Creeping flow through an annular stenosis in a pipe

The creeping flow disturbance of Poiseuille flow due to a disk can be determined by the use of a distribution of “ringlet” force singularities but the method does not readily adapt to the complementary problem involving an annular constriction. Here it is shown that a solvable Fredholm integral equation of the second kind with bounded kernel can be obtained for an Abel transform of the density function. The exponential decay associated with the biorthogonal eigenfunctions ensures that the flow adjusts to the presence of the constriction in at most a pipe length of half a radius on either side. Methods that depend on matching series at the plane of the constriction appear doomed to failure. The physical quantities of interest are the additional pressure drop and the maximum velocity. The lubricating effect of inlets is demonstrated by extending the analysis to a periodic array of constrictions.