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Continuum Mechanics - Bernouilli's Equation in a Cylindrical Tank

  1. Apr 23, 2012 #1
    1. The problem statement, all variables and given/known data

    A large cylindrical tank of radius R is full of water to a height h(t). This drains under gravity
    out of the bottom of the tank through a small hole of radius r. The acceleration due to gravity
    is g. The pressure of the air can be assumed to be the same at the top and at the bottom of the tank.

    Show that if the uniform velocity of the fluid at the top of the tank is U and the uniform
    velocity of the fluid leaving through the small hole is u then:
    [tex]u = U\;\frac{R^2}{r^2}[/tex]

    By applying Bernoulli’s Theorem on a streamline from the top of the fluid to the bottom,
    show that:
    [tex]\frac{dh}{dt} = -\sqrt{2gh}\;(\frac{R^4}{r^4} - 1)^{-\frac{1}{2}}[/tex]


    2. Relevant equations
    We've been given Bernoulli's Theorem as:
    [tex]\frac{1}{2}u\bullet u + \frac{P}{\rho} + V = constant[/tex], where P is pressure, rho is density and V is potential.

    3. The attempt at a solution
    I tried using Bernoulli's Theorem, but wasn't sure what V was. I'm really not sure what to do. :(
     
  2. jcsd
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