Unidirectional flows

Abstract

In this Chapter we consider cases where the fluid velocity is directed along a single direction, so that the continuity equation is identically satisfied and the inertial term in the Navier-Stokes equation is identically null. Consequently, these fluid flows are much easier to study, as the four equations of mass and momentum balance reduce to a single, linear equation. In Sect 7.1 we consider the most important example of unidirectional flow, namely the pipe flow of a Newtonian fluid in laminar regime, where the fluid velocity is directed along the axial direction of the pipe. This case is generalized in Sects. 7.2 and 7.3, where the fluid velocity is directed along the azimuthal or the radial directions. In the following Sect. 7.4 the velocity field induced by the sudden movement of a wall is studied, deriving the classical self-similar solution. Then, in Sect. 7.5, the first-order correction to the unidirectional pipe flow solution is determined, by studying the slider bearing problem with its related lubrication approximation. Other approximated solutions are presented at the end of the Chapter, considering first the quasi steady state hypothesis (Sect. 7.6) and then an integral approach (Sect. 7.7) that will be studied further in later Sections.