Aligned and nonaligned radial stagnation flow on a stretching cylinder

Laminar radial stagnation flow impinging on a stretching or shrinking elastic cylinder of radius a is studied. The strain rate of the stagnation flow is 2k and that of the stretching cylinder is b. The origin of stretching is in general displaced by a distance c from the inviscid stagnation circle on the cylinder. An exact similarity reduction of the Navier-Stokes equations leads to coupled ordinary differential equations describing the primary flow f (eta) and a secondary flow g(eta) with similarity variable eta = (r/a)(2). The system is governed by the Reynolds number R = ka(2)/2 nu, the dimensionless offset parameter alpha = c/a, and the dimensionless stretching parameter beta = b/2k, where v is the kinematic viscosity of the fluid. Solutions of the coupled equations only depend on R and beta, but the flow field depends crucially on alpha. Analytic solutions are found for the special values R = 2+beta and also for all beta if R = 1. For other values of R and beta, solutions are obtained numerically. We find no solutions for beta < beta(c), dual solutions when beta(c) < beta < -1, and unique solutions for beta > -1, where beta(c) depends on R. The stability of the dual primary flow solutions is determined and the effect of flow misalignment is displayed in streamfunction plots. (C) 2010 Elsevier Masson SAS. All rights reserved.