example 16-1 - Example 16-1 Oil flows in the annular space...

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Example 16-1 Oil flows in the annular space between two concentric cylinders, as shown. The velocity profile is given by () 22 12 1 1 1 2 ln 4 ln z r RR R P rR r R L R µ ⎧⎫ ⎪⎪ =− + ⎨⎬ ⎩⎭ V where R << In the cylindrical coordinate system, z d dr τµ a) Derive a general expression for shear stress as a function of r . b) Derive a general expression for r max , the location at which the velocity attains its maximum value. c) Using the following parameters, calculate the value of r max and the maximum velocity. 1 2cm R = 2 4cm R = 20cm L = 2 Ns 1.22 m = 14.4kPa P ∆=
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a) Shear stress is related to velocity profile through z d dr τµ =− V () 22 1 12 1 1 2 1 2 4 ln z R RR rR dP r R dr L R µ ⎧⎫ ⎛⎞ ⎪⎪ ⎜⎟ ⎝⎠ + ⎨⎬ ⎩⎭ 1 2 2 4 ln z d P r R dr L r R + 21 1 2 2 4 ln z d P r R dr L r R = + b) The maximum velocity occurs when the derivative of velocity is zero. 0 z d dr = when 0 z d dr = −= max 1 max 2 02 4 ln P r R L r R =+ max 1 max 2 ln r R r R
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() 22 21 2 max 1 2 02 ln RR r R R =+ 12 max 1 2 2ln r R R = d) To find the location of the maximum velocity () () max 24 2.94cm 2 4 r == ⎛⎞ ⎜⎟ ⎝⎠ The velocity at this location is max 1 max
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This note was uploaded on 04/06/2009 for the course ENGR 2250 taught by Professor Borca-tasciuc during the Spring '08 term at Rensselaer Polytechnic Institute.

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example 16-1 - Example 16-1 Oil flows in the annular space...

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