Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem

This is a study of the fluid-structure interaction between the stationary Stokes flow of an incompressible, Newtonian viscous fluid filling a three-dimensional, linearly elastic, pre-stressed hollow tube. The main motivation comes from the study of blood flow in human arteries. Most literature on fluid-structure interaction in blood flow utilizes thin structure models (shell or membrane) to describe the behavior of arterial walls. However, arterial walls are thick, three-dimensional structures with the wall thickness comparable to the vessel inner radius. In addition, arteries in vivo exhibit residual stress: when cut along the radius, arteries spring open releasing the residual strain. This work focuses on the implications of the two phenomena on the solution of the fluid-structure interaction problem, in the parameter regime corresponding to the blood flow in medium-to-large human arteries. In particular, it is assumed that the aspect ratio of the cylindrical structure $\epsilon = R/L$ is small. Using asymptotic analysis and ideas from homogenization theory for porous media flows, an effective, closed model is obtained in the limit as both the thickness of the vessel wall and the radius of the cylinder approach zero, simultaneously. The effective model satisfies the original three-dimensional, axially symmetric problem to the $\epsilon^2$-accuracy. Several novel properties of the solution are obtained using this approach. A modification of the well-known "Law of Laplace'' is derived, holding for thick elastic cylinders. A calculation of the effective longitudinal displacement is obtained, showing that the leading-order longitudinal displacement is completely determined by the external loading. Finally, it is shown that the residual stress influences the solution only at the $\epsilon$-order. More precisely, it is shown that the only place where the residual stress influences the solution of this fluid-structure interaction problem is in the calculation of the $\epsilon$-correction of the longitudinal displacement.