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Oscillatory flows in compliant conduits at arbitrary Womersley number

Shrihari D. Pande1, Xiaojia Wang1,*, and Ivan C. Christov1,2,†

  • 1School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
  • 2Department of Computer Science, University of Nicosia, CY-2417 Nicosia, Cyprus

  • *Present address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA.
  • †Corresponding author: christov@purdue.edu; http://tmnt-lab.org
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Phys. Rev. Fluids 8, 124102 – Published 20 December, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.124102

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 two-dimensional 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 or 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 a priori assumptions on the magnitude of the oscillation frequency (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.

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Phys. Rev. Fluids 8, 124102– Published 20 December, 2023
  • Received 5 April 2023
  • Accepted 13 November 2023
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