Piston Stokes flow in a semi-infinite channel

Abstract

We consider a Stokes flow of a viscous incompressible fluid in a semi-infinite two-dimensional channel $\text{x>0}$, $\text{−1<y<1}$ with rigid walls $\text{y=±1}$ and a prescribed uniform normal velocity at the end $\text{x=0}$. Recently, Katopodes, Davis and Stone have used the biorthogonal eigenfunctions expansion to construct the solution of that syringe flow. It is an analytical solution, but details of the asymptotic behaviour of the coefficients in the complex series remain unclear. We construct the analytical solution by means of the method of superposition. This solution allows us both to analytically describe the local Goodier–Taylor scraper flow and to establish the asymptotic properties of the coefficients in the eigenfunctions expansions. Knowledge of these non-decaying coeffiicents is essential for a discussion of a pointwise convergence of the non-orthogonal complex series.