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Simple Differential Equation

  1. Jan 23, 2007 #1
    1. The problem statement, all variables and given/known data

    Suppose a fluid is flowing down a pipe that has a circular cross-section of radius a. Assuming that the velocity V of the fluids depends only on its distance from the centre of the pipe, the equation satisfied by V is

    (1/r)(d/dr)(r dV /dr) = -P where P is a positive constant

    Find the expression for velocity given that:

    1. The velocity should be finite at all point in the pipe/
    2. Fluid "sticks" to boundaries (V(a) = 0 )

    Show that:

    V(r) = P/4 (a^2 - r^2)

    3. The attempt at a solution

    I integrate the first time to get:

    dV/dr = -Pr/2 + c/r

    and integrate again to get (FTC)

    V(r) - V(a) = (-Pr^2 / 4) + C ln r [with lower bound a and upper r]

    To show what I needed, it seems I only needed to get rid of that second term, but I'm unsure what assumption can justify that and where it applies.
     
  2. jcsd
  3. Jan 24, 2007 #2
    you have to apply the boundary condition at r = 0 and r = a.....
    Since log r blow up at r = 0, it cannot exist, otherwise it violate the first condition...
    Apply the second condition yourself in order to get the constant term correct.

    Good luck
     
  4. Jan 24, 2007 #3
    Thank you, that works out well.
     
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