EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7166 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Comprehensive Review on Structure of Heat and Fluid Flow in Liquid Metals at Low Grashof Number Hossam A. Nabwey1,∗, Hajra Nadeem2, Muhammad Ashraf2, A. M. Rashad3, Ali J. Chamkha4 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia 2 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 3 Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt 4 Faculty of Engineering, Kuwait College of Science and Technology, Doha District, 35004, Kuwait Abstract. The current study presents the comprehensive review of heat transfer in liquid metals at low Grashof numbers. Heat transfer in liquid metals is extremely efficient because their substantial thermal conductivity enables heat to travel swiftly, even when natural convection is minimal. At low Grashof numbers, buoyant forces are relatively weak in comparison to viscous forces, hence heat transfer in liquid metals is mainly governed by conduction instead of convection, resulting in restricted fluid motion and decreased thermal transport. The most commonly utilized liquid metals in engineering are alkali metals. Sodium, for example, is primarily employed as a coolant in fast reactors and a working fluid in high-temperature heat pipes. Potassium is an excellent working material for space power plants. Eutectic Na-K and Pb-Bi alloys, as well as Hg, Li, and Ga, may be employed in certain instances. A detailed literature review is needed to compare previous studies under various conditions and highlight the influence of controlling factors such as the Grashof number. This review also discusses the practical importance of comprehending low Grashof number heat transfer for innovative cooling technologies in nuclear, space, and solar thermal applications. 2020 Mathematics Subject Classifications: 80A19 Key Words and Phrases: Heat transfer, liquid metals, low Grashof number, thermal conduc- tivity, restricted fluid motion 1. Introduction Liquid metals have been found to be highly thermal conductive and have a unique flow behaviour that makes them good candidates to be used in future heat transfer ap- plications in areas like nuclear reactors, electronics, cooling and solar energy systems. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7166 Email addresses: eng_hossam21@yahoo.com (H. A. Nabwey) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 2 of 20 When the Grashof numbers are low, such that the influence of buoyancy is relatively low, the heat and fluid flow in liquid metals are dominated not by the strong natural convec- tional currents but by conduction and forced convection. The importance of this regime is that it demonstrates the underlying transport processes that govern thermal performance of small systems and in small geometries. Low Grashof number experiments also lay the groundwork for forecasting flow and heat transfer behaviour in mixed and turbulent convection regimes, hence facilitating the development of dependable, high-performance cooling solutions. To explain the characteristic of the liquid metals in low Grashof number more thoroughly, helps to establish the effective cooling system when the heat transfer is predicted and remains stable. Furthermore, many sophisticated thermal systems, such as micro-reactors, electronic cooling channels, and small heat exchangers, operate in con- ditions with negligible buoyancy effects. Investigating conduction-dominated heat trans- port in these situations aids in temperature homogeneity, preventing isolated hotspots, and ensuring operational reliability. The purpose of this review is to go over in depth the structure of heat and fluid movement in liquid metals under these conditions, some of the most important discoveries made by previous scientists, and the issues and directions of future research. Ahmad et al. [1] proposed a numerical analysis of unsteady flow as well as heat transfer in a viscous-type nanofluid, focusing on the effect of microorganisms on a moving surface coming from a moving slot. Akhmedagaey et al. [2] demonstrated that even with strong magnetic fields causing high Grashof and Hartmann numbers, roll instabilities and signif- icant temperature variations can also occur in flows of liquid metals, which is significant in the use of fusion reactors. Ali et al. [3] emphasized the necessity of regulating fluid flow with the Riga plate in the presence of suction and explores the behaviour of a spe- cific nanofluid under these conditions. Ali et al. [4] explored the entropy formation and the influence of magnetic field on the peristaltic motion of copper-water tiny fluid in an asymmetrical configuration saturated with porous material. Ali et al. [5] investigated the effect of Cu-Al2O3 magnetohydrodynamics hybrid nanofluid on heat transport and flow in a porous channel with heat flow and viscous dissipation effects. Ali et al. [6] examines the impact of thermal radiation and Joule heating on the magnetohydrodynamic flow of a hybrid nanofluid on a stretched cylinder. Ali et al. [7] emphasized the necessity of regulat- ing fluid flow with the Riga plate in the presence of suction and explores the