Boundary Layers for a 'Tube Law' at the Clamped End of an Elastic-Walled Tube

Fluid flows through elastic-walled tubes are common in biological and industrial systems, and have received much attention through experimental, numerical and analytical studies. Experiments show that steady flow along an elastic-walled tube can become unstable to large-amplitude oscillations involving both the tube wall and the fluid.

To study these oscillations theoretically, we need to model the fluid mechanics of the interior fluid and the solid mechanics of the elastic wall. In previous work by the author, the latter has been accomplished using a rationally-derived `tube law' valid for long-wavelength small amplitude deformations. The tube law links the pressure difference $P$ across the wall to the change $\alpha$ in cross-sectional area of the tube at each axial position $z$. Axial tension effects are included by a term involving $\partial^2\alpha/\partial z^2$.

However, this second-order tube law does not process enough axial derivatives to satisfy the full set of boundary conditions where an elastic section of tube is clamped to a rigid section. Boundary layers are needed, in which some of the neglected axial derivatives are reintroduced. Asymptotic analysis reveals a rich variety of different bending and shearing tube-end boundary layers in different regimes, some of which can have significant effects on the interior solutions.