On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within
In this chapter, we analyze the steady-state microscale fluid–structure interaction (FSI) between a generalized Newtonian fluid and a hyperelastic tube. Physiological flows, especially in hemodynamics, serve as primary examples of such FSI phenomena. The small scale of the physical system renders the flow field, under the power-law rheological model, amenable to a closed-form solution using the lubrication approximation. On the other hand, negligible shear stresses on the walls of a long vessel allow the structure to be treated as a pressure vessel. The constitutive equation for the microtube is prescribed via the strain energy functional for an incompressible, isotropic Mooney–Rivlin material. We employ both the thin- and thick-walled formulations of the pressure vessel theory, and derive the static relation between the pressure load and the deformation of the structure. We harness the latter to determine the flow rate–pressure drop relationship for non-Newtonian flow in thin- and thick-walled soft hyperelastic microtubes. Through illustrative examples, we discuss how a hyperelastic tube supports the same pressure load as a linearly elastic tube with smaller deformation, thus requiring a higher pressure drop across itself to maintain a fixed flow rate.
This research was supported by the U.S. National Science Foundation under grant No. CBET-1705637. We dedicate this work to the 70th anniversary of the director of the Institute of Problems in Mechanical Engineering of the Russian Academy of Sciences: Prof. Dr. Sc. D. I. Indeitsev. We also thank Prof. Alexey Porubov for his kind invitation to contribute to this volume, and for his efforts in organizing it.
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