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Problem 5.38

# Problem 5.38 - v x u u x y 2 This could be an...

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Problem 5.38 [Difficulty: 2] Given: The velocity field provided above Find: (a) the number of dimensions of the flow (b) if this describes a possible incompressible flow (c) the acceleration of a fluid particle at point (1,2,3) Solution: We will check this flow field against the continuity equation, and then apply the definition of acceleration 0 t w z v y u x Governing Equations: (Continuity equation) t V z V w y V v x V u Dt V D a p (Particle acceleration) Assumptions: (1) Incompressible flow ( ρ is constant) (2) Two dimensional flow (velocity is not a function of z) (3) Steady flow (velocity is not a function of t) Based on assumption (2), we may state that: The flow is two dimensional. 0 y v x u Based on assumptions (1) and (3), the continuity equation reduces to: This is the criterion against which we will check the flow field. 0 2 2
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Unformatted text preview: v x u u x y 2 This could be an incompressible flow field. v 1 3 y 3 y V v x V u a p Based on assumptions (2) and (3), the acceleration reduces to: and the partial derivatives of velocity are: k y i y x V ˆ ˆ 2 k x j y i xy y V ˆ ˆ ˆ 2 2 and Therefore the acceleration vector is equal to: k xy j y i xy k x j y i xy y k y i y xy a p ˆ 3 2 ˆ 3 1 ˆ 3 1 ˆ ˆ ˆ 2 3 1 ˆ ˆ 3 5 4 2 3 2 2 At point (1,2,3), the acceleration is: k j i k j i a p ˆ 3 16 ˆ 3 32 ˆ 3 16 ˆ 2 1 3 2 ˆ 2 3 1 ˆ 2 1 3 1 3 5 4 k j i a p ˆ 3 16 ˆ 3 32 ˆ 3 16...
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