On a diffuse interface model for incompressible viscoelastic two-phase flows

Abstract

This paper concerns a diffuse interface model for the flow of two incompressible viscoelastic fluids in a bounded domain. More specifically, the fluids are assumed to be macroscopically immiscible, but with a small transition region, where the two components are partially mixed. Considering the elasticity of both components, one ends up with a coupled Oldroyd-B/Cahn--Hilliard type system, which describes the behavior of two-phase viscoelastic fluids. We prove the existence of weak solutions to the system in two dimensions for general (unmatched) mass densities, variable viscosities, different shear moduli, and a class of physically relevant and singular free energy densities that guarantee that the order parameter stays in the physically reasonable interval. The proof relies on a combination of a regularization of the original system and a new hybrid implicit time discretization for the regularized system together with the analysis of an Oldroyd-B type equation.

53 pages, major revision, to appear at J. Nonlinear Sci

References in corpus (6)

• Global Solutions for Incompressible Viscoelastic Fluids
• Viscoelastic Cahn--Hilliard models for tumour growth
• Two-phase flows with bulk-surface interaction: thermodynamically consistent Navier-Stokes-Cahn-Hilliard models with dynamic boundary conditions
• Analysis of a viscoelastic phase separation model
• Existence, regularity and weak-strong uniqueness for the three-dimensional Peterlin viscoelastic model
• Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions