Gravitational fluid flow induced in a three layer stratified fluid by two submerged sinks or sources with stagnation points on the free surfaces

Abstract

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by two submerged sinks or sources with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing nonlinear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers, sink flow strengths and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on six parameters, namely the Froude numbers, the flow strength of the sinks and the ratio of sink heights to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.