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We investigate 3-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibers. The Brinkman equation is used to model the average fluid flow where a resistance term is inversely proportional to the permeability

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This draft was prepared using the LaTeX style ﬁle belonging to the Journal of Fluid Mechanics
1
A 3-dimensional model of ﬂagellar swimmingin a Brinkman ﬂuid
NguyenHo Ho
1
, Karin Leiderman
2
, and Sarah Olson
3
†
,
1
Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way,Cincinnati, Ohio 45221, USA.
2
Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street, Golden,Colorado 80401 USA.
3
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road,Worcester, Massachusetts 01609, USA.(Received xx; revised xx; accepted xx)
We investigate 3-dimensional ﬂagellar swimming in a ﬂuid with a sparse network of stationary obstacles or ﬁbers. The Brinkman equation is used to model the average ﬂuidﬂow where a resistance term is inversely proportional to the permeability and representsthe eﬀect due to the presence of ﬁbers. The ﬂagellum is represented as a Kirchhoﬀ rod thatcan exhibit propagating planar or spiral bending. To solve for the local ﬂuid velocity andangular velocity, we use the method of regularized Brinkmanlets and extend it to the casefor a Kirchhoﬀ rod that is discretized as point forces and torques along the centerline. Thenew numerical method is validated by comparing to asymptotic swimming speeds derivedfor an inﬁnite-length cylinder propagating lateral or spiral waves in a Brinkman ﬂuid.Similar to the asymptotics, we observe that in the case of small amplitude, swimmingspeed is enhanced relative to the Stokes case as the resistance is increased. For largeramplitude bending, the simulations show a non-monotonic change in swimming speed asthe resistance is varied, with a peak when the resistance parameter is near one. This isdue to the emergent amplitude and wavelengths; as the resistance is increased (or as thenumber of stationary ﬁbers is increased), the emergent amplitude of the swimmer has atendency to decrease.
Key words:
Authors should not enter keywords on the manuscript, as these mustbe chosen by the author during the online submission process and will then be addedduring the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfm-keywords.pdf for the full list)
1. Introduction
Microorganisms such as spermatozoa make forward progression by propagating bend-ing along their ﬂagellum. The emergent ﬂagellar curvature and beat frequency depends onthe ﬂuid properties as well as the chemical concentrations within the ﬂagellum (Gaﬀney
et al.
2011; Miki 2007; Smith
et al.
2009; Suarez & Pacey 2006; Woolley & Vernon 2001).The ﬂuid environment experienced by mammalian sperm includes complex geometriesand background ﬂows due to interactions with other sperm, cilia, and walls (Fauci& Dillon 2006; Ho & Suarez 2001; Suarez & Pacey 2006; Suarez 2010). As a sperm
†
Email address for correspondence: sdolson@wpi.edu
2
N. Ho, K. Leiderman, and S. Olson
progresses towards the egg, the ﬂuid could contain diﬀering amounts of uterine cells,sulfomucins, protein networks and other macromolecules, especially at diﬀerent times inthe menstrual cycle (Katz & Berger 1980; Katz
et al.
1989; Mattner 1968; Suarez 2010).Many experiments have examined sperm motility in gels such as methylcellulose (MC)or polyacrylamide (PA), which may be more representative of the
in vivo
environmentthan sperm motility in a culture medium. In experiments, the emergent beat frequencyand wavelength varied with viscosity in MC gels (Smith
et al.
2009) and the swimmingspeed of mouse sperm decreased in both MC and PA gels (relative to the culture medium)(Suarez & Dai 1992). This motivates the development of a 3-dimensional (3D) frameworkto study emergent properties (e.g., waveform or swimming speed) of a sperm whencoupled with this protein network.Previous computational studies of ﬁnite-length swimmers in a Newtonian ﬂuid withpreferred bending kinematics have identiﬁed that there is a non-monotonic relationshipbetween emergent swimming speeds and bending amplitude (Dillon
et al.
2006; Elgeti
et al.
2010; Fauci & McDonald 1995; Olson & Fauci 2015; Yang
et al.
2008). Onthe other hand, for inﬁnite-length ﬂagella with prescribed bending, the asymptoticswimming speeds are an increasing function with respect to the bending amplitude(Taylor 1951, 1952). Since most gels and biological ﬂuids contain proteins and othermacromolecules, recent studies have focused on swimmers in complex ﬂuids. In ﬂuids thatexhibit contributions from viscous and elastic eﬀects, the swimming speeds of inﬁnite-length ﬂagella with prescribed kinematics decrease in comparison to the Newtonian case(Fu
et al.
