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D'Alembert's paradox states that in a potential flow, the drag force is zero on a body moving with a constant velocity relative to the fluid. But what about pressure drag? Shouldn't there be drag from the pressure distribution around the airfoil? I have heard conflicting answers from multiple sources and I would like to clear up the confusion.

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Short answer: D'Alembert's paradox always holds as stated:

For incompressible and inviscid potential flow, the drag force is zero on a body moving with constant velocity relative to the fluid.

Longer answer: Pressure drag is a viscous phenomenon. Anderson section 3.18 has an extended discussion of experimental results stating that

the real flow over a circular cylinder is dominated by friction effects, namely, the separation of the flow over the rearward face of the cylinder. In turn, a finite pressure drag is created on the cylinder, and d’Alembert’s paradox is resolved.

Anderson 3.15 and any number of online resources convincingly and rigorously prove that inviscid flow around a cylinder preserves d’Alembert’s paradox (even when the cylinder is spinning and producing lift) because there will always be cancellation of aerodynamic forces due to the symmetry of the flow field across the line concurrent with the lift vector. The easiest way to convince yourself that this is also true for airfoils is the conformal mapping technique explained in this answer.

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