Steady streaming of fluid in the entrance region of a tube during oscillatory flow

To gain understanding and insight of entrance-flow phenomena, we solve a fluid dynamics problem involving an oscillating flat velocity profile (with zero mean flow) at the end (entrance) of a semi-infinite, rigid tube. A successive-approximation method is used. Analytic solutions are given for the first-order (linear) approximation. Time averages of products of these first-order solutions for velocity components and their spatial derivatives are used to approximate the nonlinear terms in the second-order equations, which are solved numerically. The time-averaged second-order results demonstrate steady bidirectional streaming in the entrance region. Solutions are given to an illustrative problem with a Reynolds number of 20 and a Womersley unsteadiness parameter of (200)(1/2). At this low Reynolds number, the streaming effects are small, with the magnitudes of the calculated streaming velocities less than three percent of the amplitude of the velocity specified at the entrance. The calculated magnitudes of streaming velocities would be expected to increase at higher Reynolds numbers; however, the convergence of the second-order solution would be less certain at higher Reynolds numbers. The steady streaming is related to first-order velocity variations that exist near the entrance of the tube. (C) 1999 American Institute of Physics. [S1070-6631(99)01210-6].