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

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Capturing the fluid dynamics of phytopl ankton: active and passive structures.

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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

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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.

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

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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. 6

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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

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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

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Are full 3D CFD calculations necessary?

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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.

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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:

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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:

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

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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

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

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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’

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

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0= TU + P 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 Steady state swimming

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

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Experimental Setup

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Counter-clockwise Clockwise

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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

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

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Dinoflagellates have 2 flagella – transverse and longitudinal Tom Fenchel, How Dinoflagellates Swim, Protist, Vol. 152:329-338, 2001

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“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

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“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.

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

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

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

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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)

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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

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

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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 )..

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Change number of pitches on ring

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Toroidal swimmer Leshansky and Kenneth, Phys. Fluids 2008 – Surface tank treading: Propulsion of Purcell’s toroidal Swimmer.

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Speed ~ log s0 / s0

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No wall. Wall. Regularized Stokeslets with images

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What if there was a cell body?

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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

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Colliding rings?

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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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