so it's a rainy gloomy day this
afternoon
and what better weather to study the
worst most baddest awful no-goods
equation in the entire history of fluid
mechanics the navier-stokes equation so
let's take a look at what it is and how
important it is for us in practice the
navier-stokes equation is a momentum
balance equation so what we do is we
balance out inside a grid the linear
momentum that's applies to particles
that are flowing through and we have to
begin with the eyes of a very well-known
scientist is Isaac Newton who tells us that
the force that's applying on particles
is the mass times their acceleration
okay so we begin with what is now called
the Cauchy equation written by a French
scientist Cauchy and for this we look at
the box little box of fluid and we are
applying our knowledge about the three
kinds of forces that apply shear
pressure and gravity so with those three
forces we add them up and this must be
says Newton equal to the mass of the
fluid particle times its acceleration so
very simply we write it out and it looks
like this we have from the point of view
of the particle if you track it
like it was an object like perhaps a
satellite or a car one solid object the
mass of the particle times is change in
velocity is the sum of three forces
weight pressure and shear but what we
want to do of course is to not use this
point of view but the point of view of
the grid the point of view of a
stationary grid through which the fluid
is flowing and so for this we rewrite
this equation by dividing
everything by the volume and
using the point of view of the grid
suppose we have a total derivative so
density times a total derivative is
equal to 1 over the volume multiplied by
the force of weight plus the force due
to pressure and the force due to shear
okay so what we want to do now to get
the Cauchy question is to rewrite the
net force due to pressure and the net
force due to shear as a function of the
fluid properties let's take a look the
pressure we dealt with in the previous
chapters a couple of chapters ago we said
that the net force due to pressure per
unit volume was minus the gradient of
pressure we said pressure is not really
important it is the change in space of
pressure which matters and so this
already solves the first problem we had
1 over the volume of the particle
multiplied by the net force due to
pressure and this is just gonna be minus
grad P let's go that's done what about
shear shear we also saw in the previous
chapter we said that shear had three
components and any point in space when
we took into account all those three
components on each of the six faces of a
tube we have a very tedious long way of
adding those up we said basically that
for a new direction for example the
direction X you had to take the change
in all three directions of the three
components of shear that points into the
X direction and so this we had written
as divergent of the shear tensor in the
X direction we had come up in the end
with this expression the volume
multiplied by a vector and this vector
is the divergent of the shear tensor and
so we wrote it like this: the divergent of
tau IJ - okay so good memories from the
previoius chapters and you take now the
puzzle pieces and we had mass times
acceleration is the sum of forces and
you replace this term and that term by
the puzzle pieces that we had developed
in the previous chapters
and you get this you get mass times
acceleration is again gravity pressure
and shear and this is just called the
Cauchy equation yes so this is the
overarching expression for the dynamics
of every fluid in every possible
situation steady flow unsteady flow
compressible flow incompressible flow
Newtonian fluid non Newtonian fluid just
every fluid flow you can conceptualize
obeys this equation and this is cool
this is very cool but it's not
immediately useful because what we would
like to get with this equation is
the vector V which is a vector field and
you can see that on the other side of
the equation we have gravity and the
gravity is fine we just know what
gravity is but we have pressure and
pressure is a field and then we have this
sheer tensor here which is also a field
and both of those here will change with
time and depend on the flow and we don't
have an equation for them so the
question is how can we eliminate those
two and especially shear how can we
eliminate the divergence of the shear
tensor from that equation well for this
we have two heroes on the Left we have
Claude-Louis Navier from the École
Polytechnique in Paris and on the right
we have George Gabriel Stokes from
Imperial College in London and both of
those people wrote up independently one
from the other without knowing that the
other was working on it both of those
wrote what is now called the
navier-stokes equation so this is what
we got we are gonna write the
navier-stokes equation for the special
case of having incompressible flow
because the mathematics are a little bit
simpler
we had Cauchy Cauchy equation was we said
Newton's second law mass is mass times
acceleration is is the sum of forces
Newton's second law applied on a field
okay and now we have incompressible
navier stokes equation it is the Cauchy
equation on which we add two
restrictions one of the restrictions is
that it's a Newtonian fluid which means
the viscosity of the fluid is just one
property of the fluid
it doesn't change with how the fluid is
strained and to this we also add the
constraint that the flow is
incompressible
this needn't be the case the
navier-stokes equation exists in a
compressible form but today we're only
interested in incompressible flow version so you
have a Newtonian fluid then in that case
good for you
