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Lecture 1: Introduction 1
Lecture 1. Introduction
Lecture 1: Introduction 2
Plan for today
— c c
— Introduction.
— Functioning of the course, resources, subject matter.
— Macroscopic description of fluid properties. (Chapter 1 of
Arzel’s notes, see also (Batchelor , 2000, Chapter 1))
how the course works 3
Practical information
— I will present 4 courses of 3 hours each.
— You can interrupt me at any time, just raise your hand.
— You can ask me questions via email :
robert.scott@univ-brest.fr
— You can find my slides on my website :
http://stockage.univ-brest.fr/~scott/ along with
other documents, including the notes in French from
previous instructor, hereafter “Arzel’s notes”.
— Other resources
Classic textbooks on fluid mechanics (much behond this
course, but useful references) :
* (Batchelor , 1967; Landau and Lifchitz , 1966)
Background on mathematics (useful for parts of this course,
how the course works 4
and as a reference for the rest of your career) :
* (Boas, 2006; Altland and von Delft , 2019)
how the course works 5
Our Perpetual Ocean !
https://www.nasa.gov/topics/earth/features/
perpetual-ocean.html
how the course works 6
Subject matter of this course
For future reference :
This short course is an introduction to homogeneous,
incompressible fluid mechanics. In particular, we will cover :
— Properties of fluids : continuum hypothesis, forces in fluids,
hydrostatic balance(pressure balances gravity), Archimedes’
principle.
— Description of fluid motion : Eulerian vs. Lagrangian
description, material and Eulerian derivatives.
— Conservation of mass.
— Incompressibility hypothesis.
— The Euler equation (material derivative + pressure force) +
mass conservation.
— The Navier-Stokes equation (I’ll explain the viscous term in
how the course works 7
detail but don’t panic if you are struggling with
mathematics).
— Bernoulli theorems.
— Momentum theorem : computation of forces exerted by a
flow on an object + application to wind propeller (example
Bet’z law
https://en.wikipedia.org/wiki/Betz%27s_law)
— (Timer permitting) Dimensional analysis, scaling, turbulence
and the difficulty of modelling flows.
how the course works 8
Notation and conventions
— I will use inertial reference frames and rectlinear (Cartesian)
coordinates denoted (x, y, z) or (x1, x2, x3) or simply xi with
it understood that the index i takes values 1, 2, 3 or
equivalently values x, y, z. Generally z will be the vertical
direction.
— Vectors are all three dimensional Cartesian vectors (no
distinction between contravariant and covariant
components) and are indicated with an arrow, ~F .
— The velocity ~u = ddt (x(t), y(t), z(t)) = (u, v, w) = (u1, u2, u3).
Fluid mechanics, how it fits into classical physics 9
Classical mechanics
— Classical mechanics is the study of the movement of material
bodies based upon Newton’s laws of motion, especially his
second law
~F = m ~a,
force = mass × acceleration , (1)
and the laws of interaction that give the forces and the
principle of conservation of mass.
— For completeness, we mention that Newton’s first law (from
a modern perspective) is really a statement that there are
special reference frames, called inertial reference frames, in
which the equation of motion takes the above form.
— Newton’s third law will intervene from time to time : for
Fluid mechanics, how it fits into classical physics 10
every reaction there is an equal and opposite reaction.
— Classical mechanics can be roughly divided into two
applications : (1) particle mechanics, and (2) continuum
mechanics.
— Fluid mechanics is a branch of continuum mechanics, where
the continuum in question is a fluid. Perhaps surprisingly
whether the fluid is a gas or liquid does not change the form
of the equations. So we group “gas” dynamics and “liquid”
dynamics into “fluid mechanics”. For this reason,
oceanography and atmospheric science are closely related
fields.
— Thermodynamics, another branch of classical physics, is
involved for the determination of the equation of state of the
fluid (which in turn is necessary to determine the fluid
density) and for calculations of heat flow. But that is not
our focus.
Fluid mechanics, how it fits into classical physics 11
— Newton’s second law, Eq(1), applied to a fluid gives the
Navier-Stokes equations, our most important dynamical
equation.
— Although we can rarely solve the Navier-Stokes equations
exactly in a given application, they are important to
understand as they represent the fundamental physics that
underlies almost everything we will learn in this course.
