1.  Amazing simulations 2:
Lisa J. Fauci Tulane University, Math Dept. New Orleans, Louisiana, USA

2.  Capturing  the  fluid  dynamics  of  phytoplankton:  active  and  passive   structures.

3. 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

4. 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
Pfiesteria piscicida
Delaware Biotechnology Institute
Goal: Use CFD to model flows around or generated by phytoplankton.

5. How do spines alter the rotational period of diatoms in shear flow? .
Thalassiosira nordenskioeldii
Copyright of the Biodiversity Institute of Ontario
Thalassiosira punctigera

6. 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 create a discretization of the structure.

7. Ellipsoid in Shear Flow
The period from the simulation is about 1.55 s, compared with the theoretical period
T = 1.59 s.
Variation of φ with time (where φ = rotation angle relative to the initial position).

8. Diatom in Shear Flow The cell body has
Our Model
(Re = )
Thalassiosira punctigera
Shear Rate = 10.0 s-1
The cell body has
diameter 4.25x10-3 cm and height 1.77x10-3 cm.
The spine length is
0.49x10-3 cm.
T = s

9. Are full 3D CFD calculations necessary?

10. 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.

11. 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 xk=0:

12. 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 xk=0:
For the choice:
the resulting velocity field is:

13. Note:
u(x) is defined everywhere
u(x) is an exact solution to the Stokes equations, and is incompressible

14. If regularized forces are exerted at “N” points, the velocities
at these points can be computed by superposition of
Regularized Stokeslets
u = A g or
Here A is a 3n by 3n matrix that depends upon the
geometry.

15. Cell body: right handed helix
Anterior helix: left handed
Posterior hook
S. Goldstein
Univ. Minnesota

16. How many rotations required to swim one body length?
Leptonema illini
Body length: microns
Body radius: 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’

18. Linear relationship:
F = TU + P W
L = PT 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
We systematically assemble these resistance matrices
by applying velocity (or angular velocity) in
each direction, and integrating the resulting forces and torques…

19. 0= TU + P W Steady state swimming
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

21. a superhelix in a Stokes fluid?
What are dynamics of
a superhelix in a Stokes fluid?
Cell body: right handed helix
Anterior helix: left handed
Posterior hook

22. Experimental Setup

23. Counter-clockwise
Clockwise

24. Jung, Mareck, Fauci
Shelley, Phys. Fluids,
2007
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…

25. Motivation: Dinoflagellates
Pfiesteria piscicida
Delaware Biotechnology Institute
Imbrickle.blogspot.com

26. Dinoflagellates have 2 flagella – transverse and longitudinal
Tom Fenchel, How Dinoflagellates Swim, Protist , Vol. 152: , 2001

27. “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

28. swimming path. The longitudinal flagellum works as a rudder, giving a
“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
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.
“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…”

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

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

31. Classical fluid dynamics examined swimming of helices with a straight axis…
Cortez, Cowen, Dillon, Fauci
Comp. Sci. Engr., 2004

32. 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)

33. 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 Nguyen, Ortiz, Cortez, Fauci

36. Wave moving counterclockwise
viewed from above
Material points of
ring progress in opposite
direction .

37. Tangential and longitudinal velocity as a function of amplitude R
For small R, tangential velocity is O(R2)..
For all R, longitudinal velocity is O(R2)..
Lighthill’s slender body theory gives an excellent approximation for the longitudinal velocity.

38. Change number of pitches on ring

39. Change number of pitches on ring

40. What if there was a cell body?

41. 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

42. Colliding rings?

43. 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.