Fluid Dynamics 3 - 2012/2013
Jens Eggers
Preliminaries
Course information
• Lecturer: Prof. Jens Eggers, Room SM2.3
• Timetable: Weeks 1-12, Monday 4.00 (SM2), Thursday 4.00 (SM2) and Friday 10.00
(SM1). Course made up of 32 lectures.
• Office hours: my office hours for this course are Tuesday 11.00 in my room, 2.3
• Prerequisites: Mechanics 1, APDE2 and Calc2. Need ideas from vector calculus,
complex function theory, separation solutions and PDE’s.
• Homework: Questions from 10 worksheets set and marked during the course. Will
be handed out each Friday, starting in the first week. Solutions to be returned the
following Friday in the box marked “Fluids3”.
• Assessment: 2.5 hours exam in April, best 4 out of 5 questions. No calculators
allowed.
• Credit points: will be awarded for serious attempts at 40% of homework. Only
comes into play for exam marks in the range 30-40.
• Web: Standard unit description includes detailed course information. Also a fluids
3 web page (http://www.maths.bris.ac.uk/~majge/fluids3.html), which contains: Homework and solution sheets as well as lecture notes (will be put up on the
web as soon as possible after each lecture).
• Lectures: There is no need to take notes during the lecture, as all material relevant
for the exam will be put on the web. The main purpose of the lecture is the live
development of the material, and a chance for you to ask questions!
Recommended texts
1. A.R. Paterson, A First Course in Fluid Dynamics, Cambridge University Press.
(The recommended text to complement this course - costs ≈ £50 from Amazon;
there are 6 copies in Queen’s building Library and 3 copies in the Physics Library)
2. D.J. Acheson, Elementary Fluid Dynamics. Oxford University Press
3. L.D. Landau and E.M. Lifshitz, Fluid Mechanics. Butterworth Heinemann
Films
There is a very good series of educational films on Fluid Mechanics available on YouTube,
produced by the National Committee for Fluid Mechanics Films in the US in the 1960’s.
Each film is also accompanied by a set of notes. I recommend them highly, and will point
out the appropriate ones throughout this course.
The following 3 sections are useful for the course. For the purposes of an examination, I
would expect you to know the definition of div, grad, curl and the Laplacian in Cartesians
and grad and the Laplacian in plane polars. Other definitions would be provided.
Revision of vector operations
Let u = (u1 , u2 , u3), v = (v1 , v2 , v3 ) be Cartesian vectors. Let φ(x) be a scalar function
and f(x) = (f1 (x), f2 (x), f3 (x)) a vector field of position x = (x, y, z) ≡ (x1 , x2 , x3 ). Then
• The dot product is u · v = u1 v1 + u2 v2 + u3 v3
• The cross (or vector) product is u×v = (u2 v3 −u3 v2 )x̂+(u3v1 −u1 v3 )ŷ+(u1 v2 −u2 v1 )ẑ
∂φ ∂φ ∂φ
,
,
• The gradient is ∇φ =
∂x1 ∂x2 ∂x3
∂f2
∂f3
∂f1
+
+
∂x1 ∂x2 ∂x3
∂f3
∂f2
∂f3
∂f1
∂f1
∂f2
• The curl is ∇ × f =
−
−
−
x̂ +
ŷ +
ẑ
∂x2 ∂x3
∂x3 ∂x1
∂x1 ∂x2
• The divergence is ∇ · f =
∂2φ ∂2φ ∂2φ
+
+
∂x21 ∂x22 ∂x23
∂f1
∂f2
∂f2
∂f3
∂f3
∂f2
∂f3
+ f3 ∂x
+
f
+
f
+
f
+
f
x̂+
f
ŷ+
f
ẑ
2
3
2
3
1
1
∂x1
∂x2
∂x3
∂x1
∂x2
∂x3
3
• The Laplacian is ∇2 φ = ∇ · ∇φ =
∂f1
∂f1
• (f·∇)f = f1 ∂x
+ f2 ∂x
1
2
Formulae in cylindrical polar coordinates
Coordinate system is x = (r, θ, z) where the relationship to Cartesians is x = r cos θ,
y = r sin θ. The unit vectors are r̂ = x̂ cos θ + ŷ sin θ, θ̂ = −x̂ sin θ + ŷ cos θ and ẑ. In the
following, f = (fr , fθ , fz ) ≡ fr r̂ + fθ θ̂ + fz ẑ.
