A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field feeds back on the flow through concentration-dependent density and viscosity, yielding a phenomenological temperature-coupled Cahn-Hilliard-Stokes-Heat system. We motivate the chemical potential by a temperature-dependent Landau free energy and derive a priori estimates for the regularized subproblems. On the analytical side, we prove local-in-time existence of weak solutions for a regularized coupled system. On the numerical side, we propose a fully discrete finite element scheme combining a convex-splitting time discretization for the Cahn-Hilliard equation with an implicit treatment of viscous and thermal diffusion terms and a an implicit Stokes solve. Under impermeable velocity boundary conditions, the Cahn-Hilliard substep conserves mass, in the purely diffusive isothermal case, the convex-splitting discretization is unconditionally energy-stable for the Cahn-Hilliard free energy. Numerical experiments in two dimensions illustrate thermally driven spinodal decomposition, wall-induced phase separation near cooled walls, and phase separation in narrow channels under imposed thermal gradients. The simulations show the qualitative influence of key nondimensional parameters (such as the mass and thermal Péclet numbers, the Cahn number, the density and viscosity ratios, and the gravitational parameter $G$) on pattern formation, interface motion, and flow structure, and confirm that the proposed framework is a robust tool for studying thermally driven phase separation in confined geometries.