behaviour of a specific nanofluid under these conditions. Ali et al. focussed on the impact of stretching, shrinking, and porous barriers on flow and thermal properties in [8] and analyzed heat transfer in MHD Jeffrey nanofluid movement across a nonlinear, variable thick elongating surface in [9]. Ali et al. [10] concluded that the Soret and Dufour number enhances the temperature of the fluid while decreasing the concentration. An et al. [11] provided a de- tailed overview of liquid metal-based mini/micro-channel heat transfer technology. Ashraf [12] investigated the thermophysical properties of heat and fluid movement, emphasizing their significance in determining environmental outcomes and implications. A thermophoretic motion, as well as temperature-dependent thermal conductivity on the natural convection flow in the surface of a sphere at multiple circumferential points was introduced by Ashraf et al. [13]. The periodic behaviour of convective heat transfer H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 3 of 20 properties of the fluid flow were discussed by Ashraf et al. [14] under scrutiny of the cone which is electrically conducting embedded in the porous medium with the presence of viscous dissipation in energy equation. Azad et al. [15] revealed that the liquify volume at the start of spontaneous heat transport falls nonlinearly as the Gr rises. Azer and Chao [16] created a novel method for studying how heat transfers in liquid metals moving through pipes, and they discovered that their estimations for temperature, heat transfer, and flow behaviour match actual observations, particularly for extremely low Prandtl number settings. Benderradji et al. [17] discussed numerically combined natural and forced convection in a cubic cavity. They demonstrated that the stream changes from natural to forced convection for different Reynolds and Grashof numbers and compared with the literature. Benderradii et al. [18] offered a numerical examination of stationary laminar mixed convection in a vented four-sided chamber. Horizontal magnetic fields were also studied by Chen et al. [19] in relation to the thermal convection of liquid metals, which demonstrated substantial impacts on the morphology of the flow and the worldwide thermal transport in the liquid. Dawood et al. [20] presented a detailed review on improving the thermal efficiency of heat swapping equipment and transporting energy at a cheap cost. Mechanisms of convection in liquid metal fluids, along with techniques used to drive them, common applications, and enhancement methods, were introduced in [21]. In addition, the scientific problems and future perspectives of liquid metal convection were also addressed. Dudek et al. [22] gave the research on the experimental evaluation and numerical computation of both the stable and transient free convective resistance around spherical shapes with low Grashof values. Molten metals have been considered as effective heat transport fluids for thermal solar power applications due to their excellent heat capacity and thermal conductivity in [23]. Fritsch et al. [24] conducted conceptual research on central receiver systems that used fluid metals as efficient heat transport fluids especially in Concentrated Solar Power plants, demonstrating their potential for better thermal performance. An experimental hypersonic air-intake leading edges liquid metal heat-pipe solution concept, numerical analysis and verification have been defined by Fusaro et al. [25]. Ganeshkumar et al. [26] investigated the exergy efficiency and entropy production of shot-blasted heat exchangers using two different nanofluids: solar glycol with activated carbon as well as solar glycol with multi-walled carbon nanotubes. Free convection in an annulated region situated between two straight eccentric cylinders was investigated by Ghernoug et al. [27], considering air as a fluid with multiple data of Grashof number. Giometto et al. [28] examined stably stratified turbulent flows over inclined smooth surfaces at high Grashof numbers using direct numerical simulation and found that slope angle significantly affects mean flow, energy distribution, and turbulence in anabatic and katabatic regimes. Guo et al. [29] presented the straight numerical calculation of combined convection unsteady heat transport in a straight channel for fluid lead. Heinzel et al. [30] presented the benefits of fluid metals for applications as heat transport fluids in terms of high thermal conductivity and low viscosity. They also pointed out that advancements in research in this field have lessened some of the technical and safety barriers seen in earlier research, which would create a foundation for more applications. The linear stability in a downward moving H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 4 of 20 liquid metal vertical duct in the existence of intense wall heating and a transverse magnetic field was studied numerically by Hu [31]. Jaeger et al. [32] discussed liquid metal thermal transfer under free to forced convection, and demonstrated the Peclet number dependency, and the need for better models and experiments. Kamiyo et al. [33] provided a detailed review on free