An asymptotic formula for the friction factor of laminar flow in pipes of varying cross section

Abstract

In this paper we study the accuracy of an asymptotic formula for the friction factor of laminar flow in pipes of varying cross-section. The asymptotic formula is based on combining the integral form of the Navier-Stokes equations with asymptotic solutions obtained via the method of slow variations. The friction factor is first expressed in terms of surface integrals over the wall of the pipe, and these integrals are then estimated by using the asymptotic solutions for the velocity and the pressure. In addition, we present a two-dimensional finite element model for computing the friction factor. With the help of our numerical model, we systematically evaluate the accuracy of the asymptotic formula, depending on three different parameters, namely, the amplitude and period of the pipe, and the Reynolds number. We finally present the regions in the parameter space, where the asymptotic formula is accurate.