This communication is devoted to the presentation of our recent results regarding the asymptotic analysis of a viscous flow in a tube with elastic walls. This study can be applied, for example, to the blood flow in an artery. With this aim, we consider the dynamic problem of the incompressible flow of a viscous fluid through a curved pipe with a smooth central curve. Our analysis leads to the obtention of an one dimensional model via singular perturbation of the Navier-Stokes system as \(\varepsilon\), a non dimensional parameter related to the radius of cross-section of the tube, tends to zero. We allow the radius depend on tangential direction and time, so a coupling with an elastic or viscoelastic law on the wall of the pipe is possible. To perform the asymptotic analysis, we take a change of variable to a reference domain where we assume the existence of asymptotic expansions on \(\varepsilon\) for both velocity and pressure which, upon substitution on Navier-Stokes equations, leads to the characterization of various terms of the expansion. This allows us to obtain an approximation of the solution of the Navier-Stokes equations.
Panasenko, G.P., Pileckas, K.: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary-layer-in-time. Nonlinear Anal. 122, 125–168 (2015)MathSciNetMATHGoogle Scholar
Panasenko, G.P., Pileckas, K.: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II. General case. Nonlinear Anal. 125, 582–607 (2015)MathSciNetCrossRefMATHGoogle Scholar
Panasenko, G.P., Stavre, R.: Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall. J. Math. Pures Appl. 85, 558–579 (2006)MathSciNetCrossRefMATHGoogle Scholar