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Lecture 11
Life in the low-Reynolds number world
Tsvi Tlusty, [email protected]
Sources:
Purcell – Life at low Reynolds number (Am. J. Phys 1977)
Nelson – Biological Physics (CH.5)
Outline
I. Friction in fluids.
II. Low Reynolds-number world.
III. Biological applications.
• Viscous friction dominates mechanics in the nanoworld.
• Friction is dissipative: converts ordered motion into thermal energy.
• Implications of symmetry.
• Biological Q: Why don't bacteria swim like fish?
• Physical idea: motion in the nanoworld have different symmetry than
motion in the macroworld.
I. Friction in fluids
Small particles can remain in suspension indefinitely
Suspension of protein
z
c z = c0
e
−
k
mnet g
zB rT
• What happens after a long time?
• Example: Myoglobin:
water water
net
Gravity:
Bouyancy force:
Net force: g Vg (Archimedes' principle)
mg Vg
m g Vg
m
water m g
mg net
0 (Smoluchowski's eq. for eqilibrium)
( ) exp (Einstein's relation / )B
B
dU dPj P D
dx dx
m gzP x k T D
k T
4 23
20
net
23
* 20
net
Myoglobin mass: 1.7 10 Da = 2.7 10 kg
net mass: / 4 0.7 10 kg
4.2 10scale height 60 m
0.7 10 10
B
m
m m
k Tz
m g
• Density in test-tube is ~ constant.
r
ω
m
2
centrifugal net f m r
Centrifuges can achieve a much higher g
• Centripetal acceleration is by frictional drag:
• Balanced by diffusion current
2
net ( ) ( )v
dUj P f P r m P r r
dr
( ) D
dP rj D
dr
2
net
2 2
net
0 ( ) 0
( ) exp
v D
B
dPj j j m P r r D
dr
m rP r
k T
• How long does it take to reach equilibrium?
Sedimentation time scale depends on solvent viscosity
• Sedimentation velocity (depends g not intrinsic property)
netdrift net
m gv m g
• Sedimentation time scale:
drift netnet
v mm
g
• Unit is Svedberg (10-13 sec)
• The sedimentation scale is determined by the size and mass
of the particles and the viscosity of the surrounding fluid.
• For a sphere
• The viscosity of water at room temperature is
6 where is the viscosity R
3 2 3 10 Pa s = 10 poise (erg/cm )s
Rheology is useful to study macromolecules
• Example: polymer size scaling.
• Assuming random walk net ~
Random walk: 0.5
Self-avoiding: 0.57
p
gR m
m m
p
p
~6
pB B
g
k T k TD m
R
1net net ~
6
p
p
g
m m ms m
R m
0.57~D m
0.44~s m
Hard to mix a viscous liquid
• Experiment: ink in glycerin
• The clockwise-counterclockwise experiment: blob will smear out but retracing
make the blob reassemble into nearly original position and shape!
• That's not what happens when you stir cream into your coffee…
Does reversibility violates 2nd law of TD?
• Ink diffuses but very slowly:
• So the blob cannot change much by diffusion
• Stirring causes organized motion: fluid layers slide over one another.
• Ink molecules spread out but not randomly (because diffusion is too slow).
• Reversing the wall motion: fluid layers slide back and reassemble the blob.
• Such organized flow is called laminar.
B Bk T k TD
0.D
In RGB
II. Low Reynolds-number world
Quantifying the friction-dominated regime (laminar vs. turbulent)
• Shear motion: moving plate feels resisting viscous force
stationary plate feels opposite force (entraining force).
• Viscous force f is proportional to area A, and speed v but
decrease with plate separation.
• Empirically, for small v, many fluids follow simple law:
Forces in laminar flow
Af v
L
Uniform flow along x
x
y
z
Force applied on y+dy layer by y layer
(Newtonian viscous formula)
Coefficient of viscosity
Newtonian fluid
( )v yv dvf dxdz
v
y dy
y
f
dv
dy
• Intuitively, flow is laminar when viscosity η is large and turbulent if η is
small. But “small” or “large” w.r.t. what?
• Dimensional analysis for Newtonian fluid:
• No dimensionless quantity from η and ρ.
• But we can make a characteristic quantity.
no intrinsic length scale: cannot tell “thick” from “thin” fluids.
Newtonian fluid is scale invariant.
• Situation dependent: fluid motion is viscous if
Critical force demarcates the friction-dominated regime
3
force[ ] ; [ ]
velocity x length
M M
LT L
2
crit critical viscous forcef
2
crit .f f
Aquatic cellular environment is viscous
• For macroscopic bodies and forces water is turbulent.
• For pN forces in the cell, water is viscous…
• For f < fcrit, fluid is thick: friction quickly damps out
inertial effects. Flow is dominated by friction.
The Reynolds number quantifies the relative
importance of friction and inertia
2
ext frict tot
ext
23
inertia
Flow past a sphere:
Velocity changes direction during ~ / .
