### Abstract

In this paper, we investigate the stability of a fluid-structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter. This model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear, or heart muscle fibers immersed in blood. Problems such as this, in which the system is subjected to internal forcing through a parameter, can give rise to "parametric resonance." We formulate the equations of motion in two dimensions using the immersed boundary formulation. Assuming small amplitude motions, we can apply Floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space. The eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable. We present direct numerical simulations of the fluid-structure interaction problem (using the immersed boundary method) that verify the existence of the parametric resonances suggested by our analysis.

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

Pages (from-to) | 494-520 |

Number of pages | 27 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 65 |

Issue number | 2 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Fluid-structure interaction
- Immersed boundary
- Parametric resonance

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*65*(2), 494-520. https://doi.org/10.1137/S003613990342534X

**Parametric resonance in immersed elastic boundaries.** / Cortez, Ricardo; Peskin, Charles; Stockie, John M.; Varela, Douglas.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 65, no. 2, pp. 494-520. https://doi.org/10.1137/S003613990342534X

}

TY - JOUR

T1 - Parametric resonance in immersed elastic boundaries

AU - Cortez, Ricardo

AU - Peskin, Charles

AU - Stockie, John M.

AU - Varela, Douglas

PY - 2005

Y1 - 2005

N2 - In this paper, we investigate the stability of a fluid-structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter. This model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear, or heart muscle fibers immersed in blood. Problems such as this, in which the system is subjected to internal forcing through a parameter, can give rise to "parametric resonance." We formulate the equations of motion in two dimensions using the immersed boundary formulation. Assuming small amplitude motions, we can apply Floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space. The eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable. We present direct numerical simulations of the fluid-structure interaction problem (using the immersed boundary method) that verify the existence of the parametric resonances suggested by our analysis.

AB - In this paper, we investigate the stability of a fluid-structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter. This model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear, or heart muscle fibers immersed in blood. Problems such as this, in which the system is subjected to internal forcing through a parameter, can give rise to "parametric resonance." We formulate the equations of motion in two dimensions using the immersed boundary formulation. Assuming small amplitude motions, we can apply Floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space. The eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable. We present direct numerical simulations of the fluid-structure interaction problem (using the immersed boundary method) that verify the existence of the parametric resonances suggested by our analysis.

KW - Fluid-structure interaction

KW - Immersed boundary

KW - Parametric resonance

UR - http://www.scopus.com/inward/record.url?scp=19944394245&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19944394245&partnerID=8YFLogxK

U2 - 10.1137/S003613990342534X

DO - 10.1137/S003613990342534X

M3 - Article

VL - 65

SP - 494

EP - 520

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -