let z V y z c d V dy but e No e w v x ω ω ν ν ω ω ω z 2 2 3 2 3 2 z 3 2 uU 1 V

Let z v y z c d v dy but e no e w v x ω ω ν ν ω

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, let z V y z c d V dy but e No e w v x ω ω −∞ ν = ν ω = → ∞ ω = = ω ( ) ( ) z 2 2 3 2 3 2 z 3 2 u=U 1 V U=c .0 0 0 0=c c V & V V y V y y V y u du y dy du c e dy u c e c u y U c c U V u U y U e e U V ν ν ν ν = − ω = − = ν = + → ∞ = + = ν = = + = − ν ω = − ν y UV ν z ω is max at y=0 z ω
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terms in eq. 3 2 2 3 2 2 V y z V y z V V U e convection y V U e diffusion y ν ν ∂ω = − ν ∂ ω ν = − ν if V vorticity moves toward wall V if V vorticity moves away from wall dif. term temdency of shear layer to grow due to viscous di Note length ν ν → / 0 ( ) ( ) ffusion term toward the wall As usual, define the B.L. thickness to be the point where u=0.99U . Air at 20 , if V=1 cm/s, 7 m V m =4.6 V V convective Eg C δ ϑ δ ν ν = D • For a plate with a leading edge (x=0), a Laminar shear layer would grow and approach this constant value. it is estimated by Iglisch (1944) 2 4 U=10 m/s, V=1 cm/s x 6m U x V ν
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Ex2: Flow through & between porous plates h Flow . dp const dx = y x . . u=0, v= V at y=0, h 2-D , N-S eqs. B C u t u u x + 2 2 1 u p u v y x x + = − + ν ρ ∂ 2 2 u y v t + v u x + v v y + 2 2 1 p v y x = − + ν ρ ∂ 2 2 v y + -V
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Assume 1) Steady 2) Fully-dev. flow ( ) 0 0 P=f x p x y = N 2 . u Cont x 2 2 2 2 part hom part part hom 1 1 0 v=const.= V 1 1 V . 1 solution = 1 , , 1 V y V y v y du dp d u d u V du dp const dy dx dy dy dy dx du d V dp Let dy dy dx DE dp A V dx c e du dp c e dy V dx µ α α α µ α α α α α α α ν ν + = = − + ν + = = ρ ν = + = ν + = = ρ = = = + ρ
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( ) ( ) ( ) 1 2 2 V 0 0 1 . . u=0 at y=0 & y=h no suc 1 h tion V=0, then 1 u 2 1 2 2 u= V 1 1 1 V y V V y V h V y V h dp u y c e y c V V dx B C s if dp y hy d e dp y dx h e V e du h dp dy V dx h x dp e y h dx µ τ µ µ τ ν ν ν ν ν = ν = − + + ρ ρ ν = = + ρ = = parabolic vel. profile
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0 0 1 1 1 1 y y h w V h y V h w V h V du h dp dy V dx h e V e h dp V dx h e µ τ µ µ τ = = = ν ν ν ν = = + ρ ν = + ρ
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dp we have fully-dev. flow, shear stress & is related dx If L h P 1 P 2 0 y w τ = y h w τ = ( ) ( ) ( ) ( ) 0 0 1 2 1 2 1 2 1 2 0 for fully-dev. flow =P . . 0 y h y y y h w w w w F h P h L L P P h L dp P P L dx P P dp dx L τ τ τ τ = = = = = + = = = − = L P 1 P 2
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