Acceleration magnitude ~ / ~ /
Newton's law
( is by fluid pressure)
=
t R v
a v t v R
f f f ma
f
vf ma l
R
0 0
23
2 2
friction force:
Net force on fluid: ( ) ( )
~friction
f dv
A dx
f f x l f x
df d v vf f l Al l
dx dx R
• Small Re: friction dominates; flow stops
immediately when force stops ("creeping flow“).
• Big Re: inertial effects dominate, coffee keeps
swirling; flow is turbulent.
3 2
nertia
3 2
friction
Reynolds number
/
/
if l v R vR
f l v R
Microbiology is viscous (low Re)
3 3
8
3
10 kg m 30 m 10 m/s3 10 1
10 Pa s
3 3 6 6
5
3
10 kg m 10 m 30 10 m/s3 10 1
10 Pa s
• Australian Pitch drop
experiment running from 1927
• Viscosity ~ 1012 of water
Tuning Re
Time-reversal properties of a dynamical law
signal its dissipative character
• Once the top plate has
returned to its initial position,
each fluid element has also
returned, regardless of the
dynamics of the return stroke.
friction force: f dv
A dx
0
crit inertia
0 0
0 0
0
0
Sheets move uniformly forces balance out:
const.
Time depndent motion: negligible
( , ) ( )
Unmixing:
: , , , ,
: , ,
dv xv v
dx d
f f f
xv x t v t
d
x y z x y zx
v v td
x xvx yt x v t
d d
xv vz
d
0 , , , ,y z x zt y
Time reversal: Newtonian mechanics
• In Newtonian physics, the time-
reversed process is a solution to the
equations of motion with the same
sign of force as the original motion.
2
2
dz dUm f mg
dt dz
2
2
dzg
dt
Time reversal:
t t
2
0
1( )
2z t v t gt
0
21( )
2v tz t gt
2
2
dzg
dt
• Time-reversed trajectory solves
Newton's law with inverse v0.
solution
solution
Time reversal: Diffusion
• Diffusion equation is
not time invariant.
Time reversal:
t t
21( , ) exp
44
xc x t
DtDt
2
2
dc d cD
dt dx
• Time-reversed solution does not
solve original diffusion equation.
2
2
d cD
x
dc
dt d
21( , ) exp
44
xc x t
DtDti
solution
solution
Viscous friction is not time reversal invariantA ball in highly viscous fluid
dz f
dt
fdz
dt
0
solution:
fz z t
0
solution:
fz z t
f
f
Time-reversed solution does not
solve original friction equation,
unless force is inversed
Frictional motion is irreversible because friction dissipates ordered motion into heat.
?
v
x
y
u
x
y
(displacement)
Fluids and solids differ in time-reversal symmetry
f duG
A dy
f d du
A dy dt
• No explicit time dependence: invariant.
• Not invariant.
• Solids have “memory” of position.
Fluids and solids differ in time-reversal symmetry
2
2
d um ku
dt
du f
dt
• 2nd order time derivative: invariant.
• 1st order time derivative: Not invariant.
• Solids have “memory” of position.
k
6 R
f ku
f
Viscous flow have other symmetry properties
Low Re flow around a stationary object
having a plane of symmetry is symmetric.
• Sensitive test for small Re flows
2
Stokes flow:
p v
, , ,
Proof: Symmetry plane 0.
Flow reversal: ( , , ) ( , , ).
Mirror symmetry:
( , , ) ( , , ) ( , , )
( , , ) ( , , ) ( , , )
x x x
y z y z y z
x
v x y z v x y z
v x y z v x y z v x y z
v x y z v x y z v x y z
The reversed flow with
obeys the same equation.
v v p p
upon reversal, fluid follows the identical
streamlines in the opposite direction.
• In low Re we can blow out a candle
either by blowing or suction.
• In high Re, we cannot blow out a
candle by suction
2
Stokes flow:
p v
Streamlines are invariant to rate of flow
2
Stokes flow:
p v 2( ) ( )p v
1 2
1 1 2 2
1 1 2 2
( , ) and ( , ) solutions
is also a solution
with
v r t v r t
v v
p p p
• Follows form linearity of the flow.
• No notion of explicit time.
III. Biological applications
Swimming and pumping
• In the low Re world: a motion can be canceled completely by applying minus the time-
reversed force. What are the implications for microorganisms?
• Flapping back and forth returns every fluid element to its original position:
No net motion!
Swimming of microorganisms: reciprocal motion
c. Repeat…
[Cartoon by Jun Zhang.]Any net motion?
a. paddles move backward at
speed v relative to the body
forward motion of the body at
speed u relative to water.
b. paddles move forward at
v' relative to the body
backward motion of body at u'
relative to water.
paddle
relative velocity of paddles w.r.t. fluid:
drage force on paddles: ( )
drag force on body:
force balance:
body velocity:
Total displacement:
p
b b
p b
p
b p
v u
f v u
f u
f f
u v
x
p
b p
ut vt
paddle
Similarly: relative velocity of paddles w.r.t. fluid: ' '
drage force on paddles: ( ' ')
drag force on body:
body velocity:
Total displacement:
p
b b
p
b p
v u
f v u
f u
u v
x u
p
b p
t v t
In reciprocal motion the paddles return to their original position:
Hence:
NO NET MOTION!
p p
b p b p
vt v t
x u t v t vt ut x
Scallop theorem forbids strictly reciprocal motion
• Scallop theorem:
Strictly reciprocating
motion won’t work for
swimming in the low-
Reynolds world [Purcell
(1977) Am. J. Phys.]
