Nonlinear gravity electro-capillary waves in two-fluid systems: solitary and periodic waves and their stability

Starting from the Euler equations governing the flow of two immiscible incompressible fluids in a horizontal channel, allowing gravity and surface tension, and imposing an electric field across the channel, a nonlinear long-wave analysis is used to derive a $$ system of evolution equations describing the interface position and a modified tangential velocity jump across it. Travelling waves of permanent form are shown to exist and are constructed in the periodic case producing wave trains and the infinite case yielding novel gravity electro-capillary solitary waves. Various regimes are analysed including a hydrodynamically passive but electrically active upper layer, pairs of perfect dielectric fluids and a perfectly conducting lower fluid. In all cases, the presence of the field produces both depression and elevation waves travelling at the same speed, for given sets of parameters. The stability of the non-uniform travelling waves is investigated by numerically solving appropriate linearised eigenvalue problems. It is found that depression waves are neutrally stable whereas elevation ones are unstable unless the surface tension is large. Stability or instability is shown to be linked mathematically to the type of local eigenvalues of the nonlinear flux matrix used to obtain travelling and solitary waves; if these are real (hyperbolic flux matrix), the system is stable, and if they are complex (elliptic), the system is unstable. The latter is a manifestation of Kelvin-Helmholtz instability in electrified flows.