JANET: In this screencast,
we'll apply certain boundary conditions
to the simplified version of
the Navier-Stokes equation
to produce a velocity profile.
If you take a look here,
we actually have two velocity profiles
between two parallel plates.
So this first top velocity
profile is parabolic in nature
which means that it is
pressure driven flow.
Therefore, the top plate
needs to be stationary.
If we look down here,
note that the velocity of the
fluid is going to the left.
Due to the no slip condition,
that means that the plate
that the fluid is attached to
also has to go to the left.
So here, we're gonna look
at a specific type of flow
and determine its velocity profile.
So this is what's known as couette flow.
Again, its steady state, one
dimensional incompressible flow
You have two horizontal
infinite parallel plates.
Here the top plate is moving at a velocity
so here, here we'll draw the top plate
and it's moving at a velocity U.
The bottom plate here is stationary
and we're gonna be looking
here in the Y direction.
We're gonna call this zero
and then the distance
between the two plates
is gonna be H.
And so now we need that velocity profile,
in other words U as a function
of Y for this couette flow.
So let's start by writing
the general, simplified
Navier-Stokes equation
for any kind of steady
state one dimensional flow,
two horizontal infinite parallel plate.
And that's going to be that
our U of Y is equal to one divided by two
times the viscosity of the fluid dPdx
which is the pressure gradient
times Y squared
plus C1
times Y plus C2.
And as I said this a
generalized simplification
and what makes it specific for
our particular type of flow
are the boundary conditions
that we're gonna use
to solve for C1 and C2.
So, let's start with our first boundary
which is at Y equals zero.
So at Y equals zero,
because the plate is stationary,
we know that the velocity of the fluid
is going to be zero.
Our other boundary is at H
so at Y equals H,
the fluid particles that are attached
to that top plate have the
velocity of the top plate.
In this case, that's
going to be equal to U.
So we're gonna start with
that first boundary condition
at Y equals zero,
so we're gonna replace every
little U in there with zero
and every Y in there with zero
and what we end up with
is that zero equals zero
plus zero plus C2
and therefore, our C2
is going to be equal to zero.
Now we're gonna do the same thing
for the second boundary condition.
So we replace U of Y with U.
So this is equal to one
divided by two Mu dPdx
and now we replace the
Y squared with H squared
plus C1 replace the Y with H
and then C2 is zero.
And so now what we can do,
is we can rearrange
and find that constant.
So our C1 is gonna be equal to minus one
over two Mu dPdx times H squared
all divided by H plus U over H.
And now we go back to our
original equation right here
and we replace our C1 with
the value that we just calculated
and that gives us our final
velocity profile U of Y
and that's gonna be equal
to one over two Mu dPdx
Y squared minus H times Y
plus U Y divided by H.
So we can apply this at any Y
to determine the velocity at that point.
So note if the pressure is constant,
the pressure gradient here is zero
which makes this entire term zero
and so what we're left with
is that U of Y is equal to this final term
that I'm gonna write big
U divided by H times Y
which is linear.
