Problem 4.14 - axdydz k c j by i ax j dxdz i dydz k c j by...

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Problem 4.14 [Difficulty: 3] Given: Data on velocity field and control volume geometry Find: Surface integrals Solution: First we define the area and velocity vectors j dxdz i dydz A d ˆ ˆ + = r k c j by i ax V ˆ ˆ ˆ + + = r or k j y i x V ˆ ˆ 2 ˆ 2 + + = r We will need the equation of the surface: x y 2 3 = or y x 3 2 = Then ( ) () () s m 24 6 6 4 3 2 3 1 2 2 3 3 2 ˆ ˆ ˆ ˆ ˆ 3 2 0 2 3 0 2 2 0 2 0 2 0 3 0 2 0 2 0 2 0 3 0 = = = = = + + = ∫∫ ∫∫ ∫∫ ∫∫ b a Q x b y a xdx dz b ydy dz a dz bydx dz axdy j dxdz i dydz k c j by i ax dA V A A r We will again need the equation of the surface: x y 2 3 = or y x 3 2 = , and also dx dy 2 3 = and b a = () ( )( ) ( ) () () () () () () 2 4 2 2 2 0 2 2 0 3 2 2 0 3 2 2 0 2 0 2 0 2 0 2 2 2 0 2 0 2 2 s m ˆ 60 ˆ 96 ˆ 64 ˆ 12 ˆ 24 ˆ 16 2 6 ˆ 3 9 ˆ 3 6 ˆ 3 ˆ 2 9 ˆ 3 3 ˆ ˆ 2 3 ˆ 2 3 2 3 ˆ ˆ 2 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ k j i k ac j a i a x ac j x a i x a k dz acxdx j dz dx x a i dz dx x a axdxdz k c j ax i ax xdxdz a dxdz ax k c j ax i ax
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Unformatted text preview: axdydz k c j by i ax j dxdz i dydz k c j by i ax k c j by i ax A d V V A A A A A = = = = + + = + + = + + = + + + + = r r r x y 5 m 4 m 3 m z...
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