convection in triangular containers. Natural heat transfer in a liquid metal warmed regionally at its top area and sub- jected to an upright magnetic field was examined by Karcher et al. [34] experimentally and numerically. Free convection in a fluid metal warmed nearby at its top surface and subjected to a upright magnetic field was studied experimentally and mathematically by Karcher et al. [35]. Ke et al. [36] focused on a temporally developing turbulent buoy- ancy driven surface layer close to an isothermally warmed upright wall with Pr = 0.71 and high Grashof number. Khan et al. [37] presented research to discuss the physical phenomenon of free convection nanofluids heat transfer down a sphere and fluid eruption through the boundary layer into a plume region above the surface of the sphere. The com- putational investigation of smooth mixed heat transfer in a lid-driven hollow with altered heated bottom wall was carried out by Khanafer [38] for different Grashof and Richardson numbers. The flow of isothermal rounded particles in a 2D upright channel, limited by warm and cold adiabatic walls and affected by thermal convection at low Grashof num- ber, was studied by Kim et al. [39]. Direct numerical simulation was used by Kis and Herwig [40], to study the turbulent natural convection of air between two upright infinite surfaces at fixed temperatures throughout a range of Grashof numbers. Kroeger and Os- trach [41] found that during the continuous metal processing, natural convection can be considerable at Grashof numbers of 105 − 106, but has no impact on the solid-liquid edge location. Laminar natural convection created by a line thermal source was investigated by Linan and Kurdyumov, revealing that heat transfer from wires [42] is still conquered by conduction at low Grashof numbers. Li with others in [43] concluded that the pres- sure loss in magnetohydrodynamics peripheral flow is comparable to that in single-state magnetohydrodynamics pipe flow under identical fluid metal flow conditions. The results in [44] showed that for a high value of Gr, the thermal rate of the nanofluid enhances dramatically with an upsurge in nanoparticle mass ratio, and that for small Gr values, the heat transport process is dominated by warm exchange, whereas for large Gr values, it is dominated by free convection. Liu and Liu [45] discussed convective cooling of liquid metals at high power density electronics, and low-melting alloys like In-Ga-Sn alloys, and pointed out thermophysical property equations, model of heat transfer and flow resistance. Liu et al. [46] quantitatively studied the thermal transfer presentation of array jet impingement using liquid metals as coolant, with an emphasis on parametric influences to create an ideal design. Lorenzin and Abanades [47] stated that fluid metals are potential heat transport fluids for next generation Concentrated Solar Power systems, but their suitability with materials, security problems, and thermophysical characteristics must all be verified at large temperatures. Lyon and Poppendiek [48] offered one of the first coordinated studies of liquid-metal heat transport, stressing their high thermal conductiv- ity and unique behavior when compared to conventional fluids. Lyu et al. [49] analysed the convective magnetohydrodynamics stream in a simple rectangular cavity model which H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 5 of 20 includes a coaxial spherical cooling pipe which is preserved at fixed temperature whilst the fluid undergoes a uniform volumetric heating. The effects of spin on the following vortex and thermal transport from a big diameter straight spinning pipe in still air were experimentally investigated by Ma et al. [50]. A detailed analysis of chip cooling with low- melting-point liquid metals and alloys has been published in [51], with a focus on thermal characteristics and prospective applications in advanced electronic cooling. Mahony [52] proposed that the heat transport from a body submerged in a liquid be estimated only by the thermal transport equation when the Grashof number, Gr, connected to the situation is low. The application of standards for the forced-to-combined convection transition in moderate to strong Pr (Prandtl number) fluids was examined by Marocco with others [53] and expanded to include low Pr fluids such as heavy liquid metals. McBain and Stephens [54] discovered that in a fluid-filled spherical cavity with small Grashof numbers, convective effects are minimal and heat transmission occurs mostly by conduction. Mebarek and Bessaïh [55] numerically examined transient magnetohydrody- namics free convection between upright coaxial pipes and reported that flow steadiness and heat transort depend on the aspect ratio, Hartmann numbers, and the orientation of the magnetic field. Medrado [56, 57] gave the experimental results that indicate a sustained stable recirculation region nearby the air-water boundary for Gr values over or equal to 50. A calculation for calculating the coefficient of thermal transport in a pipe with a oscillatory flow that takes natural convection into account was presented by Mikhailov et al. [58]. Experimental research on