Applying a second-kind boundary integral equation for surface tractions in Stokes flow

Eric E. Keaveny, Michael Shelley

Research output: Contribution to journalArticle

Abstract

A second-kind integral equation for the tractions on a rigid body moving in a Stokesian fluid is established using the Lorentz reciprocal theorem and an integral equation for a double-layer density. A second-order collocation method based on the trapezoidal rule is applied to the integral equation after appropriate singularity reduction. For translating prolate spheroids with various aspect ratios, the scheme is used to explore the effects of the choice of completion flow on the error in the numerical solution, as well as the condition number of the discretized integral operator. The approach is applied to obtain the velocity and viscous dissipation of rotating helices of circular cross-section. These results are compared with both local and non-local slender-body theories. Motivated by the design of artificial micro-swimmers, similar computations are performed on previously unstudied helices of non-circular cross-section to determine the dependence of the velocity and propulsive efficiency on the cross-section aspect ratio and orientation. Overall, we find that this formulation provides a stable numerical approach with which to solve the flow problem while simultaneously obtaining the surface tractions and that the appropriate choice of completion flow provides both increased accuracy and efficiency. Additionally, this approach naturally avails itself to known fast summation techniques and higher-order quadrature schemes.

Original languageEnglish (US)
Pages (from-to)2141-2159
Number of pages19
JournalJournal of Computational Physics
Volume230
Issue number5
DOIs
StatePublished - Mar 1 2011

Fingerprint

Stokes flow
Boundary integral equations
traction
Integral equations
integral equations
helices
aspect ratio
Aspect ratio
cross sections
propulsive efficiency
reciprocal theorems
slender bodies
prolate spheroids
translating
collocation
rigid structures
quadratures
dissipation
formulations
operators

Keywords

  • Boundary integral equations
  • Low Reynolds number locomotion
  • Second-kind integral equations
  • Slender-body theory
  • Stokes flow

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy (miscellaneous)

Cite this

Applying a second-kind boundary integral equation for surface tractions in Stokes flow. / Keaveny, Eric E.; Shelley, Michael.

In: Journal of Computational Physics, Vol. 230, No. 5, 01.03.2011, p. 2141-2159.

Research output: Contribution to journalArticle

Keaveny, Eric E. ; Shelley, Michael. / Applying a second-kind boundary integral equation for surface tractions in Stokes flow. In: Journal of Computational Physics. 2011 ; Vol. 230, No. 5. pp. 2141-2159.
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