Abstract
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.
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