On a Class of Laminar Viscous Flows Within One or Two Bounding Cones

Abstract

SummaryThe class of incompressible axially-symmetric laminar viscous flows considered are exact invariant (similarity) solutions of the Navier- Stokes equations. The functional forms of the velocity components are deduced by group-theoretic arguments. The system of governing partial differential equations is reduced to a system of ordinary differential equations and these in turn are reduced to a single first order non-linear ordinary differential equation for the dimensionless tangential velocity component. The general solution of this equation is obtained in terms of hypergeometric functions.The classes of laminar viscous flows bounded by a right conical surface or by right inner and outer conical surfaces with a common apex obtainable from the previously mentioned (similar) solutions of the Navier-Stokes equation are studied. Suction/injection velocities (inversely proportional to the distance from the apex) normal/parallel to the conical boundaries are admissible for the classes of flows considered. If the flow is bounded by impermeable conical surfaces, then it is shown that these solutions of the Navier-Stokes equation imply that it must be identically quiescent.When the bounding cones are not impermeable, closed solutions for the single-cone case are found in terms of elementary functions. A numerical example of such a flow, with fluid injected normal to the conical boundary, is given. Such a simplification of the hypergeometric functions entering into the problem, however, cannot be achieved in the two-cone case.