Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries

We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time-dependent pressure drop between the inlet and the outlet boundary. The pressure drop is assumed to be small, thereby introducing creeping flow in the tube. Stokes equations for incompressible viscous fluid are used to model the flow, and the equations of a curved, linearly elastic membrane are used to model the wall. Due to the creeping flow and to small displacements, the interface between the fluid and the lateral wall is linearized and supposed to be the initial configuration of the membrane. We study the dynamics of this coupled fluid-structure system in the limit when the ratio between the characteristic width and the characteristic length tends to zero. Using the asymptotic techniques typically used for the study of shells and plates, we obtain a set of Biot-type visco-elastic equations for the effective pressure and the effective displacements. The approximation is rigorously justified through a weak convergence result and through the error estimates for the solution of the effective equations modified by an outlet boundary layer. Applications of the model problem include blood flow in small arteries. We recover the well- known law of Laplace and obtain new improved models that hold in cases when the shear modulus of the vessel wall is not negligible and the Poisson ratio is arbitrary.