Boundary-layer growth near a rear stagnation point

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Abstract
This paper analyses the development, according to the Navier-Stokes equations, of the two-dimensional flow in the neighbourhood of the rear stagnation point on a cylinder which is set in motion impulsively with constant velocity. The local flow is idealized to the extent that the cylindrical boundary is taken to be an infinite plane bounding a semi-infinite domain of fluid. The velocity field is taken to be a linear function of the co-ordinate measured parallel to the boundary, and the initial flow is taken to be the (unique) irrotational form of such a field, namely, inviscid flow away from a stagnation point. Thereafter this irrotational flow is maintained as the outer boundary condition at a large distance from the boundary.It is suggested that, outside a viscous layer on the boundary, the asymptotic flow for large times is described by a similarity solution of the inviscid form of the governing equation, with a length scale normal to the boundary which increases exponentially with time. This inviscid solution has a steady velocity of slip along the boundary which is equal but opposite to that of the initial flow, so that the flow in the viscous layer ultimately becomes the well-known stagnation flow towards a boundary. The suggestion is supported by a numerical solution of the initial-value problem.

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