Oscillatory flows in compliant conduits at arbitrary Womersley number

Abstract

We develop a theory of fluid--structure interaction (FSI) between an oscillatory Newtonian fluid flow and a compliant conduit. We consider the canonical geometries of a 2D channel with a deformable top wall and an axisymmetric deformable tube. Focusing on the hydrodynamics, we employ a linear relationship between wall displacement and hydrodynamic pressure, which has been shown to be suitable for a leading-order-in-slenderness theory. The slenderness assumption also allows the use of lubrication theory, and the flow rate is related to the pressure gradient (and the tube/wall deformation) via the classical solutions for oscillatory flow in a channel and in a tube (attributed to Womersley). Then, by two-way coupling the oscillatory flow and the wall deformation via the continuity equation, a one-dimensional nonlinear partial differential equation (PDE) governing the instantaneous pressure distribution along the conduit is obtained, without \textit{a priori} assumptions on the magnitude of the oscillation frequency (\textit{i.e.}, at arbitrary Womersley number). We find that the cycle-averaged pressure (for harmonic pressure-controlled conditions) deviates from the expected steady pressure distribution, suggesting the presence of a streaming flow. An analytical perturbative solution for a weakly deformable conduit is obtained to rationalize how FSI induces such streaming. In the case of a compliant tube, the results obtained from the proposed reduced-order PDE and its perturbative solutions are validated against three-dimensional, two-way-coupled direct numerical simulations. We find good agreement between theory and simulations for a range of dimensionless parameters characterizing the oscillatory flow and the FSI, demonstrating the validity of the proposed theory of oscillatory flows in compliant conduits at arbitrary Womersley number.

18 pages, 4 figures; v2 makes some revisions to the presentation, introduces a Strouhal number, which is then fixed, and compares the model against new 3D simulations; v3 corrects minor typos, to appear in Phys. Rev. Fluids

References in corpus (4)

• Dynamics of pulsatile flows through elastic microtubes
• Reduced models of unidirectional flows in compliant rectangular ducts at finite Reynolds number
• Reduced modelling and global instability of finite-Reynolds-number flow in compliant rectangular channels
• Flow rate--pressure drop relations for new configurations of slender compliant tubes arising in microfluidics experiments

Cited by in corpus (4)

• Elasto-inertial rectification of oscillatory flow in an elastic tube
• Oscillatory flows in three-dimensional deformable microchannels
• Theory and simulation of elastoinertial rectification of oscillatory flows in two-dimensional deformable rectangular channels
• Reciprocal theorem for calculating the flow rate of oscillatory channel flows