performance heat sinks with optimized channel designs. Aliabadi et al.⁴³ conducted 3D numerical simulations to investigate heat transfer and flow characteristics of wavy mini-channel heat sinks using supercritical and pseudocritical CO₂ under high heat flux conditions. Their results showed that wavy channels significantly enhance thermal performance—up to 8.58 times higher heat transfer coefficient—though at the cost of increased pressure drop. The study highlights that optimizing wave amplitude, wavelength, and inlet temperature can markedly improve the overall cooling efficiency of CO₂-based miniature heat sinks. Aliabadi et al.⁴⁴ numerically analyzed advanced liquid-cooled aluminum heat sinks with various fin arrangements for thermal management in concentrated photovoltaic (CPV) systems. Their study demonstrated that interrupted fin designs improve temperature uniformity by up to 30.6% and reduce thermal stress and pumping power compared to integral fins. The optimal fin configuration ensured minimal temperature differences between cells and enhanced overall thermal–hydraulic performance under high concentration conditions.

In contrast to previous works, we present a new experimental setup by studying these structures with the help of well-placed magnetic sources (individually at x = 15 mm and x = 25 mm) and uniform magnetic field intensities of 800 G, 1000 G, 1500 G, and 2000 G. Its comprehensive approach to examining the combined impacts of source placement, magnetic field intensity, and channel shape on thermal and hydrodynamic performance is the study’s main innovation. We offer insights into optimizing heat transfer efficiency in ribbed mini-channels by investigating these yet unexplored factors. By addressing important gaps in the literature, our findings demonstrate how magnetic fields can be customized to improve convective cooling in compact systems. Thermal management solutions for high-power electronics and data center cooling applications could be greatly advanced by this research. Further research aiming at improving magnetic field-assisted cooling methods for industrial use is made possible by the novel experimental design and data produced.

Computational methodology

This numerical study explores the thermal and fluid flow behavior in a two-dimensional mini-channel with a hydraulic diameter of 4 mm and a length of 40 mm, using water as base fluid with 2% Fe₃O₄ nanoparticles as the working fluid. The mini-channel features ribbed arrangements, as illustrated in Fig. 1, designed to enhance heat transfer by generating flow disturbances. To further augment thermal performance, two magnets are strategically positioned at X = 15 mm and X = 25 mm along the channel length, creating a non-uniform magnetic field with intensities of 800 G, 1000 G, 1500 G, and 2000 G. The study investigates flow behavior across Reynolds numbers of 50, 75, 100, 150, 180, and 210. The symmetrical placement of the magnets ensures uniform magnetic field effects and balanced fluid distribution, minimizing localized hot spots and providing reliable heat transfer predictions. The combined effect of the ribs and the magnetic field induces eddies and vortices that disrupt the thermal boundary layer, significantly enhancing heat transfer rates. Although these flow disturbances promote better mixing, they remain within the laminar flow regime due to the low Reynolds numbers considered, which aligns with findings from previous validated research on similar mini-channel configurations. However, the increased mixing and vortex generation also result in significant pressure drops along the channel. Overall, this study provides practical insights for optimizing mini-channel heat sink designs for efficient thermal management in electronics and related applications.

Fig. 1.

Schematic Diagram.

Geometry and grid generation

In this numerical study, a two-dimensional mini-channel was modelled with adiabatic side walls, while the bottom wall was kept at a constant temperature of 350 K to provide a uniform heat source. The coolant used was a water-based nanofluid with a 2% volume fraction of Fe₃O₄ nanoparticles, whose thermophysical properties are listed in Table 1. This ensured a stable and predictable flow throughout the channel domain. To resolve the velocity and temperature fields with high accuracy, a structured grid with square cells was employed. A grid independence test was performed to ensure that the simulation results were not sensitive to mesh resolution. Specifically, three grid sizes were examined: 0.1 mm, 0.5 mm, and 0.75 mm, corresponding to total node counts of 16,699, 8,828, and 6,053, respectively. The grid test was based on comparing the velocity ratio at the channel centerline to the inlet velocity. The results showed that the 0.1 mm mesh produced deviations of only 3.19% compared to the reference results of Bezaatpour et al.¹⁷, confirming that this grid size was sufficiently fine to capture flow and thermal gradients accurately. Therefore, the 0.1 mm mesh was adopted for all final simulations as it provided a good balance between computational accuracy and cost, as shown in Table 2 and Fig. 2.

