How to Sketch Streamlines


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

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