• The gradient is ∇φ = r̂
1 ∂φ
∂φ
∂φ
+ θ̂
+ ẑ
∂r
r ∂θ
∂z
1 ∂(rfr ) 1 ∂fθ ∂fz
+
+
r ∂r
r ∂θ
∂z
r̂
r θ̂
ẑ
∂/∂r ∂/∂θ ∂/∂z .
fr
rfθ
fz
• The divergence is ∇ · f =
1 • The curl is ∇ × f = r
1 ∂2φ ∂2φ
∂ 2 φ 1 ∂φ
+
+
+ 2
∂r 2
r ∂r r 2 ∂θ2
∂z
f2
r
r
r
θ
θ
θ
+ frθ ∂f
+ fz ∂f
− rθ r̂ + fr ∂f
+ frθ ∂f
+ fz ∂f
+
• (f · ∇)f = fr ∂f
∂r
∂θ
∂z
∂r
∂θ
∂z
z
z
z
fr ∂f
+ frθ ∂f
+ fz ∂f
ẑ
∂r
∂θ
∂z
• The Laplacian is ∇2 φ =
fr fθ
r
θ̂ +
Formulae in spherical polar coordinates
Coordinate system is x = (r, θ, ϕ) where the relationship to Cartesians is x = r sin θ cos ϕ,
y = r sin θ sin ϕ, z = r cos θ.
The unit vectors are r̂ = x̂ sin θ cos ϕ + ŷ sin θ sin ϕ + ẑ cos θ, θ̂ = x̂ cos θ cos ϕ +
ŷ cos θ sin ϕ − ẑ sin θ and ϕ̂ = −x̂ sin ϕ + ŷ cos ϕ.
In the following, f = (fr , fθ , fϕ ) ≡ fr r̂ + fθ θ̂ + fϕ ϕ̂.
• The gradient is ∇φ = r̂
1 ∂φ
1 ∂φ
∂φ
+ θ̂
+ ϕ̂
∂r
r ∂θ
r sin θ ∂ϕ
1 ∂(sin θfθ )
1 ∂fϕ
1 ∂(r 2 fr )
+
+
r 2 ∂r
r sin θ
∂θ
r sin θ ∂ϕ
r̂
r
θ̂
r
sin
θ
ϕ̂
1 .
• The curl is ∇ × f = 2
∂/∂r
∂/∂θ
∂/∂ϕ
r sin θ fr
rfθ
r sin θfϕ ∂2φ
1
∂
∂φ
1
1 ∂
2 ∂φ
2
r
+ 2
sin θ
+ 2 2
• The Laplacian is ∇ φ = 2
r ∂r
∂r
r sin θ ∂θ
∂θ
r sin θ ∂ϕ2
2
fθ2 +fϕ
fϕ ∂fr
fϕ ∂fθ
fθ ∂fr
r
θ
θ
+
+
−
+ frθ ∂f
+ r sin
+ frrfθ −
r̂+ fr ∂f
• (f·∇)f = fr ∂f
∂r
r ∂θ
r sin θ ∂ϕ
r
∂r
∂θ
θ ∂ϕ
fϕ ∂fϕ
fr fϕ
fθ fϕ cot θ
fθ ∂fϕ
ϕ
fr ∂f
+
+
+
+
ϕ̂
∂r
r ∂θ
r sin θ ∂ϕ
r
r
• The divergence is ∇ · f =
fθ2 cot θ
r
θ̂+
1
1.1
Introduction & Basic ideas
What is a fluid ?
In this course we will treat the laws governing the motion of liquids and gases. A liquid
or gas is characterised by the fact that there is no preferred rest state for the parts it
is composed of. If a hole is made in a water bottle, the water will flow out. If a drop
is placed on a solid surface, it will spread. By contrast, a solid retains a memory of its
original state. If one deforms a piece of metal, it will relax back to its original state once
the force is no longer applied. Collecetively, we will call liquids or gases, which share this
fluid property, “fluids”.
