Viscous flow past a flat plate with uniform injection

Abstract

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 ═ xδ ≑ 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 τ0 ~ (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* →∞.