Table 1.

Thermophysical properties of Fe₃O₄ nano-particles⁴⁷.

Property	Fe3O4
ρ	4950 kg/m3
Cp	640 J/kgK
K	7w/m.K

Table 2.

Grid Independence for Re 190.

Grid size	Number of nodes	Nu (present study)	Nu (Bezaatpour et al.17)	Percentage of error
0.1	16,699	15.34	14.85	3.19%
0.5	8828	16.09	14.85	7.70%
0.75	6053	16.29	14.85	8.83%

Fig. 2.

Meshing.

In practice, resolving detailed flow features such as the vortices and recirculation zones generated by ribs and magnetic forces demands an adequately fine mesh to capture steep gradients, particularly near the channel walls and rib surfaces. However, using an excessively fine mesh can substantially increase computation time and resource requirements. Mesh resolution directly affects the prediction of key parameters such as the Nusselt number and friction factor. If the mesh is too coarse, heat transfer rates and pressure drops may be misestimated, which could lead to suboptimal rib configurations or magnet placements when the design is fabricated. Therefore, while the validated numerical results provide reliable guidance for optimal design, experimental testing remains essential to verify performance under real conditions, considering practical factors like manufacturing imperfections, material inconsistencies, and surface roughness. Selecting an appropriate mesh size is crucial for balancing accuracy and computational efficiency, and should be carefully considered alongside experimental validation when developing practical mini-channel heat sinks for advanced thermal management applications.

Governing equations

Assuming steady-state, two-dimensional, homogeneous, and laminar flow conditions, the governing equations of continuity, momentum, and energy for the fluid and solid domains are as follows.

1

2

3

represents the shear stress vector, the expansion of which is given as follows:

4

Additionally, the magnetic volume force is included to the momentum equation, derived from the given equation to take into consideration how the magnetic nanoparticles are affected by the external magnetic field.

5

where χm is the magnetic susceptibility and is temperature dependant.

Solver setting

It assumes an absolute frame of reference. Whereas the interior surface serves as the reference zone, the bottom and upper walls have been considered as stationary. A COUPLED approach is used to establish the relationship between pressure and velocity. The gradient of the second-order upwind system is based on least squares cells and controls both momentum and energy. For continuity, the x- and y-velocities are assumed to have convergence criteria of 10^–5, whereas the energy convergence criterion is 10^–6.

Boundary conditions

The particular boundary conditions that apply to the governing equations in our investigation are described below. The bottom surface of the coolant ferrofluid is continuously maintained at 350 K, while the entrance temperature is controlled at 293 K. Since the upper surface is regarded as insulated, it is assumed that there is no heat exchange at this boundary. The ferrofluid is delivered at a steady inlet velocity and maintained at atmospheric pressure at the channel’s outlet. Furthermore, we implement the no-slip rule and ensure temperature continuity at all interfaces where solid and fluid surfaces meet.

6

7

8

9

where

10

11

Here, β represents the liquid volume fraction, k the Boltzmann constant, φ the nanoparticle concentration, and g the modeling function supplied by .

12

In the current analysis, the physical characteristics of the basic fluid, water, are assumed to be temperature dependent for increased precision. These are provided by

13

14

15

Thermohydraulic parameters

The rate at which heat is transmitted is determined by the Nusselt number, which can be defined as follows ⁴⁵:

16

The coefficient of local convective heat transfer (h) is given by ⁴⁶:

17

The friction factor, which is determined by the pressure drop is ⁴⁵:

18

The Colburn j factor can be calculated ⁴⁷:

19

The thermal enhancement factor (TEF) is given by ⁴⁷:

20

Validation

The characteristics of the Fe₃O₄—water nanofluid are detailed in Table.1. The accuracy of the model was confirmed through a comparison with the research conducted by Bezaatpour et al.¹⁷, using a rectangular minichannel without turbulators as the reference case. Furthermore, the findings were compared with the experimental data presented by Kumar et al.²² and numerical data presented by Sheikholeslami et al.⁴¹ .The studies examined the changes in the Nusselt number as the Reynolds number increased, along with the effects of various placements of magnets. The current investigation demonstrated an error margin of ± 5% in relation to the validated outcomes, as illustrated in Figs. 3 and 4.