Another important property of liquids and gases is that they are “featureless”. Viewed
from a particular point in space, all directions are equivalent. Solids, on the other hand,
often have an internal (lattice) structure. As a result, it makes a difference in which
direction they are deformed relative to their internal structure.
Fluid dynamics is an example of ‘continuum’ mechanics:
Defn. A continuum is any medium whose state at a given instant can be described
in terms of a set of continuous functions of position x = (x, y, z). E.g. Density, ρ(x, t),
velocity, u(x, t), temperature T (x, t). (These functions usually also depend upon time,
t).
We know that this description fails if we observe matter on small enough length scales:
e.g. typical molecule size (∼ 10−9 m) or typical mean free path in a gas (∼ 10−7 m). The
miracle is that on a scale only slightly larger than that, all microscopic features can be
ignored, and we end up with a “universal” description of all things fluid. We will also
ignore all effects of incompressibility, so for the purposes of this course, fluids and gases
are the same thing. We will almost always speak of a fluid, but which can mean either a
fluid or a gas for the purposes of this course.
The motion of fluids and gases is governed by the same underlying principles
1.2
Viscosity
plate moving at speed U
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x
δ
u(y) = γ̇y
tangential force F
plate area, A
x
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stationary plate
A very important approximation we will be making throughout this course is the neglect
of internal friction (or viscosity) of the fluid. For example, consider two plates of area
A being moved at speed V relative to each other (see figure). It is common sense that
a certain force is necessary to maintain this motion, just like it takes a certain force to
rub one hand against the other, and there is friction between the two. In the case of a
fluid, the origin of this force is that molecules “rub” against each other. This way a force
is transmitted from one plate to the other. One expects that the necessary shear force is
proportional to
1. the speed U
2. the area A
and inversely proportional to
3. the distance δ between the plates.
Thus the force is given by
UA
,
(1)
δ
where the constant of proportionality η is called the viscosity η of the fluid. In this
course we will neglect all viscous effects, we will pretend that the viscosity is small. Note
that what constitutes a small viscosity depends on the dimensions of the system: if δ
is sufficiently small, the force can be made to be significant, no matter how small the
viscosity! There is a wonderful way to avoid further discussion of the matter: if we can
neglect all viscous effects, we speak of Ideal Fluids.
F =η
1.3
What we would like to do
In this lecture course, we will first develop an equation of motion for the velocity field
u(x, t), which gives the fluid velocity at any instant in time t, everywhere in space.
This equation (or set of equations) will necessarily have the form of a partial differential
equation. It will be based on Newton’s equations of motion, but for a continuum of
particles, distributed over space.
With the equations in hand, it is down to our ability to deal with the mathematical
complexities of solving a partial differential equation (PDE) to solve physical problems.
Some examples of problems dealt with rather successfully using the concept of ideal fluids
are the following:
There are many wonderful phe-
nomena described quite well within the “ideal” framework of this course. One of them
are impact problems. The Jesus Christ lizard (see Figure) can walk on water by hitting its
surface with its feet. At each impact, it receives an upward impulse, which compensates
its weight.
Another spectacular success is the theory of flight. The ideal flow of air around a wing is able to describe
the lift necessary for flight, and much more. What is much more difficult is the theory of
drag. Inviscid theory suggests that there should be no energy cost to flight at all!
Another huge (and hugely important) area is the theory of water waves. This includes waves from the scale of millimetres
up to huge tsunami waves. The absence of any solid boundary results in very little friction,
so the ideal theory works very well.
1.4
Outside the realm of ideal fluids
Of course in reality there is always at
least a little bit of viscosity, so a viscous description will always be more accurate than an
inviscid one. If the viscosity is large (think for example of Lyle’s golden syrup), a theory
based on neglecting viscosity is of course spectacularly wrong. We will not be able to
describe the pouring of syrup shown in the picture (note the interesting shape of bubbles
trapped inside the syrup)! The same is true if the geometry under consideration is small
and confined. Then there is a lot of shearing going on, and frictional forces are large.
