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Calculate the Flux through the Pipe

  1. Dec 29, 2008 #1
    1. The problem statement, all variables and given/known data

    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?

    [​IMG]


    2. Relevant equations
    [itex]\Phi=\int_S\mathbf{v}\cdot d\mathbf{a}[/itex]


    3. 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?
     
  2. jcsd
  3. Dec 29, 2008 #2
    Any suggestions would be appreciated. :smile:
     
  4. Dec 29, 2008 #3

    Dick

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    Homework Helper

    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?
     
  5. Dec 29, 2008 #4
    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.
     
  6. Dec 29, 2008 #5

    Dick

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    Yes, that's right. It's v*A=v*s*l. cos(theta)=1.
     
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