The transient for Stokes's oscillating plate: a solution in terms of tabulated functions

Abstract

The motion of a semi-infinite incompressible fluid caused by the sinusoidal oscillation of a plane flat plate is termed Stokes's problem. When the plate starts from rest in a still fluid a transient solution must be added to Stokes's well-known steady-state result. This paper presents a closed-form expression for the transient solution. Previous answers have contained a non-standard integral which could not be evaluated. The answer presented herein contains exponentials and error functions of a complex argument. These functions are readily available in newer mathematical tables. Graphs of the transient solution are presented for both sin (T) and – cos (T) boundary conditions. Velocity distributions in the fluid are also plotted and it is found that the transient period is essentially complete in one-half cycle for the cosine oscillation and in a full cycle for the sine wave case.