On the Representation of Stokes Flows and a Solution of the Inhomogeneous Stokes Equations in the Plane

Abstract

In the first part of this paper it is shown that the formulae u = ũ − 1 2 {(ũ x + ṽ y ) x + ( ṽ x − ũ y ) y}, and v = ṽ − 1 2 {(ũ x + ṽ y ) y − ( ṽ x − ũ y ) x} with real-analytic ũ and ṽ represent a Stokes flow v = ( u, v) (i.e., Δ v = grad p, div v = 0) in a disc around the origin if and only if ũ and ṽ are harmonic. In the second part of the paper a solution v and p of the inhomogeneous Stokes equations (i.e., Δ v grad p + h, div v = 0) in a domain G ⊂ R 2 is given by an explicit integral formula involving the conformal mapping of G onto the unit disc.