Lecture, February 4:
We discussed the structure of the course and there seemde to be a general
wish for something else than standard lecturing. So we agreed that
everybody
should read chapter 4-5 (or at least chapter 4) for next time. We shall
then discuss
the material starting from the questions raised. moreover Pierre will
give a short
derivation of "Laplaces law" (the pressure is equal in all directions)
from Chap. 4
and Rickard will derive "Archimedes law" (Chap. 5).
We discussed some properties of fields - mostly the gravitational field.
Material from
Chap. 3 and 6. The main point was, that gravitation can be formulated
very much like
electrostatics.
It was also announced that Tomas Bohr will have office hour (træffetid)
every Friday
from 13-14.
Lecture, February 11:
We discussed most of Chapter 4-5.
1. Pressure. Definition and dimension: [P] = Force/Length = Energy/Volume.
Hydrostatics. Strangeness of liquids: incompressible but deformable!
Hydrostatic paradox:
a mouse can lift an elephant. Pierre gave a nice presentation of "Pascal's"
law - the pressure
is the same in all directions.
2. The atmosphere: we discussed the evolution of physics from Horror
Vacui to the idea (Torricelli)
of an atmosphere providing a large pressure (around 10.000 kg/cubic
meter!) everywhere. We mentioned
Pascal's discovery that pressure depends on height. we discussed the
structure of the atmosphere and in
particular the variation of temperature with height. Wet and dry adiabatics.
3. Next was buoyancy. Hydrostatic equilibrium. Rickard gave a nice
derivation of Archimedesí law.
4. We then discussed stability of floating bodies. First of bodies
totally immersed in a fluid, where the
conclusion is, at the center of gravity must be vertically below the
center of buoyancy (center of the
displaced fluid). We then looked at simple experiments with a plastic
feta cup (without feta) and a glass jar-
the first was stable and the other unstable, but obviously the center
of buoyancy is way below the center of
gravity in both cases. We introduced formula (5-20) for the stability
and used if for the case of a solid cylinder
immersed in water and thereby roughly checked the observed stability
of the floating objects.
Next time we shall discuss Chapter 7. Everyone should read it. Niels
has promised to tell about
"Newton's bucket" and Iben will tell about the tides.
Lecture, February 18:
we discussed Chapter 7. First the shape of a rotating fluid "Newtonís
bucket", which Niels presented
very clearly. Then Iben presented the theory of how the moon generates
tides very nicely. After the break
Tomas went through the last part of the chapter: the flattening of
the rotating earth. We saw how the
Legendre polynomials can make life easier and heard some of the of
the investigations of shapes of rotating
objects by people like Newton, MacLaurin, Jacobi, Poincare and Cartan.
Next time we shall discuss elasticity. Please read chapters 8-9 so you're ready for stress and strain.
Lecture, February 25:
We discussed Chapters 8 and 9. We introduced a general description
of the stress by a stress tensor, which
columns consists of the components of the stress vector in cuts perpendicular
to the three coordinate
axis. It can almost always be chosen symmetric. The density of force
is given by the divergens of the
stress tensor. Note the different sign conventions for stress and pressure.
Also we introduced a general
description of deformations by a displacement field. The gradient of
this gives a complete description
of a (small) local deformation except that a stress free translation
has been removed. We split the
deformation tensor into a symmetric part (Cauchy) and an asymmetric
part. The asymmetric part
describes a stress free rotation.
Next time we shall discuss Hooke's law (linear elasticity). Please read chapters 10 and 11.
Lecture, March 4:
We discussed Chapter 10. Hooke's law: all the components of the tension
tensor are linear in all the
components of the deformation tensor and visa versa. Out of these 81
coefficients only 18 are
independent. And the number is drastically reduced if the elastic material
has some symmetry. For
instance down to 3 for a cubic crystal and to 2 for a isotropic material:
the two Lame' coefficients \mu
and \lambda and the "reciprocal" Young's modulus E and Poisson's ratio
\nu. The connection between
these two set of elastic constants were established by Peter. An expression
for elastic energy were derived
by Kristoffer. Equations of elastostatics. We combined Hooke's law
and Cauchy's deformation tensor to
give the Navier-Cauchy equation of equilibrium. It is a field equation
for the the displacement field.
The solution gives the deformation and from Hooke's law the the stress.
Cylindrical tubes and spherical
shells under pressure.
Next time we shall discuss deformation of a rod and sound in solids. Please read chapters 11 and 12.
Lecture, March 11:
We discussed bending of a beam. Geometrical moments of inertia. The
Bernoulli-Euler law. Also the
twisting of a rod. We discussed sound in isotropic elastic materials.
Navier's equation yield two (three)
possible solutions: longitudinal and (two) transverse polarized sound
waves. We considered plane
and spherical waves.
Next time we shall discuss fluids. Please read Chap 13.1-4, Chap 14, Chap 15.1-4.
Lecture, March 18.
We started the part of the course concerned with fluid motion.
First we introduced the "Eularian" velocity field: the volocity of
the fluid at a certain space point fixed in
the laboratory. We looked at the meaning of the divergence of the velocity
field: the local change in volume of
a part of the fluid which is "advected" along with the flow. We discussed
the problems of measuring fluid
velocities and defined streamlines, particle trajectories and streaklines
and went through example 13.2.1,
which clarifies matters.
We continued with the formulation of general local conservation laws
and went on to the very important
notion of a "comoving derivative". Finally, we wrote down the Euler
equation - the basic equation of "ideal"
fluid flow.
We derived the wave equation for small amplitude pressure perturbations
and the sound velocity in terms of
the compressibility and the density. In the limit, where density variations
become very small and compressibility
very large, we get the Euler equation for an incompressible fluid in
which the pressure has to be determined by
satisfying the incompressibility constraint - not easy. We showed that
this leads to a Poisson equation for the pressure,
where the right hand side was a quadratic form in the velocity derivatives.
