Static response of a soft microtube due to low Reynolds number non-Newtonian flow

Microfluidic devices are often made of polymeric materials, while biological tissue is soft. Both can deform significantly due to flow, even at vanishing Reynolds number. Due to deformation, the pressure gradient in the flow-wise direction is not constant. Deformation leads to significant enhancement of flow due to the change in cross-sectional area; hence, a nonlinear flow rate--pressure drop relation (unlike the Hagen--Poiseuille law for a rigid tube). We study steady flow in a thin, slender deformable microtube. To capture non-Newtonian effects of biofluids, we employ the power-law model. The structural problem is reduced to transverse loading of linearly-elastic cylindrical shells. A perturbative approach (in the slenderness parameter) yields analytical solutions for the flow and deformation. Then, we obtain a ``generalized Hagen--Poiseuille law'' for soft microtubes. We benchmark the analytical results against fully-3D two-way coupled numerical simulations of flow and deformation performed using the commercial CAE software ANSYS. The simulations establish the range of validity of the theory, showing excellent agreement.