Transient recirculation in a slowly varying tube impulsively rotated about its axis with constant angular velocity

Abstract

For time t̄<0, viscous fluid is in slow flow through a long straight axially symmetric tube whose radius, ā, varies slowly with axial distance, x̄. When t̄=0 the tube is impulsively rotated about its axis with angular velocity, Ω̄, at which angular speed it is thereafter maintained. During the transition from zero angular velocity, when t̄<0, to solid body rotation, when t̄→∞, the flow in the tube can briefly exhibit striking physical behavior, markedly different from the flow in the stationary tube. We present a linearization of the Navier–Stokes equations, valid when the Blasius parameter ε, which governs the magnitude of the inertial forces, tends to zero and the swirl parameter, λ, which is the ratio of a representative tube wall velocity, Ω̄ā0, to a representative axial velocity, tends to infinity, with the product ελ2≡Γ held fixed. An analytic solution suitable for computation and valid for suitably large t̄ is presented and streamlines are plotted for a typical diverging and a typical converging tube at time t̄=0.6ā02/ν when Γ=40. The relevance of the results to the phenomenon of vortex breakdown in tubes is discussed.