Lamb's solution and the stress moments for a sphere in Stokes flow

Gedi Zhou, Andrea Prosperetti*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb's general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Faxén-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the PHYSALIS method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.

Original languageEnglish
Pages (from-to)270-282
Number of pages13
JournalEuropean Journal of Mechanics, B/Fluids
Volume79
Early online date27 Sep 2019
DOIs
Publication statusPublished - 1 Jan 2020

Fingerprint

Stokes flow
Stokes Flow
Moment
moments
particulates
Reynolds number
Rotation Group
Non-uniformity
Stokes Equations
Shear Flow
Recurrence relation
shear flow
General Solution
nonuniformity
inertia
Inertia
Viscosity
viscosity
Higher Order
Fluid

Keywords

  • UT-Hybrid-D
  • Lamb solution
  • Non-uniform particulate flow
  • Particle-resolved simulations
  • Stokes multipole moments
  • Faxen theorems

Cite this

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title = "Lamb's solution and the stress moments for a sphere in Stokes flow",
abstract = "This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb's general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Fax{\'e}n-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the PHYSALIS method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.",
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Lamb's solution and the stress moments for a sphere in Stokes flow. / Zhou, Gedi; Prosperetti, Andrea.

In: European Journal of Mechanics, B/Fluids, Vol. 79, 01.01.2020, p. 270-282.

Research output: Contribution to journalArticleAcademicpeer-review

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AB - This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb's general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Faxén-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the PHYSALIS method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.

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