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We present a new hydrodynamic model for incompressible binary fluids that isthermodynamically consistent and non-isothermal. This model follows thegeneralized Onsager principle and Boussinesq approximation and preserves thevolume of each fluid phase and the positive entropy production rate underconsistent boundary conditions. To solve the governing partial differentialequations in the model numerically, we design a set of second-order, volume andentropy-production-rate preserving numerical algorithms. Using an efficientadaptive time-stepping strategy, we conduct several numerical simulations.These simulations accurately simulate the Rayleigh-B\'{e}nard convection inbinary fluids and the interfacial dynamics between two immiscible fluids underthe effects of the temperature gradient, gravity, and interfacial forces. Ournumerical results show roll cell patterns and thermally induced mixing ofbinary fluids in a rectangular computational domain with a set of specificboundary conditions: a zero-velocity boundary condition all around, theinsulation boundary condition at the lateral boundaries, and an imposedtemperature difference vertically. We also perform long-time simulations ofinterfacial dynamics, demonstrating the robustness of our newstructure-preserving schemes and reveal interesting fluid mixing phenomena.

Derivation and Efficient Entropy-Production-Rate-Preserving Algorithms  for a Thermodynamically Consistent Nonisothermal Model of Incompressible  Binary Fluids