  Motivated by problems arising in the pneumatic actuation of controllers for
micro-electromechanical systems (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 interaction (FSIs) between 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--Voigt-type linearly 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 (but implicit) solutions for the pressure for
steady flow conditions. 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 FSI coupling. The
hydrodynamic 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 deformation is similar to that of a band-pass filter.
