The laminar two-fluid wake

Abstract

The two-fluid flow behind a slender stationary symmetric object, as well as a slender symmetric self-propelled body, is investigated and governing equations are derived. These flows are also known as the classical and momentumless wakes, respectively, where the drag of the momentumless wake is zero. The upper fluid has a lower density so that instabilities do not form. As an initial study, the case where the two fluids have the same mainstream speed is considered. The fluid interface is unknown, but will be derived during the solution. Since the two fluids have the same mainstream speed, a two-fluid boundary layer upstream of the object will not form. A perturbation on the mainstream flow is considered in order to derive the far wake equations downstream of the object. The problem is formulated in terms of stream functions for the upper and lower fluids which reduces the number of unknowns in each fluid to one, while increasing the order of the partial differential equation by one. Conservation laws for the upper and lower fluid for each wake are derived using the multiplier method and conserved vectors are obtained. Conserved quantities for the two-fluid classical and momentumless wakes are calculated with the aid of the conservation laws. Lie point symmetries associated with the conserved vectors are obtained and new invariant solutions are calculated. The new solutions are compared with the existing solutions of Herczynski, Weidman and Burde [1]. Velocity profiles are plotted and analysed for different values of the parameters.