We now have governing equations in terms of the scale of velocity and
the stream function.
We don't really know what the meaning of the stream function is yet
but it turns out to be an excellent field for visualizing steady flows.
So this is the definition of the stream function in terms of the velocity field.
Now, if we draw what's called a streamline of the velocity field.
So the fluid then is flowing in the direction of the streamline,
then this is the velocity field U, this is a steady flow.
So this is independent of time.
Now, if we draw a vector along this streamline,
let me draw an infinite testable vector.
So this will be our infinite testable vector,
which I'll call DR, so DR then is a long U.
Now, let's remember our polar coordinates.
So our radio vector is equal to R times R hat.
So what is this infinite testimony vector?
So DR is equal to, We need to use the product rule
here because our hat depends on theta.
So this is DR times R hat
plus R times DR hat.
So DR hat now is DR hat D theta times D theta but
DR hat D theta is theta hat.
So this is equal to DR R Hat plus R D theta Theta hat.
Okay, DR now is in the same direction as U,
so we can make use of that.
We know then that the cross product of U and DR will be zero.
So we have a zero is equal to U cross DR and
what is this in polar coordinates?
So U is UR R hat plus you theta theta hat and
DR is what we just derived.
So DR R hat plus RD
theta theta hat.
We can do this cross product because our hat cross our hat is zero.
R hat cross theta hat is K the unit vector in the Z
direction and theta hat cross theta hat is zero.
So we need just the cross terms here, so
we'll start with theta hat cross R hat which is minus K.
So that will give us a negative sign so minus U theta DR and
then our hat cross theta hat is plus K.
So that would be plus R UR D theta and
this is all in the K direction,
the Z that access direction.
This supposed to be zero but what is this inside?
If we go back to our definition of the stream function,
U theta is just negative the side.
The r partial of the stream function with respect to R.
The minus signs cancel so
that becomes partial of cy with respect to R DR.
And R UR as partial of cy with respect to theta, so
that becomes partial of cy with respect to theta.
D theta and that's a vector in the K direction.
Okay, what is this?
Well this is just the definition of deci in the K direction.
So this is our infinite testable deci.
So because U cross DR is equal to zero
we Get deci then is equal to zero.
Okay, so on the streamline cy doesn't change Deci is equal to zero.
So therefore, cy is
constant on streamlines.
Okay, the streamlines are just the curves that follow the fluid as it flows.
So following the velocity and in a steady flow,
those curves don't change, those curves are fixed.
So this gives us an excellent tool for a visualization in a steady flow.
What we need to visualize the velocity field,
all we need to do is plot the contours of the stream function.
So this is our main visualization tool
is the contour lines of the stream function side.
Matt lab gives us built in functions that
can plot contour lines of a scale or field.
So let me summarize, we've introduced a stream function
in order to eliminate the two components of the velocity field.
It turns out the stream function has a very clear meaning
if we plot a streamline which is a curve which follows the flow of a fluid.
So the velocity is along the streamline,
then the stream function is constant along the streamline.
So by plotting curves of constant stream function we can plot the streamlines and
have a clear visualization of the flow field.
I'm Jeff Chasnov.
Thanks for watching and I'll see you in the next video