Calculate the Flux through the Pipe

  • #1
3,003
2

Homework Statement



Water flows with a speed v down a rectangular pipe of dimensions s and l as shown. What is the rate [itex]\Phi=volume\, per\, unit\, time [/itex]
at which water accumulates in the bucket?

Photo2.jpg



Homework Equations


[itex]\Phi=\int_S\mathbf{v}\cdot d\mathbf{a}[/itex]


The Attempt at a Solution



I am confused as to how to compute this integral. I do not see how could use the divergence theorem to simplify it since I only have the dimensions of the 2-dim surface of the rectangular pipe.

I am not sure how to evaluate the surface integral based solely on its definition. Can someone help to get me started?

Do I just use the definition [itex]\Phi=\int_S\mathbf{v}\cdot d\mathbf{a}[/itex] for this?
 

Answers and Replies

  • #2
3,003
2
Any suggestions would be appreciated. :smile:
 
  • #3
Dick
Science Advisor
Homework Helper
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Sure, flux is integral v.dA. If the velocity is uniform that's just v*A. Isn't it? Why do you want to use the divergence theorem? This problem sure looks simple to me. Is this some kind of a trick?
 
  • #4
3,003
2
Sure, flux is integral v.dA. If the velocity is uniform that's just v*A. Isn't it? Why do you want to use the divergence theorem? This problem sure looks simple to me. Is this some kind of a trick?
I just don't know anymore :sad: I think I am getting stupider and stupider.

I don't know what he is looking for (Griffith's Intro to Electrodynamics problem 1.32 2nd ed.).

I don't want to just memorize formulas. I want to compute the integrals. But I get all effed up when I get a surface integral no matter how simple. If I have to use the definition, I get all flustered.

How do I compute the integral?

I get,

[itex]\phi=\int_S v \cdot da[/itex]
[itex]=\int_S v*da\cos\theta[/itex]
[itex]=\int_S v*da[/itex]
[itex]=\int_s\int_l v*(dsdl)=v*s*l[/itex]

is that right?

The reason I want to do this right is that the next three parts of the problem have angles and $hit that I'll and non-uniform velocities that I will have to deal with.
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
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Yes, that's right. It's v*A=v*s*l. cos(theta)=1.
 

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