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Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

The problem of heat transfer (or mass transfer at low transfer rates) to a strip of finite length in a uniform shear flow is considered. For small values of the Péclet number (based on wall shear rate and strip length), diffusion in the flow direction cannot be neglected as in the classical Leveque solution. The mathematical problem is solved by the method of matched asymptotic expansions and expressions for the local and overall dimensionless heat-transfer rate from the strip are found. Experimental data on wall mass-transfer rates in a tube at small Péclet numbers have been obtained by the well-known limiting-current method using potassium ferrocyanide and potassium ferricyanide in sodium hydroxide solution. The Schmidt number is large, so that a uniform shear flow can be assumed near the wall. Experimental results are compared with our theoretical predictions and the work of others, and the agreement is found to be excellent.

The steady, axisymmetric, converging motion of a viscous incompressible fluid inside an infinite right circular cone is considered. It is shown that the exact solution of the Navier-Stokes equation for the stream function Ψ is of the form Ψ(r,θ) = AF(rv/A,θ), where (r, θ) are spherical polar co-ordinates chosen so r = 0 is the apex and θ = 0 is the axis of the cone, 2πA is the volumetric flow rate, and v the kinematic viscosity of the fluid. Asymptotic expansions of the stream function are found for large and small rv/A.For large rv/A, Stokes's method for slow motions is generalized to obtain a complete asymptotic expansion. Except for cones of special angles, all terms in this expansion may theoretically be found.For small rv/A a solution is constructed in two parts, namely, an inner expansion which starts from boundary-layer type equations as well as the no-slip condition at the wall, and an outer expansion in unstretched variables rv/A and cosθ which satisfies the boundary conditions at the axis of the cone. The condition that the inner solution merge with the outer solution with an exponentially small error requires an outer solution near the apex which is not potential sink flow, as might perhaps have been expected from the solution for two-dimensional flow in a wedge. The simplest outer flow satisfying the requirement is a vortex motion. Complete inner and outer expansions are developed and it is shown that they contain only six undetermined constants which must be determined by joining this solution numerically to the Stokes solution upstream. The inclusion of logarithmic terms in these expansions has not been found necessary.