PARAMETRIC RESONANCE IN
IMMERSED ELASTIC BOUNDARIES
∗
RICARDO CORTEZ
†
, CHARLES S. PESKIN
‡
,
JOHN M. STOCKIE
§
, AND DOUGLAS VARELA
¶
SIAM J. A
PPL. MATH.
c
2004 Society for Industrial and Applied Mathematics
Vol. 65, No. 2, pp. 494–520
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.
Key words. immersed boundary, fluid-structure interaction, parametric resonance
AMS subject classifications. 35B34, 35B35, 74F10, 76D05
DOI. 10.1137/S003613990342534X
1. Introduction. Fluid-structure interaction problems abound in industrial ap-
plications 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 im-
mersed 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 non-
biological problems involving flow past solid cylinders [12], flapping filaments [29],
parachutes [10], and suspensions of flexible particles [25].
∗
Received by the editors March 31, 2003; accepted for publication (in revised form) February 4,
2004; published electronically December 16, 2004. This work was partially supported by NSF grants
DMS-0094179 (Cortez) and DMS-9980069 (Peskin), NSERC grant RGP-238776-01 (Stockie), and
the Center for Computational Science at Tulane and Xavier Universities grant DE-FG02-01ER63119
(Cortez).
http://www.siam.org/journals/siap/65-2/42534.html
†
Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118
(cortez@math.tulane.edu).
‡
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York,
NY 10012 (peskin@cims.nyu.edu).
§
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A
1S6, Canada (stockie@math.sfu.ca).
¶
Department of Engineering and Applied Science, California Institute of Technology, Pasadena,
CA 91125 (vladimir@its.caltech.edu).
494
PARAMETRIC RESONANCE IN IMMERSED ELASTIC BOUNDARIES 495
While a significant body of literature has developed around numerical simula-
tions using the IB method (as well as related methods such as the blob projection
method [4]), comparatively little work has been done on analyzing the stability be-
havior of solutions to the underlying equations of motion. Beyer and LeVeque [3]
were the first to perform an analysis of the IB formulation in one spatial dimension.
Stockie and Wetton [26] investigated the linear stability of a flat fiber in two dimen-
sions 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.
In all of this previous work, the immersed boundary was assumed to be passive,
moving along with the flow and generating forces only in response to elastic defor-
mation. However, there are many problems in which the immersed boundary is an
active material, generating time-dependent forces to drive its motion. Examples in-
clude beating heart muscle fibers and flagellated cells, both of which apply periodic
forces to direct their motion and that of the surrounding fluid.
It is then natural to ask whether periodic forcing at various frequencies will give
rise to resonance. In the case of immersed boundaries, the applied force is inter-
nal, in the sense that the system is forced through a periodic variation in a system
parameter (namely, the elastic properties of the solid material) rather than through
an external body force. In contrast to externally forced systems, these internally or
parametrically forced systems are known to exhibit parametric resonance, in which
the response for linear systems (i.e., assuming small amplitude motions) can become
unbounded even in the presence of viscous damping. Once nonlinearities are taken
into account, however, the response remains bounded, but the system can still exhibit
large-amplitude motion when forced at certain resonant frequencies. Analyses of para-
metric resonance have appeared for various fluid flows involving surface or interfacial
waves wherein the system is forced through gravitational modulation [11, 14, 17] or
time-dependent heating [15]. However, to our knowledge there has been no analysis
performed on fluid-structure problems that captures the two-way interaction between
a fluid and an immersed, flexible structure. In a recent study by Wang [28], an anal-
ysis is performed of flutter and buckling instabilities in a paper making headbox, in
which a long, flexible vane is driven to vibrate under the influence of periodic forcing
from turbulent fluid jets. However, the author assumes a given fluid velocity and that
the solid structure has no influence on the fluid motion.
In this paper, we perform an analysis of parametric resonance in a two-dimensional,
circular, elastic membrane immersed in Navier–Stokes flow, which is driven by a time-
periodic variation in its elastic stiffness parameter. The elastic membrane is under
tension due to the incompressible fluid it encloses, and the changes in the elastic stiff-
ness parameter are equivalent to changes in the tension of the membrane that mimic
the contraction of biological fibers. The novelty of this work is that it includes the
full, two-way interaction between the fluid and the membrane. In sections 2 and 3, we
present the linearized equations of motion for the IB formulation and reduce them to
a suitable nondimensional form. In sections 4, 5, and 6, we perform a Floquet analysis
of the problem and derive a system of equations that relate the amplitude, frequency,
and wavenumber of the internal forcing to the physical parameters in the problem.
