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Unformatted text preview: (because f ( z ) is analytic and conformal). Applying the chain rule and using the CauchyRiemann equations, this function can be shown to satisfy 2 z ( x,y ) x 2 + 2 z ( x,y ) y 2 = 2 w ( u,v ) u 2 + 2 w ( u,v ) v 2 !  f ( z )  2 = 0 . (2) Since f is conformal in D z , f ( z ) 6 = 0, so w ( u,v ) is harmonic in D w . The Dirichlet problem Boundary conditions are required to obtain a unique solution to the Laplace equation. The Dirichlet problem is obtained by solving the Laplace equation with specied on the boundary. Potential Flow Over a Joukowski Airfoil21 1 221. 510.5 0.5 1 1.5 2 u v Stream function in the w plane21 1 221.510.5 0.5 1 1.5 2 x y Stream function in the z plane21 1 221.510.5 0.5 1 1.5 2 a b Stream function in the c plane...
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 Fall '09
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