Course intro
18.354
18.354J Nonlinear Dynamics II: Continuum Systems
Spring 2015 – Course Info
Lectures:
Instructor:
Contact:
Office Hours:
Course website:
Teaching assistant:
TR 1-2:30
in
E17-136
Jörn Dunkel
dunkel@mit.edu
253-7826 (office phone)
R 2:30-3:30 (E17-412)
math.mit.edu/⇠dunkel/Teach/18.354/
TBD, Office: TBD Email: tbd
This course introduces the basic ideas for understanding the dynamics of continuum
systems, by studying specific examples from a range of di↵erent fields. Our goal will be to
explain the general principles, and also to illustrate them via important physical e↵ects. A
parallel goal of this course is to give you an introduction to mathematical modeling.
The first part of the course will study di↵usion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion
that we derive for continuum systems are typically nonlinear partial di↵erential equations,
for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without having to solve the full equations of motion.
We shall then study the ‘calculus of variations’, which is a minimization approach to
finding solutions of continuous systems. We are familiar with these techniques for discrete
The first part of the course will study di↵usion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion
that we derive for continuum systems are typically nonlinear partial di↵erential equations,
for which it isNonlinear
very difficult to Dynamics
obtain analytical II:
solutions.
We shall therefore
briefly foray
18.354J
Continuum
Systems
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without having to
solve the
full equations
of motion.
Spring
2015
– Course
Info
We shall then study the ‘calculus of variations’, which is a minimization approach to
Lectures:
TR
1-2:30 We are
in familiar
E17-136
finding
solutions of continuous
systems.
with these techniques for discrete
Instructor:
JörntoDunkel
systems
and now adapt the ideas
help us with continuous systems. First, we examine the
Contact:
dunkel@mit.edu
253-7826
(office
phone)
classical
brachiostrome problem
posed by Bernoulli, and then
consider
an array
of problems
in many
di↵erent
physical systems
(e.g., shapes
of soap films, bending of elastic beams, etc.)
Office
Hours:
R 2:30-3:30
(E17-412)
InCourse
the second
half of the course
we will examine hydrodynamic problems. We will discuss
website:
math.mit.edu/⇠dunkel/Teach/18.354/
derivations
of Navier-Stokes-type
equations
study
particular
Teaching
assistant: TBD,
Office: and
TBD
Email:
tbd solutions. In this context,
we will introduce singular perturbation methods and survey hydrodynamic instabilities. In
the last part, we will give an outlook on the mathematical description of solitons, active
biological systems and topological defects.
This course introduces the basic ideas for understanding the dynamics of continuum
systems,
by studying specific examples from a range of di↵erent fields. Our goal will be to
GRADING
explain the general principles, and also to illustrate them via important physical e↵ects. A
Problem sets
parallel• 35%:
goal of
this course is to give you an introduction to mathematical modeling.
• 30%: Mid-term exam
The first part of the course will study di↵usion, to demonstrate how continuum de• 5%: Project proposal
scriptions
arise
from
averaging
microscopic
• 30%:
Final
project
presentation
+ reportdegrees of freedom. The equations of motion
that we derive for continuum systems are typically nonlinear partial di↵erential equations,
for which
it is very difficult to obtain analytical solutions. We shall therefore briefly foray
TEXTBOOKS
into dimensional analysis to see how it is possible to obtain qualitative information about
Although there are no textbooks which follow the precise spirit of this course, there are
a system without having to solve the full equations of motion.
literally hundreds of textbooks where the topics we will cover are discussed. For most
Welectures,
shall then
study
of variations’,
is a minimization
typed
notesthe
can‘calculus
be downloaded
from the which
course webpage.
Additionalapproach
material to
findingwill
solutions
of continuous
Wethat
are will
familiar
withfrequently
these techniques
for discrete
be handed
out in class.systems.
One book
be useful
is: D. J. Acheson,
systems and now adapt the ideas to help us with continuous systems. First, we examine the
classical brachiostrome problem posed by Bernoulli, and then consider an array of problems
in many di↵erent physical systems (e.g., shapes of soap films, bending of elastic beams, etc.)
18.354J
II: Continuum
In the second Nonlinear
half of the courseDynamics
we will examine hydrodynamic
problems.Systems
We will discuss
derivations of Navier-Stokes-type equations and study particular solutions. In this context,
Spring 2015
– Course
Info
we will introduce singular perturbation
methods
and survey
hydrodynamic instabilities. In
the last part,
we will give an outlook
on the mathematical
description of solitons, active
Lectures:
TR 1-2:30
in
E17-136
biologicalInstructor:
systems and topological
defects.
Jörn
Dunkel
Contact:
GRADING
Office Hours:
Course sets
website:
• 35%: Problem
Teachingexam
assistant:
• 30%: Mid-term
dunkel@mit.edu
253-7826 (office phone)
R 2:30-3:30 (E17-412)
math.mit.edu/⇠dunkel/Teach/18.354/
TBD, Office: TBD Email: tbd
• 5%: Project proposal
• 30%: Final project presentation + report
This course introduces the basic ideas for understanding the dynamics of continuum
TEXTBOOKS
systems,
by studying specific examples from a range of di↵erent fields. Our goal will be to
explain
the general
and which
also tofollow
illustrate
them via
important
physical
e↵ects.
Although
there areprinciples,
no textbooks
the precise
spirit
of this course,
there
are A
parallel
goal
of this course
is to give
youtheantopics
introduction
to mathematical
modeling.
literally
hundreds
of textbooks
where
we will cover
are discussed.
For most
The first
partnotes
of the
will studyfrom
di↵usion,
to demonstrate
how continuum
lectures,
typed
cancourse
be downloaded
the course
webpage. Additional
material dewill be handed
out averaging
in class. One
book thatdegrees
will be ofuseful
frequently
D. J. Acheson,
scriptions
arise from
microscopic
freedom.