What other options a microorganism has?
Motion must be periodic (to be repeated).
It can’t be of the reciprocal.
Ciliary propulsion is periodic
not reciprocal
• The difference is in the additional degrees of freedom.
• Many cells use cilia to generate net thrust.
• Each cilium contains internal filaments and motors.
• Cilia can be used for translocation and pumping
(in stationary cells).
Net motion requires breaking the back and forth symmetry
Large drag in forward motion.
Displacement:
1 /
p
b p b p
vtx ut vt
Total displacement:
01 / 1 /
b p p
b p b p b p b p
vt v tx x vt
Smaller drag in backward motion:
Displacement:
1 /
p p
p
b p b p
v tx u t v t
Cilia break symmetry by changing the direction of motion
• Effective stroke: high drag (perpendicular)
• Recovery stroke: low drag (parallel)
Bacteria use rotating
flagella for locomotion
Flagella break shape symmetry for locomotion
• The drag coefficients parallel and perpendicular
to the cylinder (helix) are not equal.
Although the velocity is in y-x plane, there is a
z-component of the force.
The net force on the helix is in z direction.
not parallel to
f f f v v
f v
v
v┴v║
f┴=-ζ┴v┴
f║ =-ζ║v║
f
• Bacteria are large enough
such that friction slows
down rotation
• Bacteria may also have
pairs of flagella rotating in
opposite directions.
How bacteria avoid rotating by torque?
Two coupled scallops can move
Strictly reciprocal
No net motion
Single pair of paddles Dimer of 2 pairs of paddles
Single pair: strictly reciprocal
Dimer: nonreciprocal
Net motion!
[Lauga and Bartolo (2008) PRE]
Why should bacteria move and stir?
• Eating? It’s anyhow hard to mix:
-- Only a few streamlines reach a moving object.
• Solution: use diffusion.
• Why stir?
-- Cilia of length d refreshes its volume every tstir = d/v
-- diffusion time scale is tdiff = d2/D
• Only if tstir < tdiff this is worthwhile.
• Peclet number:
• For D = 1000 μm2/s and d =1 μm, v = 1000 μm/s…
• Bacteria swim for other reasons, like food gradient…
Streamlines around Volvox (protozoa)
diffusion
stir
Pet d v
t D
Vesicular delivery networks are essential for
macroscopic organisms that cannot rely on diffusion
3 3
3
3 1 5
3
10 kg/m ; 10
10cm/s ; =10 Pa s
10 10 101 (within low Re)
10
R m
v
vR
Constraints:
( ) 0 (non-slip)
(0)
v R
v
2R
Balancing the forces:
Pressure applies force: 2
Viscous force from inner shell pushes shell forward:
( )2 ( / 0)
Viscous force from outer shell pulls shell backward:
2
p
in
r
out
df rdr p
dv rdf rL dv dr
dr
df r d
2
2
( ) ( ) ( )2
r dr r r
dv r dv r d v rr L r dr L dr
dr dr dr
2
22 2 2 0
p in outdf df df df
dv d vr pdr L dr rL r dr
dr dr
2
2
10 0
p dv d vdf
L r dr dr
2R
2R
Poiseuille flow:
• The flow is laminar in most blood vessels in the human body except for
the largest veins and arteries.
• Q ~ R4 flow can be controlled by small variation of radius.
2
2
10 0
p dv d vdf
L r dr dr
2
2 2
General solution:
( ) ln4
Boundary conditions:
( )4
pv r A B r r
L
pv r R r
L
4
0
Flux
2 ( )8
RR p
Q rdrv rL
Viscous force at DNA replication fork
Since the two single strands cannot pass through
each other, the original must continually rotate.
Would frictional force resisting this rotation
be enormous?
Y-shaped junction
ω
2R
Viscous force at DNA replication fork
• Small friction compared to energy consumed
by DNA helicase which unzips DNA. Y-shaped junction
ω
2R
2 2
2
The torque scales like:
The dissipation work
per turn of :
2 2
r f R RL
P R L
W R L
1 3 2
Replication rate: 1000 bp/s
DNA period: 10.5 bp/turn
1000 2 600 rad/s
10.5
(2 )(600 s )(10 Pa s)(1 nm ) ~ 0.01 L /BW L k T m
Summary
• Viscosity dominates the nano-world .
• Suspension is stabilized by diffusion at time scale:
• Hard to mix viscous fluids
• Reynolds number:
– Inertia/friction
– Force/critical force
• Symmetry:
– Newtonian dynamics: time-reversal invariant
– Viscous friction: not time-reversal invariant
• Swimming of microorganisms:
– Strictly reciprocal motion cannot translocate
– Periodic but not reciprocal motion work
in ciliary and flagellar propulsion
• Viscosity dominates flow in blood vessels.
drift netnet
v mm
g
3 2
nertia
3 2
friction
Reynolds number
/
/
if l v R vR
f l v R
