The Mullins-Sekerka model is a nonlocal evolution model for hypersurfaces, which arises as a singular limit for the Cahn-Hilliard equation. Assuming the existence of sufficiently smooth solutions we will show that the one-sided Mullins-Sekerka flow does not preserve convexity. Introduction The Mullins-Sekerka flow is a nonlocal generalization of the mean curvature flow arising from physics [10,...

Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn–Hilliard–Navier–Stokes model introduced by Abels, Garcke and Grün (Math. Models Methods Appl. Sci. 2012), which uses a volume averaged velocity, we derive a diffuse interface mo...

We review the fundamental modeling and numerical simulation for a prediction of fiber orientation during injection molding process of polymer composite. In general, the simulation of fiber orientation involves coupled analysis of flow, temperature, moving free surface, and fiber kinematics. For the governing equation of the flow, Hele-Shaw flow model along with the generalized Newtonian constit...

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure lim...

Three decades of basic cancer research have revealed that mutations in components of signalling pathways that control cell growth in primitive metazoans underlie tumour initiation in mammals. The Ras, PI(3)K and mTOR (mammalian target of rapamycin) signalling pathways form an intersecting biochemical network that, when mutated, drives cell growth in a manner unrestricted by environmental cues. ...

We compare two models describing the dynamics of phase separation of incompressible mixtures: A local model proposed by de Gennes, and a nonlocal model proposed by E and Palffy-Muhoray. We find that in the interfacial regime, the local model gives rise to interfacial motion via the Mullins-Sekerka law for moderate quenches, and surface diffusion for deep quenches. The interface dynamics is slow...