The natural modes for the unforced system are derived in section 7 and compared
to the asymptotic expressions for the natural modes in a flat fiber that were derived
in [26]. We also compare the natural modes to those observed in computations using
the blob projection method, an alternate method for solving the IB problem that
496
CORTEZ, PESKIN, STOCKIE, AND VARELA
affords higher spatial accuracy. In section 8 we consider the periodically forced prob-
lem and determine the conditions under which parametric resonance can occur. The
results are verified using numerical simulations with the IB method.
2. Problem definition. Consider an elastic fiber immersed in a two-dimensional,
viscous, incompressible fluid of infinite extent. The fiber is a closed loop with rest-
ing length of zero, so that in the absence of fluid the fiber would shrink to a point.
However, because an incompressible fluid occupies the region inside the fiber, the
equilibrium state is a circle of constant radius R, in which the pressure drop across
the fiber is balanced by the tension force. Our aim is to investigate a parametric
excitation of the fiber, in which the elastic stiffness is varied periodically in time.
The equations of motion for the fluid and immersed boundary can be written in
terms of the vorticity ξ(x,t) and stream function ψ(x,t) as [19, p. 1]:
ρ (ξ
t
+ u ·∇ξ)=µ∇
2
ξ +
ˆ
z ·∇×f ,(2.1a)
u = −
ˆ
z ×∇ψ,(2.1b)
−∇
2
ψ = ξ,(2.1c)
f(x,t)=
2π
0
(KX
s
)
s
δ(x − X) ds,(2.1d)
X
t
= u(X,t),(2.1e)
where x =(x, y) and
ˆ
z =(0, 0, 1). The fluid velocity u(x,t) and fiber force f(x,t) are
functions of position and time, whereas the fiber position X(s, t)=(X
r
(s, t),X
θ
(s, t))
is a function of time and the Lagrangian fiber parameter 0 ≤ s ≤ 2π (which is the
same as the angle θ when the fiber’s reference state is a circle). We assume in this
paper that the stiffness is a periodic function of time,
K = K(s, t)=K
c
(1+2τ sin ω
o
t),(2.2)
which has no dependence on the spatial variable s.
We point out that s is a Lagrangian parameter and is not necessarily proportional
to arclength since the fiber is allowed to stretch and shrink tangentially. The force
density (KX
s
)
s
thus includes components normal and tangential to the fiber. If we
identify the tension T (s)=KX
s
and the unit tangent vector ˆτ(s)=X
s
/X
s
,
we may write the force density term as
(KX
s
)
s
=(T (s)ˆτ(s))
s
= T
s
(s)ˆτ(s)+T (s)ˆτ
s
(s),
which shows explicitly that the force resolves itself into normal and tangential com-
ponents. We also remark that spatial dependence in K can be easily incorporated
into our analysis provided that the spatial variations are small (that is, on the order
of the parameter introduced in the following section).
Our motivation comes from active muscle fibers that may contract and relax to
generate fluid motion. For this reason we will focus on nonnegative values of the
stiffness and we will be concerned mostly with the range 0 ≤ τ ≤
1
2
in (2.2). However,
our analysis does not require this restriction and the results apply to values of τ>1/2
as well. This is important because there are instances in which a negative stiffness
plays an important role as is the case of hair bundles in the bullfrog inner ear [9].
PARAMETRIC RESONANCE IN IMMERSED ELASTIC BOUNDARIES 497
3. Nondimensionalization. We first simplify the problem by scaling the vari-
ables and forming dimensionless groups. For now, variables with a tilde will be di-
mensionless. Let
x = R ˜x,
u = U ˜u (U will be determined later),
t = R/U
˜
t,
K = K
c
˜
K,
which imply that
f(x,t)=
K
c
R
˜
f(˜x,
˜
t)=
2π
0
(
˜
K
˜
X
s
)
s
δ(˜x −
˜
X) ds,
ξ =
U
R
˜
ξ,
ψ = UR
˜
ψ.
With these scalings the vorticity equation becomes
˜
ξ
˜
t
+
˜
u ·
˜
∇
˜
ξ =
µ
ρUR
˜
∇
2
˜
ξ +
K
c
ρU
2
(
ˆ
z ·
˜
∇×
˜
f).
