Two-fluid jets and wakes

Abstract

Similarity solutions for laminar two-fluid jets and wakes are derived in the boundary-layer approximation. Planar and axisymmetric fan jets as well as classical and momentumless planar wakes are considered. The interface between the immiscible fluids is stabilized by the action of gravity, with the heavier fluid, taken to be a liquid, placed beneath the lighter fluid. Velocity profiles for the jets and the classical wake depend intimately, but differently, on the parameter ϭ 1 1 / 2 2 , where i and i are, respectively, the density and absolute viscosity of the fluid in the upper (iϭ1) and lower (iϭ2) fluid domains, while the momentumless wake profile depends on the parameter ⍀ϭ 1 2 3 / 2 1 3. Generally, all interfaces deflect from horizontal except the fan jet. However, while the interface for the classical planar two-fluid wake is never flat, the interfaces for the planar jet and the momentumless wake become flat in the particular case 1 ϭ 2. Velocity profiles illustrating the strongly asymmetrical jet and wake profiles that arise in air-over-water, oil-over-water, and air-over-oil flows are presented.

Boundary layers due to shear flow over a still fluid: A direct integration approach

“…The analysis of the former class of problems was extended by Magyari et al [3] to account for lateral injection or suction through the permeable wall. Herczynski et al [4] found similarity solutions for two-fluid jets and wakes by matching the fluid velocity and shear stresses at the two-fluid interface. Narasimhamurthy et al [5] studied the mixing-layer between two parallel Couette flows.…”

Solutions for the turbulent classical wake using Lie symmetry methods

“…A study of the momentumless laminar wake behind a thin symmetric self-propelled body was first investigated by Birkhoff and Zorantello [4]. The two-fluid laminar classical wake was later investigated by Herczynski, Weidman and Burde [5]. The partial differential equation (PDE) for the flow in the wake was derived from the Navier-Stokes equation in the boundary layer approximation.…”

“…. .Þ ¼ D y T 2 ; ð3:8ÞIntegrating (3.9) across the wake from y ¼ À1 to y ¼ þ1 at a fixed point x gives where D is a dimensionless constant proportional to the drag on the body[1,5]. We now calculate the Lie point symmetry associated with the elementary conserved vector (3.6).…”