# 7334347 - 1 Starting with the u momentum equation for a...

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Unformatted text preview: 1. Starting with the u- momentum equation for a fully developed axi-symmetric viscous flow, given by: dp µ d du = r dz r dr dr Evaluate the flow in the annulus between two concentric cylinders of radius ‘a’ and ‘b’. Here the pressure gradient dp/dz is a constant. Show that the velocity field is given by: ( )( ) () log r 1 dp 2 b − r 2 − b2 a − b2 u (r ) = − 4 µ dz log a b ( ) Hint Integrate the above equation twice with respect to r to get an expression for u as a function or r. Apply u=0 at r=a, and r=b. Solution: = 1 1 = . + + Let, 1 1 = + = − 1 1 1 = 1 =0 + = =′ Let, += Solving the exact equation: Integrating factor, . . = ..= = = = 2 = 2 + 2 = + + + + 2 .. ln + ln + = =0 =− Solving, 2 =− Hence, =− 4 1 4 − 4 ln − ln + = Boundary conditions: + = = + =0 ln + =− − ln − ln − ln − ln 4 + + − − =0 − 4 − ln ln ...
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