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

In the first part of this paper it is shown that the formulae u = ũ − 12{(ũx + ṽy)x + (ṽx − ũy)y}, and v = ṽ − 12{(ũx + ṽy)y − (ṽx − ũy)x} with real-analytic ũ and ṽ represent a Stokes flow v = (u, v) (i.e., Δv = 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., Δv grad p + h, div v = 0) in a domain G ⊂ R2 is given by an explicit integral formula involving the conformal mapping of G onto the unit disc.