NaK and mercury flowing via rod bundles were utilized by Mikityuk [59] to enhance the heat transfer correlation, resulting in better estimates for liquid-metal-cooled reactor and heat exchanger designs. Mohamad and Viskanta [60] examined the turbulent free convection in a chamber occupied with fluids having a low Prandtl value. The effects of Reynolds as well as Grashof numbers on the steady natu- ral circulation flow were experimentally studied with the FASSIP-02 important test loop by Moniaga et al. [61]. Moundjia and Samah [62] investigated a mathematical analysis of buoyancy driven in permeable media described by the Darcy-Brinkman-Forchheimer comparison at small Grashof numbers, in a permeable square inclusion. The impact of nanoparticle-induced Brownian and thermophoresis diffusion, as well as boundary defor- mation, on several physical variables were investigated in [63]. Direct computational simulations were used in [64] to study mixed convection in liquid metal flow in a straight pipe. Nabwey et al. [65] gave a comprehensive review that covers the essential ideas of entropy production reduction and their significance in the growth of renewable energy systems. Nabwey et al. [66] summarized the most important published works on improving convective thermal transfer in the occurrence of humid air, as well as its physical implications. The impact of dimensionless operating parameters on the dimensionless thermal transfer coefficient in a liquid metal flow in a circular tube was investigated by Ogneruboy and Listratov [67]. Oyinloye et al. [68] described the findings from an experimental examination of CO2 desublimation on a cylindrical, highly conduc- tive surface. Pacio et al. [69] undertook a careful assessment of the experimental data and heat transfer correlations published in the literature for fully developed, forced-convective liquid metals in pipes. Pacio and Wetzel [70] presented liquid metals as enhanced heat H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 6 of 20 transport fluids, identifying sodium (Na), tin (Sn), and lead-bismuth (PbBi), as promis- ing possibilities and examining their different benefits and limits. An inter-wrapper flow and heat transfer study was conducted on liquid metal-cooled fast reactors and presented information on thermal behaviour and safety in [71]. Rahimi et al. [72] discussed natural convection in enclosures with different geometries, boundary conditions and fluids, with emphasis on the effect on heat transfer and applications in industrial processes. Rong et al. [73] examined the flow structure of persuaded airflow with respect to the liquid flow state, the influence of changes in Grashof number and flow rate, and the primary variable spatial series of the persuaded airflow. The application of liquid gallium in metal inserts was proposed by Salyan et al. [74] as a different method for improving thermal transmission in solar thermal energy storage systems. Samanta and Chattopadhyay [75] extended the use of the lattice Boltzmann process to tackle solid-to-liquid phase change issues using low Prandtl number materials. Sarowar [76] examined the thermal presentation of a small network heat descend using five distinct substrate materials (aluminum nitride, titanium diboride, beryllium oxide, zirconium diboride, and hafnium diboride) and four distinct gallium alloys. Shah et al. [77] systematically reviewed the experimental studies on the influence of Grashof number and free to forced convection parameter and concluded that an rise in buoyancy causes an upsurge in velocity and a reduction in temperature with varying impacts on the heat with mass transfer characteristics. The production of heat convection flow in the molten metal battery, a technology described as a viable explanation to the problem of temporary energy storage, was investigated by Shen and Zikanov [78] using a numerical model. Shevaladze et al. [79] studied Grashof number computation in large-scale passive cooling structures, with a focus on temperature-induced fluctuations. Free convection between spheres that are concentric was investigated by Singh and Chen [80] for Grashof numbers that varied from 1, 000 to 20, 000, revealing similar flow and heat transmission trends with previous results. Stewart and Weinberg gave the experimental [81] and theoretical [82] analysis on fluid flows in liquid metals at low Grashof number. Their results showed that flow rates rise with growing temperature differential among the liquid cell, regular temperature, and liquid layer thickness. Suri et al. [83] examined numerically smooth heat transfer in power- law fluids spanning the region of vanishingly tiny Grashof number (10−4 ≤ Gr ≤ 10) for a solitary sphere and twin spheres with changing center-to-center distances. Theoretical models, design issues and applications of high-temperature liquid-metal heat pipes were summarized in a review by Tian et al. [84], in which performance limits and pragmatic guidelines were outlined. Ullah et al. [85] investigated the effects of de- creased gravity and magnetohydrodynamics on alternating mixed-convective electrically conducting fluid flow over a heated, non-conducting straight circular