We derive a thermodynamically consistent, non-isothermal, hydrodynamic model for incompressible binary fluids following the generalized Onsager principle and Boussinesq approximation. This model preserves not only the volume of each fluid phase but also the positive entropy production rate under thermodynamically consistent boundary conditions. Guided by the thermodynamical consistency of the model, a set of second order structure-preserving numerical algorithms are devised to solve the governing partial differential equations along with consistent boundary conditions in the model, which preserve the entropy production rate as well as the volume of each fluid phase at the discrete level. Several numerical simulations are carried out using an efficient adaptive time-stepping strategy based on one of the structure-preserving schemes to simulate the Rayleigh-B\'{e}nard convection in the binary fluid and interfacial dynamics between two immiscible fluids under competing effects of the temperature gradient, gravity, and interfacial forces. Roll cell patterns and thermally induced mixing of binary fluids are observed in a rectangular region with insulated lateral boundaries and vertical ones with imposed temperature difference. Long time simulations of interfacial dynamics are performed demonstrating robust results of new structure-preserving schemes.

翻译：基于广义Onsager原理和Boussinesq近似，我们推导了不可压缩二元流体的热力学一致非等温流体动力学模型。该模型不仅保持各流体相体积，还在热力学一致边界条件下保持正熵产率。以模型的热力学一致性为指导，我们设计了一套二阶结构保持数值算法，用于求解模型中的控制偏微分方程及相应的一致边界条件，这些算法在离散层面上同时保持熵产率和各流体相体积。基于其中一种结构保持方案，采用高效自适应时间步进策略进行了多个数值模拟，模拟了二元流体中的Rayleigh-Bénard对流以及温度梯度、重力和界面力竞争作用下两种不混溶流体的界面动力学。在具有绝热侧边界和上下定温差边界的矩形区域中，观察到滚胞图案和热致二元流体混合。界面动力学的长时间模拟展示了新结构保持方案的鲁棒性结果。