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**Unformatted text preview: **Using formula (7.19) for the complex derivative, d dz = x i y = u i v, so x = u, y = v. Thus, = v , and hence the real part ( x, y ) of the complex function ( z ) defines a velocity potential for the fluid flow. For this reason, the anti-derivative ( z ) is known as a complex potential function for the given fluid velocity field. Since the complex potential is analytic, its real part the potential function is harmonic, and therefore satisfies the Laplace equation = 0. Conversely, any harmonic function can be viewed as the potential function for some fluid flow. The real fluid velocity is its gradient v = . The harmonic conjugate ( x, y ) to the velocity potential also plays an important role, and, in fluid mechanics, is known as the stream function . It also satisfies the Laplace equation = 0, and the potential and stream function are related by the CauchyRiemann equations (7.18): x = u = y , y = v = x . (7 . 44) The level sets of the velocity potential, { ( x, y ) = c } , where c R is fixed, are known as equipotential curves . The velocity vector v = points in the normal direction to the equipotentials. On the other hand, as we noted above, v = is tangent to the level...

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