Parametric Resonance in Immersed Elastic Boundaries

Abstract

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].

Computing spectra of linear operators using the Floquet–Fourier–Hill method

“…The method is somewhat popular in the area of fluid stability, e.g. [8,26,40,16], but its connection to Hill's work and its generality are unacknowledged. The FFHM was rediscovered in the Russian literature by Pavlenko and Petviashvili [43], and successively used by others [23].…”

Convergence proof of the velocity field for a stokes flow immersed boundary method

“…It has since proved to be a generally useful computational framework for solving problems in fluid-structure interaction. The method has found many applications, including blood flow in the heart [17], vibrations of the cochlear basilar membrane [1,10], blood clotting [7,39,42], aquatic locomotion [4,6,8,14], insect flight [18,19], flow with suspended particles [9,33], and other physical problems [5,15,16,29]. We refer to [25] for a more extensive list of applications.…”

The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow

“…(13), is stiff. Tu and Peskin [34], Stockie and Wetton [32], and Cortez et al [4] have conducted stability analysis for the immersed boundary method. To improve on the numerical stability, an implicit treatment was implemented in the immersed interface method [18,16], which is…”