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FL07Ch08

# FL07Ch08 - EQUATION OF MOTION one dimensiona l steady dp d...

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dP P dp P P p P p P dX L dx dX, L dx X L x L x X U d V u d dU, U V du u U V u V u U p, u, x, ess dimensionl dx u d μ dx dp steady l, dimensiona one MOTION OF EQUATION O O O 2 2 O 2 O O O 2 O 2 2 O O 2 2 = × = = = = × = = = = × = = = 2 2 RE 2 O o O O 2 2 O O 2 O o O 2 O 2 2 2 O O O o dX U d N 1 dX dP ρ V P Number Reynolds parameter ess dimensionl the is μ L ρ V dX U d μ L ρ V 1 dX dP ρ V P , L ρ V by divide dX U d L μV dX dP L P p, and x, u, for ng substituti = = =

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PIPE FLOW ( ) Groups ess Dimensionl 4 Expect 4 Units 3 - Parameters 7 0 0 0 1 - 1 - 0 2 - T 1 1 1 1 1 - 3 - 1 - L 0 0 0 0 1 1 1 M e D L V μ p e , D , L , V , , f P : hypothesis = ρ µ ρ = ( ) ( ) ( ) 31 22 32 21 13 31 23 33 21 12 32 23 33 23 11 21 12 22 11 33 32 31 23 22 21 13 12 11 a a a a a a a a a a a a a a a 3 order of t determinan a a a a 2 order of t determinan a a a a a a a a a + ( ) ( ) ( ) ( ) ( ) Groups ess Dimensionl 4 Expect 4 Rank) Matrix ( 3 Parameters 7 3 Rank atrix M 1 0 0 1 1 0 1 1 D V, , select check, t Determinan t determinan positive largest of order - Rank Matrix Exponent = = + = + + × × ρ 2 D y x e L

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( ) ( ) ( ) ( ) f e d 2 f e d f e d 2 f e d f e d 1 f e d c b a Ve VD VL V p ) e D L V ( f p e D L V f p f e d 2 b 2 c f e d b f e d b 2 b b 3 -3 1 - equation, L the into ng substituti b 2 c fromT b 1 a M from c - -b 2 - T, for f e d c b - -3a 1 - L, for b a 1 M, for Exponents Unit Balance µ ρ µ ρ µ ρ ρ = × × × × µ × ρ = × × × × µ × ρ = + + + = = = + + + + + = = = = + + + + = + = + + + + + + PIPE FLOW ( ) 0 0 0 1 - 1 - 0 2 - T 1 1 1 1 1 - 3 - 1 - L 0 0 0 0 1 1 1 M e D L V μ p e , D , L , V , , f P : hypothesis ρ µ ρ = Multiplication or division of one dimensionless number by another result in a dimensionless number µ ρ = ρ µ ρ µ ρ
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FL07Ch08 - EQUATION OF MOTION one dimensiona l steady dp d...

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