On backward boundary layers and flow in converging passages

In a backward boundary layer the fluid has, in the mathematical model, been flowing along a solid wall through an infinite distance. The co-ordinate distance x along the boundary is measured upstream, and the velocity U of the flow outside the boundary layer is taken as negative. The main application is to flow in converging passages. The existence of similar solutions is considered, with emphasis on the correct asymptotic behaviour for large values of the stretched co-ordinate normal to the wall. This emphasis is shown to be necessary in considering backward boundary layers. For two-dimensional flow in converging passages the requirement that a boundary layer should be possible for vanishingly small viscosity with a potential core flow is shown to lead directly to Hamel's spirals as the shape of the boundary streamlines. Flow in axisymmetric converging passages is considered. For flow in a cone there is no limit as the viscosity tends to zero, and no potential core flow with a boundary layer is possible. The nature of a solution of the Navier-Stokes equations for laminar flow is considered.