The is:
equations
of motion
Elementary
Dynamics, Oxford
Pressnonlinear
(1990). partial di↵erential equations,
that
we deriveFluid
for continuum
systemsUniversity
are typically
for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without having to solve the full equations of motion.
We shall then study the ‘calculus of variations’, which is a minimization approach to
finding solutions of continuous systems. We are familiar with these techniques for discrete
18.354J Nonlinear Dynamics II: Continuum Systems
Spring 2015 – Course Info
Lectures:
Instructor:
Contact:
Office Hours:
Course website:
Teaching assistant:
TR 1-2:30
in
E17-136
Jörn Dunkel
dunkel@mit.edu
253-7826 (office phone)
R 2:30-3:30 (E17-412)
math.mit.edu/⇠dunkel/Teach/18.354/
TBD, Office: TBD Email: tbd
HOMEWORK
- PROBLEM
This
course introduces
the basic SETS
ideas for understanding the dynamics of continuum
systems,
by studying
specific roughly
examples
from
a rangeweeks.
of di↵erent
fields. Our
will be to
Homework
will be assigned
every
two-three
Each homework
setgoal
will contain
analytical
and computational
problems,
even the
oddvia
experiment.
must A
explain
the general
principles, and
also to and
illustrate
them
importantAssignments
physical e↵ects.
be handed
by 1pm
(start
of give
class)you
on the
due date. First
late homework
score
parallel
goal ofinthis
course
is to
an introduction
tounexcused
mathematical
modeling.
will be
multiplied
No subsequent
homework is accepted.
You are deThe
first
part of by
the3/4.
course
will study unexcused
di↵usion, late
to demonstrate
how continuum
welcome
to discuss
solution strategies
and even
solutions,
but please
write
up the solution
scriptions
arise
from averaging
microscopic
degrees
of freedom.
The
equations
of motion
own.
Be sure to support
answer bynonlinear
listing any
relevant
Theoremsequations,
or by
that on
weyour
derive
for continuum
systems your
are typically
partial
di↵erential
explaining important steps. Be as clear and concise as possible. I strongly encourage the
for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray
computational problems to be written in MATLAB.
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without having to solve the full equations of motion.
MID-TERM EXAM
We shall then study the ‘calculus of variations’, which is a minimization approach to
There
will be of
a mid-term
exam,
which will
place about
of techniques
the way through
the
finding
solutions
continuous
systems.
We take
are familiar
with3/4’s
these
for discrete
course. The exam will be a take home, and will be in place of a homework set. There will
18.354J Nonlinear Dynamics II: Continuum Systems
Spring 2015 – Course Info
HOMEWORK - PROBLEM SETS
Lectures:
TR 1-2:30
in weeks.E17-136
Homework
will be assigned roughly
every two-three
Each homework set will contain
Instructor:
Dunkel
analytical
and computationalJörn
problems,
and even the odd experiment. Assignments must
beContact:
handed in by 1pm (start of
class) on the due date. First 253-7826
unexcused late
homework
score
dunkel@mit.edu
(office
phone)
will be multiplied by 3/4. No subsequent unexcused late homework is accepted. You are
Office Hours:
R 2:30-3:30 (E17-412)
welcome to discuss solution strategies and even solutions, but please write up the solution
website:
math.mit.edu/⇠dunkel/Teach/18.354/
onCourse
your own.
Be sure to support
your answer by listing any relevant Theorems or by
explaining
important
steps. TBD,
Be as clear
and concise
possible.
I strongly encourage the
Teaching
assistant:
Office:
TBD as
Email:
tbd
computational problems to be written in MATLAB.
MID-TERM EXAM
This course introduces the basic ideas for understanding the dynamics of continuum
There will be a mid-term exam, which will take place about 3/4’s of the way through the
systems, by
studying specific examples from a range of di↵erent fields. Our goal will be to
course. The exam will be a take home, and will be in place of a homework set. There will
explain the
principles, and also to illustrate them via important physical e↵ects. A
be general
no final exam.
parallel goal of this course is to give you an introduction to mathematical modeling.
FINAL
The first
partPROJECT
of the course will study di↵usion, to demonstrate how continuum descriptionsThe
arise
from
averaging
microscopic
degrees
of fields,
freedom.
The
equations
ideas
we will
be discussing
have applications
to many
many of
which
we will notof motion
cover. for
To give
you a chance
to explore
area of interest
to you,partial
the course
will requireequations,
that we derive
continuum
systems
arean
typically
nonlinear
di↵erential
project, in which you explore in depth something of interest to you and within the
for which aitfinal
is very
difficult to obtain analytical solutions. We shall therefore briefly foray
course’s scope. Final projects will be presented in class during the final two classes.
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without
havingDATES
to solve the full equations of motion.
IMPORTANT
We shall
then study the ‘calculus of variations’, which is a minimization approach to
• Tues Feb 24 - Problem Set 1: DUE
finding solutions
of 12
continuous
systems.
with(1these
techniques
for discrete
• Thur Mar
- Problem Set
2: DUE •We
Thurare
Marfamiliar
12 - Proposal
page) for
final project.
will be multiplied by 3/4. No subsequent unexcused late homework is accepted. You are
welcome to discuss solution strategies and even solutions, but please write up the solution
on your own. Be sure to support your answer by listing any relevant Theorems or by
explaining important steps. Be as clear and concise as possible. I strongly encourage the
computational problems to be written in MATLAB.
18.354J Nonlinear Dynamics II: Continuum Systems
MID-TERM EXAM
Spring 2015 – Course Info
There will be a mid-term exam, which will take place about 3/4’s of the way through the
Lectures:
TR 1-2:30
in
E17-136
course. The exam will be a take home, and will be in place of a homework set. There will
Instructor:
Jörn Dunkel
be no
final exam.
