Simultaneous solution of the Navier‐Stokes and elastic membrane equations by a finite element method

Abstract

AbstractFluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self‐excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non‐linear equations governing even the most simplified two‐dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier‐Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton‐Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time‐dependent problem might be approached.