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Applied mathematics, Fluids and plasma physics, Applied mechanics
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 modes to determine the flow near the apex.
This paper studies the relative dominance of these two effects and maps out regions of parameter space, with attention to how shear and overall rotation can destroy the infinite sequence of eddies that may be present when only the lid is rotated. A qualitative picture of the number of eddies visible in the meridional circulation is obtained as a function of the rotation rates of cones and lid, for various choices of angles. The results are discussed in the context of previous work, including their significance for applications to the mixing of viscous fluids in this geometry.
Hall, O., Hills, C.P. and Gilbert, A.D.: Slow Flow Between Concentric Cones. Quarterly Journal of Mechanics and Applied Mathematics,Vol, 60, Issue 1, 2007, pp.27-48.