Locomotion in ViscoElastic Fluids

1
Locomotion in Visco-Elastic Fluids
Teran, Fauci, Shelley 07
Undulatory swimming at low Re in Newtonian fluids
is fairly well understood
Resistive force thry Taylor, Lighthill, Purcell,
and many, many others, e.g. Hosoi et al on
optimization of stroke for speed and
eff. Standard tools Singularity and boundary
integral methods for Stokes Eqs.
Undulatory slender-body swimmer
C. elegans nematode swimming in water
Biofluids typically non-Newtonian and
viscoelastic. Fauci Dillon Ann. Rev. Fluids 2006
Right panels stroke pattern of bull sperm in
cervical mucus
Left panel sinusoidal stroke pattern of bull
sperm in Newtonian fluid
Ho and Suarez, 2007
2

• Stokes-Oldroyd-B
• Standard viscoelastic flow models balance of
  solvent and polymer stresses.
• Derives from a microscopic thry of dilute
  suspension of polymer coils acting as Hookean
  springs
• Model of a Boger elastic fluid (normal
  stresses, no shear thinning),
• but can excessively strain harden in
  extensional flow

momentum and mass balance
transport and damping of polymer stress
with
3
The Stokes Eqs. are reversible
contours vorticity
4
The Stokes-OB Eqs are not
contours trace of elastic stress
5
Nonlinearity of transport of polymer stress
allows for symmetry-breaking
Non-Newtonian flow rectifier at low Re
pressure gradient
Flow in positive direction twice flow in negative
direction
- pressure gradient
Groisman and Quake, 2004
6

• Kinematics from energetics move a geometric
  deformation in a sheet via curvature-based
  energy, couple sheet to Stokes-OB Eqs. via
  Immersed Boundary Method Peskin McQueen 89

contours of trace(S)
7
Wi5, beta1/10.
8
The classical problem Swimming of a period sheet
O(a2) scaling of swimming speed with
sinusoidal profile amplitude a Stokes -- G.I.
Taylor, 1951 Stokes-OB -- E. Lauga, 2006 small
amplitude analysis Same scaling Stokes
wins.
swimming speed
stokes
stokes-OB
time
9
Modified Swimming Kinematicsa Stokes-OB winner
Stokes
Stokes Oldroyd-B
10
Modified Swimming Kinematics
Forward
Motion Modified Kinematics
Stokes Stokes OB
peak forward velocity
recoil phase