The revision tutorial for the course is on Thursday May 10, 12:00 in
Hall D in the Arts building (usual time and place for the
Thursday lecture)
A schedule of content covered and exercises
set will
gradually appear
below (by week).
Here are some links to last year's version of the
lecture notes (djvu/PDF)
The djvu file is smaller and higher quality than the PDF file, so use that
if possible (the content is the same).
These notes are not a replacement for your own notes, but hopefully
they are useful. Any update to these notes will be posted here and given
out in a lecture or tutorial.
If you discover any errors in the notes, or if you think something is
missing or unclear, please let me know
(joost-dot-slingerland-at-thphys-dot-nuim-dot-ie).
Solutions to the first 5 assignments are
available here (pdf)
Solutions assignments 6 to 8 are here (pdf)
Week 1:
Notes:
We introduce the most basic variables for a fluid (density and velocity),
and derive the continuity equation which relates these and
ensures mass conservation. We then introduce body forces and surface
forces and the concept of stress.
This material is covered on pages 1-6 of my hand-written notes - see
the top of this page for a link to these notes.
Week 2:
Notes:
We analysed stress in some detail, showing how it depends on the direction
of the surface element that it acts on, and introducing the stress tensor.
We found a general equation for the total
force on a region in the liquid. This material is covered on pages 6-9 of my hand-written notes - see
the top of this page for a link to these notes.
Exercises: Please find assignment 1
here
Please hand in your solutions no later than Friday, February 17, 12:05 am.
Week 3:
Notes:
We noted that the stress reduces to pressure when the fluid is stationary
or inviscid (frictionless).
We looked at the force balance equation
for static fluids, which simply says that the total force is zero, or
equivalently that the surface forces (due to pressure variations) cancel
the external body forces. We applied the equation to find the pressure in
some important examples: Notably, a liquid in the absence of body forces
(gives constant pressure), an incompressible liquid in the presence of a
constant downward force, e.g. gravity, which is the simplest model of the ocean
(this give pressure linearly increasing with depth) and an ideal gas in
the presence of a uniform downward force, which is the simplest model of
the atmosphere (This gives exponentially decreasing pressure and density
with altitude). This material is covered on pages 10-11 of my hand-written notes - see
the top of this page for a link to these notes.
Exercises: Please find assignment 2
here
Please hand in your solutions no later than Friday, February 24, 12:05 am.
Week 4:
Notes:
We show that in the general case of a static fluid subject to a
conservative body force (which is the gradient of a potential), the
surfaces of constant density, constant pressure and constant potential
in the fluid coincide.
We started on moving fluids, introducing the
stream derivative and deriving the equations of motion for
fluids, with special attention for Euler's equation for inviscid fluids.
This material is covered on pages 12-13 and 16-19 of my hand-written
notes - see
the top of this page for a link to these notes.
Notes:
As a last hurrah on static fluids, we derived a very general version
of Archimedes' principle (relating the buoyant force on an immersed
object to the weight of the displaced fluid).
We then continued with moving inviscid fluids. Bernoulli's principle,
(relating velocity to pressure) was introduced and some of its
applications discussed. We then proved a Bernoulli equation quantifying
the principle for steady incompressible flow.
We also introduced the vorticity (curl of
the velocity) and the concept of irrotational flow (aka potential flow,
aka vorticity free flow) and the velocity potential.
This material is covered on pages 14-15 and 20-22 of my hand-written
notes - see the top of this page for a link to these notes.
Notes:
We spent some more time with potential (irrotational) flow. For
incompressible fluids, we saw that the velocity
potential satisfies Laplace's equation (due to continuity).
We also proved a very general Bernoulli equation for irrotational flow.
Given the velocity potential this fixes the pressure. We
noted some properties of potential flow, such as
the fact that closed flow lines do not occur for potential flow.
We then started studying the time dependence of the vorticity. This material is covered on pages 20-26 and 28 of my hand-written
notes - see the top of this page for a link to these notes.
Week 7:
Notes:
We derived the vorticity equation for inviscid fluids by taking the curl
of the Euler equations and we looked at various simplifications which
occur when the fluid is barotropic, incompressible, subject to
conservative forces etc. In particular, we showed that, in
two-dimensional incompressible flow subject to conservative forces, the
vorticity is conserved.
We also introduced circulation and its relation to vorticity and
we stated Kelvin's theorem on the conservation of vorticity in
inviscid fluids. This material is covered on pages 29-32 of my hand-written
notes - see the top of this page for a link to these notes.
