Thermomagnetic convection is based on the use of external magnetic fields to better control heat transfer fluxes in ferrofluids, finding important applications in engineering and many related areas. The improvement of such methods relies on fundamentally understanding the flow of ferrofluids under temperature gradients and external magnetic fields. However, the underlying physics of this phenomenon is very complex and not yet well characterized. The problem we analyze in this paper consists of a ferrofluid confined in a square cavity heated from the top and subjected to an external magnetic field applied in the horizontal direction. Differently from earlier investigations, this scenario leads to a clear competition between stabilizing gravitational forces and destabilizing magnetic forces; the unbalance between them drives the flow and dictates the mechanisms of heat transfer in the system. The problem is described by the equations of conservation of mass, momentum, and energy; both gravitational and magnetic effects are accounted for through the definition of the body force following the well-known Boussinesq approximation to express the ferrofluid’s density and magnetization as functions of temperature. The resulting set of fully coupled, nonlinear equations of the model is solved with a fully implicit finite element method. The results show that thermomagnetic convection increases convective heat transfer fluxes in the flow, but its net effects essentially depend on a complex balance among viscous, magnetic, and gravitational forces which determines the pattern of recirculation regions inside the cavity. There are two critical values associated with the external field’s intensity: the first marks the onset of thermomagnetic convection when the destabilizing magnetic effects become comparable to the stabilizing gravitational ones and the second corresponds to the external field’s strength that suffices to make the effects of gravity nearly negligible in the flow dynamics. The latter regime is dictated by a balance between viscous and magnetic forces only, and the corresponding numerical predictions agree notoriously well with a scaling analysis which suggests that $${\overline{\mathrm{Nu}}} \sim \mathrm{Ra}_\mathrm{m}^{1/4}$$ Nu ¯ ∼ Ra m 1 / 4 , where $${\overline{\mathrm{Nu}}}$$ Nu ¯ is the average Nusselt number at the isothermal walls and $$\mathrm{Ra}_\mathrm{m}$$ Ra m is the magnetic Rayleigh number, a dimensionless parameter associated with the intensity of the external field.

中文翻译：

---

重磁对流作用下方腔铁磁流体传热的数值研究

热磁对流基于使用外部磁场来更好地控制铁磁流体中的传热通量，在工程和许多相关领域有重要应用。这种方法的改进依赖于从根本上理解铁磁流体在温度梯度和外部磁场下的流动。然而，这种现象的基础物理学非常复杂，尚未得到很好的表征。我们在本文中分析的问题包括限制在从顶部加热的方腔中的铁磁流体，并受到水平方向施加的外部磁场的影响。与之前的研究不同，这种情况导致稳定重力和不稳定磁力之间存在明显的竞争。它们之间的不平衡会驱动流动并决定系统中的热传递机制。该问题由质量、动量和能量守恒方程描述；引力和磁效应均通过遵循众所周知的 Boussinesq 近似的体力定义来解释，以将铁磁流体的密度和磁化强度表示为温度的函数。使用完全隐式有限元方法求解该模型的一组完全耦合的非线性方程。结果表明，热磁对流增加了流动中的对流传热通量，但其净效应基本上取决于粘性力、磁力和重力之间的复杂平衡，这决定了腔内再循环区域的模式。有两个与外场强度相关的临界值：第一个标志着热磁对流的开始，当不稳定的磁效应变得与稳定的引力效应相当时，第二个对应于足以使重力影响接近的外场强度。在流动动力学中可以忽略不计。后一种状态仅由粘性力和磁力之间的平衡决定，相应的数值预测与标度分析非常吻合，这表明 $${\overline{\mathrm{Nu}}} \sim \mathrm{Ra} _\mathrm{m}^{1/4}$$ Nu ¯ ∼ Ram 1 / 4 ，其中 $${\overline{\mathrm{Nu}}}$$ Nu ¯ 是等温壁的平均努塞尔数$$\mathrm{Ra}_\mathrm{m}$$ Ra m 是磁瑞利数，