This shows that even distant velocity
changes will influence the pressure, they will contribute like 1/r.
This works as long as all velocities are much smaller
than the speed of sound.
We used the Euler equation to derive Bernoullis theorem and used it
to calculate the relation between velocity and
height increase in a Pitot tube (p. 231). Then we introduced "potential
flow" and solved the case of potential flow around
a sphere by the general methods for azimuthally symmetric solutions
of Laplaces equation using the notes on Legendre
Polynomials with the general solution (30). The boundary conditions
were 1. The flow shopuld approach a constant flow
in the z-direction for large r 2. That there can be no flow through
the sphere. This leads to (14-42). We noted
"díAlemberts paradox": this flow leads to no drag on the sphere.
Finally, we mentioned the control volume methods of chapter 15
and the transport theorem (15-8),
where the most important chaoices of Q are 1. The mass 2 The momentum.
These methods are very
practical if the problem has some symmetry which allows the calculation
of the integrals in (15.8).
We went very rapidly through the example "Jet on a wall" on. p. 247.
For next time, please read chapters 16-17. Christian will go through
the planar flow, p. 283-284 and
Laurits will go through pipe flow, p. 286-290.
Lecture, April 8.
Introduction of viscosity. Ideal fluids lack something, which will
smooth out velocity differences.
We went through Stokesí formula for the viscous stress in a
fluid and intorduced the coefficient of
"dynamic viscosity" m with the dimension of Pascal s. Usually the more
relevant one is the "kinematic viscosity"
n = m/r. This has the dimension m2/s and
thus acts like a diffusion coefficient. We deirived the Navier-Stokes
equation and the corresponding boundary conditions: no-slip on a solid
boundary and only normal stress on a
free surface. We included surface tension.We noted that these are different
from the ones for the Euler equations
and they will give riser to "boundary layers".
We introduced the dimensionless Reynolds number Re=UL/n
and gave examples of how we can use it to make
strong predictions e.g. that the drag force on an object can be written
FD = r U2 S f(Re). We
looked at the flow
around a circular cylinder for varius Reynolds numbers and saw how
the drag suddenly drops at the "drag crisis".
We started looking at exact solutions of the Navier-Stokes equation.
Christian discussed planar flow and Laurits
dicussed pipe flows. Next time we shall continue, and especially discuss
flows at low Reynolds numbers.
Lecture, April 15.
We covered 17.5, 18.1 and 19 + the paper "Life at low Reynolds numbers"
by Purcell.
That was alot and it was quite strenuous - at least for the lecturer.
The coverage of the
Purcell paper was very incomplete and we urge everyone to read that
delightful work.
Main points of lecure:
1. Taylor-Couette flows: an exact solution of the Navier-Stokes equations
for the flow between two concentric rotating cylinders.
Note that in this case (in contrast to e. g. pipe flows) the nonlinear
term does not vanish.
Gymnastics in cylindcoordinates. The Centrifulgal Instability: The
Rayleigh criterion. Taylor vortices.
2. Stokes 2nd problem: tranverse oscillations in fluids. Again an exact
solution of the full Navier-Stokes equations. Result:
Transverse oscillations are strongly damped and there is a specific
phase relation betwee drive and friction.
3. Stokes 1st problem: the drag force on a sphere moving slowly through
a fluid. In this case, we do not have a solution of the
full Navier-Stokes equations, but only their linearization valid for
Re -> 0. We showed that the solution (19-7) satisfied the
bondary conditions and calculate the drag-force to get Stokes' famous
result F = 6 p h r U .
4. Life at low Reynolds numbers. Linearity. The "Scallop theorem":
you cant swim at low Reynolds numbers by reciprocal motion.
Swimming by a rotating flagellum: slewing. You use the fact that the
drag force on a thin cylinder is larger transverse that along a flow,
even in the limit Re -> 0. The ratio is only around 1.5 so the effeciency
of this kind of motion is extremely low - around 1%. Still an
oranism like E. Coli uses only a small fraction of its energy
on swimming. It's like "driving a Datsun in Saudi Arabia".
Lecture, April 22.
We discussed surface waves. First we talked a little about wave vector
and wave length, frequency, phase velocity
and group velocity. Dispersion. Gave a simple derivation of the velocity
for shallow water (gravitational)
waves. Could explain why waves allways arrive at a beach with the wave
front parallel to the beach. Considered the
theory of Gerstner for Gravitational water waves on deep water.
Lecture, April 29.
We discussed surface waves again. This time including surface tension.
Capillary waves. We derived the phase
velocity for the surface waves assuming that the flow was a potential flow.
We solved the Laplace equation connected
to the flow in two different ways. The method of separating the variables
and the method of conformal mapping.
Surface tension. Minimal surfaces. Rayleigh-Plateau instability.
Lecture, May 6.
We discussed boundary layer theory following Chapter 22. After quite
a thorough derivation of Prandtlís boundary layer
equations, we looked at the behaviour of the boundary layer over a
flat plate. The scaling assumption (which leads to some nasty
substitutions) gives a single, very nonlinear ODE leading to the "Blasius"
boundary layer. We discussed the concept of separation,
and noted that it is a very wide-spread phenomenon, but not very well
understood. Finally we very biefly discussed von Karmanís
averaging technique (Chap. 22.5), which, even using a very crude profile
like (22-30) leads to excellent results.
Lecture, May 13.
We discussed whirls and vortices. Circulation and vorticity. Kelvins
circulation theorem. Helmholtz vortex theorems.
The Rankine vortex. Einsteins tea cup problem.
Rotating fluids. The Coriolis force. The weather and sea currents.
The smoke ring machine.