Solutions for Steady and Nonsteady Entrance Flow in a Semi-Infinite Circular Tube at Very Low Reynolds Numbers

Analytic solutions are found to the time-dependent Stokes and continuity equations describing entrance flow of an incompressible fluid into a circular tube at very low Reynolds numbers. Fourier transform methods are used, and entrance-flow solutions are obtained in the form of normal-mode expansions. Complex dispersion relations are obtained for each mode, and analytic expressions are determined for the normal-mode coefficients in terms of entrance boundary conditions. Results for velocity and pressure near the entrance are given. The generalized method is applied to obtain solutions to specific nonsteady-flow problems with a sinusoidal time dependence. In the last section of the paper, the steady-flow solutions are obtained by taking the limits of the nonsteady-flow solutions as the frequency approaches zero.