chall_2 - v in order to x obtain this approximate solution?...

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Challenge Problem 2 (OPTIONAL) Name: ____________________ ( ) Consider steady flow in a circular tube whose radius ax varies slowly with x along the centerline. A constant pressure difference is maintained between the two ends of the tube, dp and the axial pressure gradient = ± Gx () also varies slowly with x . The flow is K dx , ( ) ( ) axisymmetric so that V = ( v v ) . Because ax and Gx vary slowly, the local flow x r near some point x in the tube is approximately described by Poiseuille flow and the axial velocity component is given by () (, ) = Gx ( ( ) 2 ± r 2 ) vx r ax x 4 P
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Unformatted text preview: v in order to x obtain this approximate solution? 2. What is the constant volume flux Q along the tube in terms of G x and a x ? ( ) ( ) 3. Using your answer for Q , show that x 2 4 P P = 8 Q a dx 1 2 S x 1 where P and P are the pressures at x and x . 1 2 1 2 4. Assume the tube radius is given by a x 1 ( ) = + D x , where is a constant. Obtain an equation for the pressure gradient G x ( ) assuming the pressures P and P are known at 1 2 points x and x . Express your answer in terms of x 1 , x 2 , P , P 2 , and x . 1 2 1 5. If = 0 , what is the pressure gradient? 1...
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