2009; Lauga 2007). In contrast, swimming speeds increase for certain parameterchoices for a ﬁnite-length swimmer in a nonlinear viscoelastic ﬂuid and a Carreau ﬂuid(Newtonian ﬂuid with shear-dependent viscosity) (Montenegro-Johnson
et al.
2012; Teran
et al.
2010; Thomases & Guy 2014). An enhancement in swimming speed relative to aNewtonian ﬂuid has also been observed in a model of a two-phase ﬂuid for a gel whenthe elastic network is stationary (Fu
et al.
2010).The average ﬂuid ﬂow through an array of sparse, spherical particles can be modeledvia the Brinkman equation (Auriault 2009; Brinkman 1947; Durlofsky & Brady 1987;Howells 1974; Spielman & Goren 1968). A ﬂow dependent resistance term with resistanceparameter accounts for the presence of the particles in the ﬂuid. In the case of an inﬁnite-length ﬂagellum in a Brinkman ﬂuid with prescribed bending, in both 2D and 3D, theswimming speed increases as the resistance increases (Ho
et al.
2016; Leshansky 2009).This increase in swimming speed is an enhancement relative to the Newtonian case; thepresence of particles or ﬁbers actually aids in forward progression. In contrast, for a ﬁnite-length swimmer with preferred planar bending, there was a non-monotonic relationshipbetween swimming speed and the resistance parameter (Cortez
et al.
2010; Olson &Leiderman 2015; Leiderman & Olson 2016).To explore emergent properties of ﬂagellar swimming in a ﬂuid with a sparse andstationary protein network, we use the incompressible Brinkman equations to govern theﬂuid motion (Brinkman 1947; Howells 1974):
−∇
p
+
µ∆
u
−
µα
2
u
+
f
b
=
0
,
(1.1)
∇·
u
= 0
.
(1.2)Here,
p
is the average ﬂuid pressure (force per area),
u
is the average ﬂuid velocity (lengthper time),
f
b
represents the body force (force per unit volume) applied on the ﬂuid bythe immersed structure,
µ
is the viscosity (force time per area), and
α
= 1
/
√
γ
is theresistance parameter (inverse length), which is inversely proportional to the square rootof the permeability
γ
. One can think of the Brinkman equation as the addition of a lower-
Regularized Kirchhoﬀ Rod model
3
0 0.05 0.1 0.15 0.200.0050.010.0150.020.025
a
f
/
√
γ
ϕ
Volume FractionCollagen GelCervical Mucus
Figure 1.
The volume fraction
ϕ
solved for in Eq. (1.3) is plotted as a function of
a
f
/
√
γ
where
a
f
is ﬁber radius and
γ
is permeability. The green and red markers represent the volume fraction for a collagen gel (
γ
= 8
.
6
) and cervical mucus (
γ
= 0
.
0085
), respectively.
order resistance term to the Stokes equations for low Reynolds number ﬂow (since thelength scale of these swimmers is small, they live in a viscosity dominated environmentwhere inertia can be neglected). As
α
→
0 (no resistance), the incompressible Stokesequations are recovered and as
α
→∞
(high resistance), the term
µ∆
u
becomes negligibleand Eq. (1.1) behaves like Darcy’s law. An important characteristic of a Brinkman ﬂuidis the Brinkman screening length,
√
γ
, which marks the approximate length over whicha disturbance to the velocity would decay.To consider a microorganism swimming in this environment, we assume that theobstacles are at a low enough volume fraction that the distance between ﬁbers islarger than the diameter of the microorganism. In the case of randomly oriented ﬁbers,Spielman & Goren (1968) have derived a relationship between the volume fraction
ϕ
, thepermeability
γ
, and the radius of the ﬁber
a
f
as
a
2
f
γ
= 4
ϕ
13
a
2
f
γ
+ 56
a
f
√
γ K
1
(
a
f
/
√
γ
)
K
0
(
a
f
/
√
γ
)
.