Fig. 3.

Validation of the present study with Sheikholeslami et al.⁴¹.

Fig. 4.

Validation of the present study with Kumar et al.²².

Result and discussion

The main objective of this work was to examine how a magnetic field affected the flow characteristics and heat transfer rate of a nanofluid containing Fe3O4 nanoparticles. Numerical simulations using the finite volume approach were performed in order to gain a better understanding of the hydro-thermal dynamics. Important factors including the Nusselt number (Nu), friction factor (f), and Thermal Enhancement Factor (TEF) were then used to analyze the results.

a. Magnetic force-induced vortex formation

Figure 5 shows the creation of the primary and secondary vortices in the ribbed mini channel with a magnetic intensity of 800G and 2000G and a distance (X) of 15 mm and 25 mm. The Reynolds number (Re) at which the occurrence happens is 180. As seen in Fig. 5a,b and the external magnetic force (800G, 1000G, 1500G, and 2000G) applied at two distinct places, X = 15 mm and X = 25 mm, functions as an imagined rib that pulls the fluid upward. Vortices and flow separation are caused by this force. As the strength or intensity of the magnetic pull increases, so does the size of these eddies. The flow separation and vortex size increase with the strength of the magnetic field. When the fluid travels upward and strikes the higher wall, it creates another secondary vortex that goes downward. This graphic demonstrates how the magnetic field can improve heat transmission and fluid mixing while also acting as a source of vortex formation. As the temperature rises, the density of the nanofluid falls. Heat transfer causes the fluid’s temperature to increase as it passes through the micro channel, which lowers the fluid’s density. It should be mentioned that because the heated wall is closer to the thermal boundary layer, the density change is greater there. The density of the nanofluid close to the heated wall decreases significantly as it travels through the inclined channel. As the flow enters the channel, a buoyant force is applied to the fluid nearer the wall in a similar direction; as the inclination angle increases, the flow velocity drops. The rearward component of the gravitational force acting on the flow is the cause this increases the flow velocity near the wall.

Fig. 5.

a Temperature contours of 800G at 15 mm and 25 mm (Re 180). b Temperature contours of 800G at 15 mm and 25 mm (Re 180). c Velocity contours of 800G at 15 mm and 25 mm (Re 180). d Velocity contours of 2000G at 15 mm and 25 mm (Re 180).

The enhanced heat transfer performance in the mini-channel is due to two key mechanisms: (i) ribs act as passive vortex generators, disrupting the boundary layer and promoting fluid mixing near the walls, and (ii) the applied magnetic fields interact with the Fe₃O₄ nanoparticles, inducing Lorentz forces that generate additional swirling flows. Together, these passive and active effects intensify flow mixing, thin the thermal boundary layer, and significantly increase the heat transfer rate while keeping pressure drops within acceptable limits, as illustrated in Fig. 5c,d.

b. Influence of the nusselt number on the magnetic field in mini-channels

The ratio of convective to conductive heat transport across a fluid boundary is represented by the dimensionless Nusselt number (Nu). It is a crucial sign of how well heat transfer works in fluid flow systems. The Reynolds number (Re), which describes the flow regime based on the ratio of inertial to viscous forces, is frequently associated with the Nusselt number. Higher magnetic strength and quantity make this effect more noticeable. As previously mentioned, the presence of magnets causes swirl to form where they are positioned at different positions, which increases heat transmission. The improved heat transfer caused by the higher fluid velocity close to the heated wall is responsible for this improvement. Different flow patterns are produced by different inclination angles, and the heat transfer process is greatly impacted by the Reynolds number (Re).

