Systematic weakly nonlinear analysis of interfacial instabilities in Hele-Shaw flows - art. no. 016302

We develop a systematic method to derive all orders of mode couplings in a weakly nonlinear approach to the dynamics of the interface between two immi scible viscous fluids in a Hele-Shaw cell. The method is completely general : it applies to arbitrary geometry and driving. Here we apply it to the cha nnel geometry driven by gravity and pressure. The finite radius of converge nce of the mode-coupling expansion is found. Calculation up to third-order couplings is done, which is necessary to account for the time-dependent Saf fman-Taylor finger solution and the case of zero viscosity contrast. The ex plicit results provide relevant analytical information about the role that the viscosity contrast and the surface tension play in the dynamics of the system. We finally check the quantitative validity of different orders of a pproximation and a resummation scheme against a physically relevant, exact time-dependent solution. The agreement between the low-order approximations and the exact solution is excellent within the radius of convergence, and is even reasonably good beyond this radius.