New solutions for interfacial capillary waves of permanent form

Abstract

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.