Unsteady Nonuniform Flow in the Entrance of a Pipe

Abstract

The laminar incompressible flow started from rest in the entrance of a pipe is analyzed using the integral form of the equations of continuity, momentum, and energy, together with a similarity hypothesis for the velocity profiles across the boundary layer. A first-order partial-differential equation for the growth of the boundary-layer thickness is obtained and solved by the method of characteristics. The flow is shown to depend critically on a dimensionless parameter which can be interpreted as the ratio of a characteristic time of the variation of the discharge to a time characteristic of the diffusion of viscous effects.