SummaryThis 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.

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.

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/ sin α. 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 non-parallel case is discussed and the angles (α, β) identified where it breaks down, analogously to flow in a wedge geometry. For the case of convergent flow, Q < 0, solutions governed by the Navier-Stokes equation are discussed for both parallel and non-parallel geometries. At large distances the flow is in a viscous regime and takes a Poiseuille profile. As the origin is approached the inertial terms become important and a plug flow emerges, with constant radial velocity in the core, sandwiched between thin boundary layers. By systematic approximation, PDEs are derived that describe the transition from viscous to high Reynolds number flow, and solutions describing the plug flow and boundary layers are obtained using matched asymptotic expansions.

SUMMARYIn this paper, a high-resolution ÿnite volume method for calculating solutions to a hyperbolic conservation law is presented. The method works in two space dimensions on general domains and uses curvilinear meshes. A non-trivial estimation of gradients needed for the reconstructions is presented. The paper contains examples of numerical solutions of the Euler gas dynamics equations.