On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities
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Submission date: 08. Feb. 2007 (revised version: October 2007)
published in: Archive for rational mechanics and analysis, 194 (2009) 2, p. 463-506
DOI number (of the published article): 10.1007/s00205-008-0160-2
MSC-Numbers: 76T99, 76D27, 76D03, 76D45, 35B40, 35B65, 35Q30, 35Q35
Keywords and phrases: two-phase flow, free boundary value problems, diffuse interface model, mixtures of viscous fluids, Cahn-Hilliard equation, Navier-Stokes equation
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We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density 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. This leads to a coupled Navier-Stokes/Cahn-Hilliard system, which is capable to describe the evolution of droplet formation and collision during the flow. We prove existence of weak solutions of the non-stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities. Moreover, we present some results on regularity and uniqueness of weak solutions. In particular, we obtain that unique ``strong'' solutions exist in two dimensions globally in time and in three dimensions locally in time. Finally, we prove that any weak solution converges as to a solution of the stationary system.