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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online