Steady finite-Reynolds-number flows in three-dimensional collapsible tubes

Abstract

A fully coupled finite-element method is used to investigate the steady flow of a viscous fluid through a thin-walled elastic tube mounted between two rigid tubes. The steady three-dimensional Navier–Stokes equations are solved simultaneously with the equations of geometrically nonlinear Kirchhoff–Love shell theory. If the transmural (internal minus external) pressure acting on the tube is sufficiently negative then the tube buckles non-axisymmetrically and the subsequent large deformations lead to a strong interaction between the fluid and solid mechanics. The main effect of fluid inertia on the macroscopic behaviour of the system is due to the Bernoulli effect, which induces an additional local pressure drop when the tube buckles and its cross-sectional area is reduced. Thus, the tube collapses more strongly than it would in the absence of fluid inertia. Typical tube shapes and flow fields are presented. In strongly collapsed tubes, at finite values of the Reynolds number, two ’jets‘ develop downstream of the region of strongest collapse and persist for considerable axial distances. For sufficiently high values of the Reynolds number, these jets impact upon the sidewalls and spread azimuthally. The consequent azimuthal transport of momentum dramatically changes the axial velocity profiles, which become approximately $\uTheta$-shaped when the flow enters the rigid downstream pipe. Further convection of momentum causes the development of a ring-shaped velocity profile before the ultimate return to a parabolic profile far downstream.