Elementary_irrotational_flows

Elementary_irrotational_flows - Elementary Planar...

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Elementary Planar Irrotational Flows in Complex Variables Author: John M. Cimbala, Penn State University Latest revision: 17 October 2007 Note : Consider steady, incompressible, irrotational, Newtonian fluid flow in which gravity is neglected. The flow is assumed to be two-dimensional in the x-y or r- θ plane. Summary of the Equations Complex potential: ( ) ( ) () , , wz x y ix y φψ =+ or ( ) ( ) , , r i r φ θψ , where i zxi yr e =+ = , u x y ψ ∂∂ == , v yx , 1 r u rr , and 1 u . Complex velocity: i (, ) (,) (, i r dw uxy i vxy q e u r iu r e dz α θθ =− = = where 22 qu v , 1 tan v u ⎛⎞ = ⎜⎟ ⎝⎠ . Elementary Planar Irrotational Flows x = 0 1 y U 2 - 2 - 1 3 = 0 1 2 - 2 - 1 a. Uniform stream in the x -direction : 0 uU v = = , w z Uz Ux iUy ==+ , dw U dz = , Ux = , Uy = x = 0 1 y U 2 - 2 - 1 = 0 1 2 - 2 - 1 b. Uniform stream in an arbitrary direction : cos sin vU = = , i U ze = , cos sin i dw Ui U dz e αα = , cos sin Ux y , ( ) cos sin Uy x . c. Line source at the origin
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