Summary: To appear in J. Fluid Mech. 1
The Singular Perturbation of Surface
Tension in Hele­Shaw Flows
By HECTOR D. CEN I CEROS AND THOMAS Y. HOU
Applied Mathematics. California Institute of Technology, Pasadena CA 91125
(Received 7 September 1999)
Morphological instabilities are common to pattern formation problems such as the non­
equilibrium growth of crystals and directional solidification. Very small perturbations
caused by noise originate convoluted interfacial patterns when surface tension is small.
The generic mechanisms in the formation of these complex patterns are present in the
simpler problem of a Hele­Shaw interface. Amid this extreme noise sensitivity, what is
then the role played by small surface tension in the dynamic formation and selection of
these patterns? What is the asymptotic behavior of the interface in the limit as surface
tension tends to zero? The ill­posedness of the zero­surface­tension problem and the
singular nature of surface tension pose challenging difficulties to the investigation of these
questions. Here, we design a novel numerical method that greatly reduces the impact of
noise, and allows us to accurately capture and identify the singular contributions of
extremely small surface tensions. The numerical method combines the use of a compact
interface parametrization, a rescaling of the governing equations, and very high precision.
Our numerical results demonstrate clearly that the zero­surface­tension limit is indeed

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics