Aligned and nonaligned radial stagnation flow on a stretching cylinder

Laminar radial stagnation flow impinging on a stretching or shrinking elastic cylinder of radius aa is studied. The strain rate of the stagnation flow is 2k2k and that of the stretching cylinder is bb. The origin of stretching is in general displaced by a distance cc 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(η)f(η) and a secondary flow g(η)g(η) with similarity variable η=(r/a)2η=(r/a)2. The system is governed by the Reynolds number R=ka2/2νR=ka2/2ν, the dimensionless offset parameter α=c/aα=c/a, and the dimensionless stretching parameter β=b/2kβ=b/2k, where νν is the kinematic viscosity of the fluid. Solutions of the coupled equations only depend on RR and ββ, but the flow field depends crucially on αα. Analytic solutions are found for the special values R=2+βR=2+β and also for all ββ if R=1R=1. For other values of RR and ββ, solutions are obtained numerically. We find no solutions for β<βcβ<βc, dual solutions when βc≤β<−1βc≤β<−1, and unique solutions for β>−1β>−1, where βcβc depends on RR. The stability of the dual primary flow solutions is determined and the effect of flow misalignment is displayed in streamfunction plots.