The general solution of Stokes flow in a half-space as an integral of the velocity on the boundary

Abstract

The motion in a half‐space of a viscous fluid governed by the Stokes flow equations and driven by the instantaneous velocity on the boundary is considered. It is demonstrated that the flow field can be represented as an integral of the boundary velocity distribution, and the simple kernel function is derived. Previously it was thought necessary for the flow to be represented as an integral of the force distribution on the boundary; the velocity distribution was then related to the force distribution through an integral equation, which was solved numerically. Expressions for the stress field in terms of the velocity distribution on the boundary are also determined, and some technical difficulties involving the convergence of the integrals are discussed.