Problem 5.74

# Problem 5.74 - V s d V s d V s d V From the definition of...

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(a) show that the velocity field represents a possible incompressible flow (b) Rotation at (x, y) = (1, 1) (c) Circulation about the unit square shown above Problem 5.74 [Difficulty: 2] Solution: Given: Two-dimensional flow field Find: Γ 0.500 ft 2 s We will apply the definition of circulation to the given velocity field.    0 t w z v y u x Governing Equations: (Continuity equation) V 2 1 (Definition of rotation) s d V (Definition of circulation) Assumptions: (1) Steady flow (2) Incompressible flow (3) Two dimensional flow (velocity is not a function of z) Based on the assumptions listed above, the continuity equation reduces to: x u y v 0 This is the criterion against which we will check the flow field. x u y v 2A x Bx 2 1 2f t s x 1 ft s x 0 This could be an incompressible flow field. k By Bxy Ax z y x k j i ˆ 2 1 0 ˆ ˆ ˆ 2 1 2 s rad ˆ 5 . 0 k From the definition of rotation: At (x, y) = (1, 1) da cd bc ab s
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Unformatted text preview: V s d V s d V s d V From the definition of circulation we break up the integral: Bxydy dx Ax j dy i dx j Bxy i Ax s d V 2 2 The integrand is equal to: Therefore, the circulation is equal to: x a x b x A x 2 d y b y c y B x y d x c x d x A x 2 d y d y a y B x y d Evaluating the integrals: A 3 x b 3 x a 3 x d 3 x c 3 B 2 x c y c 2 y b 2 x a y a 2 y d 2 Since x a x d and x b x c we can simplify: B 2 x c y c 2 y b 2 Substituting given values: 1 2 1 ft s 1 ft 1 2 2 ft 2...
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## This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.

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