Interface evolution: the Hele-Shaw and Muskat problems

Interface evolution: the Hele-Shaw and Muskat problems

We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, ...

A Hybrid Method for Moving Interface Problems with Application to the Hele-Shaw Flow

A Hybrid Method for Moving Interface Problems with Application to the Hele-shaw Flow

In this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method Li, 1995], is used to solve the diierential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coeecients or/and singular sources along the interfaces. The...

Imperfect Hele - Shaw cells

2014 The covering surfaces of a Hele-Shaw cell may be roughened mechanically, or modified by random chemical patches which change the wetting properties. We discuss first the quasi-static injection of a single fluid (with a finite contact angle 03B8e) in an imperfect cell. The shape of the injected region depends critically on a dimensionless number i (ratio of the boundary energy 03C4 over the...

On Hele-shaw Problems Arising as Scaling Limits

This work is motivated by the paper [LP]. In this paper, the authors consider the scaling limits of three discrete aggregation models with multiple sources the internal DLA (diffusion-limited aggregation), the rotor-router, and the divisible sandpile model. They show that for all three models, the scaling limit is the same, and it is a solution of the Hele-Shaw injection problem with multiple s...