In view of this and (2.2), we now define
U = Rω
o
,ν=
µ
ρUR
=
µ
ρR
2
ω
o
,κ=
K
c
ρU
2
=
K
c
ρR
2
ω
2
o
,(3.1)
so that the unperturbed fiber configuration is the unit circle (X =
ˆ
r(s)) and we can
write the dimensionless equations (omitting the tildes) as
ξ
t
+ u ·∇ξ = ν∇
2
ξ + κ(
ˆ
z ·∇×f ),(3.2a)
u = −
ˆ
z ×∇ψ,(3.2b)
−∇
2
ψ = ξ,(3.2c)
f(x,t)=
2π
0
(KX
s
)
s
δ(x − X) ds,(3.2d)
X
t
= u(X,t),(3.2e)
with
K =(1+2τ sin t),(3.3a)
X =
ˆ
r(s)+X
(1)
1
(s, t)+···.(3.3b)
Our aim is now to solve (3.2a)–(3.2e) with the only two remaining parameters, ν and
κ, defined in (3.1). We point out that ν is the reciprocal of the Reynolds number and
κ is the square of the natural frequency divided by the driving frequency.
4. Small-amplitude approximation. For the linear analysis, assume the fiber
force can be written as
f = f
(0)
(x,t)+f
(1)
(x,t)+···,
498
CORTEZ, PESKIN, STOCKIE, AND VARELA
so that the term (
ˆ
z ·∇×f ) in (3.2a) can be expanded as
(
ˆ
z ·∇×f )=(
ˆ
z ·∇×f
(0)
)+(
ˆ
z ·∇×f
(1)
)+···.
Claim 1. Given the expansions in (3.3a) and (3.3b),
(
ˆ
z ·∇×f
(0)
)=0 and
(
ˆ
z ·∇×f
(1)
)=K(t)
X
θ
ss
+ X
r
s
δ(r − 1)
r
r
− K(t)
X
r
sss
− X
θ
ss
δ(r − 1)
r
.
The proof is found in Appendix B.
Since the leading-order term in the curl of the force is zero, the corresponding
flow is simply the trivial solution ξ
(0)
= ψ
(0)
= 0. Therefore, we look for a series
solution expanded in terms of powers of a small parameter :
ξ = ξ
(1)
(x,t)+···,
ψ = ψ
(1)
(x,t)+···.
In the remainder of this work, we assume is small and consider the O() equa-
tions
ξ
t
= ν∇
2
ξ + κK(t)
X
θ
ss
+ X
r
s
δ(r − 1)
r
r
− κK(t)
X
r
sss
− X
θ
ss
δ(r − 1)
r
,(4.1a)
∇
2
ψ = −ξ,(4.1b)
X
r
t
= ψ
θ
r=1
,(4.1c)
X
θ
t
= −ψ
r
r=1
,(4.1d)
where we have omitted the superscript
(1)
on most variables because we will be working
solely with these quantities from now on.
It is worthwhile mentioning that in the above expansions, the influence of the
time-dependent stiffness coefficient is felt solely at first order in , and so we need
only consider the forcing terms up to O(). The derivation can be easily extended to
include the case where the stiffness is spatially dependent, for which the s-variation
appears as an order perturbation of a time-dependent stiffness,
K = K(s, t)=K
(0)
(t)+K
(1)
(s, t)+O(
2
).
When a Floquet-type expansion is assumed for K
(1)
(s, t) (refer to the next section),
correction terms appear in the O() equations, and these corrections complicate the
analysis somewhat by coupling together the various spatial modes. We do not consider
a spatially dependent stiffness in this paper.
It is quite striking that a spatially homogeneous K(t) can have any interesting
effects in this problem. This is because such K(t) would have no effect at all on the
equilibrium state in which the fiber is a circle. Starting with that equilibrium state
as initial data and imposing time-periodic K(t) will produce only an oscillation of
the pressure and no motion of the fiber at all. But if instead we start with an initial
condition that is arbitrarily close but not exactly equal to a circular equilibrium
state, we can nevertheless build up a large-amplitude and spatially inhomogeneous
oscillation of the fiber through the application of spatially homogeneous but time-
periodic K(t) at the right driving frequency. This is actually typical of parametric
