Transient compressible flow in a compliant viscoelastic tube

Abstract

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.

arXiv comment30 pages, 11 figures; v2 revision removes Knudsen number effects and clarifies a few points; accepted for publication in Physics of Fluids; v3: update to REVTeX 4.2; v4: corrections per Erratum, (see "Note added" before Acknowledgements); Figs. 4(b), 5(b), 7(b), 9, 11 replaced

References in corpus (6)

• Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory
• Start-up flow in shallow deformable microchannels
• Viscous-elastic dynamics of power-law fluids within an elastic cylinder
• Dynamics of pulsatile flows through elastic microtubes
• On some model equations for pulsatile flow in viscoelastic vessels
• On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within

Cited by in corpus (8)

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• Three-dimensional soft streaming
• Elasto-inertial rectification of oscillatory flow in an elastic tube
• An electrokinetic route to giant augmentation in load bearing capacity of compliant microfluidic channels
• Oscillatory flows in three-dimensional deformable microchannels
• Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels
• Lattice Boltzmann method for fluid-structure interaction in compressible flow