— Fluid mechanics is a very well developed subject (dating
back several hundred years), with very wide applications.
— In this course we restrict attention to simple fluids
(“Newtonian (simple viscous properties), homogeneous and
incompressible fluids).
Fluid mechanics, how it fits into classical physics 12
Physical properties of fluids
Fluid mechanics, how it fits into classical physics 13
Fluids, liquids, and gases
— Fluid is a general term that includes both liquids and gases.
— Most substances have both stable gas and liquid states, with
the density of the liquid typically 1000 times greater than
the gas at atmospheric pressure. For our purposes the
distinction between a liquid and gas is usually not
fundamental, since liquid and gas flows with small pressure
variation have very similar properties (they are both
incompressible fluid flow).
— A fluid is distinguished from a solid in that solids have their
own shape, but a fluid does not. A fluid rather takes the
shape of its container. The reason we will explain later is
that in a solid there are internal static “surface forces” that
can balance the “volume forces” such as gravity and thereby
Fluid mechanics, how it fits into classical physics 14
hold the solid together. (Surface and volume forces will be
explained in detail below.) A fluid also has internal surfaces
forces but, with the exception of a pure compression, they
only arise when there is relative motion, when the fluid is
deforming.
— Detail : The distinction between solids and fluids
is not a sharp one because there are some materials,
such as paints and jellies, that behave as liquids and
elastic solids depending on the situation.
Fluid mechanics, how it fits into classical physics 15
The continuum hypothesis
— Fluids are composed of molecules, the basic stable unit
(baring chemical reactions).
— We want to apply Eq(1) to a fluid, and in principle we could
apply it directly to the individual molecules. But the vast
number of particles involved makes this impractical.
— Fluid mechanics is greatly fascilitated by the continuum
hypothesis in which we ignore the molecular particle nature
of the fluid and approximate it as a continuous medium.
— To be more precise, consider an imaginary cube with sides of
length `. We could in principle define mean values to
quantities to describe key properties such as the density,
Fluid mechanics, how it fits into classical physics 16
pressure, velocity. For instance density ρ would be defined
ρ =1
`3
N∑i=1
mi, where mi is mass of molecule i
u =1
ρ`3
N∑i=1
mivi, where vi is velocity of molecule i (2)
and N is the number of molecules in the cube. But if the
cube is small relative to, d, the mean free path between
molecules, then N will be small and we expect the mean
quantities to fluctuate wildly because of the random motion
of the molecules. So we want d� ` so that N � 1.
— But if ` ∼ L, where L is the length scale of the fluid flow,
then our averages will smear out important information
about the flow.
— Thus the continuum hypothesis requires that there exists an
Fluid mechanics, how it fits into classical physics 17
` such that
d� `� L, (3)
— In short, when Eq(3) applies, then there will be a very large
number of molecules in the cube, and we expect stable
averages to exist and give well-defined, average, local
quantities describing the properties of the fluid. Assigning
the average to the centre of the cube, say the spatial point P
at time t, we obtain point-like continuous functions
ρ(P, t) density
ui(P, t) velocity
p(P, t) pressure (4)
— The dimensionless number Kn = dL , the Knudsen number,
must be small for the continuum hypothesis to be valid. An
often cited tolerance is Kn . 0.01.
Fluid mechanics, how it fits into classical physics 18
— We are exclusively concerned with the continuum regime.
Detail : In the upper atmosphere Kn . 0.01 is no longer
valid and one is dealing with a rarefied gas.
Body (or volume) and surface forces 19
Body (or volume) forces
— Body (or sometimes “volume”) forces are most easily
described as forces per unit volume.
— These forces are capable of penetrating into the interior of
the fluid and extend over long distances and vary slowly.
— Because the force is slowly varying it acts almost identically
throughout a fluid parcel and indeed on all fluid parcels in a
small volume and thus these forces are most easily described
as forces per unit volume.
— For us the most important example is the force of gravity ~Fg
per unit volume, which, in a gravitational field of g N/kg in
the vertical direction ~k would be at a point P where the fluid
density is ρ(P, t), at time t :
~Fg(P, t) = −gρ(P, t)~k. (5)
Body (or volume) and surface forces 20
— Detail :
In rotating fluid mechanics, the inertial Coriolis force
is another body force of great interest. In plasma
physics, electromagnetic forces are another possible
body force.
