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Problem 5.14 - Problem 5.14[Difficulty 3 Steady...

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Problem 5.14 [Difficulty: 3] Given: Steady, incompressible flow in x-y plane: u A x 2 y 2 A 0.3 m 3 s 1 Find: (a) a possible y component of velocity for this flow field (b) if the result is valid for unsteady, incompressible flow (c) number of possible y components for velocity (d) equation of the streamlines for the flow (e) plot streamlines through points (1,4) and (2,4) Solution: We will check this flow field using 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) 0 y v x u Based on the two assumptions listed above, the continuity equation reduces to: 2 2 Axy x u 2 2 Axy x u y v The partial of u with respect to x is: Therefore from continuity: Integrating this expression will yield the y-component of velocity: v x 2 A x y 2 d f x ( ) v 2 3 A x 2
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