Unfortunately, there
are also many cases where inviscid theory fails even if the viscosity is small, as illustrated
in the figure showing the flow around an obstacle. The fluid flow does not remain attached
to the body, but rather separates from it at the rear. We will see that ideal theory usually predicts the flow lines to hug the body, even in the back. The reason for the failure
of ideal fluid theory is very subtle. There is a very small region near the body (called
the boundary layer), where viscosity becomes important. It is in this small region, not
described properly by inviscid theory, where separation occurs. More on this in Fluids
4...
1.5
Lagrangian and Eulerian descriptions of the flow
We now begin to develop a dynamical description of fluid flow, which will lead us to
formulate a PDE for fluid motion, known as the Euler equation. Before we can do
that, we must understand the motion of fluids a little better. The description of motion
is called kinematics. In this chapter, we will deal with kinematics. I encourage you to
look at the film “Fluid Mechanics (Eulerian and Lagrangian description) parts 1-3” on
YouTube.
There are two very different ways of describing fluid motion, known as the Eulerian
and Lagrangian description. Ultimately, they are equivalent, as they describe the same
thing. However, they serve different purposes, so we need them both.
Eulerian description of the flow.
This is what the stationary observer sees. Choose a fixed point, x to measure, for e.g. the
velocity u(x, t). This provides a spatial distribution of the flow at each instant in time.
This is the way continuum equations are usually formulated, and our equation of motion
will indeed be an equation for the Eulerian field u(x, t). If the flow is steady, then u does
not depend on time, t: u = u(x).
Why do we need anything else? The reason is that to make contact with Newton’s
equations, we need to describe the flow as a moving particle would see it. This is the
Lagrangian description of the flow.
The observer moves with the fluid. Choose a fluid particle (for example, we can place a
small drop of ink in the fluid), and follow it through the fluid. Measuring its velocity at
a given time, t gives its ‘Lagrangian velocity’. Now we describe the whole velocity field
this way, by labelling all material points. A convenient way of doing so is to choose an
initial time t0 , and to label all fluid particle by their position x = a at that time. Then
at time t > t0 , the particle is at
x = x(a, t), where x(a, t0 ) = a.
Of course, x(a, t) is very interesting in its own right. For example, it describes the course
of a balloon, launched at time t0 and at position a into the atmosphere.
The Lagrangian velocity is defined as
dx .
(2)
v(a, t) =
dt af ixed
By definition, it is the velocity of a particle going with the flow. This is precisely how
velocity is defined in Newtonian Mechanics. As we said earlier, the Eulerian and the
Lagrangian velocity fields contain the same information; the relation between the two is:
v(a, t) = u(x(a, t), t).
(3)
An example: Consider logs flowing along a narrowing section of river. A fixed observer
measures the velocity by observing the velocity of logs at a given point in space. By
observing many logs at different positions, he will be able to obtain the entire Eulerian
velocity field u(x, t). Unless the flow conditions are changing, this field will be timeindependent, u(x, t) = u(x). Now imagine each log being ridden by a moving observer,
each of whom reports his velocity as time goes by. If all observers are labelled by their
position a at some reference time t0 , this will produce the Lagrangian velocity field v(a, t).
The logs travel with the fluid and will see the flow accelerating, as the river bed becomes
narrower. Thus v(a, t) is manifestly time-dependent although the Eulerian field is not!
Observer moving with the flow sees the logs
accelerate
Stationary observer sees logs passing at constant speed.
Defn: A flow is two-dimensional if it is independent of one of its components (in some
fixed frame of reference). E.g. u = (u, v, 0).
Example 1: Let us consider a very simple model flow for the river shown above.
Consider the two-dimensional, stationary Eulerian velocity field u = (u, v, 0), defined by
u = kx,
v = −ky.
As can be seen in the Figure, the flow speeds up as the river
contracts, which we can think of as being confined by the river banks, shown as the red
lines.
1.6
Particle paths and streamlines
Defn. The particle paths (pathlines) are the paths followed by individual particles. They
are determined by the solution of the differential equation:
dx
= u(x, t),
dt
with initial condition x(t0 ) = a = (a1 , a2 , a3 )
(4)
where u is assumed given.
