The viscous incompressible flow inside a cone

Abstract

The steady, axisymmetric, converging motion of a viscous incompressible fluid inside an infinite right circular cone is considered. It is shown that the exact solution of the Navier-Stokes equation for the stream function Ψ is of the form Ψ(r,θ) = AF(rv/A,θ), where (r, θ) are spherical polar co-ordinates chosen so r = 0 is the apex and θ = 0 is the axis of the cone, 2πA is the volumetric flow rate, and v the kinematic viscosity of the fluid. Asymptotic expansions of the stream function are found for large and small rv/A. For large rv/A, Stokes's method for slow motions is generalized to obtain a complete asymptotic expansion. Except for cones of special angles, all terms in this expansion may theoretically be found. For small rv/A a solution is constructed in two parts, namely, an inner expansion which starts from boundary-layer type equations as well as the no-slip condition at the wall, and an outer expansion in unstretched variables rv/A and cosθ which satisfies the boundary conditions at the axis of the cone. The condition that the inner solution merge with the outer solution with an exponentially small error requires an outer solution near the apex which is not potential sink flow, as might perhaps have been expected from the solution for two-dimensional flow in a wedge. The simplest outer flow satisfying the requirement is a vortex motion. Complete inner and outer expansions are developed and it is shown that they contain only six undetermined constants which must be determined by joining this solution numerically to the Stokes solution upstream. The inclusion of logarithmic terms in these expansions has not been found necessary.