Body (or volume) and surface forces 21
Surface forces
— Surface forces are most easily described as forces per unit
area.
— These forces are of molecular origin, decrease rapidly with
distance and require direct molecular contact between the
elements in question for the force to be transmitted.
— For example, two volumes of gas separated by an imaginary
surface will exert a force on each other proportional to the
rate of transport of the momentum across the surface by
migrating molecules.
— In liquids the origins of the surface forces can be more
complex, but the details of the molecular origins need not
concern us here.
— Detail
Body (or volume) and surface forces 22
Because these forces are so short range, a fluid volume
interacting with an adjacent volume will be exposed
to short-range force acting significantly only on fluid
parcels in a thin boundary layer, thickness of order d,
on the surface between the two volumes. Thus these
forces are most easily described as forces per unit
area acting on a planar surface between fluid parcels.
— The total surface force on a fluid volume is obtained by
integrating the surface force per unit area over the surface
bounding the fluid volume.
— The most important examples are pressure and viscosity,
which can both be formulated as stresses (forces per unit
area) using a mathematical tool called “Cartesian second
rank tensors”.
Body (or volume) and surface forces 23
Aside on Cartesian tensors
— Tensors are generalizations of vectors that are indispensible
in fluid mechanics, solid mechanics, and more generally in
mathematical physics and geometry.
— Detail : Cartesian tensors make no distinction between
covariant and contravariant components ; we only need
Cartesian tensors.
— A rank 2 Cartesian tensor has two indices. A rank 1
Cartesian tensor has one index. These are just the Cartesian
vectors you are familiar with, e.g. the velocity ui in a
Cartesian coordinate system, is a Cartesian tensor of rank
one (it has only one index) and can be represented by a
Body (or volume) and surface forces 24
3× 1 matrix,
ui =
ux
uy
uz
(6)
— Rank 2 Cartesian tensors in 3D are represented with a 3× 3
matrix, the first index being the row index and the second
the column index.
— So rank 2 Cartesian tensors look just like matrices and are
multiplied like matrices. The only distinction is that when
the coordinate system changes, say by rotating the axes, the
tensor components change but the tensor itself
remains the same. Recall this is the same distinction we
make between a column of 3 numbers and a Cartesian
vector, say ~Fg = −g~k = (0, 0,−g). If I chose Ox in the
vertical then ~Fg = −g~i′ = (−g, 0, 0). It’s the same force and
Body (or volume) and surface forces 25
the same vector with different components in different
coordinate systems. The same principle applies to tensors.
— The Kronecker delta, δij = 1 when i = j and zero when
i 6= j is a Rank 2 Cartesian tensor represented by the
identity matrix, diag(1, 1, 1).
— The alternator tensor (totally antisymmetric tensor) εijk
takes values +1 or -1 when the indices are an even or odd
permutation of (1,2,3). When the indices are not all different,
εijk = 0. This is most useful to present the cross product
~w = ~u× ~v,
wk = uivjεijk, (7)
where there is an implicit summation over repeated indices,
Body (or volume) and surface forces 26
here i and j. This is the Einstein summation convention,
wk = uivjεijk =3∑
i=1
3∑j=1
uivjεijk. (8)
Body (or volume) and surface forces 27
Table 1 – Vectors and scalars are also tensors – they are quanti-
ties described with real numbers and they are independent of our
coordinate system.
type example tensor rank
scalar temperature, T zero
vector force, Fi one
vector surface element, δAi one
rank 2 tensor stress tensor, σij two
rank 2 tensor Kronecker delta, δij two
rank 2 tensor rate or strain, ∂ui/∂xj two
rank 3 tensor alternating tensor, εijk three
Body (or volume) and surface forces 28
Exercise
Find the 3 components of wk = uivjεijk.
Body (or volume) and surface forces 29
Viscose forces
— We consider a planar surface element in the fluid of area δAj
normal to the direction nja, and specify the i-th component
of the local short-range force exerted on the fluid across the
surface element
δFi(P, t) = σij(P, t)δAj , i = 1, 2, 3, and j = 1, 2, 3. (9)
Here σij is the stress tensor, a second rank Cartesian tensor.
