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.

Converging flow between coaxial cones

“…This is solved numerically and shows the transition to a plug flow, as found in the study of Vatistas & Ghaly (1999). The same PDE is obtained by Ackerberg (1965) for flow in a single cone, but not solved numerically. Further asymptotic approximation gives the structure of the boundary layers bounding the plug flow in terms of the solution of a third order ODE (again obtained by Ackerberg (1965)), which we solve numerically.…”

“…The resulting converging or diverging flows are taken to be steady, axisymmetric, with no swirl component. This problem was considered for the case of a single cone by Bond (1927) and Ackerberg (1965), who suggested that a vortex motion might occur near the apex; this prediction was later discussed critically by Brown & Stewartson (1965) and has not been observed experimentally. Nonetheless the approach by Ackerberg (1965) has close links with our study as we indicate below.…”

“…Our work below will confirm many of these general features, in the case of converging flow with a narrow gap; however as Vatistas & Ghaly (1999) state, their analysis is based on a local linearisation procedure which is somewhat ad hoc and so their results do not constitute a solution of the Navier-Stokes equation. The other two studies, Fohr & Mallet (1975) and Renaud (1980), focus on the Stokes regime and then bring in the effects of inertia by means of an expansion in inverse powers of distance from the apex (as does Ackerberg (1965) for a single cone). They obtain terms in their expansion, and comment on some singular behaviour for certain cone openings, a topic we will return to in this paper.…”

Reduced Models for Thick Liquid Layers with Inertia on Highly Curved Substrates

“…The coordinate system used on the underlying conical substrate is shown on the right-hand side of Figure 2. This is essentially working in a polar coordinate system where the polar angle is fixed at φ as used in other studies [1,14,43].…”

Branching of the Falkner-Skan solutions for λ<0

“…From that time applications have been presented by various authors [13][14][15][16][17]. Goldstein [13] has given the first, heuristic, discussion of existence and uniqueness.…”