Vector fields question; not sure how to approach?

[Elmer Correa]

Homework Statement

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.

Homework 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

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

 

[jedishrfu]

Try the Desmos calculator, it might be able to plot it:

https://www.desmos.com/calculator/eijhparfmd

 

[Ray Vickson]

Homework Statement

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.

Homework 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

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

What are "consitive integers"?

 

[NFuller]

I have no idea how to graph this and every online grapher that I've used has been unable to graph it.

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

 

[Elmer Correa]

What are "consitive integers"?

Positive

 

[Elmer Correa]

Try the Desmos calculator, it might be able to plot it:

https://www.desmos.com/calculator/eijhparfmd

I don't need to graph a vector field, just level curves in 2D

 

[NFuller]

I don't need to graph a vector field, just level curves in 2D

The streamlines should take the equation C=10sin(πx)sin(πy) where C is some constant.

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.

 

[Elmer Correa]

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.

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

 

[Ray Vickson]

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

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);

 

[Orodruin]

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.

 

[NFuller]

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

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