— There are 9 components of the stress tensor, σij , which can
a. recall a surface can be oriented by a unit vector ~n normal to that surface
Body (or volume) and surface forces 30
be represented in a 3× 3 matrix
σij =
σxx σxy σxz
σyx σyy σyz
σzx σzy σzz
. (10)
— Eq(9) implies that σxy is the x component of the force per
unit area on a surface with normal in the y direction. This is
a tangential stress because the force is acting along the
surface.
Contrast this to σxx is the x component of the force per unit
area on a surface with normal in the x direction. This is a
normal stress because the force is acting along the surface.
— More generally, the diagonal components of the stress
tensor, σii, are the normal stresses – the force on the surface
is in the direction of the normal to the surface.
— The off-diagonal components of the stress tensor, σij with
Body (or volume) and surface forces 31
i 6= j, are the tangential stresses – the force on the surface is
orthogonal to the normal to the surface.
[Draw diagram, Arzel Fig 1.5 or draw (Batchelor , 2000)
Figs. 1.3.2]
— But the stress tensor must be symmetric σij = σji (for
otherwise infinitesimal fluid parcels would experience infinite
torque/volume, (Batchelor , 2000, §1.3)), so there are in fact
only six independent components in this matrix.
— It is always possible to choose the orientation of the
orthogonal axes Ox,Oy,Oz such that (a symmetric tensor
such as) the stress tensor has diagonal matrix, say σ′ijb in
b. prime indicates specially choosen coordinate system
Body (or volume) and surface forces 32
this specially choosen coordinate system :
σ′ij =
σ′xx 0 0
0 σ′yy 0
0 0 σ′zz
. (11)
But this special orientation of the orthogonal axes
Ox,Oy,Oz will in general depend upon location within the
fluid domain, so that for a given coordinate system the stress
tensor will generally have non-zero off-diagonal components.
— Conceptually it is useful to know that at a given point in the
fluid, at a given instant in time, the surface forces result in a
superposition of stretching (tension) if σ′ii > 0 and
compression if σ′ii < 0 along three mutually orthogonal
directions.
— The sum of the diagonal components of a tensor (i.e. the
trace of the matrix) is independent of choice of orientation of
Body (or volume) and surface forces 33
the orthogonal axes Ox,Oy,Oz of the coordinate system.
We define the static fluid pressure or thermodynamic
pressure, p, as minus a third of the trace of the stress tensor
for a static fluid :
p = −1
3(σxx + σyy + σzz). (12)
— The static fluid pressure is a normal stress.
[Draw (Batchelor , 2000) Fig. 1.3.3.]
Body (or volume) and surface forces 34
Exercises
1. A static fluid has a completely isotropic stress tensor.
Consider a fluid at rest with uniform static fluid pressure p.
Write the matrix of the stress-tensor that applies
everywhere.
Body (or volume) and surface forces 35
Viscose forces
— The static and mechanical fluid pressures are not the only
stresses in a fluid. The stress tensor in a moving fluid is
generally not isotropic.
— The non-isotropic terms arise from viscosity, a momentum
transportant phenomena that arises from the molecular
motions that were “averaged out” in the continuum
description of a fluid but can be parameterized in terms of
the continuous properties of the macroscopic fluid. The
effect of these viscose terms is to resist the relative motion of
adjacent fluid parcels.
— Consider a simple experiment with two flat parallel plates
separated by a homogeneous layer of fluid of thickness L.
The vertical dimension is suppressed because it plays no
Body (or volume) and surface forces 36
role. Suppose the north plate moves to the east relative to
our coordinate system fixed to the south plate with fixed
velocity U .
[Note to self : draw plane Couette flow.]
