  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.
