We present a new hydrodynamic model for incompressible binary fluids that is thermodynamically consistent and non-isothermal. This model follows the generalized Onsager principle and Boussinesq approximation and preserves the volume of each fluid phase and the positive entropy production rate under consistent boundary conditions. To solve the governing partial differential equations in the model numerically, we design a set of second-order, volume and entropy-production-rate preserving numerical algorithms. Using an efficient adaptive time-stepping strategy, we conduct several numerical simulations. These simulations accurately simulate the Rayleigh-B\'{e}nard convection in binary fluids and the interfacial dynamics between two immiscible fluids under the effects of the temperature gradient, gravity, and interfacial forces. Our numerical results show roll cell patterns and thermally induced mixing of binary fluids in a rectangular computational domain with a set of specific boundary conditions: a zero-velocity boundary condition all around, the insulation boundary condition at the lateral boundaries, and an imposed temperature difference vertically. We also perform long-time simulations of interfacial dynamics, demonstrating the robustness of our new structure-preserving schemes and reveal interesting fluid mixing phenomena.

翻译：我们针对不可压缩二元流体提出了一种热力学一致且非等温的新型流体动力学模型。该模型遵循广义昂萨格原理和布辛涅斯克近似，在一致边界条件下保持各流体相的体积和正熵产率。为数值求解模型中的偏微分方程控制方程组，我们设计了一套二阶精度且保持体积和熵产率的数值算法。采用高效的自适应时间步长策略，我们进行了若干数值模拟。这些模拟精确再现了温度梯度、重力和界面力作用下二元流体中的瑞利-贝纳德对流以及两种不混溶流体间的界面动力学。数值结果显示，在特定边界条件下（全边界零速度条件、侧边界绝热条件、垂直方向施加温差），矩形计算域内出现了滚动涡胞图案和热致二元流体混合现象。我们还进行了界面动力学的长时间模拟，验证了新型结构保持算法的鲁棒性，并揭示了有趣的流体混合现象。