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Integration with variables Notes_Part_8

Integration with variables Notes_Part_8 - Using...

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Unformatted text preview: Using formula (7.19) for the complex derivative, dχ dz = ∂ϕ ∂x − i ∂ϕ ∂y = u − i v, so ∂ϕ ∂x = u, ∂ϕ ∂y = v. Thus, ∇ ϕ = v , and hence the real part ϕ ( x, y ) of the complex function χ ( z ) defines a velocity potential for the fluid flow. For this reason, the anti-derivative χ ( 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 Δ ϕ = 0. Conversely, any harmonic function can be viewed as the potential function for some fluid flow. The real fluid velocity is its gradient v = ∇ ϕ . The harmonic conjugate ψ ( 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 Δ ψ = 0, and the potential and stream function are related by the Cauchy–Riemann equations (7.18): ∂ϕ ∂x = u = ∂ψ ∂y , ∂ϕ ∂y = v = − ∂ψ ∂x . (7 . 44) The level sets of the velocity potential, { ϕ ( x, y ) = c } , where c ∈ R is fixed, are known as equipotential curves . The velocity vector v = ∇ ϕ points in the normal direction to the equipotentials. On the other hand, as we noted above, v = ∇ ϕ is tangent to the level...
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Integration with variables Notes_Part_8 - Using...

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