Slow viscous motion round a cylinder in a simple shear

Abstract

The two-dimensional steady flow of an incompressible viscous liquid round a circular cylinder is described in terms of matched expansions valid asymptotically at low Reynolds number, the velocity field at large distances being a combination of a uniform simple shear and a uniform stream, relative to axes moving with the centre of the cylinder (but not rotating with it).

An infinite number of terms are computed and summed. There is a transverse force on the cylinder, independent of its rate of rotation to the approximation considered here. At moderate and large distances the balance between convection and diffusion of vorticity is dominated by the shear, being quite different from that in a uniform stream alone.

The rate of longitudinal diffusion of a substance released from an instantaneous line source in a simple shear alone is enhanced. Round a maintained line source for which the local convection velocity is parallel to that of the shear the concentration falls only algebraically with distance in all directions, though it is largest in twin wakes extending directly both upstream and downstream. If there is superposed a lateral convection, however small, at sufficiently large distances the concentration is exponentially small outside a wake centred on half a parabola.

The two-dimensional perturbation velocity round any obstacle held in an unbounded simple shear at any Reynolds number is, at a sufficient distance, an irrotational cross flow decreasing as the inverse two-thirds power, associated with twin shear layers extending upstream and downstream. If there is a uniform lateral motion at large distances this conclusion is completely altered.