Contact:
dunkel@mit.edu
253-7826 (office phone)
FINAL PROJECT
Office Hours:
R 2:30-3:30 (E17-412)
The ideas we will be discussing have applications to many fields, many of which we will not
Course website:
math.mit.edu/⇠dunkel/Teach/18.354/
cover. To give you a chance to explore an area of interest to you, the course will require
Teaching
TBD,
Office:
TBD
Email:to tbd
a final
project, inassistant:
which you explore
in depth
something
of interest
you and within the
course’s scope. Final projects will be presented in class during the final two classes.
IMPORTANT DATES
This course introduces the basic ideas for understanding the dynamics of continuum
• Tues Feb 24 - Problem Set 1: DUE
systems,• by
specific
from
of (1di↵erent
fields.
Our goal will be to
Thurstudying
Mar 12 - Problem
Set examples
2: DUE • Thur
Mar a
12 range
- Proposal
page) for final
project.
DUEgeneral principles, and also to illustrate them via important physical e↵ects. A
explain the
• Thur Apr 2 - Problem Set 3: DUE
parallel goal
of this course is to give you an introduction to mathematical modeling.
• Tues Apr 14 - Problem Set 4: DUE
The •first
the course
will
study
di↵usion, to demonstrate how continuum deThur part
Apr 16 of
- Take-home
Midterm
exam.
POSTED
• Thur
Aprfrom
23 - Take-home
Midterm
exam. DUE degrees of freedom. The equations of motion
scriptions
arise
averaging
microscopic
• May 5/7 - Final project presentations.
that we •derive
continuum
are typically nonlinear partial di↵erential equations,
May 7 -for
Final
project report.systems
DUE
for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray
Note: The exact due dates for the P-sets may be subject to change
into dimensional analysis to see how it is possible to obtain qualitative information about
a system without having to solve the full equations of motion.
We shall then study the ‘calculus of variations’, which is a minimization approach to
finding solutions of continuous systems. We are familiar with these techniques for discrete
Email:
Phone:
Office:
Office hours:
Course website:
dunkel@mit.edu
253-7826
E17-412
Thursday 2:30-3:30 (E17-412)
http://math.mit.edu/⇠dunkel/Teach/18.354/
1.
2.
3.
4.
—
5.
6.
7.
T
R
T
R
T
R
T
R
Feb
Feb
Feb
Feb
Feb
Feb
Feb
Feb
3
5
10
12
17
19
24
26
Introduction, overview & mathematical basics
Dimensional analysis & scalings
Hamiltonian dynamics & Kepler’s Laws
Random walkers
MIT MONDAY (PRESIDENTS DAY)
Di↵usion equation: Fourier method
Di↵usion equation: Green’s function method
Linear stability analysis & pattern formation
8.
9.
10.
11.
12.
13.
—
T
R
T
R
T
R
TR
Mar
Mar
Mar
Mar
Mar
Mar
Mar
3
5
10
12
17
19
23-27
Calculus of variations
Surface tension
Elasticity
Deformation of a thin beam
Towards hydrodynamics
Navier-Stokes equations I
SPRING VACATION
14.
15.
16.
17.
18.
—
19.
20.
21.
R
T
R
T
R
T
R
T
R
Apr
Apr
Apr
Apr
Apr
Apr
Apr
Apr
Apr
2
7
9
14
16
20,21
23
28
30
Stokes limit & Oseen tensor
Navier-Stokes equations II
Singular perturbations
Rotating flows & Taylor columns
2D hydrodynamics & conformal maps
MIT HOLIDAY (PATRIOTS DAY)
Hydrodynamic instabilities (overview)
Solitons
Active Matter
PS1 due
PS2 & proposal due
PS3 due
PS4 due
Mid-term posted
Mid-term due
9.
10.
11.
12.
13.
—
R
T
R
T
R
TR
Mar
Mar
Mar
Mar
Mar
Mar
5
10
12
17
19
23-27
Surface tension
Elasticity
Deformation of a thin beam
Towards hydrodynamics
Navier-Stokes equations I
SPRING VACATION
14.
15.
16.
17.
18.
—
19.
20.
21.
R
T
R
T
R
T
R
T
R
Apr
Apr
Apr
Apr
Apr
Apr
Apr
Apr
Apr
2
7
9
14
16
20,21
23
28
30
Stokes limit & Oseen tensor
Navier-Stokes equations II
Singular perturbations
Rotating flows & Taylor columns
2D hydrodynamics & conformal maps
MIT HOLIDAY (PATRIOTS DAY)
Hydrodynamic instabilities (overview)
Solitons
Active Matter
22.
23.
24.
25.
T
R
T
R
May
May
May
May
5
7
12
14
Final projects: student presentations
Final projects: student presentations
Bouncing droplets
Topological defects & summary
PS2 & proposal due
PS3 due
PS4 due
Mid-term posted
Mid-term due
Project report due
Q: What is Physical Applied Maths?
A:
PAM is like cooking...
Often the ingredients (physical principles) are already known but
not the way (mathematics/equations/couplings) to turn them into a
nice dinner
With some creativity, many new dishes (novel phenomena) can be
created (discovered/understood)
Wednesday, February 5, 14
Why study Applied Maths?
• intellectual challenge
• obtain general understanding of physical
phenomena and the world around us
• be able to make prediction about physical
processes
• development of general tools to be applied
to other fields
Wednesday, February 5, 14
Historical backdrop
Some famous thoughts on
Applied Maths
'Eureka, Eureka'
(Archimedes)
c. 287 BC - c. 212 BC Wednesday, February 5, 14
Some famous thoughts on
Applied Maths
'Mathematic is written for mathematicians'
(Nicolaus Copernicus)
19 February 1473 – 24 May 1543
1543
Some famous thoughts on
Applied Maths?
