Solution%20to%20HW#2

Solution%20to%20HW#2 - EGM6812/FluidMechanicsI Homework#2...

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EGM 6812/ Fluid Mechanics I Homework # 2      Due Sept. 16, 2009 1. Given a 2-D unsteady velocity field u=ax, v=(t-by) in which a and b are constants, find  the streamline and the pathline passing through (x 0 , y 0 , t 0 )=(1, 2, 1). Solution: i) Streamline:  / / dx u dy v =    / ( ) / ( ) dx ax dy t by = - 1 1 ln ln( ) x t by c a b = - - + / ( ) a b x c t by - = - % Since x=1 when t=1, y=2, we have  / 1 (1 2 ) a b c b - = - % / (1 2 ) a b c b = - % / 1 1 2 a b by x b - - = - Or ( 29 / 1 1 (1 2 ) b a y b x b - = - - (1) ii) Pathline: / dx u dt = / dy v dt = / dx dt ax = / dy dt t by = - 1 at x c e = 2 2 1 bt bt y c e b - - = + At t=1, (x, y)=(1, 2). Thus  1 a c e - = 2 2 1 (2 ) b b c e b - = - Finally,  ( 1) a t x e - = ,   ( 1) 2 2 1 1 (2 ) b t b bt y e b b - - - - = - + (2) Note:  (1) is valid at t=1 only. It is for the particular streamline passing (1, 2). (2) is valid for all t but only for a particular fluid element that passes through (1, 2) at  t=1.
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2. Consider a small rectangular element ABCD shown below in a flow   A B D C A' B' C' D' dy dx D'.  Show that for a continuous velocity, A'B' || D'C' and A'D' || B'C' to the leading order in  O (dt).
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This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

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Solution%20to%20HW#2 - EGM6812/FluidMechanicsI Homework#2...

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