1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find the equilibrium solution for an autonomous equation.

  1. Aug 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From Torcelli’s principle in hydrodynamics, it follows that the rate at which water flows through the hole is (alpha)(a)squareroot((2)(g)(h)) , where h is the current depth of water in the tank, g is the acceleration due to gravity, and alpha is a contraction coefficient that satisfies 0.5 < alpha < 1.0.

    1. Show that the depth of water in the tank at any time satisfies the equation
    dh/dt = ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A
    2. Determine the equilibrium depth he , of water, and show that it is asymptotically stable. Observe that he does not depend on A.


    2. Relevant equations

    We covered how to solve a DE using the integrating factor method after putting the eq. in to standard form but now we have moved on to autonomous eq.'s and I'm a little unsure how to go about solving this.

    The problem looks more complex than it is because of all the brackets I had to use but I think it is pretty straight forward for someone familiar with the subject.


    3. The attempt at a solution

    1.

    dh/dt = rate in - rate out

    = (volume in/min)(1/area) - (volume out/min)(1/area)
    = [k(m3/min) / A(m2)] - [((alpha)(a)squareroot{(2)(g)(h)}(m3/min) / A(m2)]
    = ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A

    (Seems straight forward have I done this wrong?)

    2.

    dh/dt = m(h) h

    => [(k/hA - (alpha)(a)squareroot{(2)(g)}/sqrt{h}A] h

    (Skipped the intermediate steps. Is this the right form? What do I do now?)


    Thanks in advance :)
     
    Last edited: Aug 2, 2011
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted