Lisa J. Fauci
Tulane University, Math Dept.
New Orleans, Louisiana, USA
Amazing simulations 2:
Capturing
the
fluid
dynamics
of
phytopl
ankton:
active
and
passive
structures.
Collaborators:
Hoa Nguyen
Tulane University
Lee Karp

Boss
University of Maine
Pete Jumars
University of Maine
Ricardo Ortiz
University of North Carolina
Ricardo Cortez
Tulane University
Diatoms, dinoflagellates
Plankton are the foundation of the oceanic food chain and are responsible
for much of the oxygen present in the Earth’s atmosphere.
Thalassiosira nordenskioeldii
Copyright of the Biodiversity
Institute of Ontario
Thalassiosira punctigera
image by Ashley Young,
University of Maine
4
Pfiesteria piscicida
Delaware Biotechnology
Institute
Goal: Use CFD to model flows around or generated by phytoplankton.
•
How do spines alter the
rotational period of diatoms in
shear flow? .
Thalassiosira nordenskioeldii
Copyright of the Biodiversity Institute of Ontario
Thalassiosira punctigera
5
Discretization:
Spherical Centroidal Voronoi Tessellation
The triangulation on the unit sphere is the dual mesh of the Spherical Centroidal Voronoi Tessellation (SCVT), as coded
by Lili Ju [6]. We map this triangulation to a surface (such as an ellipsoid, a flat disc or a plankter’s cell body) to creat
e a
discretization of the structure.
6
Ellipsoid in Shear Flow
Variation of
φ
with time (where
φ
= rotation angle relative to the initial
position).
The period from the simulation is about
1.55 s
, compared with the
theoretical
period
T = 1.59 s
.
7
Diatom in Shear Flow
The cell body has
diameter
4.25x10

3
cm
and
height
1.77x10

3
cm
.
The spine length is
0.49x10

3
cm.
Thalassiosira punctigera
Our Model
(Re = 0.0181)
Shear Rate = 10.0 s

1
T = 1.323 s
8
Are full 3D CFD calculations necessary?
•
Motion of spined cells can be predicted
from simple theory by examining the
smallest spheroid that inscribes the cell.
•
Spines thus can achieve motion
associated with shape change that greatly
alters rotational frequency with
substantially less material than would be
needed to fill the inscribing spheroid.
Hydrodynamics of spines: a different spin.
Limnology & Oceanograpy :Fluids, Nguyen,
Karp

Boss, Jumars, Fauci 2011, Vol. 1
.
Grid
–
free numerical method for
zero Reynolds number
Steady Stokes equations:
Method of regularized Stokeslets
(R. Cortez, SIAM SISC 2001;
Cortez, Fauci,Medovikov, Phys. Fluids, 2004)
Forces are spread over a small ball

in the case x
k
=0:
Grid
–
free numerical method for
zero Reynolds number
Steady Stokes equations:
Method of regularized Stokeslets
(R. Cortez, SIAM SISC 2001;
Cortez, Fauci,Medovikov, Phys. Fluids, 2004)
Forces are spread over a small ball

in the case x
k
=0:
For the choice
:
the resulting velocity field is:
Note:
u(x) is defined everywhere
u(x) is an exact solution to the Stokes equations, and is incompressible
If regularized forces are exerted at “N” points, the velocities
at these points can be computed by superposition of
Regularized Stokeslets
u
= A
g
Here
A
is a 3n by 3n matrix that depends upon the
geometry
.
or
S. Goldstein
Univ. Minnesota
Cell body: right handed helix
Anterior helix: left handed
Posterior hook
Leptonema illini
Body length: 11.93 microns
Body radius: .0735 microns
Helix radius: .088 microns
How many rotations required
to swim one body length?
We assume steady swimming
–
rigid body
and use computed ‘resistance matrices’
F =
T
U +
P
W
L =
P
T
U +
R
W
Where F is the total hydrodynamic force,
L is the total hydrodynamic torque,
T, P, R
are resistance matrices acting on
velocity U and angular velocity
W
Linear relationship:
D
)
(
ds
)
(
x
x
f
F
D
)
(
ds
)
(
x
x
f
x
L
We systematically assemble these resistance matrices
by applying velocity (or angular velocity) in
each direction, and integrating the resulting forces and torques…
0=
T
U +
P
W
Where F is the total hydrodynamic force,
L is the total hydrodynamic torque,
T, P, R
are resistance matrices acting on
velocity U and angular velocity
W
Steady state swimming
D
)
(
ds
)
(
x
x
f
F
D
)
(
ds
)
(
x
x
f
x
L
Cell body: right handed helix
Anterior helix: left handed
Posterior hook
What are dynamics of
a superhelix in a Stokes
fluid?
Experimental Setup
Counter