'It would be better for the true physics
if there were no mathematicians on
earth'
(Daniel Bernoulli)
8 February 1700 – 17 March 1782
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths
'Now I will have less distraction'
(Leonhard Euler, upon losing the use of his right eye)
15 April 1707 – 18 September 1783
Some famous thoughts on
Applied Maths
'I do not know'
(Joseph-Louis Lagrange, summarizing his life's work)
25 January 1736 - 10 April 1813
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths
'Nature laughs at the difficulties of integration'
(Pierre-Simon Laplace)
23 March 1749 – 5 March 1827
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths?
'Mathematicians are born, not made'
(Henri Poincaré)
29 April 1854 – 17 July 1912
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths?
'Prediction is very difficult, especially about the future'
(Niels Bohr)
7 October 1885 – 18 November 1962
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths?
'It is more important to have beauty in
one's equations than to have them fit
experiment' and 'This result is too
beautiful to be false'
(Paul Dirac)
8 August 1902 – 20 October 1984
Wednesday, February 5, 14
Some famous thoughts on
Applied Maths?
'To those who do not know mathematics it is difficult to
get across a real feeling as to the beauty, the deepest
beauty, of nature'
(Richard Feynman)
May 11, 1918 – February 15, 1988
BSc 1939
Wednesday, February 5, 14
Course overview
18.354
•
ODEs
•
linear/nonlinear PDEs for scalar & vector fields
•
perturbation theory
•
calculus of variations
•
Fourier transformations
•
complex numbers, conformal maps
Dimensional analysis
G.I. Taylor
1886-1975
Trinity nuclear test, July 1945
Life Magazine, August 20, 1945
The formation of a blast wave by a very intense explosion.
II. The atomic explosion of 1945
BY SIR GEOFFREYTAYLOR,F.R.S.
(Received 10 November 1949)
[Plates
7 to 9]
Photographs by J. E. Mack of the first atomic explosion in New Mexico were measured, and the
radius, R, of the luminous globe or 'ball of fire' which spread out from the centre was determined for a large range of values of t, the time measured from the start of the explosion. The
relationship predicted in part I, namely, that RI would be proportional to t, is surprisingly
accurately verified over a range from R = 20 to 185 m. The value of R~t-1 so found was used
in conjunction with the formulae of part I to estimate the energy E which was generated in
the explosion. The amount of this estimate depends on what value is assumed for y, the
ratio of the specific heats of air.
Two estimates are given in terms of the number of tons of the chemical explosive T.N.T.
which would release the same energy. The first is probably the more accurate and is 16,800
tons. The second, which is 23,700 tons, probably overestimates the energy, but is included
to show the amount of error which might be expected if the effect of radiation were neglected
and that of high temperature on the specific heat of air were taken into account. Reasons
are, given for believing that these two effects neutralize one another.
After the explosion a hemispherical volume of very hot gas is left behind and Mack's
photographs were used to measure the velocity of rise of the glowing centre of the heated
volume. This velocity was found to be 35 m./sec.
Until the hot air suffers turbulent mixing with the surrounding cold air it may be expected
to rise like a large bubble in water. The radius of the 'equivalent bubble' is calculated and
found to be 293 m. The vertical velocity of a bubble of this radius is I /(g 29300) or 35-7 m./sec.
The agreement with the measured value, 35 m./sec., is better than the nature of the measurements permits one to expect.
COMPARISON WITH PHOTOGRAPHIC RECORDS OF
THE FIRST ATOMIC EXPLOSION
Two years ago some motion picture records by Mack (I947) of the first atomic
explosion in New Mexico were declassified. These pictures show not only the shape
of the luminous globe which rapidly spread out from the detonation centre, but also
gave the time, t, of each exposure after the instant of initiation. On each series of
photographs a scale is also marked so that the rate of expansion of the globe, or
'ball of fire', can be found. Two series of declassified photographs are shown in
figure 6, plate 7.
These photographs show that the ball of fire assumes at first the form of a rough
sphere, but that its surface rapidly becomes smooth. The atomic explosive was
fired at a height of 100 ft. above the ground and the bottom of the ball of fire reached
the ground in less than 1 msec. The impact on the ground does not appear to have
disturbed the conditions in the upper half of the globe
which continued to expand as
1/5
a nearly perfect luminous hemisphere bounded by a sharp edge which must be taken
as a shock wave. This stage of the expansion is shown in figure 7, plate 8 which
corresponds with t = 15 msec. When the radius R of the ball of fire reached about
130 m., the intensity of the light was less at the outer surface than in the interior. At
⇤
ln R = ln c
Vol
201.
A.
E
[ 175 ]
⇥
⌅
2
+ ln t
5
12
the two objects that is proportional to the product of t
y proportional
to
the
square
of
the
separation
of
the
tw
Hamiltonian dynamics &
e statement, itKepler’s
is possible toproblem
derive Kepler’s laws.
s
nd
2
law
aw is the simplest to derive, and is a statement that the
ving under a central force, such as gravity, is constant.
um L of a particle with mass m and velocity u is
dr
L=r⇥m ,
dt
ector position of the particle. The rate of change of ang
Brownian motion
By dividing both sides by A ⌧ the number of particles per un
is seen to increase at the rate1
Random walks & diffusion
n(x, t + ⌧ )
⌧
In the limit ⌧ ! 0 and
n(x, t)
=
@Jx
@2n
= D 2,
@x
@x
For strictly one-dimensional systems, the boundary area is just a poin
7
Mark Haw
David Walker
Jx
! 0, this becomes
@n
=
@t
1
[Jx (x + /2, t)
Pattern formation
dunkel@math.mit.edu
e to the static angle of repose and subseque
Zebra vs. granular media
dunkel@math.mit.edu
arxiv: 1208.4464
2d Swift-Hohenberg model
tropic fourth-order model for a non-conserved scalar or !
reflection-symmetry ⇤(t, x), given by
b
=
2
⇤
⇤,
(1)
2 2
0
⇤t ⇥ =2 U (⇥) + 0 ⇥2 ⇥
(⇥
) ⇥
2
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a 2 b 3 c 4
U (⇤) = ⇤ + ⇤ + ⇤ ,
2
3
4
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
0
trajectory of a swimming cell can
arxiv:exhibit
1208.4464a pre
2d Swift-Hohenberg
example,model
the bacteria Escherichia coli [40] an
broken
tropic fourth-order model for
a
non-conserved
scalar
or
⌃ ij denotes the Cartesian components of the Levi-Civ
reflection-symmetry
⇤(t, x), given by
a summation convention
for equal indices throughout.
