Homework Help: Vector fields question; not sure how to approach?

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1. Nov 30, 2017

Elmer Correa

1. The problem statement, all variables and given/known data
The stream function Ψ(x,y) = Asin(πnx)*sin(πmy) where m and n are consitive integers and A is a constant, describes circular flow in the region R = {(x,y): 0≤x≤1, 0≤y≤1 }. Graph several streamlines with A=10 and m=n=1 and describe the flow. Explain why the flow is confined to the region R.

2. Relevant equations
The y partial derivative of the stream function equals the x-component of the vector field
The x partial derivative of the stream function equals the y-component of the vector field
I'm assuming the vector field is source-free
3. The attempt at a solution

The streamlines should take the equation C=10sin(πx)sin(πy) where C is some constant. I have no idea how to graph this and every online grapher that I've used has been unable to graph it. I imagine I need to have some idea of how to do this before I can determine why the flow is confined to the region R

2. Nov 30, 2017

Staff: Mentor

3. Nov 30, 2017

Ray Vickson

What are "consitive integers"?

4. Nov 30, 2017

NFuller

If you have access to it, I would recommend MATLAB for doing this. It has a specialized function for plotting streamlines. See the description here https://www.mathworks.com/help/matlab/ref/streamline.html

5. Nov 30, 2017

Elmer Correa

Positive

6. Nov 30, 2017

7. Dec 1, 2017

NFuller

You can use desmos if you just set C to some numbers like 0, 1, 2, etc. and graph the set of equations with different C's simultaneously.

8. Dec 1, 2017

Elmer Correa

I've tried, the formula is too complicated to graph

9. Dec 1, 2017

Ray Vickson

If you have access to Maple the task is easy. I plotted over the region $0 \leq x,y \leq 1.2$, just to see what happens if you go outside the region $R$ that you specified. It seems that the streamlines do not "cross" the boundary of $R$, but rather, you get a whole new set of orbits for a whole new flow regime. I guess the question is asking you to explain this behavior theoretically.

I assume Mathematica has similar capabilities, but I do not have access to it so cannot say for sure.

>P:=10*sin(Pi*x)*sin(Pi*y);
P := 10 sin(Pi x) sin(Pi y)

> with(plots):
> contourplot(P,x=0..1.2,y=0..1.2,contours=20);

10. Dec 1, 2017

Orodruin

Staff Emeritus
Any of the set Mathematica, Matlab, Maple, or many others should do fine to be honest.

Also, you do not need to graph anything to show that the flow is contained. It is sufficient to verify that the boundary of the region is a level curve of the flow function.

11. Dec 1, 2017

NFuller

How? I literally just typed in the formula and it worked fine.