  A flow vessel with an elastic wall can deform significantly due to viscous
fluid flow within it, even at vanishing Reynolds number (no fluid inertia).
Deformation leads to an enhancement of throughput due to the change in
cross-sectional area. The latter gives rise to a non-constant pressure gradient
in the flow-wise direction and, hence, to a nonlinear flow rate--pressure drop
relation (unlike the Hagen--Poiseuille law for a rigid tube). Many biofluids
are non-Newtonian, and are well approximated by generalized Newtonian (say,
power-law) rheological models. Consequently, we analyze the problem of steady
low Reynolds number flow of a generalized Newtonian fluid through a slender
elastic tube by coupling fluid lubrication theory to a structural problem posed
in terms of Donnell shell theory. A perturbative approach (in the slenderness
parameter) yields analytical solutions for both the flow and the deformation.
Using matched asymptotics, we obtain a uniformly valid solution for the tube's
radial displacement, which features both a boundary layer and a corner layer
caused by localized bending near the clamped ends. In doing so, we obtain a
``generalized Hagen--Poiseuille law'' for soft microtubes. We benchmark the
mathematical predictions against three-dimensional two-way coupled direct
numerical simulations (DNS) of flow and deformation performed using the
commercial computational engineering platform by ANSYS. The simulations show
good agreement and establish the range of validity of the theory. Finally, we
discuss the implications of the theory on the problem of the flow-induced
deformation of a blood vessel, which is featured in some textbooks.