Notes:
We proved Kelvin's theorem on the conservation of the circulation of a
curve that moves with an inviscid barotropic fluid, subject to
conservative forces. Kelvin was very interested in vortex rings (he
thought they could be used to describe atoms). His theorem
implies that such rings are very stable in nearly inviscid fluids.
Examples are smoke
rings, such as those blown
by volcanoes, and the bubble
rings that dolphins like to blow and play
with. In both cases the fluid (are or water) rotates around the
ring. In a smoke ring, the smoke just serves to make the ring visible.
Other types of vortices can also remain stable for quite a long time.
Have a look at this tornado-like
sea spout recently observed near Bray!
We also showed that, if the vorticity is zero
everywhere, the circulation vanishes for any curve that bounds a surface
in the fluid where the flow is differentiable, and vice versa, if the
circulation of all curves vanishes then the vorticity is zero
everywhere. In combination with Kelvin's theorem, we then see that
vorticity free flow stays vorticity free (in inviscid barotropic
fluids).
We also studied two-dimensional
incompressible flow and introduced the stream function.
We showed that the stream lines of the flow are the
level curves of this stream function ψ
(the curves given by the equation ψ(x,y)=c for some constant c).
We also showed that the mass flow between two points in the plane is
determined by the density of the fluid and by the difference of the values
of the stream potential in those points.
We then considered
two-dimensional incompressible and irrotational flow. For such a flow,
both a stream function and a velocity potential exist and we showed that
these two are related by the Cauchy-Riemann equations, the same equations
that relate the real and imaginary components of a complex-differentable
function. This motivated the introduction of the complex velocity
potential, which combines the ordinary velocity potential and the
stream function.
in last year's notes (linked above), you can find some simple examples of
flows described by complex potentials, for example flow between two walls at an angle, flow
from a source or into a sink, flow in a vortex (with the special property
that its vorticity is zero everywhere except at the center) and flow
around a cylinder.
This material is covered on pages 33-38 of my hand-written
notes - see the top of this page for a link to these notes.
Notes:
We worked on the flow of fluids with friction, or in other words, viscous
fluids. We introduced the shear viscosity η and bulk viscosity ξ.
We gave an expression for
the stress tensor of a Newtonian fluid in terms of the velocity and
η and ξ. We then
derived the Navier Stokes equations from the general equations of motion
for a fluid by substituting the Newtonian stress tensor. We also looked
at the special form the Navier-Stokes equation takes for
incompressible fluids (where the bulk viscosity term drops out). This material is covered on pages 42-46 of my hand-written
notes - see the top of this page for a link to these notes.
Exercises:
I recommend problem 6 from Charles Nash's notes. This can be
found here.
After week 9, we had our Easter break (there was also no Friday lecture in
week 8 due to Good Friday)
Week 10:
Notes:
We show how the vorticity equation is modified in the presence of
viscosity; there is an extra term in the vorticity equation
which makes the vorticity diffuse (spread out and even out).
We gave a general treatment of unidirectional viscous incompressible flow,
reducing the Navier-Stokes equation to as simple a form as possible. We
also introduced the non-slip boundary condition for viscous flow,
which says that the fluid at any solid boundary has the same velocity as
the solid boundary (in particular the velocity is zero at stationary
boundaries). We then looked at specific examples, notably flow in
an infinitely deep channel with potentially one moving boundary
(the simplest case where the boundaries are both stationary and the
pressure is contant is called Couette flow). This material is covered on pages 46-50 of my hand-written
notes - see the top of this page for a link to these notes.
Notes:
We consider viscous flow through a cylindrical tube (also called
Poiseuille flow). We then introduce the Reynolds number and its relation
to turbulence. This material is covered on pages 51-57 of my hand-written
notes - see the top of this page for a link to these notes.
Notes:
We spent a bit more time with the Reynolds number and discussed its
relation to turbulence. We also calculated the velocity of sound for a
barotropic inviscid fluid, such as air (the last subject will not be
examined). Some interesting stuff
related to the Reynolds number can be found online, for example, here is a discussion on life at low Reynolds number. There are lots
of videos on flow at high Reynolds number, such as this one, which
shows the transition to turbulence, as well as some great ones about low
Reynolds number laminar flow, like this one
(must be seen to be believed)
The material on the Reynolds number is covered on pages 51-57 in my
hand-written notes - available using the link at the top of this
page.
For an alternative treatment of some of this material, I
recommend paragraphs 13, 14 and 15 of the
lecture notes by Charles Nash.