(1.3)Here,
K
0
(
·
) and
K
1
(
·
) are the zeroth and ﬁrst order modiﬁed Bessel functions of thesecond kind. Figure 1 shows the volume fraction
ϕ
as a function of the ratio
a
f
/
√
γ
,indicating a relevant biological range for
ϕ
. For reference, we also indicate the volumefraction of cervical mucus and a collagen gel based on experimental values for
ϕ
and
a
f
(Saltzman
et al.
1994). Fibers in vaginal ﬂuid have been reported to be 1–20 micronsapart, whereas the ﬂagellar diameter for many species of sperm is in the range of 0.2–0.6 microns (Bahr & Zeitler 1964; Bloodgood 1990; Hafez & Kenemans 2012; Rutllant
et al.
2001, 2005). The interﬁber spacing (or distance between the ﬁbers) can also beapproximated as (Leshansky 2009)
D
≈
2
a
f
12
3
πϕ
−
1
,
(1.4)based on a known volume fraction
ϕ
and ﬁber radius
a
f
. In the case where the ratio
D/a
f
≫
1, there are little or no interactions between a stationary network of ﬁbers andthe swimmers. Thus, it is assumed that the ﬁbers do not impart any additional stressonto the ﬁlament.A fundamental solution of the incompressible Brinkman equations given in (1.1)–(1.2),is well-known (Durlofsky & Brady 1987; Pozrikidis 1989). It represents the velocity due toa concentrated external force acting on the ﬂuid at a single point. However, the velocity
4
N. Ho, K. Leiderman, and S. Olson
becomes singular when the point forces are concentrated along curves in 3D. To eliminatethese singular solutions in Stokes ﬂow, the method of regularized Stokeslets (Cortez 2001;Cortez
et al.
2005) is employed while the method of regularized Brinkmanlets (Cortez
et al.
2010) is introduced to deal with these situations in a Brinkman ﬂuid.In order to model emergent waveforms of swimmers that can be either planar orhelical, as observed in experiments (Woolley & Vernon 2001), we use a Kirchhoﬀ rodmodel to represent the elastic ﬂagellum. The propagation of bending along the ﬁlamentis given as a time-dependent preferred curvature function, where deviations from thispreferred conﬁguration lead to forces and moments (body forces). An immersed boundaryformulation of the Kirchhoﬀ rod model was ﬁrst developed by Lim
et al.
(2008) and hasbeen extended to a regularized Stokes formulation (Lee
et al.
2014; Olson
et al.
2013;Olson 2014). Here, we extend the regularized method to now study ﬂagellar swimmingin a ﬂuid governed by the Brinkman equation where we account for the local linear andangular velocity due to point forces and torques along the length of the ﬂagellum. Twoapproaches for the numerical method are derived in Section 3.3, where in the limit asresistance
α
→
0, the solutions approach those of Stokes equations (detailed in AppendixB). We are able to match emergent swimming speeds with asymptotic swimming speedsfor both the planar and spiral bending cases where swimming speed increases withamplitude and beat frequency for a ﬁxed resistance. The numerical results show that forthe planar and helical bending cases, there is an optimal range of resistance
α
, around
α
= 1, that allows the swimmer to achieve a large bending amplitude while receivingan extra boost in propulsion from the presence of the ﬁber network. In addition, as
α
increases, the emergent waveform of the swimmer has a decreased amplitude (relative tothe preferred amplitude), resulting in a decreased swimming speed.
2. Kirchhoﬀ Rod Model
With the Kirchhoﬀ Rod (KR) formulation, a ﬂagellum is described by a 3D space curve
X
(
s
) for 0
< s < L
, where
s
is a Lagrangian parameter initialized to be the arclength and
L
is the length of the unstressed rod. Here, we assume the rod length is much greater thanthe radius and that the rod is isotropic and homogeneous. The associated orthonormaltriads
{
D
1
(
s
)
,
D
2
(
s
)
,
D
3
(
s
)
}
follow the right-handed rule. The triad
D
3
(
s
) is eﬀectivelyin the direction of the tangent vector while
D
1
(
s
) and
D
2
(
s
) are rotations of the normaland binormal vectors, respectively, coinciding with the principle axes of the rod crosssection.Figure 2 shows the centerline of a ﬂagellum discretized as a helix using the centerlineapproximation with the associated orthonormal triads plotted at one point on the spacecurve. In the standard KR model,
D
3
(
s
) is enforced to be the tangent vector and the rodis inextensible. We employ an unconstrained version whereby an elastic energy penaltyis used to numerically maintain the inextensibility of the rod and keep
D
3
(
s
) as a unittangent vector along the rod (Lim
et al.
