Viscous stratified flow towards a line sink

Abstract

A theoretical and experimental investigation has been made for the problem of two-dimensional, viscous, incompressible, steady, slightly-stratified flow towards a line sink. The analytical solution was obtained from the Navier Stokes equations, the continuity equation, and the diffusion equation by making a boundary-layer-type assumption and by using a small perturbation technique based on a perturbation parameter proportional to the sink strength q. The effects of viscosity, diffusivity, and gravity have been included while the inertia effect is neglected in the zeroth order solution. The solution indicates that there exists a withdrawal layer which grows in thickness with the distance x from the sink at the rate x^(1/3) and that the velocity distributions u(y) are similar from one station x to another. Twenty-five tank experiments were performed using water stratified by means of either salt or temperature. Detailed measurements of the velocity field were made by means of photographs of vertical dye lines. The experiments verify the shape of the velocity profiles as well as their similarity in x as predicted by the theory. The applicability of these results to the problem of selective withdrawal from a reservoir is discussed and compared with nonviscous solutions by Yih (6) and Kao (7).