### 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 language | English (US) |
---|---|

Pages (from-to) | 2141-2159 |

Number of pages | 19 |

Journal | Journal of Computational Physics |

Volume | 230 |

Issue number | 5 |

DOIs | |

State | Published - Mar 1 2011 |

### Fingerprint

### 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

*Journal of Computational Physics*,

*230*(5), 2141-2159. https://doi.org/10.1016/j.jcp.2010.12.010

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

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 230, no. 5, pp. 2141-2159. https://doi.org/10.1016/j.jcp.2010.12.010

}

TY - JOUR

T1 - Applying a second-kind boundary integral equation for surface tractions in Stokes flow

AU - Keaveny, Eric E.

AU - Shelley, Michael

PY - 2011/3/1

Y1 - 2011/3/1

N2 - 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.

AB - 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.

KW - Boundary integral equations

KW - Low Reynolds number locomotion

KW - Second-kind integral equations

KW - Slender-body theory

KW - Stokes flow

UR - http://www.scopus.com/inward/record.url?scp=78651444426&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651444426&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2010.12.010

DO - 10.1016/j.jcp.2010.12.010

M3 - Article

AN - SCOPUS:78651444426

VL - 230

SP - 2141

EP - 2159

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 5

ER -