The Nusselt number (Nu) is significantly influenced by both magnetic intensity and channel design, according to the numerical study of heat transfer enhancement in mini-channels under various magnetic field strengths (800 G, 1000 G, 1500 G, and 2000 G). In order to show that heat transmission improves with increased Reynolds number (Re) and magnetic field strength, the study investigates at three different designs of mini-channels: plane parallel, staggered, and ribbed stepped. The Nusselt number increases significantly with increasing magnetic field strength in both plane and staggered arrangements. As shown in Fig. 6a,b the Nu rises 42% at 800 G (X = 15 mm) and reaches 112% at 2000 G (X = 25 mm) in the parallel arrangement, but the staggered structure exhibits a much lesser boost, peaking a 43% (X = 15 mm) and 99% (X = 25 mm) in the same circumstances. This variation implies that flow disturbance is mostly dependent on magnetic field alignment, with parallel setups promoting more consistent interaction as shown in Fig. 6e-h. Even though the ribbed stepped mini-channels only show slight improvements at 800 G (1.25% at X = 15 mm and 3.76% at X = 25 mm), they show the most noticeable improvement at higher magnetic strengths. Significant enhancement occurs at 2000 G, reaching 32.15% at X = 15 mm and 65% at X = 25 mm as shown in Fig. 6i,j. This is explained by increased vortex formation and secondary flows brought on by the magnetic forces and ribs. With two different magnetic positions, X = 25 mm and X = 15 mm, which show cumulative magnetic effects along the flow route, the axial position also has a substantial impact on performance. Additionally, the function of the Reynolds number is clear in all possible situations since convective heat transfer is amplified by larger flow rates, especially when paired with stronger magnetic fields, demonstrating a synergistic link. The parallel design outperforms the staggered one in terms of heat transfer augmentation, most likely as a result of more effective magnetic field interaction. The ribbed stepped channels’ promise for high-performance thermal applications is shown by the fact that they exhibit the greatest relative improvement at 2000 G. According to the results, magnetic fields can greatly improve heat transmission; the most important optimization factors are field strength and geometry.

Fig. 6.

a Variation of Nusselt number with Re for parallel at 800G. b Variation of Nusselt number with Re for parallel at 1000G. c Variation of Nusselt number with Re for parallel at 1500G. d Variation of Nusselt number with Re for parallel at 2000G. e Variation of Nusselt number with Re for staggered at 800G. f Variation of Nusselt number with Re for staggered at 1000G. g Variation of Nusselt number with Re for staggered at 1500G. h Variation of Nusselt number with Re for staggered at 2000G. i Variation of Nusselt number with Re for ribbed at X = 15 mm. j Variation of Nusselt number with Re for ribbed at X = 25 mm.

c. Influence of the magnetic field on the friction factor in mini-channels

For fluid flow in a pipe, the Reynolds number (Re) and the friction factor (f) are related by the Darcy-Weisbach equation. The simplified version f = 64/Re provides the friction factor for laminar flow (Re ≤ 2300Re). The friction factor falls as Re rises, indicating the decrease in flow resistance at higher velocities, according to this dimensionless relationship that characterizes the resistance caused by viscous forces in the fluid. At two axial positions (x = 15 mm and x = 25 mm), the experimental study investigated heat transfer characteristics under different magnetic field strengths (800G to 2000G) over Reynolds numbers (Re) ranging from 50 to 210.

The application of magnetic fields showed a consistent reduction in friction at both measurement points in the parallel channel design as shown Fig. 7a-d. With friction factor reductions of 3.57% and 7.14%, respectively at 800G position X = 25 mm and magnetic position at X = 15 mm. At higher field strengths, this performance inequality improved considerably, reaching 80.8% decrease at X = 25 mm for 2000G and 75.2% reduction at X = 15 mm. This behavior points to two key findings first, that flow development along the channel length improves the efficiency of applying a magnetic field and second, that there is a nonlinear relationship between magnetic field strength and friction reduction, with higher field intensities yielding disproportionately greater benefits. According to the results boundary layer growth may be altered by magnetic fields in ways that become more noticeable as flow development increases and magnetic impact increases.More detailed behavior was shown by the staggered channel layout as shown in Fig. 7e-h, highlighting the significance of geometric aspects of magnetic flow regulation. Remarkably at X = 15 mm demonstrated a higher initial improvement of 7.01% than at X = 25 mm of 3.4% at lower field strengths 800G. At 2000G, however, the trend reversed, at position of magnets eventually performed better 79% (X = 25 mm) decrease against 73% (X = 15 mm). This reversal implies that localized vortex structures are produced by the staggered geometry, and that these structures’ interactions with magnetic fields change as the flow direction does. The position of magnets probably comes from changing vortex interactions in more established flow zones and emerging flow structures.Surface characteristics produced notable synergies with applied magnetic fields in the ribbed channel arrangement as shown Fig. 7i,j, which produced the most striking results. The ribbed channel demonstrated notable improvements of 17.8% at X = 15 mm and 26% at X = 25 mm, even at low 800G strength. With increasing field strength, these advantages increased significantly, with notable decreases of 67% and 86% at 2000G, respectively. Localized areas of modified flow that significantly lower overall friction are created by the ribbed surface’s apparent generation of controlled vortex forms that interact strongly with magnetic fields. The particularly large downstream effect indicates that developed flow zones, where the vortices are fully formed and may be maximally controlled by magnetic forces, are where the combination of rib-induced vortices and magnetic fields works effectively.