This system of equations specifies a unique curve. For some (simple) u, can be integrated using elementary methods, but in general not. Let u = (u(x, t), v(x, t), w(x, t)),
then in components, (4) is

dx

= u(x, y, z, t) 


dt


dy
= v(x, y, z, t)

dt



dz

= w(x, y, z, t) 
dt
with x(t0 ) = a1 , y(t0 ) = a2 , z(t0 ) = a3 .
Example 1: In the example above, the equations become
ẋ = kx,
ẏ = −ky.
The solutions, with initial positions a = (a1 , a2 ) at time t0 = 0, are
x(t) = a1 ekt ,
y(t) = a2 e−kt ,
which describes the path of particles in the flow. Using the relation (3) between Eulerian
and Lagrangian fields, we find
v(a, t) = ka1 ekt , −ka2 e−kt .
In particular, the Lagrangian velocity field is indeed time-dependent, while the Eulerian
field was steady. Now let us consider a case where the Eulerian field is also time dependent.
Example 2: Two-dimensional flow,
u = (y, −x) cos t,
with x(t0 = 0) = (1, 0). As seen in the Figure, this is a vortex whose strength (and sense
of rotation) oscillates.
Then
dx
= y cos t, (∗)
dt
dy
= −x cos t,
dt
which we can combine as.
0=x
dx
dy
d
+y
= ( 21 x2 + 12 y 2 ),
dt
dt
dt
so that the solution is
x2 + y 2 = 1.
Now we know that the particle paths lie on a circle, which is not a surprise when looking
at the velocity field. We still need to find the time dependence x(t) and y(t).
We can describe the curve by
x(t) = cos θ(t),
y(t) = sin θ(t),
and so dx/dt = − sin θθ̇ = −y θ̇ and from (∗) must have θ̇ = − cos t or θ = − sin t + const.
The constant is sin 0 = 0, since θ = 0 at t = 0. Thus the particle paths oscillate around
a circle, with angle
θ(t) = − sin t.
As shown in the Figure below, the particle position oscillates back and forth in a section
y
θ=1, t=3π/2
t=0,π,2π
x
of the circle.
θ=−1,t=π/2
Defn: A streamline of a flow u(x, t) at a given instant in time t0 , is a curve which is
everywhere parallel to u(x, t0 ).
Thus, along a streamline,
dx
= u(x, t0 ).
(5)
dt
Eliminating t, we find
dx
dy
dz
=
= (= dt),
t = t0 .
(6)
u
v
w
Notice that time doesn’t go into the definition of a streamline, and we could have taken
any quantity to parameterise the curve.
In general, a streamline varies with time, but are time-independent for a steady flow.
Streamlines can be visualised by taking a short-time exposure of illuminated particles
in a flow. Taking an arbitrary starting point, one always continues in the local flow
direction. For steady flows, streamlines and particle paths coincide. Evidently, the
defining equations (5) and (4) become the same, if u does not depend on time.
Example 1: u = (kx, −ky). Since this is a steady flow, we could eliminate t from the
pathline to obtain the streamline. Alternatively, we have
dy
dx
=−
kx
kv
according to (6). Integrating, we find ln |x| + ln |y| = C, or
|x||y| = eC ,
which is the equation for a hyperbola. Any such hyperbola is a possible equation for the
river bank, which must be a streamline of the flow.
Example 2: u = (y, −x, 0) cos t.
dy
dx
y
dx
=−
so that
= − and then integrating, x2 + y 2 = const.
At t = t0 ,
y cos t0
x cos t0
dy
x
Thus streamlines are circles, and are different from pathlines, which just cover a section
of a the circle. It is also important to draw arrows into the streamlines, to indicate the
local direction of the flow. This has to be done only once for each streamline, since the
direction cannot reverse; it is enough to compute the velocity at one point along the
streamline. As seen in the Figure below, the direction of flow reverses periodically in
our example. Namely, for θ̇ >, the motion is counterclockwise, otherwise it is clockwise.
0 < t < π/2
y
π/2 < t < 3π/2
x
y
x