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Lamb-type solution and properties of unsteady Stokes equations

Itzhak Fouxon1,2,3,*, Alexander Leshansky1,†, Boris Rubinstein4,‡, and Yizhar Or2,§

  • 1Department of Chemical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel
  • 2Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 3200003, Israel
  • 3Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea
  • 4Stowers Institute for Medical Research, 1000 East 50th Street, Kansas City, Missouri 64110, USA

  • *itzhak8@gmail.com
  • †lisha@technion.ac.il
  • ‡bru@stowers.org
  • §izi@me.technion.ac.il
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Phys. Rev. Fluids 7, 094103 – Published 21 September, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.094103

Abstract

In this paper, we derive the general solution of the unsteady Stokes equations for an unbounded fluid in spherical polar coordinates, in both time and frequency domains. The solution is an expansion in vector spherical harmonics and given as a sum of a particular solution, proportional to a pressure gradient exhibiting power-law spatial dependence, and a solution of the vector Helmholtz equation decaying exponentially in the far field, the decomposition originally introduced by Lamb. The proposed solution representation resembles the classical Lamb decomposition of the solution of steady Stokes equations, with the series coefficients being projections of the radial component, divergence, and curl of the boundary velocity on scalar spherical harmonics. It can be applied to construct the transient exterior flow induced by an arbitrary velocity distribution at the spherical boundary, such as arising in the squirmer model of a microswimmer. The approach can also be used to construct solutions for transient flows driven by initial conditions, unbounded flows driven by volumetric forces, and disturbance to the unsteady flow due to a stationary spherical particle and to study particle-particle and particle-wall interactions in oscillatory flows. The long-time behavior of the solution is controlled by the flow component corresponding to an average (or collective) motion of the boundary. This conclusion is illustrated by the study of decay of a transversal wave in the presence of a fixed sphere. We further show that the general representation reduces to the well-known solutions for unsteady flow around a sphere undergoing oscillatory rigid-body motion. The proposed solution representation provides an explicit form of the velocity potential far from an oscillating body and high- and low-frequency expansions. The leading-order high-frequency expansion yields the well-known ideal (inviscid) flow approximation, and the leading-order low-frequency expansion yields the steady Stokes equations. We derive the higher-order corrections to these approximations and discuss the d'Alembert paradox. Continuation of the general solution to imaginary frequency yields the general solution of the Brinkman equations describing viscous flow in a porous medium.

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Phys. Rev. Fluids 7, 094103– Published 21 September, 2022
  • Received 21 December 2021
  • Revised 30 May 2022
  • Accepted 29 August 2022
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