Fig. 7.

a Variation of friction factor with Re for parallel at 800G. b Variation of friction factor with Re for parallel at 1000G. c Variation of friction factor with Re for parallel at 1500G. d Variation of friction factor with Re for parallel at 2000G. e Variation of friction factor with Re for staggered at 800G. f Variation of friction factor with Re for staggered at 1000G. g Variation of friction factor with Re for staggered at 1500G. h Variation of friction factor with Re for staggered at 2000G. i Variation of friction factor with Re for ribbed at 15 mm. j Variation of friction factor with Re for ribbed at 25 mm.

d. Magnetic field thermal enhancement factor (TEF) in mini channels

The thermal enhancement factor, also known as TEF, is utilized to evaluate the efficiency of heat transmission. The ratio of the heat transfer rate to the friction factor is known as the thermal enhancement factor, or TEF. It suggests that the friction factor will rise and cause greater pressure loss when the TEF falls at higher Re numbers. When it comes to heat transfer enhancement techniques, having a TEF value that is bigger than one is absolutely necessary as in all the cases whether it is parallel mini channel, staggered mini channel and ribbed mini channel all the values are greater than 1 in all the cases which shows the better heat transfer enchancement. Behavior characteristics in the parallel channel setup were greater than 1 as shown in Fig. 8a,b. TEF measurements at the magnetic strength 800G revealed average values of 1.14 at X = 15 mm and 1.11 at X = 15 mm. At 2000G, this connection increased , with average values to 1.77 when position of magnets at X = 15 mm and 1.56 at X = 25 mm, signifying remarkable increases of 55.3% and 40.5%, respectively. The magnetic field application favourably affects parallel channels, probably because it disrupts developing boundary layers more effectively. Similar but more amplification patterns were observed in the staggered channel design as shown in Fig. 8c,d. TEF values for the 800G measurements were 1.12 at X = 15 mm and 1.08 at X = 25 mm, increasing to 1.52 at X = 15 mm and 1.50 at X = 25 mm when the magnetic strength is 2000G respectively. Although considerable, the improvements of 35.7% and 38.9% were noticeably less noticeable than in parallel channels, indicating that the geometric complexity of staggered layouts may exacerbate flow disruption and partially offset magnetic benefits. The most impressive results were obtained with the ribbed channel structure as shown in Fig. 8e-f, particularly at the X = 25 mm. With 800G values of 1.11 at X = 15 mm and 1.69 at X = 25 mm, the configuration reached 1.42 and 2.06 at 2000G, respectively. A benchmark in thermal enhancement performance is represented by the average value of TEF value of 2.06 at the X = 25 mm magnet position, which implies that ribbed surfaces provide optimal circumstances for magnetic field interaction with flow structures. This behaviour is probably caused by the rib-induced vortices creating ordered, predictable flow disruptions that are easily controlled by magnetic fields. Allowing flow patterns to partially evolve prior to magnetic intervention is beneficial for ribbed channels as an improvement of 21.9% compared to 27.9%.

Fig. 8.

a Thermal enhancement factor (TEF) for parallel at 800G. b Thermal enhancement factor (TEF) for parallel at 2000G. c Thermal enhancement factor (TEF) for staggered at 800G. d Thermal enhancement factor (TEF) for staggered at 2000G. e Thermal enhancement factor (TEF) for ribbed at 15 mm. f Thermal enhancement factor (TEF) for ribbed at 25 mm.

