The oscillatory pipe flow with arbitrary wall injection

Abstract

The linearized Navier–Stokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube. Asymptotic solutions of these equations have been found and here we extend the class of known solutions by solving the problem for an arbitrary mean–flow function of the Berman type. In the process, we show how not only do we recover, confirm, or correct some of the previously known solutions, but also find some completely new forms. It is interesting that, for sufficiently small injection, the Sexl profile can be restored from ours. Furthermore, we find that analytical, numerical and experimental results obtained by other investigators compare favourably with ours. The methods we apply provide accurate expressions for the main flow variables and help describe the ensuing oscillatory field. By appealing to a space–reductive multiple–scale technique, the problem's underlying length–scale is rigorously derived. Our results indicate that, irrespective of the mean–flow details, the unsteady component of vorticity initiated by small pressure disturbances can be more intense than its mean counterpart. No vortical study in porous tubes can therefore be complete unless it incorporates the unsteady field contribution.