Fluid Dynamics-D'Alembert's paradox

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SUMMARY

D'Alembert's paradox addresses the behavior of nonviscous, irrotational flow around a cylinder, highlighting the lack of drag in such conditions. The discussion references Equation (2.38) from a fluid dynamics resource, which derives from Equations 2.36 and 2.37. The key point is that at the cylinder's surface, the radial velocity is zero, leaving only the tangential velocity to contribute to the total velocity. The paradox emphasizes that with zero viscosity, flow patterns are indeterminate due to the absence of interaction between streamlines.

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hhhmortal
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Hi, I'm trying to prove D'alembert's paradox, but considering a nonviscous, irrotational flow of a fluid around a cylinder.

I used the following webpage to help me, which was very good, but I got to a part where I can't seem to solve, perhaps cause my maths needs some brushing up.

http://galileo.phys.virginia.edu/classes/311/notes/fluids1/fluids11/node19.html


It's on Equation (2.38). How do you get this from Eq 2.36 and Eq 2.37 ?

What component of velocity is being used on Eq 2.38?


Thanks.
 
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At the surface of the cylinder (stated measurement point)

R = r so substituting

The radial velocity (cosine term) is zero.

So total velocity is given by tangential velocity (sine term) alone

substitute R = r and square and you have your next equation (2.38)
 
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Note - D'Alembert's paradox relies on a specific flow pattern in a fluid with zero viscosity. With zero viscosity, there's no interaction between adjacent "streamlines" so flow patterns aren't determinate. An alternate, but just as valid flow pattern would consists of a long rectangle of fluid flowing at the same speed as the cylinder, with no interaction of the surrounding fluid. It all depends on the initial state of the fluid. It's not clear to me, how any mathematical model can be made to predict a flow in fluid with zero viscosity.
 

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