2
b(1)=
⇤
⇤,
2
2 2
0
2
⇤t ⇥ = U (⇥) + 0 ⇥ ⇥
2 (⇥ ) ⇥
he time derivative, and ⇤ = ⌅2 is the d-dimensional
ved from a Landau-potental U (⇤)
a 2 b 3 c 4
U (⇤) = ⇤ + ⇤ + ⇤ ,
2
3
4
(2)
e rhs. of Equation (1) can also be obtained by variational
ed energy functional. In the context of active suspensions,
x) =fluctuations,
⇥ v
tify local (t,
energy
local alignment, phase
will assume throughout that the system is confined to
d
of volume
(3)
0
Calculus of variations
What is the di↵erential equation satisfied by Y (x)? To answer this question, we compute
the functional derivative
What is the di↵erential equation satisfied by Y (x)? To answer this question, we compute
he functional derivativeI[Y ] = lim 1 {I[f (x) + ✏ (x y)] I[f (x)]}
✏!0 ✏
Y
Z x12 
I[Y ]
@f (x) + ✏ (x @fy)]0 I[f (x)]}
== lim {I[f
(x y) +
(x y) dx
✏!0 ✏
Y
0
@Y
Z xx21  @Y
Z x2 @f
@f 0
@f (x d y)
@f+
=
(x y) dx
0
= x1 @Y
(x y)dx.
(17)
@Y
0
dx @Y
Z xx21  @Y
@f
d @f
= the Euler-Lagrange equations
(x y)dx.
(17)
Equating this to zero, yields
0
@Y
dx @Y
x1
@f
d @f
0 =
Equating this to zero, yields the Euler-Lagrange
equations
@Y
dx @Y 0
(18)
@fY =d0 @f
It should be noted that the condition
I/
alone is not a sufficient condition for
0 =
(18)
0
dx @Y a maximum. It is often possible,
a minimum. In fact, the relation might @Y
even indicate
however,
convince
that no maximum
for is
thenot
integral
(e.g., the
distancefor
t should
betonoted
thatoneself
the condition
I/ Y = exists
0 alone
a sufficient
condition
along a smooth path can be made as long as we like), and that our solution is a minimum.
minimum.
In fact, the relation might even indicate a maximum. It is often possible,
To be rigorous, however, one should also consider the possibility that the minimum is merely
owever, to convince oneself that no maximum exists for the integral (e.g., the distance
a local minimum, or perhaps the relation I/ Y = 0 indicates a point of inflexion.
long a smooth path can be made as long as we like), and that our solution is a minimum.
Elasticity
Elasticity
photo: Andrej Kosmrlj
http://newsoffice.mit.edu/2015/predicting-wrinkles-fingerprints-curved-surfaces-0202
collaboration with Reis lab (MechE)
Surface tension
Goldstein lab, Cambridge
Large drop in microgravity
e V is
Z
Z d/dt inside the integral sign we must take
account
pndS =
rpdV.
(7) of this. The Reyno
⇢u · ndS,
⇢dV.so, and it can be shown that for a deforming,
(1)incompressible flu
does
V
Z
S
Z
S(t) are being deformed by the motion of the fluid, so if we want
to take the
d
Du
eaving
this volume through
the
bounding
surface
S
is
where u(x, t) is the velocity of the fluid
and=n is the
norm
⇢udV
⇢ outward
dV
ntegral sign we must take account of this. The Reynolds transport
theorem
dt V (t)
V (t) Dt
Z
Z
Z
can be shown that for a deforming, incompressibleZfluid
element
@⇢
⇢u
·
ndS,
Here
dV =
⇢u · ndS(2)
=
r · (⇢u)d
Z
Z
S
V @t
S
V
d
Du
Hydrodynamics
D
@ (8)
⇢udV =
⇢
dV
= have
+ (u · r)
Dt
(t)
V (t)
the velocity ofdttheV fluid
and
is
thehold
outward
normal.
Hence
we
Thisn must
for any
arbitrary
fluid
element
dV , thus
Dt
@t
Z
Z
Z
is called the convective derivative, and
@⇢we shall discuss it’s significance
@⇢
+ r · (⇢u)
dV =
⇢u · assuming
ndS = thatr
(3) = 0.
F ·is(⇢u)dV.
solely given by gravity,
@t
D S@
V @t
V
=
+ (u · r)
Z
Z(9)
Dt
@tThis is called the continuity equation.
Du
⇢
dV =
( rp + ⇢g)dV
for any arbitrary fluid element dV , thus
V (t) Dt
(t)
For fluids like water, the density
does not Vchange
very much and
vective derivative, and we shall discuss it’s significance in a moment. Hence,
@⇢ to neglect the density variations. If we make this approximation
F is solely given by gravity,
+ r Since
· (⇢u)this
= 0.must hold for any arbitrary fluid element
(4) we arrive at
to the incompressibility condition
@t reduces
Z
Z
Du
Du (10)
rp
⇢
dV
=
(
rp
+
⇢g)dV
=
+ g.
e continuity Vequation.
