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Simple Differential Equation

  • Thread starter Menisto
  • Start date
18
0
1. Homework Statement

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.
 

Answers and Replies

you have to apply the boundary condition at r = 0 and r = a.....
1. The velocity should be finite at all point in the pipe/
2. Fluid "sticks" to boundaries (V(a) = 0 )
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
 
18
0
Thank you, that works out well.
 

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