Steady flow between two reservoirs of fluid with different densities and levels through a rectangular channel section of varying depth and width

Abstract

Abstract The steady hydrostatic flow through a channel of rectangular cross section connecting reservoirs of infinite width and depth and containing inviscid fluids of different densities and levels is studied. The main goal is the determination of the discharges of the lighter and denser fluids in terms of the external conditions (reservoir levels, fluid densities and variation of width and depth along a channel). It is shown that the key parameter is δ , which is the ratio of relative reservoir level difference, γ , to relative density difference, e . If δ δ > δ * (1 δ * δ * depends on the geometry of the constriction. The solutions describing these regimes are stated. If 0 δ δ * then both layers are in motion. A qualitative analysis of the solution for arbitrary bottom shape and channel width and arbitrary e is presented and the problem is reduced to a system of two equations which can be easily solved numerically for any particular channel profile. We give detailed analyses for the following two cases: 1) the narrowest width of the channel is on the side of the heavier fluid and the top of the sill is on the side of lighter fluid; 2) the minima in channel depth and width coincide. In the second case the discharges for one class of geometries in the Boussinesq approximation are calculated and discussed.