Dt
Dt r · u⇢ = 0.
(t)
V (t)
e water, the density does not change very much and we will often be tempted
This,
combined
with
equation
(4),
constitutes
Eu
hold forvariations.
any arbitrary
fluid
element
we
arrive
at the the
ensity
If we
make
this
approximation
thecontinuity
continuity
equation
Like
all
approximations,
this
one
is sometimes
very
good andthesom
can be tidied up a little if we realise that the gravitational force, being
ncompressibility condition
will
have to figure out where it fails.
Du
rp
= written
+ g.as the gradient of a scalar potential r
(11). It is therefore usual t
Dt
p⇢+
! p. This implies that gravity simply modifies
the pressure di
r6.1.2
·u
= 0.Momentum
(5)
equations
and does nothing to change
the velocity. However, we cannot do this
with the the continuity equation
(4),have
constitutes
the Euler
equations.
Things
if
we
a
free
surface
(as
we
shall
see
later with
water waves).
mations,
this
one
is
sometimes
very
good
and
sometimes
not
so
good.
We
So
far
we
have
more
unknowns
than
equations
(three
velocity co
p a little if we realise that the gravitational
force,
being
conservative,
can
be
Assuming the density is constant means we now have four equatio
er Summer School 2011: Introduction to Low Reynolds Number Locomotion
from Peko Hosoi’s Lecture)
Typical Reynolds numbers
Reynolds Numbers in Biology
ynolds number is dimensionless group that characterizes the ratio of inertial to viscous
t is defined as
Macro⇥U L
UL
Re =
=
µ
world
is the density of the medium the organism is moving through; µ is the dynamic viscosity
medium; is the kinematic viscosity; U is a characteristic velocity of the organism; and L
racteristic length scale. When we discuss swimming biological organisms, we are usually
to creatures that are moving through water (or through a fluid with material properties
se to those of water). This means that the material properties µ and ⇥ are fixed1 and the
s number is roughly determined by the size of the organism.
al, the characteristic size of the organism and the characteristic swimming velocity are
As a rule-of-thumb, the characteristic locomotion velocity, U , in biological organisms is
o L by U L/second e.g. for people L 1 m and we move at U 1 m/s; bugs are about
mm, and they move at about U
1 mm/s; for microorganisms L 100 µm and U
100
Obviously this is a very very very very rough estimate and one does not have to think very
come up with exceptions (as is always the case in biology!). However, it serves as a good
point to estimate the Reynolds numbers for various biological organisms as illustrated in
ch in Figure ??. Note that even for organisms as small as ants, the Reynolds number is
the order of 1 (which is not very low). In this lecture we will focus on Re ⇥dunkel@math.mit.edu
1 which is
meters
lds Numbers in Biology
What happens at low 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
; 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
Low-Re (laminar) flow
ertial (dynamic) pressure ⇤U 2 and viscous shearing str
µU/L can be characterized by the Reynolds number4
Swimming at low Reynolds number
R ⌅ U L⇤/µ = U L/⇥.
Example:
Swimming in water with ⇥ = 10
(
6
m2 /s
• fish/human: L ⌅ 1 m, U ⌅ 1 m/s ⇧ R ⌅ 106 .
R
• bacteria: L ⌅ 1 µm, U ⌅ 10 µm/s ⇧ R ⌅ 10
U L⇥/ ⇥ 1
Geoffrey Ingram Taylor
5
James Lighthill
If the Reynolds number is very small, R ⇥ 1, t
NSE (8) can be approximated by the Stokes equations
0 = µ ⌥2 u
0 = ⌥ · u.
⌥p + f ,
(10
(10
These equations+must
still be endowed
time-dependent
BCs with appropria
initial and boundary conditions, such as ,e.g.,6
Edward Purcell
u(t, x) = 0,
as
Shapere
& Wilczek
p(t, x)
= p⇥1987
, PRL
|x| ⇤⌃ .
(1
Superposition of singularities
2x stokeslet =
symmetric dipole
stokeslet
rotlet
-F
F
r̂ · F
p(r) =
+ p0
2
4⇥r
(8⇥µ) 1
vi (r) =
[ ij + r̂i r̂j ]Fj
r
flow ~
r
1
F
r
2
‘pusher’
r
2
E.coli (non-tumbling HCB 437)
Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS
Results
w
of itswe
“puller”
image.
d. field
Atwalls,
distances
rfocused
<6µ
mon
theadipole
model
overestimates
the
bacterial
flow
field.
(E)
Experimentally
measured flow to
plane 50 µm from the top and bottom
tancesurfaces
2 µm parallel
to sample
the wall. chamber,
(F) Best fitand
force-dipole
model,
(G) residual
field.
Notewhere
the
Bacterial
cell
body,
of the
recorded
∼ 2 and
terabytes
of flow
4
flowmovie
fieldResults
ofdata.
an E. In
coli
“pusher”
decays
much faster,
when
swims close
thecule
surface,
fortothe
length
of t
theflow
mea
this
data
we
identified
∼
10
events when
(non-tumbling HCB 437)rarea bacterium
achieved
by
fittin
theByTo
measured
andminisbest-fit
force
cellsBacterial
swam in the
for > surfaces.
1.5 s.
tracking
decays
labeled,
n
flowfocal
fieldplane
far from
resolve the
the
at
variable
locatio
fluidcule
tracers
in
each
of
the
rare
events,
relating
their
position
of
the
c
decays
of
the
flow
speed
u
with
flow
field
created
by
individual bacteria, we tracked gfp- sion of
flu
m surfaces.
To
resolve
the
minisfield (r >field
8 µm).
and labeled,
velocity to
the
position
and
orientation
of
the
bacterium,
dis
non-tumbling
E.
coli
as
they
swam
through
a
suspenof
the
cell
body
(Fig.