cylinder. Wang [86] summarized the heat exchanger tube rupture accidents in reactors, which would be of importance for the safety evaluation. The study found that although some phenomena are well researched, others need more research and better tools in the field. Wang et al. [87] examined liquid metals due to their special thermal characteristics, their uses as thermal interface materials, phase change materials and heat transfer fluids as well as the challenges facing their uses and directions of future uses. [88] concluded that alloy thermal H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 7 of 20 interface materials (TIMs), with higher conduction, flexibility, and lower melting points than conventional Thermal Interface Materials, show promise in electronics, soft devices, and bio-heat transfer, though barriers to widespread implementation remain. Wang et al. [89] numerically examined the liquid metal free convection on a vertical plate and proved that Gr and magnetic field strength have a great effect on the heat transport and flow features. Wolff et al. [90] did a collective experimental and mathematical study on free convection of liquid metals in upright hollows. They concluded that free convection in liquid metals differs significantly from that in strong Pr fluids. Xie et al. [91] investigated the hydrodynamic reply of liquids, focusing on the creation instruments of splashing and hollow shape, as well as their connections with motion of particles, while taking contact line motion into account. Yao et al. [92] summarized the features, heat transfer models, and application research developments of liquid metal phase change materials. Zhang et al. [93] solved low thermal volume of liquid metals by introducing manifold structure where they demonstrated that a liquid metal manifold temperature sink has high cooling capac- ity and efficiency. Also, Zhang [94] concise the single-phase convective thermal transport of liquid metal, that is helpful for scientists as well as engineers involved in liquid metal convection thermal transfer. Zurner et al. [95] experimentally investigated turbulent Rayleigh-Benard heat trans- port in liquid metal (GaInSn) at small Prandtl number, and found complicated large-scale circulation structure, Reynolds number dependence and heat transfer scaling in good agreement with simulations. Zwirner et al. [96] considered the effect of cell elevation on thermal transfer and important rotation in liquid metallic convection. The novelty of the review is that it focuses only on heat and fluid movement in liquid metals with low Grashof values. This regime is sometimes overlooked when compared to high Gr investigations, although it is critical for understanding conduction-dominated heat transport. The pa- per also applies these findings to contemporary micro-scale and high-performance cooling systems. 2. Thermophysical Properties of Liquid Metals Table 1 evaluates the viscosity η of several liquid metals, as reported in the original source. While the provided statistics are consistent with the relevant literature, inconsis- tencies between the calculated and experimental values suggest that Kim and Chair’s [97] correlation given in equation (1) is an estimate. These distinctions underscore the diffi- culties in adequately simulating the thermophysical characteristics of liquid metals, which are extremely sensitive to temperature and impurities. η = 2(MT )1/2 V [ CV ( 1− 1 γ )]1/2 λ, (1) and compared them with the experimental values. Here, M is the molar mass, T is the temperature in kelvin, V is the specific volume, CV is the specific heat at constant volume, and γ is the ratio of specific heats. The calculated and experimental numbers are given below: H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 8 of 20 Figure 1: Flowchart of the role of heat transfer in liquid metals at low Grashof number. 3. Dimensionless numbers and flow characteristics 3.1. Grashof number One important dimensionless ratio that determines the behaviour of fluid flow and heat in liquid metals is the Grashof number (Gr). It shows how much more significant are buoyant forces compared to viscous forces in determining whether it is conduction, laminar convection or turbulent convection which dominates the heat exchange process. Heat transport at very low Grashof values is largely conduction based whereas at intermediate values, natural convection occurs. The higher the Grashof number the more important is convection and at numbers large enough completely developed turbulent convection is seen. Table 1 identifies these different flow regions and demonstrates the change in behaviour of heat transfer in liquid metals with change in Grashof value. 3.2. Prandtl number The Prandtl number Pr = ν α is the ratio of diffusivity to momentum. The value of Pr in liquid metals is very low, with a range of 103 − 102. This implies that heat diffuses slower compared to momentum. This causes the velocity boundary layer to be very thin compared with the thermal boundary layer and the flow patterns and transition behaviours are very different as found in high-pressure fluids, such as air or water. