Viscous flow past a flat plate with uniform injection

Abstract

The boundary-layer equations for an incompressible fluid in motion past a flat plate are examined, numerically and analytically, in the special case when the pressure gradient vanishes and there is a uniform injection of fluid from the plate. In the numerical study the principal properties of the boundary layer are computed as far as separation ( x ═ x δ ≑ 0.7456) with a high degree of accuracy. In the analytic study the structure of the singularity at separation is determined. It is of a new kind in boundary layer theory and its elucidation requires the division of the boundary layer into three zones—an outer zone in which the non-dimensional velocity u is much larger than x * (the non-dimensional distance from separation), a central zone in which u ~ x * and an inner zone in which u ≪ x *. A match is effected between solutions in the central and inner zones from which it is inferred that the skin friction τ 0 ~ ( x * / In (1/ x *) 2 as x * → 0. A completely satisfactory agreement between the numerical and analytic studies was not possible. The reason is that the analytic study is only valid when ln ( 1 / x *) ≫ 1 which means that for the analytic and numerical studies to have a common region of validity, the numerical integration must be extended to much smaller values of x * than is possible at present. It was also not possible to effect a match between the central and outer zones in the analytic solution due to the difficulty of finding the properties of the stress τ in the central zone as u / x * →∞.