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p1122 - y and comparing to the equation for ∂φ ∂ y...

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28 CHAPTER 11. POTENTIAL FLOW 11.22 Chapter 11, Problem 22 Problem: Determine the velocity potential for a flow whose velocity vector is u = U [ i + ( y/a ) j ] , where U and a are constant velocity and length scales, respec- tively. Is this flow incompressible? Solution: Since the velocity vector is u = U i + U ( y/a ) j , the velocity potential satisfies the following differential equations. u = ∂φ x = U and v = ∂φ y = U y a Integrating the first equation over x , we find φ ( x, y ) = Ux + f ( y ) where f ( y ) is a function of integration. Then, differentiating with respect to
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Unformatted text preview: y and comparing to the equation for ∂φ / ∂ y above, ∂φ ∂ y = f I ( y ) = ⇒ f I ( y ) = U y a = ⇒ f ( y ) = Uy 2 2 a + constant Therefore, the velocity potential is φ ( x, y ) = U ^ x + y 2 2 a ± + constant The flow is incompressible if ∇ · u vanishes. To check, ∇ · u = ∂ u ∂ x + ∂ v ∂ y = ∂ ∂ x ( U ) + ∂ ∂ y w U y a W = U a W = 0 Thus, the flow is not incompressible ....
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