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Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis

Published online by Cambridge University Press:  09 October 2017

KEI FONG LAM  and
HAO WU
KEI FONG LAM
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong email: [email protected]
HAO WU
Affiliation:
School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, 220 Han Dan Road, Shanghai 20043, China email: [email protected], [email protected]
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Abstract

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

Keywords

Cahn–Hilliard equationNavier–Stokes equationsmass transferchemotaxisvolume-averaged velocitymass-averaged velocitywell-posedness
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 4 , August 2018 , pp. 595 - 644
Copyright
Copyright © Cambridge University Press 2017 

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