Low Reynolds

Particle Dynamics Basics Motivation Jonas Einarsson Presentation by Göteborg, December 2010 Present work 1: "Low Reynolds" 2: "Particle Dynamics" Forces

Hydrodynamic

Electromagnetic

Interactions

Brownian Experiments Future prospects Fluid dynamics Particle dynamics Our analysis so far Oladiran, Hanstorp (2009) Stokes equation Navier-Stokes equations Low Reynolds number Present case: Pressure driven

channel flow No-slip boundary Interesting flows Rheological properties Many-particle dynamics Flexible polymers - No inertia approximation Force + Torque balance Present case: Ellipsoid in channel flow Equation of motion - Hydrodynamic force View particle as disturbance in flow: Solve Stokes equations with "moving boundary" Have ambient flow Velocity Rotation Linearizing the ambient flow In general - difficult! Introduce the symmetric, and antisymmetric

parts of the flow Jacobian A Now the boundary condition is Studied extensively in

1960's by H. Brenner,

leads to general formulation for forces: Interesting: For orthotropic particles,

then

and "Material tensors" Flow in z-direction Solved by separation of variables Jeffery (1922 (!)) For axisymmetric ellipsoid Aspect ratio

parameter Solutions are closed orbits Solution of the

Jeffery equations New coordinates Comparing to

experiments In progress... Insight! Normalization Normalization For time-independent flow, explicit solution Consider non-normalized motion in shear Project back onto sphere then With eigensystem of B-matrix Write time evolution of an initial value Neutral

direction Flow direction Principal shear direction This is an ellipsoid! orbit constant orbit speed aspect ratio Axes determined by flow This is Jeffery's 1922 result! Expressing the normalized solution in

the usual spherical coordinates: But our equation of motion is now

Linear

Constant Extracting angles from the movie Observation:

Particle follows Jeffery orbit

But switches orbit! Noise Asymmetry Rotational diffusion Disturbances in flow Elliptic coordinate system Further intuition?

Can it simplify calculations? Lateral displacement Particle imperfections

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# Dynamics of particles in viscous flows

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