Lecture 1. Introductionstockage.univ-brest.fr/~scott/ENSTA/RAN3/2019/ENSTA_RAN3_1.pdf · This short course is an introduction to homogeneous, incompressible uid mechanics. In particular,

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 :

[email protected]

— 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,


and as a reference for the rest of your career) :

* (Boas, 2006; Altland and von Delft , 2019)


Our Perpetual Ocean !

https://www.nasa.gov/topics/earth/features/

perpetual-ocean.html


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


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.


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


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.


— 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).


Physical properties of fluids


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


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.


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,


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


` 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.


— 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)


— 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.


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


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”.


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


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


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,


here i and j. This is the Einstein summation convention,

wk = uivjεijk =3∑

i=1

3∑j=1

uivjεijk. (8)


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


Exercise

Find the 3 components of wk = uivjεijk.


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


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


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


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


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.]


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.


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


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


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


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


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


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)


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


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


of Theoretical Physics, vol. 6, 3rd impression of English

translation ed., Pergamon Press, Oxford U.K.

Documents

Lecture 1. Introductionstockage.univ-brest.fr/~scott/ENSTA/RAN3/2019/ENSTA_RAN3_1.pdf · This short course is an introduction to homogeneous, incompressible uid mechanics. In particular,