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phys documents (dragged) 40

phys documents (dragged) 40 - Chapter 9 Transport phenomena...

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Chapter 9: Transport phenomena 43 and the definitions v x = ∂ψ / y , v y = - ∂ψ / x holds: Φ AB = ψ ( B ) - ψ ( A ) . In general holds: 2 ψ x 2 + 2 ψ y 2 = - ω z In polar coordinates holds: v r = 1 r ∂ψ ∂θ = ∂φ r , v θ = - ∂ψ r = 1 r ∂φ ∂θ For source flows with power Q in ( x, y ) = (0 , 0) holds: φ = Q 2 π ln( r ) so that v r = Q/ 2 π r , v θ = 0 . For a dipole of strength Q in x = a and strength - Q in x = - a follows from superposition: φ = - Qax/ 2 π r 2 where Qa is the dipole strength. For a vortex holds: φ = Γ θ / 2 π . If an object is surrounded by an uniform main flow with v = ve x and such a large Re that viscous effects are limited to the boundary layer holds: F x = 0 and F y = - Γ v . The statement that F x = 0 is d’Alembert’s paradox and originates from the neglection of viscous effects. The lift F y is also created by η because Γ = 0 due to viscous effects. Henxe rotating bodies also create a force perpendicular to their direction of motion: the Magnus effect .
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