Existence of Weak solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities
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Submission date: 19. Jan. 2008
published in: Communications in mathematical physics, 289 (2009) 1, p. 45-73
DOI number (of the published article): 10.1007/s00220-009-0806-4
MSC-Numbers: 76T99, 76D27, 76D03, 76D05, 76D45, 35Q30, 35Q35
Keywords and phrases: two-phase flow, free boundary value problems, diffuse interface model, mixtures of viscous fluids, Cahn-Hilliard equation, inhomogeneous Navier-Stokes equation
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We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study the general case that the fluids have different densities. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of weak solutions for the non-stationary system in two and three space dimensions.