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 9 of 20 Table 1: Values of different liquid metals at different temperatures Liquid Metal Temperature T (K) ηcalculated ηobserved Na 376.6 6628 6860 440.6 4205 5040 523.0 3308 3810 623.0 2844 2690 K 342.9 5146 5150 440.4 3265 3310 523.0 2760 2680 623.0 2473 1910 Rb 311.0 6789 6734 320.9 6180 6258 371.7 4643 4844 413.5 4016 4133 Pb 729.0 21378 20590 842.0 17964 17000 976.0 15995 13490 1117.0 14831 11850 Cs 316.4 6258 6299 371.6 4555 4753 413.5 3972 4065 441.0 3730 3760 483.0 3462 3430 Hg 253.0 18379 18500 273.0 17430 16800 293.0 16879 15500 373.0 15422 12700 473.0 14446 10100 4. Comparative Analysis of Reported Studies on Liquid Metal Heat Transfer A number of experiments have been conducted over the years aiming at the heat trans- fer properties of liquid metals at various temperatures and flow rates. Various scholars have come up with models, experimented and published data to comprehend their thermo- physical properties, as well as their convective behaviour better. These works are useful in terms of performance of liquid metals in the broad field of technological applications such as nuclear systems and cooling of electronics. In order to capture the diversity of contribu- tions, Table 4 presents and contrasts the main findings of various researchers and focuses on the differences in the approach, the qualities studied, and the important conclusions made by them. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 10 of 20 Table 2: Grashof number ranges and domain of flow of liquid metals from reference [36, 98]. Grashof Number (Gr) Flow Domain (in Liquid Metals) Heat Transfer Be- haviour Gr < 103 Conduction dominated Natural convection negligi- ble 103 < Gr < 105 Onset of weak free convec- tion Small circulation but con- duction still dominant 105 < Gr < 107 Onset of measurable convec- tion Convection begins to domi- nate heat transfer Gr ≈ 107 and above Turbulent convection Strong buoyancy flow Table 3: Assessment of liquid metal heat transfer from various studies Reference Fluid Geometry Parameter Range Method Key Findings Lyon & Poppen- dick [48] Na, Hg Pipes Low– medium Grashof Experimental/ handbook summary Early systematic studies of liquid-metal heat transfer estab- lished baseline data and demon- strated conduction supremacy at low Gr. Mahony [52] General liq- uid metals Various bod- ies in fluid Low Grashof number Theory/ analysis Conduction-only models can es- timate the disputed heat transfer for suitably low Gr levels. Azer & Chao [16] Liquid met- als Pipe/tube flows Low Pr Theory/ ex- perimental A redesigned mixing-length pro- cess for turbulent heat transport in liquid metals was proposed. Fendell [98] Generic Isothermal sphere Small Gr Analytical Laminar natural convection around a sphere proved conduction-dominated be- haviour and yielded low-Gr asymptotic solutions. Stewart & Weinberg [81, 82] Liquid met- als Crustal de- velopment setups Low– moderate Gr Experimental/ theoretical Flow structures in liquid metal melts were seen experimentally, and a theoretical companion study was written to examine driving processes. Kroeger & Os- trach [41] Pure met- als 2D freezing with convec- tion Solidifying Gr ranges Numerical Convection changes the interface during freezing, as demonstrated by coupled solid–liquid models that include modest convection effects. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 11 of 20 Table 3 (continued): Assessment of liquid metal heat transfer from various studies Reference Fluid Geometry Parameter Range Method Key Findings Singh & Chen [80] Fluids Concentric spheres Moderate Gr Numerical Numerical results for free convec- tion between concentric spheres, including flow structure and Nu– Gr data for comparison to exper- iments. Dudek et al. [22] Liquid met- als Spheres Small Gr Experimental/ theory Measured convection heat transfer on low-Gr spheres; theory–experiment comparisons confirmed conduction dominance tendencies. Wolff et al. [90] Liquid met- als Vertical cavi- ties Moderate Gr Experimental Extensive natural convection in- vestigations in vertical cavities yield data valuable for validating low-Pr models. Mohammad & Viskanta [60] Liquid met- als Various Low Pr Modelling/ review Modelling issues for turbulent buoyant flow in liquid metals were discussed, along with the requirement for modified turbu- lence closures at low Pr. Mikityuk [59] Pb, Na, NaK Tubes Wide Gr ranges Review/data analysis Compiled statistics and correla- tions for tube-bundle transfer of heat in liquid metals, as well as engineering correlations for reac- tor builders. Ke et al. [36] Liquid metal Infinite ver- tical wall High Gr Numerical Exhibited turbulent natural con- vection scaling at high Gr; useful as high-Gr benchmarks. Zwirner et al. [96] Ga or Ga-based alloys Laboratory cells Low– moderate Gr Experimental Demonstrated integrated velocity–temperature mea- suring techniques for low-Pr liquids. Salyan et al. [74] Ga Thermal en- ergy package with metal inserts System tem- perature Experimental Displayed Ga in metal inserts improves thermal performance and charging/discharging efficiency—a useful heat transfer enhancer. Zhang et al. [94] Liquid met- als Manifold mi- crochannel heat sink Low Pr Experimental/ numerical A manifold microchannel de- sign was created employing liq- uid metal for ultra-high heat flux cooling. Yao et al. [92] Ga-based alloys PCM mod- ules Low-Pr regime Review A summary of current break- throughs in liquid-metal phase change materials for electronics cooling. