### Abstract

We present a new method for the evolution of inextensible vesicles immersed in a Stokesian fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the non-local hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method (FMM) to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme.

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

Pages (from-to) | 2334-2353 |

Number of pages | 20 |

Journal | Journal of Computational Physics |

Volume | 228 |

Issue number | 7 |

DOIs | |

State | Published - Apr 20 2009 |

### Fingerprint

### Keywords

- Fast summation methods
- Fluid membranes
- Inextensible vesicles
- Integral equations
- Moving boundaries
- Numerical methods
- Particulate flows

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

*Journal of Computational Physics*,

*228*(7), 2334-2353. https://doi.org/10.1016/j.jcp.2008.11.036

**A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D.** / Veerapaneni, Shravan K.; Gueyffier, Denis; Zorin, Denis; Biros, George.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 228, no. 7, pp. 2334-2353. https://doi.org/10.1016/j.jcp.2008.11.036

}

TY - JOUR

T1 - A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D

AU - Veerapaneni, Shravan K.

AU - Gueyffier, Denis

AU - Zorin, Denis

AU - Biros, George

PY - 2009/4/20

Y1 - 2009/4/20

N2 - We present a new method for the evolution of inextensible vesicles immersed in a Stokesian fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the non-local hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method (FMM) to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme.

AB - We present a new method for the evolution of inextensible vesicles immersed in a Stokesian fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the non-local hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method (FMM) to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme.

KW - Fast summation methods

KW - Fluid membranes

KW - Inextensible vesicles

KW - Integral equations

KW - Moving boundaries

KW - Numerical methods

KW - Particulate flows

UR - http://www.scopus.com/inward/record.url?scp=60049101665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60049101665&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2008.11.036

DO - 10.1016/j.jcp.2008.11.036

M3 - Article

VL - 228

SP - 2334

EP - 2353

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 7

ER -