Cross-coupling in hydrodynamic phase-field models for nonisothermal binary fluids

In this paper, we present a general thermodynamically consistent hydrodynamic phase-field model for nonisothermal binary viscous fluids. This model incorporates cross-coupling effects among phase, velocity and temperature, while adhering to the generalized Onsager principle and conservation laws. We systematically explore its validity across the model parameter space and provide guidelines for determining consistent, dissipative physical boundary conditions. This model preserves both fluid phase volumes and ensures positive entropy production under adiabatic and dissipative boundary conditions. We then specifically investigate the phase-temperature coupling in detail, elucidating roles played by additional reversible (nondissipative) and irreversible(dissipative) processes due to the cross-coupling. Leveraging the entropy quadratization strategy, we develop a theoretical framework for devising second-order, entropy-production-rate-preserving numerical schemes for the model. We validate the schemes through mesh refinement tests and demonstrate its efficacy using an adaptive time-stepping strategy. Four distinct phase-temperature coupling scenarios are examined to illustrate the model’s capacity to capture complex interfacial dynamics in the context of Rayleigh–Bénard convection in nonisothermal binary viscous fluids. The role of cross-coupling parameters in promoting or retarding heat convection and fluid mixing via either the entropy-enhancing or entropy-preserving cross-coupling is qualitatively identified. This work advances the field by providing a robust, physically consistent framework for modeling nonisothermal binary fluid systems, with potential applications in materials science, geophysics, and engineering.