Converging flow between coaxial cones

Abstract

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, partial differential equations (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.