Chapter 9: Transport phenomena43and the definitionsvx=∂ψ/∂y,vy=-∂ψ/∂xholds:ΦAB=ψ(B)-ψ(A). In general holds:∂2ψ∂x2+∂2ψ∂y2=-ωzIn polar coordinates holds:vr=1r∂ψ∂θ=∂φ∂r,vθ=-∂ψ∂r=1r∂φ∂θFor source flows with powerQin(x, y) = (0,0)holds:φ=Q2πln(r)so thatvr=Q/2πr,vθ= 0.For a dipole of strengthQinx=aand strength-Qinx=-afollows from superposition:φ=-Qax/2πr2whereQais the dipole strength. For a vortex holds:φ=Γθ/2π.If an object is surrounded by an uniform main flow withv=vexand such a large Re that viscous effects arelimited to the boundary layer holds:Fx= 0andFy=-Γv. The statement thatFx= 0is d’Alembert’sparadox and originates from the neglection of viscous effects. The liftFyis also created byηbecauseΓ= 0due to viscous effects. Henxe rotating bodies also create a force perpendicular to their direction of motion: theMagnus effect.
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