Title: Longwave Models of Solidification: Self-similar Blow-up and its Regularization
Speaker: Professor A. Bernoff
Speaker Info: Northwestern U.
In this talk I consider three long-wave models of solidification; the Modified Kuramoto-Sivashinsky equation (MKSE) which describes solidification of a hypercooled melt, the Sivashinsky equation (SE) and the Riley-Davis equation (RDE) both of which describe directional solidification. I demonstrate numerically and analytically that the MKSE and the SE both exhibit self-similar blow-up, corresponding to the formation of deep roots in their solutions. A formalism for analyzing the linear stability of self-similar blow-up is developed and used to verify the stability of the blow-up. Finally, when the RDE is properly scaled, it reduces to the SE plus a small, regularizing perturbation. The results suggest that the RDE is well-posed and typified by periodic solutions which resemble fingers separated by deep roots. We present matched asymptotics for these solutions. Stability analysis suggests that these fingers will coarsen until a lengthscale determined by the segregation coefficient.Date: Friday, April 25, 1997