Stationary bathtub vortices and a critical regime of liquid discharge

Abstract

A modified Lundgren model is applied for the description of stationary bathtub vortices in a viscous liquid with a free surface. Laminar liquid flow through the circular bottom orifice is considered in the horizontally unbounded domain. The liquid is assumed to be undisturbed at infinity and its depth is taken to be constant. Three different drainage regimes are studied: (i) subcritical, where whirlpool dents are less than the fluid depth; (ii) critical, where the whirlpool tips touch the outlet orifice; and (iii) supercritical, where surface vortices entrain air into the intake pipe. Particular attention is paid to critical vortices; the condition for their existence is determined and analysed. The influence of surface tension on subcritical whirlpools is investigated. Comparison of results with known experimental data is discussed.