From the Navier-Stokes equations,
we derive the vorticity equation in two dimensions
that reduces to an equation
for the scalar vorticity field.
But that equation depends on
two components of the velocity field.
We want to now reduce the two components of
the velocity field to a single scalar field.
That scalar field will be called the stream function.
We haven't used the continuity equation yet,
so we'll start with the continuity equation
Del.u in polar coordinates,
that's 1/r d/dr of
ru_r plus du_Theta/ dTheta.
We know that this is equal to
zero for incompressible fluid.
We can replace now the velocity field by scalar field,
that's called the stream function in order to
directly enforce the zero of Del.u.
How do we do that?
Well we can let this ru_r
equals the partial derivative of
stream function divided by
Theta so that we have a mixed partial derivative here.
Then we can let u_Theta equals
partial derivative of Psi with
respect to r with a negative sign.
Then the sum of these two terms will be the sum of
the two mixed partial derivatives
with one that has a negative sign.
This side then is called the stream function.
Now the continuity equation is
satisfied and we can eliminate
the two components of
the velocity field by
the derivatives of the stream function.
What we still need to do is determine a relationship
between the scalar vorticity and the stream function.
The scalar vorticity,
Omega is in polar coordinates,
is 1/r times d/dr
of ru_Theta minus du_r/dTheta.
We've already derived the scalar vorticity
in terms of the two components of the velocity field,
but now we want to replace
u_Theta and u_r by the stream function.
This is 1/r d/dr
and then ru_Theta is using this second definition here,
ru_Theta then is minus
r dPsi/dr minus du_r/dTheta,
so u_r is 1/r d/d_Theta,
so minus d/dTheta,
and then this term u_r is 1/r,
so
1/r dPsi/dTheta.
It looks a lot more complicated than it is.
We can factor out 1/r,
so we have a 1/r squared.
Then we need to multiply this by
r. We also have a negative sign,
we can factor out the negative sign
and the 1/r from inside here.
So we get a negative 1/r squared
and then we put in the r from here,
so we have an r d/dr and then an r d/dr of Psi,
and then this term is a plus d
squared Psi/ dTheta squared.
That gives us the vorticity in
terms of the derivatives of the stream function.
This whole operator here operating on
Psi is actually the Laplacian.
This equation is a complicated way of
writing the Laplacian of Psi,
and then the minus sign goes on
the left is negative Omega.
This is valid in
both polar coordinates and in Cartesian coordinates.
This is the polar coordinate form of
what this is called a two dimensional Poisson equation.
Let me summarize.
The idea here then is that we're replacing
the two component velocity field by
a scalar vorticity field and a scalar stream function,
we do that here.
We've already have the equation
for the scalar vorticity field,
but it's still depends on
the two components of the velocity field.
So we can make use of the continuity equation,
Del.u equals 0 to replace
the two components of
the velocity field by a stream function.
If we define the stream function like this,
then we automatically satisfy the continuity equation.
Then we can go back and look at our definition
of the scalar vorticity field and
replace the two components of
the velocity field by
our new definition of the stream function,
and we show that
the stream function satisfies the equation;
Del squared Psi equals negative Omega.
I'm Jeff Chasnov.
Thanks for watching and I'll see you in the next video.