— We can anticipate the steady-state result from a few simple
considerations. The fluid within a scale d to the wall will
exchange momentum with the wall via its molecular motion
until eventually the fluid in this layer is at rest relative to
the wall. On a macroscopic scale we conclude the so-called
no-slip boundary condition applies giving
u(x, L) = U,
u(x, 0) = 0. (13)
— What is the velocity profile u(x, y) in between ? We could
equally have chosen our reference frame to be attached to
the moving plate. The only velocity profile that gives a
Body (or volume) and surface forces 37
consistent description in the two reference frames is a profile
of constant shear,
∂u
∂y= constant =
U
L. (14)
— All fluid parcels in the interior find themselves in the same
situation of having fluid moving past them, overtaking them
on their left (facing downstream) and they overtaking fluid
on their right. The microsopic molecular motions will lead to
exchange of momentum between adjacent fluid parcels
producing a stress σxy that ultimately leads to an exchange
of momentum between the two plates, causing a drag-force F
per unit area A that resists the relative motion of the plates.
— The only stress profile that would not tend to change the
velocity profile (or equivalently, the only stress profile
consistent between the two reference frames) is a constant
Body (or volume) and surface forces 38
stress profile σxy :
σxy =F
A. (15)
— We define the dynamic viscosity as the proportionality factor
µ between these
F
A= µ
U
L. (16)
— Experimental data shows that for a diverse range of fluids,
that we shall call Newtonian, this proportionality factor µ is
independent of the velocity shear and other quantities
directly related to the flow (but it can depend on physical
quantities such as the temperature).
— Generalizing this to a non-uniform velocity profile we assume
there is no stress associated with higher-order derivatives of
Body (or volume) and surface forces 39
the velocity so that Eq(16) applies at the fluid parcel level :
σxy = µ∂u
∂y, (17)
— A fluid for which the tensor relation in Eq(17) applies is
called a Newtonian fluid. More complicated relations exist
for non-Newtonian fluids, the study of which is the field of
rheology. We will only consider Newtonian fluids, which
includes water and air at normal temperatures and pressures.
— Returning to the complete stress tensor, the most general
form of relation for a Newtonian fluid consistent with the
analysis thus far is
σij = Aijk`∂uk∂x`
(18)
The second rank tensor ∂uk
∂x`is called the rate of strain
tensor. It will appear again in the term for the acceleration
Body (or volume) and surface forces 40
of a fluid parcel. The fourth rank tensor Aijk` can be shown
(see (Batchelor , 2000) or Arzel’s notes) to be restricted by
symmetry considerations such that Eq(18) for a Newtonian
fluid must be
σij = −pδij + 2µeij −2
3µδijekk, (19)
where eij is the symmetric part of the rate of strain tensor
eij ≡1
2
(∂ui∂xj
+∂uj∂xi
). (20)
— We define the kinematic viscosity ν by
ν =µ
ρ(21)
Body (or volume) and surface forces 41
Table 2 – Physical properties of pure water and air
fluid temp.
(◦C)
density
(kg/m3)
dynamic visco-
sity kg/(m s)
kinematic visco-
sity (m2/s)
water 0 999.9 1.787× 10−3 1.787× 10−6
water 5 1000.0 1.787× 10−3 1.787× 10−6
water 20 998.2 1.002× 10−3 1.004× 10−6
water 100 958.4 0.283× 10−3 0.285× 10−6
air -50 2.04 1.16× 10−5 5.7× 10−6
air 0 1.293 1.71× 10−5 9.2× 10−6
air 20 1.205 1.81× 10−5 15.0× 10−6
air 100 0.946 2.18× 10−5 23.0× 10−6
Body (or volume) and surface forces 42
References
Altland, A., and J. von Delft (2019), Mathematics for Physicists :
Introductory Concepts and Methods, xvi+700 pp pp., Cambridge
University Press.
Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, 615
pp., Cambridge University Press, Cambridge, UK, 615 + xviii pp
+ 24 plates.
Batchelor, G. K. (2000), An Introduction to Fluid Dynamics, 615
pp., Cambridge University Press, Cambridge, UK, 615 + xviii pp
+ 24 plates.
Boas, M. L. (2006), Mathematical methods in the physical sciences,
3rd ed., John Wiley and Sons, New York, 839 + xviii pp.
Landau, L. D., and E. M. Lifchitz (1966), Fluid Mechanics, Course
Body (or volume) and surface forces 43
of Theoretical Physics, vol. 6, 3rd impression of English
translation ed., Pergamon Press, Oxford U.K.
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