18.354 Nonlinear Dynamics II
Towards
hydrodynamics
Jörn Dunkel
Monday, February 24, 14
Hydrodynamics works at all these scales!
µm
Monday, February 24, 14
mm
m
km
ly
lds Numbers in Biology
Some typical Reynolds numbers
number is dimensionless group that characterizes the ratio o
fined as
⇥U L
UL
Re =
=
µ
density of the medium the organism is moving through; µ is t
m; is the kinematic viscosity; U is a characteristic velocity of
stic length scale. When we discuss swimming biological organ
eatures that are moving through water (or through a fluid with
hose of water). This means that the material properties µ and
ber is roughly determined by the size of the organism.
e characteristic size of the organism and the characteristic sw
rule-of-thumb, the characteristic locomotion velocity, U , in bi
y U L/second e.g. for people L 1 m and we move at U 1
d they move at about U
1 mm/s; for microorganisms
L 1
dunkel@math.mit.edu
sly this is a very very very very rough estimate and one does no
Monday, February 24, 14
High-Re vs Low-Re
dunkel@math.mit.edu
Monday, February 24, 14
VERY High-Re
dunkel@math.mit.edu
Monday, February 24, 14
Low-Re (laminar) flow
Monday, February 24, 14
Flow & transport in cells
Drosophila
embryo
Goldstein lab (Cambridge)
Monday, February 24, 14
Flow & transport in cells
Drosophila
embryo
Goldstein lab (Cambridge)
Monday, February 24, 14
Intracellular transport
Chara corralina
Giant cell
Monday, February 24, 14
http://damtp.cam.ac.uk/user/gold/movies.html
Vesicles in a shear flow
model for
blood cells
dynamics
Vasily Kantsler
Monday, February 24, 14
Swimming bacteria
movie: V. Kantsler
~20 parts
20 nm
Berg (1999) Physics Today
source: wiki
Chen et al (2011) EMBO Journal
dunkel@math.mit.edu
Monday, February 24, 14
E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
Monday, February 24, 14
Results
At distances
r
<
6
µ
m the dipole model overestimates the bacterial flow field. (E)
Experimentally
wd. field
of
its
“puller”
image.
measured
flow to
walls, we focused on a plane 50 µm from the top and bottom
ance 2 µm parallel to the wall. (F) Best fit force-dipole model, and (G) residual flow
field.
Notewhere
the
cell
body,
Bacterial
surfaces of the sample chamber, and recorded ∼ 2 terabytes of
flow fieldResults
an E. coli “pusher” decays much faster, when
a bacterium swims close
tothe
the
surface,
4
for
length
of t
theflow
mea
cule
movieofdata.
In(non-tumbling
this data we identified
∼
10
rare
events
when
HCB 437)
achieved
by
fittin
the
measured
and
best-fit
force
cellsBacterial
swam in the
focal
plane
for
>
1.5
s.
By
tracking
the
decays
labeled,
n
flow field far from surfaces. To resolve the minisat
variable
locatio
fluid
tracers
in
each
of
the
rare
events,
relating
their
position
of
the
c
decays
of
the
flow
speed
u
with
cule flow
created
individual bacteria, we trackedfield
gfp- (rsion
of
flu
m surfaces.
To field
resolve
the by
minis>field
8 µm).
and labeled,
velocity to
the
position
and
orientation
of
the
bacterium,
dis
of swam
thedipole
cell
body
(Fig.
1D)
illustr
non-tumbling
E. coli
as they
through
a (Fig.
suspenwalls,
we
the
measured
and
best-fit
force
field
1C).
The
dividual
bacteria,
we
tracked
gfpthe specific fitting
and performing an ensemble average over all tracers, we reHoweve
sion
of
fluorescent
tracer
particles.
For
measurements
far
from
field
displays
the
characteristic
surfaces
of theaflow
flow
speed
u with
distance
r from
the length
center! =o1
dipole
i asministhey swam
through
suspenhe
solved
thedecays
time-averaged
field
in the
E. coli
swimming
measure
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is cons
oftothe
cell
(Fig.
1D) However,
illustrate
that
flow
movie
dat
the
force
dipole
flow
sig
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±the
5 measured
ckedplane
gfpticles.
For
measurements
far
from
cell force
bod
2 ∼ 2 terabytes
surfaces
of
the
sample
chamber,
and
recorded
of
and
resistive
µm/s.
As field
E.the
colidisplays
rotate
about
their
swimming
direction,
their
the
characteristic
1/r4 decay
oftoa the
forceside
dipole.
cells
measured
flow
ofswam
thel
a 50
suspenµmmovie
from
top
and
bottom
for
the
data.
In field
this
data
we
identified
∼is10cylindrically
rare events
when
note that
in
the b
time-averaged
flow
in
three
dimensions
However,
the
force
dipole
flow
significantly
overestimates
the
fluid
trace
cell
body,
where
the
flow
magnit
far from
ber,
and
recorded
∼
2
terabytes
of
achieved
µm
behind
the ce
cells swam
in
the
focal
plane
for
>
1.5
s.
By
tracking
the
symmetric.
Our
measurements
capture
all
components
of
this
measured
flow to the side of
the
cell
body,ofand
behind
the
4
and
veloci
didentified
bottom
for
the
length
the
flagellar
bun
at
varia
fluid
drag
on
the
∼
10
rare
events
when
fluid tracers
in each
of the
rare
events,
relating
their
position
cylindrically
symmetric
flow,
except
the
azimuthal
flow
due
to
cell body, where the flow magnitude
u(r)
is nearly
constant
and
perfo
abytes
of
field
(r f
achieved
by
fitting
two
opposite
velocity
tocell
the
position
and orientation
of the bacterium,
the and
rotation
of the
about
its the
body
axis. The topology
of
ne for
> 1.5
s.
