Asymptotic flow in the depth of narrow cavity

Abstract

It is shown that the viscous fluid motion in the depth of a narrow cavity is a counterflow moving and periodically varying along the cavity length. The obtained exact solutions of the Stokes equations describe unsteady three-dimensional flows, developing as the distance from the cavity surface increases. The found asymptotic motion is common for a variety of boundary conditions and cavity shapes. Creeping flows between parallel and inclined planes, coaxial cylinders, and concentric cones are particularly addressed. For these flows, the analytical asymptotes and numerical results merge as the gap width decreases.

Low-to-moderate Reynolds number swirling flow in an annular channel with a rotating end wall

“…Most existing studies on annular channel flows have been performed numerically or experimentally [13,[15][16][17], quite often with the aim of investigating the impact of physicochemical contamination along the upper liquid surface. To our knowledge, except for the recent analytical modeling performed by Shtern [18,19] for an annular cavity considered as semi-infinite along the vertical direction, all existing analytical studies devoted to this configuration only focus on the azimuthal flow either when the liquid surface is free of contamination [20] or when it is contaminated [21,22],…”

Patterns of a slow air–water flow in a semispherical container

“…It is interesting that the entire problem formally is nonlinear despite the motion is creeping, but can be reduced to the two linear problems: one for the swirl velocity (5) and the other for the meridional motion (6)- (8). After solving problem (5), the "source" term, v 2 /r in equation (7), is prescribed, so the problem for the meridional motion also is linear.…”

“…The number of variables involved in the problem is reduced by introducing a streamfunction-vorticity-circulation form. System (1)-(4) is transformed into three equations for the 8 Stokes stream function Ψ, u = − r −1 ∂Ψ/∂z, w = r −1 ∂Ψ/∂r, the azimuthal vorticity component, η = ∂u/∂z-∂w/∂r, and circulation, Γ = rv:…”

“…The mechanism of the Moffatt eddies is based on the dead-end geometry of a cavity and the dissipation of kinetic energy. The dead end induces a counterflow and the dissipation is smaller in a set of eddies than that in a global counterflow [8]. A different mechanism of eddy formation occurs in two-fluid swirling flows.…”