The Dynamics of an Elastic Membrane Using the Impulse Method

Cortez, Ricardo; Varela, Douglas

doi:10.1006/jcph.1997.5842

Cited by 13 publications

(13 citation statements)

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“…In the limit as ν → 0, these become α = 0 and β = ω N √ κ, which is precisely the linear dispersion relation found by other methods in [5] for a closed membrane in Euler flow.…”

Section: Collapse Of the Curves The Condition D(ωsupporting

confidence: 77%

“…Stockie and Wetton [26] investigated the linear stability of a flat fiber in two dimensions which is subjected to a small sinusoidal perturbation, and presented asymptotic results on the frequency and rate of decay for the resulting oscillations. Cortez and Varela [5] performed a nonlinear analysis for a perturbed circular elastic membrane immersed in an inviscid fluid.…”

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confidence: 99%

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Parametric Resonance in Immersed Elastic Boundaries

Cortez¹,

Peskin²,

Varela³

2004

SIAM J. Appl. Math.

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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. 1. Introduction. Fluid-structure interaction problems abound in industrial applications as well as in many natural phenomena. Owing to the potentially complex, time-varying geometry and the nonlinear interactions that arise between fluid and solid, such problems represent a major challenge to both mathematical modelers and computational scientists.This paper deals with a model for fluid-structure interaction known as the immersed boundary (IB) formulation [20], which has proven particularly effective for dealing with complex problems in biological fluid mechanics. In this formulation, the immersed structure is treated as an elastic surface or interwoven mesh of elastic fibers that exert a singular force on a surrounding viscous fluid, while also moving with the velocity of adjacent fluid particles. The associated immersed boundary method has been used to solve a wide range of problems such as blood flow in the heart and arteries [2,21], swimming microorganisms [8], biofilms [7], and insect flight [16]. More recently, the IB method has also been extended to a much wider range of nonbiological problems involving flow past solid cylinders [12], flapping filaments [29], parachutes [10], and suspensions of flexible particles [25].

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“…In the limit as ν → 0, these become α = 0 and β = ω N √ κ, which is precisely the linear dispersion relation found by other methods in [5] for a closed membrane in Euler flow.…”

Section: Collapse Of the Curves The Condition D(ωsupporting

confidence: 77%

mentioning

confidence: 99%

Parametric Resonance in Immersed Elastic Boundaries

Cortez¹,

Peskin²,

Varela³

2004

SIAM J. Appl. Math.

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“…These comparisons are shown in Figure 8.2. We see that the computed Fourier modes have good agreement with both A(t) and B(t),-somewhat better than the results shown by Cortez and Varela [5]. They use an impulse method with a projected force PF analogous to that used in the blob projection method.…”

Section: Numerical Resultsmentioning

confidence: 56%

“…Example 1. We first consider an example used by Cortez and Varela [5], who present an asymptotic solution that can be used for comparison purposes. A closed elastic membrane in an inviscid fluid is perturbed slightly from a circular steady state and allowed to oscillate.…”

Section: Numerical Resultsmentioning

confidence: 99%

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An Immersed Interface Method for Incompressible Navier--Stokes Equations

Lee¹,

LeVeque²

2003

SIAM J. Sci. Comput.

290

279

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Abstract. The method developed in this paper is motivated by Peskin's immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the two-dimensional incompressible Navier-Stokes equations using a high-resolution finite-volume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included. The IB method represents the flexible membrane using Lagrangian markers at which force-densities are computed, based on the position of the membrane, and spread to a Cartesian fluid grid by a set of discrete delta functions. The support of the delta function is comparable to the mesh width. The Navier-Stokes equations with this forcing are then advanced one time step on the Cartesian grid using a projection method. The velocities are first updated, ignoring the pressure gradient term in the Navier-Stokes equations in a transport step, and then a projection step is applied to compute the pressure and impose the incompressibility constraint on the velocity field. The resulting velocities are interpolated back to the Lagrangian markers using the same set of discrete delta functions. The markers are moved based on these velocities and the process is repeated in the next time step. This algorithm is written out in more detail in section 3.

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