Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube

We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666.