because the shear the norm of the shear
tensor is just the viscosity multiplied
by the gradient of velocity and that's
kind of cool because if you take the
velocity in your direction you find its
change perpendicular to the plane in
which you're looking for then you get
the shear in that direction and so if
you now look at all the faces six faces
of the cube that we have every time in
incompressible flow we're able to
express the value of that vector using
the the local velocity gradients and the
viscosity this is just it and so if you add
up all of those components to get the
net effect you want to have the
divergent of the shear tensor which is
the sum of three components 1 X 1 Y and
1 Z then every time you take the
change in X of something and that shear
vector here is itself viscosity times
the change in X of another thing first
let me switch the slide so if we can see
the equations better I guess so
the shear vector pointing in the X
Direction perpendicular to the X
Direction okay
this shear vector is the change in X of
mu times the change in X of the velocity
okay and the same thing happens for this
year in the X Direction perpendicular to
the Y plane in this time it's a change in
Y of viscosity multiplied by the change
in Y of velocity and same thing happens
happens for Z
if you rewrite this you get something
that looks like this you get the
viscosity which you can group out of all
those three derivatives and every time
you take the second derivative with
respect to space of the velocity
component so the net effect of shear in
the X direction is viscosity times the
second derivative with respect to space
of the X component of velocity like so this is pointing in
the X direction this is cool but as
usual in fluid mechanics in engineering
courses you write this on the board and
soon enough somebody raises their hand
and says this is very tedious to copy
and write down there must be a better
way to write this and of course there is
and for this we bring in a new cool tool
that's going to help you impress your
date when you're on a date and
you have nothing to say you can bring it
up and have success we call this the
laplacian the laplacian operator and the
laplacian operator is the dot product
of two operators one of this is going to
be divergent and the other is going to
be a gradient so if you apply it to a
scalar field then you take the
divergent of the gradient of that
operator so changing space of the
changing space of that operator so if
you apply it now to a vector field you
get a vector field and this vector
field is every time made out of the
laplacian of the component of that
vector field in in the direction you're
looking at so it looks like this and so
now coming back to the shear equation
we had the shear pointing in the X
direction was a vector pointing in the X
direction with a magnitude that was
every time the second derivative of the
velocity in that direction with respect
to space and this was tedious to write
and now we can just sum it up as being
the laplacian of the velocity field yeah
the laplacian of the X component of
the velocity field U like so okay so we have
all the ingredients we need now we just
looked at the X direction but we do the
same thing in Y and the Z directions
so let's do let's take a look
X Direction y direction is that
direction and here we have it now the
divergent of shear so the vector field
that's pointing everywhere in the
direction in which shear is acting on
the fluid particles this divergent we
express it with three components and
every time the component is
the divergent of the component of shear
in that direction and so every time we
apply this as the laplacian of the component of
velocity in this direction and look this
we can just sum up in one nice
expression which is the divergent of
shear is viscosity multiplied by the
laplacian
of velocity this is it we're finished we're done
this leads us to the absolutely amazing
glorious incompressible navier stokes
equation acceleration of the particle is
due to three things gravity pressure and
shear and this we expressed some
mastan's acceleration
yeah there's the effect of gravity the
effect of pressure and the effect of
shear and we replace this this very
inconvenient term that said
the divergent of shear we replace this
with a laplacian of velocity so that our
unknown velocity field appears two times
now in this equation it's gonna be appearing one time here and
one time there this describes all
ordinary flow two conditions for this
are again incompressible flow so low
Mach number as long as you're lower than
a thousand kilometers per hour you're
fine and second condition is Newtonian
fluid which means mu has only one value
viscosity has only one value for
the fluid water air most fluids have this property
this is awesome this is really cool but there
remains a question what is the
velocity field itself so we apply our
two laws one conservation of mass what
balance of man's
continuty equation divergent of velocity
is equal to zero and the second is the
navier-stokes equation for
incompressible flow which is there then
the question is what is V okay you have
two conditions and you want the solution
this is the problem it's not the
solution so what is the solution well
I'd like you to think about it a little
bit because it's a worthwhile problem to
spend your time on if you find the solution
well you will win a little mug that we
have in my lab ISUT/LSS for fluid dynamics and
we have a little mug that says "I found
the general solution to the Stokes equation"
and it is really cool because people
will ask you about the lab when you tell
them you solved the navier-stokes equation
and they will see your mug and you can
