HW#6 Solutions - Problem 5.1[Difficulty 1 The list of velocity fields provided above Given Which of these fields possibly represent two-dimensional

HW#6 Solutions - Problem 5.1[Difficulty 1 The list of...

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Problem 5.1 [Difficulty: 1] Given: The list of velocity fields provided above Find: Which of these fields possibly represent two-dimensional, incompressible flow Solution: We will check these flow fields against the continuity equation 0 t w z v y u x Governing Equations: (Continuity equation) Assumptions: (1) Incompressible flow ( ρ is constant) (2) Two dimensional flow (velocity is not a function of z) Based on the two assumptions listed above, the continuity equation reduces to: x u y v 0 This is the criterion against which we will check all of the flow fields. a) u x y t ( ) 2 x 2 y 2 x 2 y v x y t ( ) x 3 x y 2 4 y x u x y t ( ) 4 x 2 x y y v x y t ( ) x 2 y 4 ( ) Hence x u y v 0 INCOMPRESSIBLE b) u x y t ( ) 2 x y x 2 y v x y t ( ) 2 x y y 2 x 2 x u x y t ( ) 2 y 2 x y y v x y t ( ) 2 x 2 y Hence x u y v 0 NOT INCOMPRESSIBLE c) u x y t ( ) x 2 t 2 y v x y t ( ) x t 2 y t x u x y t ( ) 2 t x y v x y t ( ) t Hence x u y v 0 NOT INCOMPRESSIBLE d) u x y t ( ) 2 x 4 y ( ) x t v x y t ( ) 3 x y ( ) y t x u x y t ( ) t 2 x 4 y ( ) 2 t x y v x y t ( ) t 3 x 3 y ( ) 3 t y Hence x u y v 0 NOT INCOMPRESSIBLE
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