Boundary-Layer Flow on a Vertical Plate

Abstract

The motion of a film of viscous incompressible fluid running down a semi-infinite vertical plate under the influence of gravity is considered. The problem is to determine how the final asymptotic shear flow is achieved when the fluid is introduced at the leading edge with uniform velocity. The Reynolds number Re and the Froude number Fr, based on the initial film thickness and velocity, characterize the flow and are assumed large compared to unity. For these cases the motion is determined uniformly to first order by solutions of the boundary-layer equations in terms of a single parameter ε = (Fr/Re)12 which is assumed small. Numerical solutions of the boundary-layer equations are obtained using Mises variables for 0 ≤ ε ≤ 0.5. The results indicate that all solutions, for sufficiently small ε, become indistinguishable (in terms of scaled variables) from the solution obtained for ε ≡ 0 after a short distance downstream, and this occurs before the potential flow has been absorbed by the boundary layer. This “forgetfulness” of the initial conditions is found to be associated with a nonuniform limit at the leading edge when ε → 0.