one weakness of the navier-stokes
equation is that it presents us the
problem of finding the velocity field V
and it expresses the change in time of this
velocity field as a function of V itself
over there but it also has a term for
pressure and this term for pressure is a
gradient it's the change in space of
pressure and tells us nothing about how
pressure should change in time so that
in itself the Navier-Stokes equation
is not enough to be able to
find the velocity field V and so because
of this it happens from time to time
that students come to me and ask well if
we don't have the pressure P through the
navier-stokes equation could we not use
the Bernoulli equation in there and
plug it in to be able to compute the
velocity field V it could be like a
makeshift solution to this and the
answer is no you cannot do that and let
me show you how to do this by showing
you how to go from the navier-stokes
equation to the Bernoulli equation and
this helps us hopefully understand what
the relationship between the two is so
we have overall it this situation
the Bernoulli equation a question that
never dies based on the navier-stokes equation
in short you have a
philosophical discussion between three
people you have on the right Daniel
Bernoulli who came up vaguely with the
Bernoulli equation and on the left you have
George Gabriel Stokes on the top and
Claude-Louis Navier on
the bottom left and you can see it as a
Greek philosophical difference point of
view between those three people so
Bernoulli comes up and says hey Bros I hear
you want to calculate pressure and then
Stokes says no go away Bernoulli and then Navier
says and we're not your bros by the
way so
Bernoulli says dudes pressure is like
easy to qualify and then of course
Stokes gets angry says "no Olivier said you are
only valid for steady incompressible
frictionless flow with no energy
transfer and only one dimension" yeah and
of course Navier adds up "yes and we are
valid for all flows by the way" and so
Bernoulli says but hey but maybe you can
just take the square of velocity and
then of course Navier and Stokes are pretty
angry so Stokes says "no we do vectors
with the arrows it's a three dimensional
equation" and then Navier says "yes
do you even do the thing with the
upside down deltas"
Nablas in so Bernoulli says are you
nervous because you can't calculate
pressure and then Stokes says dude chill
out and yes why don't you chill out by
the way well why don't you chill out no
you chill out yes chill out Bernoulli and
so we are left with a problem that
Bernoulli just will not chill out Bernoulli
wants to help Navier-Stokes but Bernoulli
can't help
Navier-Stokes and why because both
equations the Navier Stokes equation and the
Bernoulli equation they are the same
equation and so for this to show this
and to understand what the relationship
between the two are we need to go
through how to get from Navier-Stokes
to Bernoulli so we're going to take
inside any arbitrary flow we're gonna take a
piece of a flow that goes from one to two
here and we're gonna follow a little bit
of trajectory a little piece of
trajectory which we call DS here and
we're gonna follow the flow from one to
two when I'm applying navier stokes on
there and then we're going to apply the
conditions of the Bernoulli equation to
that  and we're gonna come up with
with the Bernoulli equation so let's start
with you remember now there are five
conditions for the Bernoulli equation
let's start with last one first and put
in one dimension so we take the
three-dimensional navier-stokes equation
which is here this is the complete
three-dimensional Navier-Stokes equation
and you take this equation here and you
project it onto a tiny bit of element D
s so we take the dot product of this
vector with a tiny bit of element in D
s so we every time every every bit here
we take a dot product dot product so that we have just scalars
so we can remove all the vectors and we
get we get this the details don't matter
what matters is that we just removed all
the vectors so this is just the numbers
now it's the series of numbers added up
one to another once we've done this we
turn to the first condition which is
steady flow and so inside this equation
here that we had we can just remove the
unsteady part the change in time of the
velocity field si we'rejust left with
this now is one two three and four term
equation four term equation then when
we get frictionless and if the flow is
frictionless then you can remove
viscosity viscosity will have no
effect so we're just left with three
terms here that's convenient so we write
it like this and then we can just remove
the dsds which are there and the DSDS
which are here so we get something that
looks like mmm look velocity times
change in velocity
is Rho G times the change in
altitude - a little bit of delta pressure
so it looks something like something
familiar already then we apply the
fourth condition which is no heat and
work transfer and if you do that then
the density will become constant and you
can integrate the previous term which is
Rho you can put the rho out of the
integral and integrate V DV and then put
the rho out of this and integrate G DZ
and then integrate DP between 1 and 2
and so you get at the Delta of the
square velocity plus the Delta of Z
altitude multiplied by G Plus 1 over Rho
of the Delta pressure or the sum of this
is going to be 0 so you see this is how
you get to the to the Bernoulli equation
from the navier-stokes equation
understand how those two are
built and you will feel a lot more
comfortable manipulating computational
fluid dynamics software later on