show it to your friends it's kind of cool
a second consequence
is that you will receive the Nobel Prize
the Fields Medal and the MIllenium
prize which is the $1,000,000 prize
that's waiting for the person who could
find the general solution to the Navier-Stokes equation and the consequences for
this is if you find the solution the general
solution to this equation well all of
the computational fluid dynamics software
companies will just instantly collapse
there's no need to carry out
computational fluid dynamics simulations
if you can calculate by hand the answer
already there's no more need for wind
tunnels for water channels for all this
extremely expensive technical equipment
that we use to make measurements because
what need is there to measure a flow if
you can just compute the solution
already and of course academics will be
busy revisiting everything they know yeah
so matching their descriptions of flow
with your solution and finding out
whether they were right or wrong so this
is quite a major feat to come up with
the solution also important consequence
is we have to buy a new mug which
with the would be I guess a positive consequence
when I say we're looking for this
solution we really mean the solution
not just a solution almost any flow you
describe is a solution of the navier
stokes equation and by to make it clear
what I mean with the solution imagine
you take a look at a different problem
and this problem is the problem of the
cannonball in this the problems is what
is the trajectory of the cannonball when
you shoot it at one way or the other it
turns out depending on the angle of the
cannon and the speed
at which you start the ball will have
different trajectories and so if you
have this law which is the physical
law that applies to the cannonball as
it's flying around the answer to this
"what is V" is not "it depends" it
looks like it's a lot of different
trajectories but as it turns out the
general solution which means all of the
solutions at once they are all of the
form of a vector V with the horizontal
velocity being constant and the vertical
velocity
being a function of time and based on
this you can redraw every one of those
trajectories now we're looking for the
same general family of solution for the
Navier Stokes equation yeah so if you
write now an equation that's just a
little bit more complicated but not that
much if you look into it if you have
those two conditions and what is V the
solution to that describes almost every
flow that you can encounter in real life
it describes the laminar flow passing
over this weir but also the completely
turbulent flow that crashes after the
water has passed the weir
it describes the rain falling on the
roof above my head it describes every
splash of water inside a little wave of
water splashing on the beach
it describes the water flow around the
hull of this catamaran but also the flow
of air on the sails and all the forces
that result from those it describes the
flow of air around the wings of this
vulture as it comes in to land on the
ground
it describes the aerodynamics of the air
as it flows around this jet aircraft
it describes the cloud of dust that's
pushed up by the downwash of the rotor
of this helicopter and so all of those
are solutions all of those flows are
solutions to the navier-stokes equation
and what we're looking for what we have
been looking for for the last 150 years
is the general solution that
describes them all at once yeah so the
incompressible navier stokes equation is
quite complex perhaps you see better why
it's so complex once you get rid of
this very convenient compact very nice
and elegant vector notation and you
write it in three Cartesian coordinates
if you do this then you see immediately
that it's quite a complicated mass of
terms and perhaps an important feature
in this is that you see that you cannot
separate them very clearly you cannot
for example take the first term in the X
direction and say isolate U from
this because in there you have U yes
but you also have V and W appearing in
this term if you switch of course to
V to the V equation then you see that
you have also U and W appearing there that
so it's a very coupled equation makes it
particularly hard to solve
if Cartesian coordinates don't work for you
you can also use cylindrical coordinates or
angular coordinates which make the problem
perhaps even more interesting but in any
case it's a difficult problem it's a
difficult equation we do not of course
have the general solution to this
because it's not the Bernoulli equation it's
quite a stumper so what is
it for then let's let's conclude on this slide
why do we even study the navier
Stokes equation if we don't have the
general solution well it helps us as
engineers it helps us in several ways
the first way is that it helps us
understand and quantify the influence of
different forces on fluids and so it
enables us to quantify what is important
and what is not important inside any
given flow and we'll work on this a lot
more in the coming chapters
the other thing is it helps us
it allows us to find analytical
solutions in very simple flows in very
very simple cases sometimes we find an
analytical solutions but this is mostly
academic what navier-stokes equation is
really really good for is computational
fluid dynamics practically every
software you use to compute fluid flow will
have some form of a simplified or
approximate model for the navier-stokes
equation built-in so if you want to
understand what the software is doing
and why it's so difficult to
manipulate computational fluid dynamics
software then you need to work with
the navier-stokes equation so here you are
the momentum balance for incompressible flow