Conclusion

With the growing use of electronic devices, efficient cooling has become increasingly important. A numerical study on mini-channels revealed that heat transfer performance is significantly affected by the position and strength of the applied magnetic field. Introducing a magnetic field within the flow domain enhances heat transfer and improves the energy efficiency of cooling systems. This study analyzed the effects of varying magnetic field intensities in an inclined 2D mini-channel using nanofluids, evaluating key performance metrics such as the thermal enhancement factor (TEF), friction factor, and Nusselt number. The main findings of this investigation are as follows:

1. The ribbed stepped configuration demonstrated the highest heat transfer enhancement at stronger magnetic fields (up to 2000 G). This is primarily due to intensified vortex formation and enhanced fluid mixing, which effectively disrupt the thermal boundary layer and maintain lower surface temperatures making this design highly suitable for compact, high-power electronics cooling.

2. Parallel mini-channels produced higher Nusselt numbers than staggered configurations because the magnetic field aligns more uniformly with the structured flow. Nevertheless, ribbed channels showed the greatest overall improvement relative to a smooth channel, indicating their advantage when high heat removal is required within space-constrained systems.

3. Placing the magnetic source at X = 25 mm, where the flow is more developed, consistently reduced friction losses for all geometries. Ribbed channels benefited the most, achieving up to 86% friction reduction at 2000 G, underscoring the importance of optimal magnet positioning for balancing thermal performance and flow resistance.

4. Ribbed mini-channels generated controlled vortices that synergize with magnetically induced secondary flows, resulting in both higher heat transfer rates and moderate pressure drops. This dual effect enhances cooling performance for components with high heat flux demands.

5. Performance improvements plateaued between 1500 and 2000 G, indicating a practical upper limit for field strength. This suggests that beyond a certain point, increasing magnetic intensity yields diminishing returns relative to energy input, guiding designers to select an efficient field strength range.

6. The study establishes a clear ranking: ribbed channels outperform parallel, which in turn outperform staggered layouts. Additionally, the effectiveness of magnetic placement is highly dependent on channel geometry. These insights provide practical design guidelines for customizing magnet-assisted mini-channel heat sinks for specific cooling applications.

Future work

Further work should optimize rib geometry to minimize pressure drops while maximizing heat transfer, and assess the feasibility of scaling these designs for practical electronics cooling. Such advancements will enable the development of compact, energy-efficient mini-channel systems integrated with magnetic field enhancement for next-generation thermal management solutions.

Abbreviations

Re

Reynolds number

Nu

Nusselt number

J

Colburn j factor

TEF

Thermal enhancement factor

D

Diameter(mm)

T

Temperature (K)

u

Velocity (m/s)

k

Thermal conductivity (Wm^-1 K^-1 )

∆p

Pressure difference

X

Distance from the inlet (mm)

G

Magnetic field strengths (Gauss)

Pr

Prandtl number

C_p

Specific heat capacity (J kg⁻¹ K⁻¹)

f

Friction Factor

h

Heat transfer coefficient (W/m² K)

Nanoparticle Concentration

ρ

Density (kgm⁻³)

Author contributions

Suvanjan Bhattacharyya: S.B.; Nada Al Taisan: N.A.T.; Sumit Khatri: S.K.; Basma Souayeh: B.S.; Huda Alfannakh: H.A.; Devendra Kumar Vishwakarma: D.K.V. Conceptualization, S.B., S.K., and D.K.V. ; methodology, N.A.T., S.K., and D.K.V.; software, N.A.T., B.S., and H.A.; validation, S.B., S.K., and D.K.V.; formal analysis, S.B., B.S., and D.K.V.; investigation, S.B., N.A.T., S.K., B.S., H.A., and D.K.V.; resources, S.B., B.S., and D.K.V.; data curation, S.K., and D.K.V.; writing—original draft preparation, S.B., S.K., B.S., and D.K.V.; writing—review and editing, S.B., N.A.T., B.S., H.A., and D.K.V.; supervision, S.B.

Funding

Open access funding provided by Manipal University Jaipur.

Data availability

Data sets generated during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.