Impedance of circular tubes with slowly varying cross-sections for unsteady laminar flow

This thesis describes the resistance and the phase lag for an unsteady flow driven by a sinusoidally fluctuating pressure difference inside a rigid circular channel with slow axial variations in cross-sections. The amplitude ratio between the flow rate and the pressure difference along with the phase lag between these is represented by complex impedance as seen in an electrical circuit. A general non-dimensional formulation based on lubrication analysis yields the real and imaginary values of the impedance as functions of arbitrary frequency of fluctuation. Then, two asymptotic theories are presented to derive simplified expressions under low and high frequency limits, respectively. The former uses a regular perturbation method to express the desired quantities in Taylor series expansion, whereas the latter pursues boundary-layer approach revealing the trends near infinity. A novel functional mapping shows that the results from the general simulation accurately match with the asymptotic behaviors at both extremities. Such independent corroborations validate the correctness of all involved derivations and computations. Finally, two specific geometries in the form of a tapering conduit and a tube with wavy wall are considered. Their impedance is calculated as function of frequency for a wide range of system-defining geometric parameters.