On the Rayleigh–Taylor instability for two uniform viscous incompressible flows
Fei Jiang, Song Jiang and Weiwei Wang
Abstract: We study the Rayleigh–Taylor instability for two incompressible immiscible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transform of Lagrangian coordinates, we transform the perturbed equations in Eulerian coordinates to an integral form and formulate the two-fluids flow in a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with an free interface separating the two fluids, and both fluids being in (unstable) equilibrium. By a general method of studying a family of modes, we construct smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces, thus leading to an global instability result for the linearized problem. Then, using these pathological solutions, we demonstrate the global instability for the corresponding nonlinear problem in an appropriate sense.