Integral Equations of the Linear Sloshing in an Infinite Chute and Their Discretization

Abstract

AbstractA mathematical model of the linear sloshing in an infinite chute with a rectangular section filled by an inviscid incompressible liquid is considered. This model is an abstract second order differential equation in time with a singular integral operator coefficient on a free surface. In the case of a chute with a rib this operator coefficient is defined semi-explicitely through an auxiliary singular integral equation. Mathematical properties of the solution of the model are studied. The fundamental frequencies of the liquid as functions of the width and depth of the rib are investigated. The mathematical model is discretized by using the collocation quadrature method for the operator on the free surface and the Cayley transform method for the time derivative. This fully discrete model is investigated and the error estimates are obtained showing the spectral accuracy with respect to both the spatial and the time discretization parameters.