Surface-tension driven flows: self-similar DNS

We revisit the scale-invariant recoil of an inviscid liquid wedge driven by surface tension, which was first theoretically studied and numerically solved with boundary integral methods by Keller and Miksis (KM) [SIAM J. Appl. Math, vol. 43, n°2, 1983, pp. 268-277].

We reformulate the problem in self-similar coordinates X = (ρ/σt²)^1/3 x and Y = (ρ/σt²)^1/3 y, where σ is surface tension, ρ mass density, x and y the physical space variables, t time and using τ = ln t as time variable. This transformation simply adds two new terms to the usual Navier-Stokes equations: a second advection term, and a source term.

The direct numerical simulation of this reformulated problem actually converges towards KM solution, which is stationary in (X, Y) coordinates, and demonstrates the stability of the scale-invariant solution. The numerical advantage of the use of (X, Y, τ) variables over (x, y, t) variables is that the characteristic length and time scales displayed by the solution are O(1) over the whole computation time span. This allows us to study the perturbation of the route towards self-similar behavior, caused by viscosity.