Nonaxisymmetric stokes flow between concentric cones

Abstract

SummaryWe study the fully three-dimensional Stokes flow within a geometry consisting of two infinite cones with coincident apices. The Stokes approximation is valid near the apex and we consider the dominant flow features as it is approached. The cones are assumed to be stationary and the flow to be driven by an arbitrary far-field disturbance. We express the flow quantities in terms of eigenfunction expansions and allow for the first time for non-axisymmetric flow regimes through an azimuthal wavenumber. The eigenvalue problem is solved numerically for successive wavenumbers. Both real and complex sequences of eigenvalues are found, their relative dominance affecting the flow features observed. The implications for the presence of eddy-like structures (analogous to those found in other corner geometries) are discussed and we find that these flow features depend, not only upon the internal angles of the two cones, but also upon the symmetry of the driving mechanism. For an arbitrary disturbance the dominant flow mode is not axisymmetric but rather is associated with wavenumber one and, by breaking axisymmetry, eddies can be avoided in this geometry.

A flow in the depth of infinite annular cylindrical cavity

“…This work extends the studies by Moffatt (1964), Shankar (1998), Hall, Hills & Gilbert (2009) and other researchers to flow in an infinitely deep annular cavity. Moffatt discovered that a flow in a narrow corner between two planes has an infinite number of vortices, whose scale and intensity decrease as the corner edge is approached.…”

“…In more general flows, swirl and meridional motions are coupled; the asymptotic pattern corresponds to modes with azimuthal wavenumber m = ±1 and is typically not cellular. This feature of annular cylindrical flow is similar to that in annular conical flows (Hall et al 2009). …”

“…The vortex scale does not decrease in the plane cavity, in contrast to the corner and cone flows. Reviews of early studies are in Shankar's book (2007) and the paper by Hall et al (2009).…”

Moffatt-type flows in a trihedral cone

“…Cases which have been successfully treated include three-dimensional flow in a two-dimensional corner (see e.g. Moffatt & Mak (1998), who showed that eddies need not occur for three-dimensional flow), the circular cone (Wakiya 1976;Liu & Joseph 1978;Malyuga 2005;Shankar 2005) and two concentric circular cones (Malhotra, Weidman & Davis 2005;Hall, Hills & Gilbert 2009).…”