clockwise
Clockwise
Circles
–
Reg. Stokeslets
Squares
–
Resistive force theory
Triangles
–
Experiments
Transition from clockwise to counter

clockwise rotations is observed in
experiment and Reg. Stokeslet
calculations
–
but missed with
resistive force theory…
Jung, Mareck, Fauci
Shelley,
Phys. Fluids
,
2007
Motivation: Dinoflagellates
Pfiesteria piscicida
Delaware Biotechnology Institute
Imbrickle.blogspot.com
Dinoflagellates have 2 flagella
–
transverse and
longitudinal
Tom Fenchel,
How Dinoflagellates Swim
, Protist , Vol. 152:329

338, 2001
“The transversal flagellum causes the cell to rotate around its length axis. The
trailing flagellum is responsible for the translation of the cell; ” Fenchel 2001
“The transversal flagellum causes the cell to rotate around its length axis. The
trailing flagellum is responsible for the translation of the cell; ” Fenchel 2001
“The transverse flagellum works as a propelling device that provides the
main driving force or thrust to move the cell along the longitudinal axis of its
swimming path. The longitudinal flagellum works as a rudder, giving a
lateral force to the cell…”
Miyasaka, K. Nanba, K. Furuya, Y. Nimura, A. Azuma,
Functional roles of the
transverse and longitudinal flagella in the swimming motility of
Prorecentrum minimum (Dinophyceae
), J. Exp. Biol., 2004.
So, which is it?
Does the transverse, helically

beating
flagellum cause rotational or longitudinal
motion, or both?
So, which is it?
Does the transverse, helically

beating
flagellum cause rotational or longitudinal
motion, or both?
Classical fluid dynamics examined swimming of
helices with a straight axis…
Cortez, Cowen, Dillon, Fauci
Comp. Sci. Engr., 2004
X(s,t) = [r
–
R sin (2 π s / λ
–
ω t)] cos (s/r)
Y(s,t) = [r
–
R sin (2 π s / λ
–
ω t)] sin (s/r)
Z(s,t) = R cos (2 π s / λ
–
ω t)
Dual approach
Solve the kinematic problem using
Lighthill’s slender body theory and
regularized Stokeslets.
Solve the full Stokes equations coupled to a
ring that is actuated by elastic links whose
rest lengths change dynamically over the
wave period.
The action of waving cylindrical rings in a viscous fluid
.
J. Fluid Mech. 2010 Nguyen, Ortiz, Cortez, Fauci
Wave moving counterclockwise
viewed from above
Material points of
ring progress in opposite
direction .
Tangential and longitudinal velocity as a
function of amplitude R
Lighthill’s slender body theory
gives an excellent approximation
for the longitudinal velocity.
For small R, tangential velocity is O(R
2
).
.
For all R, longitudinal velocity is O(R
2
).
.
Change number of pitches on ring
Change number of pitches on ring
What if there was a cell body?
Interactions between a helical ring and spherical cell
using
IBAMR
top view
side view
Sphere:
VB =
2.71x10

3
Helical ring:
VB =
1.04x10

3
Colliding rings?
Conclusions
Undulating helical rings exhibit both
rotational and translational velocity in a
Stokes fluid.
These helical rings provide an interesting
kinematic problem to validate the method
of regularized Stokeslets used for
complex fluid

structure interaction problems.
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