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 12 of 20 Table 3 (continued): Assessment of liquid metal heat transfer from various studies Reference Fluid Geometry Parameter Range Method Key Findings Xie et al. [91] General liq- uids Particle– liquid inter- face High Re Review Dynamic analysis of a single par- ticle contacting a liquid surface. Comparisons of several geometries (pipes, spheres, and vertical cavities) show that, while geometry influences local temperature gradients, the dominant conduction mode stays con- stant under low Gr circumstances. These investigations confirm the unique low-buoyancy, high-conduction behaviour of liquid metals, setting the framework for exact modelling in advanced applications. 5. Emerging applications and geometries Liquid metals are employed in many new technologies in lacking natural convection which has a small Grashof number, and small systems. Thermal transfer in these situations is conducted and forced through conduction and is not brought about by the buoyant flow. It is important to know the behaviour of liquid metals in such low-Grashof environments in order to develop effective thermal systems. • The existing electronics and heat exchangers contain narrow channels that reduce the effects of buoyancy. This indicates that flow operates in a low Grashof number. Liquid metals such as gallium and Na-K alloys are able to remove exceptionally large levels of heat because of their speed in heat transport and their smooth flow in these tiny cavities. Nevertheless, such issues as oxidation, clogging, and the compatibility with other materials require consideration. • Low melting liquid metals including In-Ga-Sn alloys are adopted as thermal interface materials to enhance the heat transfer between surfaces. The distance between surfaces is very low, hence heat conduction is carried out mainly through conduction with low Grashof number. Low-Gr behaviour analysis improves contact performance, reduces corrosion and improves the life of some materials. • The liquid metal phase transition materials are able to store and dissipate heat in small-scale energy systems. During melting and solidification, the conduction is the principle mode of heat transmission because convection is weak. Knowledge of low Grashof number flow in these materials would allow development of safe and efficient energy storage systems. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 13 of 20 6. Conclusion This review gave a thorough examination of the structure of heat and fluid flow in liquid metals with low Grashof values. The gathered studies show that liquid metals be- have differently than typical fluids when convection is weak, owing to their high thermal conductivity and distinctive thermophysical features. At extremely low Grashof num- bers, conduction dominates heat transfer, but buoyancy-induced convection progressively alters the flow pattern as Gr rises. The accumulated correlations, experimental data, and numerical analyses emphasize the necessity of precisely describing flow regimes and transport processes for designing dependable systems for nuclear reactors, solar thermal applications, and electronic cooling. Despite significant advances, there are still hurdles in building predictive models that account for the coupled effects of buoyancy, instability, and magnetic fields under actual settings. More systematic practical research and high- fidelity simulations are required to validate theoretical models, especially in mini- and microchannel systems where liquid metals are increasingly being used. In conclusion, liquid metals have a high potential as heat transfer medium in next- generation thermal systems, provided that their low Grashof number behaviour is well understood and precisely characterized. To fully exploit the benefits of liquid-metal-based cooling systems, future research should focus on connecting experimental data with so- phisticated simulations, improving design techniques, and resolving material compatibility and safety concerns. Acknowledgements The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (2025/RV/14). Author Contributions All authors have equal work. All authors have read and agreed to the published version of the manuscript. Funding The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (2025/RV/14). Data Availability Statement Data available upon request. H. A. Nabwey et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7166 14 of 20 Conflicts of Interest The authors declare no conflict of interest. References [1] A. Ahmad, R. Nawaz, A. Ali, and A. D. Alruwaili. Impact of microorganisms on blasius–rayleigh–stokes flow and heat transfer of nanofluid: A numerical study. ZAMM—Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 104(12):e202300287, 2024. [2] R. Akhmedagaev, O. Zikanov, and Y. 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