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

• Published in: European journal of applied mathematics Vol. 29; no. 4; pp. 595 - 644
• Main Authors: LAM, KEI FONG, WU, HAO
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.08.2018
• Subjects: Applied mathematics; Chemical potential; Computational fluid dynamics; Data transfer (computers); Dependence; Fluid flow; Mass transfer; Mathematical models; Navier-Stokes equations; Norms; Order parameters; Organic chemistry; Two phase flow; Well posed problems; Navier–Stokes equations; well-posedness; mass transfer; Cahn–Hilliard equation; volume-averaged velocity; mass-averaged velocity; chemotaxis
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792517000298

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.