1D)
illustr
walls,
we
dividual
bacteria,
tracked
gfp- overforce
the measured
andaverage
best-fit
dipole we
field
The
the 1C).
specific
fitting
and performing
anwe
ensemble
all tracers,
re- (Fig.
Howeve
sionswam
ofdecays
fluorescent
tracer
particles.
For
measurements
farcharacteristic
from
field
displays
the
1
dipole
length
ℓ =o
of
the
flow
speed
u
with
distance
r
from
thesurfaces
center
i
as
they
through
a
suspensolved
the
time-averaged
flow
field
in
the
E.
coli
swimming
he minismeasure
walls,
we
focused
on
a
plane
50
µm
from
the
top
and
bottom
value
of
F
is cons
movie
dat
However,
the
force
dipole
flow
sig
oftothe
cell
(Fig.
1D) illustrate
that
the
flow
down
0.1%
of body
the mean
swimming
speed V0 =
22 ±
5 measured
ticles.
For
measurements
far
from
ckedplane
gfpcell force
bod
surfaces
of
the
sample
chamber,
and
recorded
∼
2
terabytes
of
2
and
resistive
µm/s.
As E.
colidisplays
rotate
about
their
swimming
direction,
their
cells
field
the
characteristic
1/r4 decay
oftoa the
forceside
dipole.
measured
flow
ofswam
thel
a 50
suspenµmmovie
from
the
top
and
bottom
for
the
data.
In field
this in
data
we dimensions
identified ∼is10cylindrically
rare events when
note that
in
the b
time-averaged
flow
three
fluid
trace
However,
the
force
dipole
flow
significantly
overestimates
the
cell1.5body,
where
the
flow
magnit
far from
achieved
ber,
and
recorded
∼
2
terabytes
of
cells
swam
in
the
focal
plane
for
>
s.
By
tracking
the
µm behind the ce
symmetric.measured
Our
measurements
capture
all
components
of
this
flow to when
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
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
and
velocity
to
the
position
and
orientation
of
the
bacterium,
the rotation
of the
cell
about its the
body axis. The topology of
ne for
> 1.5fors.
By
tracking
the
length
of the
flagellar
bundle.
The
force
dipole
fit
was
solved
the
nts
when
the
spec
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)
dipole
le
plane
kingforce
the
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
bacterium,
dipole
flow
descri
at
variable
locations
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
ℓF 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,
[ℓ
1 ]= we
dipole
1.9
and
dip
ow
field
E.
swimming
2
the
specific
fitting
and
obtain
the
note
tha2
|r|
8πηdimensions
|r| is cylindrically
s, we retime-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 ℓ==22
5 capture
symmetric.
Our
measurements
allof
components
this
wimming
and applying
the
cylindrica
and
resistive
force
theory
calculat
fluid
dra
of
Fforce,
is consistent
with
optical
trap
measurements
[45]
where
F isvalue
the dipole
ℓ the
distance
separating
the force
ut
direction,
their
symmetric
flow,
except
the
azimuthal
flow due
to the
= their
22
±cylindrically
5swimming
resulted
in
a
sligh
rotati
note
thatThe
in
the
best
fit,
the
cell
and
resistive
force
theory
calculations
[46].
It is
interesting
to
η the
viscosity
the
fluid,
dˆ the
orientation
vector
theof
flow
field
struc
the
rotation
ofisof
the
cell
about
itsunit
body
axis.
topology
on, pair,
their
three
dimensions
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 offl
dipole
close
to
the
cell
body
as
shown
by
the
residual
of
outward
streamlin
fluid
drag
on
the
flagellar
bundle.
flow. (B) Best fit force-dipole model, and (C) residual flow field, obtained by subtracting the best-fit dipole from the experimentally measured field. The presence of the flagella
E.coli
w due to
weak ‘pusher’ dipole
Volvox carteri
somatic cell
cilia
200 ㎛
daughter colony
Drescher et al (2010) PRL
dunkel@math.mit.edu
Singular
perturbations
y di↵erential equation (Acheson, pp
ntial equation
2
d u
du
✏ 2+
= 1.
dx
dx
t argue that we can neglect this term, the s
u = x + C.
e,the
using
the Cauchy-Riemann
we see that
Z-plane
uniform flow pastequations
a more wing-like
shape? (Note that we have
d some technical details here,
such2as the requirement that dF/dz 6= 0 at any
2
@
@
@
@
would cause+a blow-up
of the +
velocity).= 0.
=
(6)
2
2
@x
@y
@x@y @y@x
Conformal mappings
le
conformal
maps
e used
for . We
can therefore consider any analytic function (e.g.,
he real
map
is and imaginary parts and both of them satisfy Laplace’s equaZ = F (z) = z + b,
(12)
onents u and v are directly related to dw/dz, which is conveniently
onds to a translation. Then there is
dw
@
@
=
+Zi = F=
u= ze
iv.i↵ ,
(7) (13)
(z)
dz
@x
@x
consider
at anangle
angle↵.↵ to
The
corresponding
onds to auniform
rotationflow
through
In the
this x-axis.
case, the
complex
potential for
i↵ . Using the above relation,
wast
= au0cylinder
ze i↵ . In
this
case
dw/dz
=
u
e
0
making angle ↵ with the stream is
u0 cos ↵ and v = u0 sin✓↵.
◆
2
R past
i
mine the complex
potential
for
since
we
know
that
i↵ flow
i↵ a cylinder
W (Z) = u0 Ze
+
e
ln Z.
(14)
Z
2⇡
✓
◆
R2
=
u
r
+
cos
✓,
(8)
i↵
0
s expression could also include
the term ln e = i↵ which I have neglected.
r
constant however and doesn’t change the velocity.
al
the complex
potential
re part
is theofnon-trivial
Joukowski
transformation,
✓
◆
2
2
completely controls the dynamics. In the process of deriving this result we
rn about a rather remarkable phenomenon in rotating fluid dynamics.
