On almost rigid rotations

Abstract

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

Spin-up problems of stratified rotating flows inside containers

“…These are thin layers, predominantly steady in nature, involving a balance between the pressure gradient, Coriolis force and the viscous drag. With time, in the axisymmetric case, fluid in the upper (lower) Ekman layer moves towards the vertical sidewall, and then is transported down (up) the sidewall in the '1/4' and '1/3' Stewartson layers (Stewartson 1957). The upper and lower Stewartson layers then collide to form a shear layer, directed towards the container centreline, leading to two meridional-plane circulation regions.…”

Rotating flow in a cylinder with a circular barrier on the bottom

“…At sufficiently small values of the differential rotation the flow is one of almost rigid rotation and can in a number of situations be studied analytically. A central phenomenon is the axial shear layer, first studied by Stewartson [7], which possesses a complex multi-layered structure. Such shear layers may be free, as in the work of Stewartson [7,8] and Moore & Saffman [6], or attached to some solid boundary.…”

“…A central phenomenon is the axial shear layer, first studied by Stewartson [7], which possesses a complex multi-layered structure. Such shear layers may be free, as in the work of Stewartson [7,8] and Moore & Saffman [6], or attached to some solid boundary. Foster [2,3] has considered the structure of a Stewartson layer that is partially free, partially attached to a solid boundary.…”

Wave field and zonal flow of a librating disk

“…These internal shear layers can be generated from singularities of the oscillating source too, such as from the corners of a cylinder (McEwan 1970) or from angular topography (St Laurent et al 2003). They are the equivalent of the shear layer (Stewartson layer) limiting Taylor-Proudman columns in steady rotating flows for an oscillating flow in a stratified rotating fluid (Stewartson 1957).…”

“…A comprehensive list of references, especially for stratified fluids, can be found in Voisin (2003) and Voisin, Ermanyuk & Flór (2011). The case of the disk has been considered in numerous works, for steady displacements in a rotating fluid (Stewartson 1957;Moore & Saffman 1969; Vedensky & Ungarish 1994; Tanzosh & Stone 1995), for oscillating displacements in a stratified fluid (Il'inyhk & Chashechkin 2004;Bardakov, Vasil'ev & Chashechkin 2007;Davis & Llewellyn Smith 2010) or for more complicated surface fluctuations (Walton 1975;Kerswell 1995). The method of resolution is based on the use of the Hankel transform, which leads to a system of dual integral equations.…”