Conical vortices: A class of exact solutions of the Navier–Stokes equations

Abstract

A two‐parameter family of exact axially symmetric solutions of the Navier–Stokes equations for vortices contained within conical boundaries is found. The solutions depend upon the same similarity variable, equivalent to the polar angle ϕ measured from the symmetry axis, as flows previously discussed by Long and by Serrin, but are distinct from the cases they treated. The conical bounding stream surfaces of the present solution can be located at any angle ϕ=ϕ0, where 0<ϕ0<π. The flows in all of these cases, when solutions exist, are finite everywhere except at the cone vertex which is a source of axial momentum, but not of volume. Solutions are of three types, flow may be (a) towards the vertex on the axis and away from the vertex at the conical boundary, (b) towards the vertex both on the axis and at the cone, or (c) away from the vertex on the axis and towards it at the bounding cone. In the first and second case, strong shear layers form on the cone walls for high Reynolds numbers. In case (c), a region of strong axial shear and strong axial vorticity forms near the axis, even for low Reynolds numbers. The qualitative nature of the possible solutions is deduced, using methods of argument due to Serrin, and examples of flows are numerically computed for cone half‐angles of π/4, π/2 (flows above the plane z=0), and 3π/4. Regions of the parameter space where solutions are proven not to exist are given for the cone half‐angles given above, as well as regions where solutions are proven to exist.