Rational approximation schemes for unstable Hele-Shaw flows (Q1200186)

A rational approximation method or spectral method is developed and applied to the numerical calculation of unstable free surface flows in a Hele-Shaw cell. It is used to solve the potential problem. Both Eulerian and Lagrangian representations of the moving boundary have been used. The evolution equation for the moving boundary is reduced to a system of coupled ordinary differential equations which are solved numerically by a second-order implicit procedure with adaptive time steps. The unstable Hele-Shaw problem has two types of exact solution, one, the Saffman finger, remaining analytic for all time and the other leading to a cusped profile in a finite time period.