p1122

# p1122 - I y = ⇒ f I y = U y a = ⇒ f y = Uy 2 2 a...

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11.22. CHAPTER 11, PROBLEM 22 25 11.22 Chapter 11, Problem 22 Since the given velocity vector is u = U i + Uy/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 ,wef ind φ ( x, y )= Ux + f ( y ) where f ( y ) is a function of integration. Then, differentiating with respect to y and comparing to the equation for ∂φ / y above, ∂φ y =
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Unformatted text preview: 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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