Double Degeneracy in the Immiscible Wilton Ripple Phenomenon: A Three Perturbation Parameter Problem in Bifurcation Theory

Abstract

A study is made of the waves which may occur at the interface of two fluids and which arise from an interaction between the two lowest adjacent harmonics of the motion. The problem is studied by reformulating it, via a conformal mapping transformation, into a pair of coupled nonlinear functional differential equations which are invariant under the action of the group [Formula: see text] The classical method of Lyapunov–Schmidt is then employed to replace this formulation by a pair of algebraic equations, known as the bifurcation equations. Throughout our analysis we impose a physical condition which essentially means that the densities and phase speeds in each fluid are approximately equal. This means that the bifurcation equations assume a cubic form rather than the quadratic form which they generically possess in problems of this type. The bifurcation equations are then analyzed by means of singularity theory by which means we are able to draw the solution curves of the system. A large variety of solution diagrams are found to exhibit such behavior as multiple primary bifurcations, secondary bifurcations and bifurcation from infinity.