On a Nonlocal Two-Phase Flow with Convective Heat Transfer

Abstract

We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse interface and is known as the Cahn–Hilliard model. We shall consider a nonlocal version of the Cahn–Hilliard model which replaces the gradient term in the free energy functional into a spatial convolution operator acting on the order parameter and incorporate with it a potential that is assumed to satisfy an arbitrary polynomial growth. The order parameter is influenced by the fluid velocity by means of convection; the temperature affects the interface via a modification of the Landau–Ginzburg free energy. The fluid is governed by the Navier–Stokes equations which is affected by the order parameter and the temperature by virtue of the capillarity between the two fluids. The temperature on the other hand satisfies a parabolic equation that considers latent heat due to phase transition and is influenced by the fluid via convection. The goal of this paper is to prove the global existence of weak solutions and show that, for an appropriate choice of sequence of convolutional kernels, the solutions of the nonlocal system converge to its local version.