Entropy-Production-Rate-Preserving Algorithms for a Hydrodynamical Model of Binary Fluids

We present a hydrodynamic model for nonisothermal, incompressible binary fluids that incorporates buoyancy effects using the Boussinesq approximation. This model adheres to the generalized Onsager principle, ensuring both volume preservation for each fluid phase and a positive entropy production rate under thermodynamically consistent boundary conditions, maintaining overall thermodynamic consistency. We then develop a set of second-order numerical algorithms that preserve both volume and entropy production rates to solve the model in finite domains, subject to physically appropriate boundary conditions. By implementing an efficient adaptive time-stepping strategy, we perform several numerical simulations that validate the accuracy and second-order convergence of the schemes. These simulations successfully capture Rayleigh-Bénard convection in binary fluids and the interfacial dynamics between immiscible fluids under the influence of temperature gradients, buoyancy, and interfacial forces, demonstrating the accuracy and practical utility of the model and the numerical methods.