So how can we use the velocity components, and the stream function to actually sketch
what the streamlines are in an actual flow.
Well lets look at an example.
Lets say we have velocity components such that u equals 4y+3, and v equals 2x+6, and
remember when we are defining theses stream functions there are in 2-D. So the first thing
we want to know.
Is this physically possible flow.
What that means is, is this in-compressible, and that for that to be the case du/dx plus
dv/dy, as to equal 0, and just by looking at it you can tell that is the case.
Now lets introduce our stream function.
So by definition u is the derivative of the stream function with respect to y, which is
4y plus 3.
So we separate and integrate in order to find this part of the stream function, and when
we do that we come up with.
That the stream function equals 2y^2+3y+f(x).
Now lets do it for v. So v is the negative derivative of the stream function with respect
to x, which equals 2x+6.
Again we separate and integrate, and this time it is (2x+6) dx, but lets not forget
that negative sign.
So now our stream function equals negative x^2-6x+f(y), and when we combine these two.
We come up with a stream function that is 2y^2 plus 3y minus x^2 minus 6x plus a constant.
So how do we go from there to actually drawing stream lines.
Well the first thing we are going to do is set this constant equal to 0, and in order
to get different stream lines we change the number of the stream function and change it
to different constants.
So we can set this equal to 0, we can set it equal to 1.
We can set it equal to 2.
We can set it equal to any constant we want.
So let me show you what happens, and how you work this if we set this stream function equal
to 0.
Now you have to solve this in terms of y, and so our y is going to equal minus 3/4 plus
or minus 1/4 times the square root of 9 plus 8, x^2 plus 48 times x.
By the way I did not figure this out on my own.
I put it in a computer program, and let it solve for me.
So now that you have this equation, what do you do with it.
Well you take something like excel, and you have an x column, a y1 column, and a y2 column,
and your x's can go from any number that you want generally you are going to want to start
with 0.
So what I did here is I went from 0, 0.1 all the way to 1.
Now you solve for your y1, such that you use the positive, minus 3/4 plus 1/4, etc, and
then you use it for the negative, and then you plot it.
What you will end up with is a plot that looks like this.
So these are your streamlines, and you can do it negative x's, etc.
These are your streamlines for our phi, excuse me our streamline equals 0.
Now instead of setting it equal to 0.
You can use this stream function to equal for example 2 or 4, and you solve it for y,
except now your equation is going to be for example 2 equals 2y^2 plus 3y, minus x^2,
minus 6x.
So you solve for y, and then plot this particular series or families of streamlines