A long wave approximation to peristaltic motion

Abstract

The problem of motion of a viscous incompressible fluid induced by travelling wave motions of the confining walls has been studied analytically for the two-dimensional geometry. The analysis is aimed at the possible application to urine flow in human ureters. The wave length, λ, of the peristaltic waves is assumed to be large compared to the half channel width, d, whereas the amplitude of the wave, a, need not be small compared to d. A systematic approach based on an asymptotic expansion of the solution in terms of the small parameter d λ has been used and solutions up to O( d 2 λ 2 ) have been presented in closed forms. The limiting solution has been studied including the effects of externally applied pressure gradients, and the criteria for backward flow have been established and discussed in detail. Higher order solutions which include the effects of non-linear inertial terms of the Navier-Stokes equations, have been studied for the case of zero mean volume flow. The results indicate that the peristaltic waves on the wall do not contribute to the time-averaged flow quantities to O( d λ ) , but they do give rise to a mean axial velocity of O( d 2 λ 2 ) , and a mean pressure gradient of O( d 2 λ 2 ) is associated with the motion. Comparisons with some earlier results have been made and certain agreements are brought out.