Rotating flows
The Taylor-Proudman theorem
r a fluid rotating with angular velocity ⌦. The equation of motion in the fram
e rotating with the fluid is
@u
1
+ u · ru + ⌦ ⇥ (⌦ ⇥ r) =
rp⌦ + ⌫r2 u
@t
⇢
r · u = 0.
2⌦ ⇥ u,
re two additional terms: the first ⌦ ⇥ (⌦ ⇥ r) is the centrifugal acceleration, w
e discussed before. This can be thought of as an augmentation to the pre
tion, using the identity
Taylor columns,
1 etc
⌦ ⇥ (⌦ ⇥ r) =
2
r(⌦ ⇥ r)2 .
rth, we will simply absorb this into the pressure by writing
p = p⌦
⇢
r(⌦ ⇥ r)2 .
http://ocw.mit.edu/
Taylor - column
Solitons
KdV equation
Solitons
credit: Christophe Finot
the droplet surfaces to produce highly active two-dimensional
nematic liquid crystals whose streaming flows are controlled by
internally generated fractures and self-healing, as well as unbinding
and annihilation of oppositely charged disclination defects. The
resulting active emulsions exhibit unexpected properties, such as
autonomous motility, which are not observed in their passive analogues. Taken together, these observations exemplify how assemblages of animate microscopic objects exhibit collective biomimetic
Active matter
a
+
+
b
PEG
Depletion
force
Microtubules
Kinesin clusters
+
Motor Time
force
c
Dogic lab (Brandeis) Nature 2012
d
to these d
bundled m
ters simul
thus enha
Motorrelative p
microtub
between m
Active matter
Dogic lab (Brandeis) Nature 2012
Active nematics
REVIEW LETTERS
week ending
31 MAY 2013
FIG. 2 (color online). Defect pair production in an active
suspension of -1/2
microtubules and kinesin +1/2
(top) and the same
phenomenon observed in our numerical simulation of an extensile nematic fluid with " ¼ 100 and ! ¼ "0:5. The experimental picture is reprinted with permission from T. Sanchez et al.,
Giomi
et
al
PRL
2012
Nature (London) 491, 431 (2012). Copyright 2012, Macmillan.
•
optical effects
•
work hardening, etc
Π2 (S 2 ) = Z.
(7)
Here, the 2 subscript says that we’re studying the second
Homotopy group. It represents the fact that we are surrounding the defect with a 2-D spherical surface, rather
number:
FIG. 12: (a) Hedgehog defect. Magnets have no line defects (you can’t lasso a basketball), but do have point defects.
⃗ (x) = M0 x̂. You can’t
Here is shown the hedgehog defect, M
surround a point defect in three dimensions with a loop, but
you can enclose it in a sphere. The order parameter space, remember, is also a sphere. The order parameter field takes the
enclosing sphere and maps it onto the order parameter space,
wrapping it exactly once. The point defects in magnets are
categorized by this wrapping number: the second Homotopy
group of the sphere is Z, the integers.
(b) Defect line in a nematic liquid crystal. You can’t
lasso the sphere, but you can lasso a hemisphere! Here is the
defect corresponding to the path shown in figure 5. As you
pass clockwise around the defect line, the order parameter rotates counterclockwise by 180◦ .
This path on figure 5 would actually have wrapped around
the right–hand side of the hemisphere. Wrapping around the
left–hand side would have produced a defect which rotated
clockwise by 180◦ . (Imagine that!) The path in figure 5 is
halfway in between, and illustrates that these two defects are
really not different topologically.
(
Finally, why are these defect categories a group?
group is a set with a multiplication law, not necess
FIG. 13: Multiplying two loops. The product of two loo
is given by starting from their intersection, traversing the fi
loop, and then traversing the second. The inverse of a lo
is clearly the same loop travelled backward: compose the t
and one can shrink them continuously back to nothing. T
definition makes the homotopy classes into a group.
This multiplication law has a physical interpretation. If t
defect lines coalesce, their homotopy class must of course
given by the loop enclosing both. This large loop can
deformed into two little loops, so the homotopy class of t
coalesced line defect is the product of the homotopy clas
of the individual defects.
Two parallel defects can coalesce and heal, even thou
each one individually is stable: each goes halfway arou
the sphere, and the whole loop can be shrunk to zero.
Π1 (RP 2 ) = Z2 .
than the 1-D curve we used in the crystal.[13]
You might get the impression that a strength 7 def
is really just seven strength 1 defects, stuffed togeth
You’d be quite right: occasionally, they do bunch
but usually big ones decompose into small ones. T
doesn’t mean, though, that adding two defects alwa
gives a bigger one. In nematic liquid crystals, two li
defects are as good as none! Magnets didn’t have a
line defects: a loop in real space never surrounds so
thing it can’t smooth out. Formally, the first homoto
group of the sphere is zero: you can’t loop a basketb
For a nematic liquid crystal, though, the order para
ter space was a hemisphere (figure 5). There is a lo
on the hemisphere in figure 5 that you can’t get rid
by twisting and stretching. It doesn’t look like a lo
but you have to remember that the two opposing poi
on the equater really represent the same nematic orie
tation. The corresponding defect has a director field
which rotates 180◦ as the defect is orbited: figure 1
shows one typical configuration (called an s = −1/2
fect). Now, if you put two of these defects together, th
cancel. (I can’t draw the pictures, but consider it a ch
lenging exercise in geometric visualization.) Nematic li
defects add modulo 2, like clock arithmetic in elementa
school:
Topological defects
are discontinuities in
order-parameter fields
"umbilic defects" in a nematic liquid crystal
“Quantum” HD
Couder lab (Paris)
Bush group