Exact and limiting solutions of fluid flow for axially oscillating cylindrical pipe and annulus

The Navier–Stokes equations have been solved to derive the expressions of the velocity distributions for two cases: (1) oscillatory flows inside and outside of an axially oscillating cylindrical pipe, and (2) oscillatory flow inside an axially oscillating cylindrical annulus. In both the cases, in addition to the exact expressions for the velocity profiles, particular emphasis has been given for the determination of approximate velocity distributions for the high frequency and low frequency or quasi-static limits. It is shown that, for sufficiently large value of an appropriate frequency parameter, the velocity distribution inside the axially or longitudinally oscillating cylindrical annulus can be approximated as a superposition of the velocity distribution inside an axially oscillating cylindrical pipe of radius $${\bar R_o}$$ and the velocity distribution outside an axially oscillating cylindrical pipe of radius $${\bar R_i}$$ , where $${\bar R_i}$$ and $${\bar R_o}$$ are the inner and outer radii of the axially oscillating annulus, respectively.