Slow flow between concentric cones

Hall, Oskar; Hills, C.; Gilbert, Andrew D.

doi:10.1093/qjmam/hbl024

Cited by 14 publications

(16 citation statements)

References 18 publications

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“…For this reason, the discovery of a chain of eddies in a slow flow between two inclined planes (Moffatt 1964) has attracted much attention and initiated numerous studies of creeping cellular motions. In steady planar or axisymmetric flows, similar eddies have been found in a plane cavity (Moffatt 1964;Shankar & Deshpande 2000), cone (Wakiya 1976), cylinder (Blake 1979;Hills 2001), in cavities with oppositely moving walls (Gürcan et al 2003;Wilson, Gaskell & Savage 2005), between concentric cones (Hall, Hills & Gilbert 2007) and coaxial cylinders (Shtern 2012a).…”

supporting

confidence: 57%

Patterns of a creeping water-spout flow

Herrada

Shtern²

2014

J. Fluid Mech.

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This paper explains a mechanism of eddy formation in a slow air-water motion, driven by the rotating top disk, in a vertical sealed cylinder. The numerical simulations reveal nine changes in the flow topology as water volume fraction H w varies from 0 to 1. At H w around 0.8, there are two large regions of the clockwise meridional circulation, one in air and one in water. These regions are separated by two small cells of the anticlockwise circulation adjacent to the interface near the sidewall in water and near the axis in air. The air cell is a thin layer and topologically is a bubble-ring for 0.745 < H w < 0.785. Alterations of this flow pattern are explored as (i) pressure increases, (ii) the bottom disk co-rotates and (iii) the top-disk rotation speeds up. This paper provides the physical reasoning behind the flow transformations; the results are of fundamental interest and can be utilized in bioreactors.

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confidence: 57%

Patterns of a creeping water-spout flow

Herrada

Shtern²

2014

J. Fluid Mech.

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“…In common with the axisymmetric case (see (17)), the effect of introducing a slender inner cone is therefore marked.…”

Section: Numerical Results and Flow Visualisation 41 Eigenvalue Spectramentioning

confidence: 99%

“…We extend the analysis of Hall et al (17) to allow for non-axisymmetric flow modes and hence fully three-dimensional flows. We incorporate, for the first time in this geometry, azimuthal dependence via a wavenumber and analyse the form of, and competition between, particular eigenmodes for individual wavenumbers and eigenvalues across the parameter range.…”

Section: Introductionmentioning

confidence: 96%

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Nonaxisymmetric stokes flow between concentric cones

Hall

Hills²,

Gilbert

2009

The Quarterly Journal of Mechanics and Applied Mathematics

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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.

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“…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. An axisymmetric flow between concentric cones has vortices similar to those in Moffatt's problem (Hall, Hills & Gilbert 2007). A motion in a plane infinitely deep cavity (Shankar & Deshpande 2000) can be considered to be a limiting case of that in the corner whose angle tends to zero.…”

Section: Introductionmentioning

confidence: 99%

A flow in the depth of infinite annular cylindrical cavity

Shtern¹

2012

J. Fluid Mech.

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The paper describes an asymptotic flow of a viscous fluid in an infinite annular cylindrical cavity as the distance from the flow source tends to infinity. If the driving flow near the source is axisymmetric then the asymptotic pattern is cellular; otherwise it is typically not. Boundary conditions are derived to match the asymptotic axisymmetric flow with that near the source. For a narrow cavity, the asymptotic solutions for the axisymmetric and three-dimensional flows are obtained analytically. For any gap, the flow is described by a numerical solution of an eigenvalue problem. The least decaying mode corresponds to azimuthal wavenumber m = 1.

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Slow flow between concentric cones

Cited by 14 publications

References 18 publications

Patterns of a creeping water-spout flow

Patterns of a creeping water-spout flow

Nonaxisymmetric stokes flow between concentric cones

A flow in the depth of infinite annular cylindrical cavity

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