Transient compressible flow in a compliant viscoelastic tube

Abstract

Motivated by problems arising in the pneumatic actuation of controllers (for MEMS, labs-on-a-chip or biomimetic soft robots) and the study of microrheology of both gases and soft solids, we analyze the transient fluid-structure interactions (FSIs) in a viscoelastic tube conveying compressible flow at low Reynolds number. We express the density of the fluid as a linear function of the pressure, and we use the lubrication approximation to further simplify the fluid dynamics problem. On the other hand, the structural mechanics is governed by a modified Donnell shell theory accounting for Kelvin-Voigttype linear viscoelastic mechanical response. The fluid and structural mechanics problems are coupled through the tube's radial deformation and the hydrodynamic pressure. For small compressibility numbers and weak coupling, the equations are solved analytically via a perturbation expansion. Three illustrative problems are analyzed. First, we obtain exact implicit solutions for the pressure under steady conditions, taking into account flow rarefaction. Second , we solve the transient problem of impulsive pressurization of the tube's inlet. Third, we analyze the transient response to an oscillatory inlet pressure. We show that an oscillatory inlet pressure leads to acoustic streaming in the tube, attributed to the nonlinear pressure gradient induced by the interplay of FSI and compressibility. Furthermore, we demonstrate an enhancement in the volumetric flow rate due to the coupling. The hydrodynamics pressure oscillations are shown to exhibit a low-pass frequency response (when averaging over the period of oscillations), while the frequency response of the tube itself is similar to that of a band-pass filter.