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## Full-Text Articles in Applied Mechanics

Nonaxisymmetric Stokes Flow Between Concentric Cones, O. Hall, C. P. Hills, A. D. Gilbert

#### Nonaxisymmetric Stokes Flow Between Concentric Cones, O. Hall, C. P. Hills, A. D. Gilbert

*Articles*

We 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 nonaxisymmetric flow regimes through an azimuthal wave number. The eigenvalue problem is solved numerically for successive wave numbers. Both real and complex sequences of eigenvalues are found, their ...

Converging Flow Between Coaxial Cones, O. Hall, A. D. Gilbert, C. P. Hills

#### Converging Flow Between Coaxial Cones, O. Hall, A. D. Gilbert, C. P. Hills

*Articles*

Fluid flow governed by the Navier-Stokes equation is considered in a domain bounded by two cones with the same axis. In the first, 'non-parallel' case, the two cones have the same apex and different angles θ = α and β in spherical polar coordinates (r, θ, φ). In the second, 'parallel' case, the two cones have the same opening angle α, parallel walls separated by a gap h and apices separated by a distance h/sinα. Flows are driven by a source Q at the origin, the apex of the lower cone in the parallel case. The Stokes solution for the ...

Slow Flow Between Concentric Cones, O. Hall, C. P. Hills, A. D. Gilbert

#### Slow Flow Between Concentric Cones, O. Hall, C. P. Hills, A. D. Gilbert

*Articles*

This paper considers the low-Reynolds-number flow of an incompressible fluid contained in the gap between two coaxial cones with coincident apices and bounded by a spherical lid. The two cones and the lid are allowed to rotate independently about their common axis, generating a swirling motion. The swirl induces a secondary, meridional circulation through inertial effects. For specific configurations complex eigenmodes representing an infinite sequence of eddies, analogous to those found in two-dimensional corner flows and some three-dimensional geometries, form a component of this secondary circulation. When the cones rotate these eigenmodes, arising from the geometry, compete with the forced ...

Flow Patterns In A Two-Roll Mill, Christopher Hills

#### Flow Patterns In A Two-Roll Mill, Christopher Hills

*Articles*

The two-dimensional flow of a Newtonian fluid in a rectangular box that contains two disjoint, independently-rotating, circular boundaries is studied. The flow field for this two-roll mill is determined numerically using a finite-difference scheme over a Cartesian grid with variable horizontal and vertical spacing to accommodate satisfactorily the circular boundaries. To make the streamfunction numerically determinate we insist that the pressure field is everywhere single-valued. The physical character, streamline topology and transitions of the flow are discussed for a range of geometries, rotation rates and Reynolds numbers in the underlying seven-parameter space. An account of a preliminary experimental study of ...

Eddies Induced In Cylindrical Containers By A Rotating End Wall, Christopher Hills

#### Eddies Induced In Cylindrical Containers By A Rotating End Wall, Christopher Hills

*Articles*

The flow generated in a viscous liquid contained in a cylindrical geometry by a rotating end wall is considered. Recent numerical and experimental work has established several distinct phases of the motion when fluid inertia plays a significant role. The current paper, however, establishes the nature of the flow in the thus far neglected low Reynolds number regime. Explicitly, by employing biorthogonality relations appropriate to the current geometry, it is shown that a sequence of exponentially decaying eddies extends outward from the rotating end wall. The cellular structure is a manifestation of the dominance of complex eigensolutions to the homogeneous ...

Eddy Structures Induced Within A Wedge By A Honing Circular Arc, C. P. Hills

#### Eddy Structures Induced Within A Wedge By A Honing Circular Arc, C. P. Hills

*Articles*

In this paper we outline an expeditious numerical procedure to calculate the Stokes flow in a corner due to the rotation of a scraping circular boundary. The method is also applicable to other wedge geometries. We employ a collocation technique utilising a basis of eddy (similarity) functions introduced by Moffatt (1964) that allows us to satisfy automatically the governing equations for the streamfunction and all the boundary conditions on the surface of the wedge. The circular honing problem thereby becomes one-dimensional requiring only the satisfaction of conditions on the circular boundary. The advantage of using the Moffatt eddy functions as ...

Rotary Honing: A Variant Of The Taylor Paint-Scraper Problem, Christopher Hills, H. Moffatt

#### Rotary Honing: A Variant Of The Taylor Paint-Scraper Problem, Christopher Hills, H. Moffatt

*Articles*

The three-dimensional Row in a corner of fixed angle α induced by the rotation in its plane of one of the boundaries is considered. A local similarity solution valid in a neighbourhood of the centre of rotation is obtained and the streamlines are shown to be closed curves. The effects of inertia are considered and are shown to be significant in a small neighbourhood of the plane of symmetry of the flow. A simple experiment confirms that the streamlines are indeed nearly closed; their projections on planes normal to the line of intersection of the boundaries are precisely the 'Taylor ...