A pure and incompressible material is confined between two plates such that it is heated from below and cooled from above. When its melting temperature is comprised between these two imposed temperatures, an interface separating liquid and solid phases appears. Depending on the initial conditions, freezing or melting occurs until the interface eventually converges toward a stationary state. This evolution is studied numerically in a two-dimensional configuration using a phase-field method coupled with the Navier-Stokes equations. Varying the control parameters of the model, we exhibit two types of equilibria: diffusive and convective. In the latter case, Rayleigh-Bénard convection in the liquid phase shapes the solid-liquid front, and a macroscopic topography is observed. A simple way of predicting these equilibrium positions is discussed and then compared with the numerical simulations. In some parameter regimes, we show that multiple equilibria can coexist depending on the initial conditions. We also demonstrate that, in this bistable regime, transitioning from the diffusive to the convective equilibrium is inherently a nonlinear mechanism involving finite-amplitude perturbations.

中文翻译：

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具有熔化边界的瑞利-贝纳德对流的双稳态

纯净且不可压缩的材料被限制在两个板之间，以便从下方加热并从上方冷却。当其熔融温度介于这两个施加的温度之间时，会出现分离液相和固相的界面。取决于初始条件，会发生冻结或熔化，直到界面最终收敛到静止状态为止。使用相场方法和Navier-Stokes方程在二维配置中对这种演化进行了数值研究。改变模型的控制参数，我们表现出两种类型的平衡：扩散和对流。在后一种情况下，液相中的瑞利-贝纳德对流形成固-液前沿，并观察到宏观形貌。讨论了预测这些平衡位置的一种简单方法，然后将其与数值模拟进行了比较。在某些参数方案中，我们表明，取决于初始条件，多个平衡可以共存。我们还证明，在这种双稳态状态下，从扩散平衡过渡到对流平衡本质上是一种涉及有限幅值扰动的非线性机制。