2008; Olson
et al.
2013).The derivation of the internal force and torque in terms of the associated orthonormaltriads has previously been described in detail (Lim
et al.
2008; Olson
et al.
2013). Here,we summarize the main equations of the KR model which are utilized later. The balanceof force and torque on a cross section of the rod are0 =
f
+
∂
F
∂s ,
(2.1)0 =
m
+
∂
M
∂s
+
∂
X
∂s
×
F
,
(2.2)
Regularized Kirchhoﬀ Rod model
5
2 3
Figure 2.
The Kirchhoﬀ rod representation of a ﬂagellum is discretized as a helix using acenterline approximation with orthonormal triads
{
D
1
,
D
2
,
D
3
}
plotted at one point on thespace curve.
where
f
(units of force per length) and
m
(units of force) are part of the externalforces applied on the rod. Whereas,
F
and
M
are the internal components of the forcetransmitted across each section of the rod and are given in terms of
X
(
s
) and its triads.The components of
F
and
M
can be expanded in the basis of the triads:
F
=
3
i
=1
F
i
D
i
,
M
=
3
i
=1
M
i
D
i
,
(2.3)for
i
= 1
,
2
,
3 where both
F
and
D
i
are 3 by 1 vectors at a given
s
along the rod centerline.The constitutive relations for the unconstrained KR are (Olson
et al.
2013; Lim
et al.
2008)
M
1
=
a
1
∂
D
2
∂s
·
D
3
−
Ω
1
, M
2
=
a
2
∂
D
3
∂s
·
D
1
−
Ω
2
, M
3
=
a
3
∂
D
1
∂s
·
D
2
−
Ω
3
,
(2.4)
F
1
=
b
1
∂
X
∂s
·
D
1
, F
2
=
b
2
∂
X
∂s
·
D
2
, F
3
=
b
3
∂
X
∂s
·
D
3
−
1
,
(2.5)where the material properties of the rod are characterized through the parameters
a
i
and
b
i
for
i
= 1
,
2
,
3. The bending moduli are
a
1
,
a
2
and
a
3
is the twisting modulus while
b
1
,b
2
arethe shear moduli, and
b
3
is the extensional modulus. The strain-twist vector is represented by
{
Ω
1
,Ω
2
,Ω
3
}
where
Ω
3
is the intrinsic twist and
Ω
1
,Ω
2
are the geodesic and normal curvatures,respectively, associated with the intrinsic curvature
Ω
through the equation
Ω
=
Ω
21
+
Ω
22
.This vector determines the preferred conﬁguration of the rod where internal force and torqueare generated by diﬀerences from the actual and preferred conﬁguration. The preferred strainand twist of the rod can be varied (in time
t
and with respect to arc length parameter
s
), topropagate planar or spiral bending that are representative of sperm ﬂagellar beatforms observedin experiments (Smith
et al.
2009; Woolley & Vernon 2001).
3. Method of Regularized Brinkmanlets for the Kirchhoﬀ Rod
Given a 3D elastic structure immersed in a Brinkman ﬂuid, the equations of motion include theexternal forces and torques of the structure on the ﬂuid. The solutions can be calculated exactlyin terms of fundamental solutions due to the linearity of the Brinkman equation. However,singular solutions are obtained when evaluating the ﬂow at the location of a point force ortorque on the centerline of the structure. Eliminating these singularities requires a regularizationmethod and we utilize the Method of Regularized Brinkmanlets (MRB) (Cortez
et al.
2010).The idea is to use a smooth approximation to the singular point force or point torque. Thesmooth approximation is called a “blob” or regularization function,
φ
ε
(
r
), and is a radiallysymmetric function whose width is determined by the regularization parameter
ε
≪
1. In thelimit as
ε
→
0, the singular solutions are recovered.Since we want to capture both the bending and twisting motions of the rod in a 3D inﬁniteﬂuid, the expression of the force density
f
b
at a point
x
in the ﬂuid is a contribution of both
f
and
m
, given as
f
b
(
x
,t
) =
Γ
−
f
(
s,t
) + 12
∇×
(
−
m
(
s,t
))
φ
ε
(
r
)
ds
(3.1)

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