In this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface... Show moreIn this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface problems. The idea of rescaling is to map the current time-space onto a new time-space frame such that the interfaces evolve at a chosen speed in the new frame. We couple the rescaling idea with boundary integral method to demonstrate the efficiency of the rescaling idea, though it can be applied to Cartesian-grid based method in general. As an example, we use the Hele-Shaw problem to examine the efficiency of the rescaling scheme. First, we apply the rescaling scheme to a slowly expanding interface. In the new frame, the evolution is dramatically accelerated, while the underlying physics remains unchanged. In particular, at long times numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies. The rescaling idea can also be used to simulate the fast shrinking interface, e.g. the Hele-Shaw problem with a time dependent gap. In this case, the rescaling scheme slows down the interface evolution in the new frame to remove the severe time step constraint that makes the long-time simulations prohibitive. Finally, we study an analytical solution to the stability of the interface of the Hele-Shaw problem, assuming a small surface tension under a time dependent flux Q(t). Following [116, 109], we find the motions of daughter singularity ζd and simple singularity ζ0 do not depend on the flux Q(t). We also find a criterion to identify the relation between ζ0 and ζd. Ph.D. in Applied Mathematics, July 2017 Show less

In this paper, we study a moving interface problem in a Hele-Shaw cell, where two immiscible reactive fluids meet at the interface and... Show moreIn this paper, we study a moving interface problem in a Hele-Shaw cell, where two immiscible reactive fluids meet at the interface and initiate chemical reactions. A new gel-like phase is produced at the interface and may modify the elastic bending property there. We model the interface as an elastic membrane with a local curva- ture dependent bending rigidity. In the first part of this paper, we review the linear stability analysis on a curvature weakening model, and derive critical flux conditions such that a Hele-Shaw bubble can develop unstable fingering pattern and self-similar morphology. In the second part of this report, we develop a boundary integral nu- merical algorithm to perform nonlinear simulations. Preliminary numerical results show that in the nonlinear regime, there also exist stable self-similar solutions. M.S. in Applied Mathematics, December 2013 Show less