phys documents (dragged) 49

phys documents (dragged) 49 - Chapter 9 Transport phenomena...

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Unformatted text preview: Chapter 9: Transport phenomena 43 and the defnitions 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 ¡ows 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 o¢ strength Q in x = a and strength- Q in x =- a ¢ollows ¢rom superposition: φ =- Qax/ 2 π r 2 where Qa is the dipole strength. For a vortex holds: φ = Γ θ / 2 π . I¢ an object is surrounded by an uni¢orm main ¡ow with v = ve x and such a large Re that viscous e¢¢ects 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 ¢rom the neglection o¢ viscous e¢¢ects. The li¢t F y is also created by η because Γ = 0 due to viscous e¢¢ects. Henxe rotating bodies also create a ¢orce perpendicular to their direction o¢ motion: thedue to viscous e¢¢ects....
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