By
tracking
for the
length
of the
flagellar
bundle.
The
force
dipole
fitthe
wasspec
nts when
solved
the
the
measured
flow
field
(Fig.
1A)
is
the
same
as
that
of
a
and
performing
an
ensemble
average
over
all
tracers,
we
reat variable locations
along
the
sw
are events, relating
their
position
Bacterial
flowdow
fiel
achieved
by1B),
fitting
two
opposite
force
monopoles
(Stokeslets)
kingforce
the
dipole
le
plane
dipole
flowtime-averaged
(Fig.
defined
by
solved
the
flow
field
in
the
E.
coli
swimming
field
(r
>
8
µm).
From
the
best
and
orientation
of the locations
bacterium,
dipole
flow
descri
at
variable
along
the
swimming
direction
to
the
far
value
of
position
µm/s.
As
plane down to 0.1% of the mean swimming
speed
V
=
22
±
5
0
the
specific
fitting
routines
fi
with good and
accura
h
i
e
average
over
all
tracers,
we
refield
(r
>
8
µm).
From
the
best
fit,
which
is
insensitive
to
and resi
A E. coli
r direction, their time-avera
!F swimming
acterium,
µm/s.
As
2rotate about their
ˆ
thisµm
approximatio
u(r)in= the
3(r̂.coli
d) −
1 r̂, routines
A=
, r̂fitting
= length
, regions,
[!
1 ]= 1.9
dipole
and
dip
ow
field
E.
swimming
2
the
specific
fitting
and
we
obtain
the
note
tha2
s, we re|r|
8πηdimensions
|r| is cylindrically
time-averaged
flow field in three
symmetric
a
wall.
Focusing
value
F is Fconsistent
with
opt
dipole
length
1.9±µm
and
dipole
force
= of
0.42
pN.
This
µm
beh
mean
swimming
speed
V0 !==22
5 capture
wimming
symmetric.
Our
measurements
allof
components
this
and applying
the
cylindrica
and
resistive
force
theory
calculat
fluid
dra
of
Fforce,
is consistent
with
optical
trapforce
measurements
[45]
ut
direction,
their
where
F isvalue
the dipole
! the
distance
separating
the
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a
sligh
rotati
note
thatThe
invector
the
best
fit,
the
cell
and
resistive
force
theory
calculations
[46].
It is
interesting
to
the
rotation
of
the
cell
about
itsunit
body
axis.
topology
η the
viscosity
the
fluid,
dˆ the
orientation
theof
flow
field
struc
on, pair,
their
three
dimensions
isof
cylindrically
the measu
(swimming
direction)
the best
bacterium,
and
rbehind
the
distance
surfaces,
the
note
thatflow
inofthe
fit, 1A)
theµm
cell
drag
Stokeslet
isfrom
0.1
the measured
field
(Fig.
is the
same
as that
oflocated
aof
the
center
the
cell
ndrically
nts
capture
all
components
of
this
force
dipo
Bacteria
vector
relative
to
the
center
of
the
dipole.
Yet
there
are
some
ity
of
a
no-slip
sur
µm
behind
the
center
of
the
cell
body,
possibly
reflecting
the
force
dipole
flow
(Fig.
1B),
defined
by
fluid
drag
on
the
flagellar
bundle
tsexcept
of
this
flow
due
Dunkel,
Ganguly,
Cisneros,
Goldstein
(2011) to
PNAS
Fig. differences
1.Drescher,
Averagethe
flow
fieldazimuthal
created
by a single
freely-swimming
bacterium.
(A)
Experimentally measured flow field far from a surface. Stream lines indicate local
direction of fl
dipole
close
to
the
cell
body
as
shown
by
the
residual
of
outward
streamlin
fluid
drag
onflowthe
flagellar
flow. (B) Best fit force-dipole
model, and
(C) residual
field, obtained
by subtractingbundle.
the best-fit dipole from the experimentally measured field. The presence of the flagella
E.coli
weak ‘pusher’ dipole
due
to
utwinduces
its
body
The
topology
go
a anterior-posterioraxis.
asymmetry. (D)
Radial
of the flow field.iAtof
distances r < 6 µm the dipole model overestimates the bacterial flow field. (E)with
Experimentally
h decay
Monday, February 24, 14
Chlamydomonas
PRL 105, 168101 (2010)
Movie: Jeff Guasto (TUFTS)
‘puller’
Monday, February 24, 14
PHYSICAL
FIG. 4 (color online). Time- and azimuthally-averaged fl
Drescher et al PRL 2010
from velocity vectors (blue [dark
The
spiraling near
Guastogray]).
et al PRL
2010
velocity field. A color scheme indicates flow speed magnitu
model: flagellar thrust is distributed among two Stokesle
arrows), whose sum balances drag on the cell body (cen
separate colors in the inset, compared to results from the
size ~ 20µm
speed ~ 100µm/s
beat frequency ~30 Hz
flow may be important [30]. We are currently investig
Volvox carteri
somatic
cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
Monday, February 24, 14
dunkel@math.mit.edu
Volvox
Goldstein lab (Cambridge)
Monday, February 24, 14
Volvox
meta-chronal waves
Brumley et al (2012) PRL
Monday, February 24, 14
meters
Monday, February 24, 14
kilometers (miles)
http://svs.gsfc.nasa.gov/site_usage/site_reqts.html
Monday, February 24, 14
Galactic & intergalactic gas dynamics
horsehead nebula (hubble)
Monday, February 24, 14
Water knots
Irvine lab (Chicago)
Monday, February 24, 14
Surface effects
http://web.mit.edu/ehl/www/Home.html
Monday, February 24, 14
Bush group
“Quantum” HD
Couder lab (Paris)
Monday, February 24, 14
Bush group