Self-Consistent Effective Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries

Abstract

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 head difference. The Navier--Stokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a curved, linearly elastic membrane are used to model the wall. Employing the asymptotic techniques typically used in thin domains, we derive a set of effective equations that hold in medium-to-large compliant vessels for laminar flow regimes. The main novelty is the derivation of the effective equations that do not assume any ad hoc closure, typically assumed in the derivation of one-dimensional models. Using ideas from homogenization theory for porous media flows, we obtain a closed system of effective equations that are of Biot type with memory. Memory accounts for the wave-like phenomena in the problem. Although the equations are two-dimensional, their simple structure enables a design of a numerical algorithm that has the complexity of a one-dimensional solver. Our numerical simulations show that our model captures two-dimensional effects that cannot be captured using standard one-dimensional methods.