The Richardson's Annular effect and a transient solution of oscillating pressure-driven flow in circular pipes

Abstract

In this paper it is shown that the location of the characteristic overshoot of the Richardson's annular effect changes with the Kinematic Reynolds number $\omega^*=\omega r_0^2 /\nu$ in the range of frequencies within the laminar regime. From the study of the Richardson's overshoot at different times it was identified the existence of aparent transverse damped waves similar to those ones observed in the famous Stokes second problem, the physical analysis of this waves was used for the establishment of a semi-empirical law that gives the functional relation of the mean overshoot maximum with the kinematic reynolds number, say $B_0(\omega^*)=2.28 + 0.51{\omega^*}^{-1/2}$. Finally a transient solution was constructed and verified asymptotically for large times, and the tipical time for which the transient solution resembles the steady oscillating one was identified to be dependent of the viscosity of the fluid and of the radius of the pipe.