### Abstract

The immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid. The immersed elastic material is represented in the fluid equations by a system or field of applied forces. The particular case of Stokes flow with applied forces on a periodic domain involves two related mathematical complications. One of these is that an arbitrary constant vector may be added to the fluid velocity, and the other is the constraint that the integral of the applied force must be zero. Typically, forces defined on a freely floating elastic immersed boundary or body satisfy this constraint, but there are many important classes of forces that do not. For example, the so-called tether forces that are used to prescribe the simulated configuration of an immersed boundary, possibly in a time-dependent manner, typically do not sum to zero. Another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system. We present a method for periodic Stokes flow that when used with tether points, admits the use of all forces irrespective of their integral over the domain. A byproduct of this method is that the additive constant velocity associated with periodic Stokes flow is uniquely determined. Indeed, the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied.

Original language | English (US) |
---|---|

Pages (from-to) | 3404-3416 |

Number of pages | 13 |

Journal | SIAM Journal on Scientific Computing |

Volume | 31 |

Issue number | 5 |

DOIs | |

State | Published - 2009 |

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### Keywords

- Immersed boundary method
- Numerical methods
- Stokes equations

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*31*(5), 3404-3416. https://doi.org/10.1137/080720217

**Tether force constraints in Stokes flow by the immersed boundary method on a periodic domain.** / Teran, Joseph M.; Peskin, Charles S.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 31, no. 5, pp. 3404-3416. https://doi.org/10.1137/080720217

}

TY - JOUR

T1 - Tether force constraints in Stokes flow by the immersed boundary method on a periodic domain

AU - Teran, Joseph M.

AU - Peskin, Charles S.

PY - 2009

Y1 - 2009

N2 - The immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid. The immersed elastic material is represented in the fluid equations by a system or field of applied forces. The particular case of Stokes flow with applied forces on a periodic domain involves two related mathematical complications. One of these is that an arbitrary constant vector may be added to the fluid velocity, and the other is the constraint that the integral of the applied force must be zero. Typically, forces defined on a freely floating elastic immersed boundary or body satisfy this constraint, but there are many important classes of forces that do not. For example, the so-called tether forces that are used to prescribe the simulated configuration of an immersed boundary, possibly in a time-dependent manner, typically do not sum to zero. Another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system. We present a method for periodic Stokes flow that when used with tether points, admits the use of all forces irrespective of their integral over the domain. A byproduct of this method is that the additive constant velocity associated with periodic Stokes flow is uniquely determined. Indeed, the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied.

AB - The immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid. The immersed elastic material is represented in the fluid equations by a system or field of applied forces. The particular case of Stokes flow with applied forces on a periodic domain involves two related mathematical complications. One of these is that an arbitrary constant vector may be added to the fluid velocity, and the other is the constraint that the integral of the applied force must be zero. Typically, forces defined on a freely floating elastic immersed boundary or body satisfy this constraint, but there are many important classes of forces that do not. For example, the so-called tether forces that are used to prescribe the simulated configuration of an immersed boundary, possibly in a time-dependent manner, typically do not sum to zero. Another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system. We present a method for periodic Stokes flow that when used with tether points, admits the use of all forces irrespective of their integral over the domain. A byproduct of this method is that the additive constant velocity associated with periodic Stokes flow is uniquely determined. Indeed, the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied.

KW - Immersed boundary method

KW - Numerical methods

KW - Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=79957462278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957462278&partnerID=8YFLogxK

U2 - 10.1137/080720217

DO - 10.1137/080720217

M3 - Article

VL - 31

SP - 3404

EP - 3416

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -