D’Alembert’s paradox

Since the pressure field depends only on the magnitude of the velocity (see Eq. (1-22)) and since the flow field has fore-and-aft symmetry, the modified pressure field forward from the equator of the sphere is the mirror image of that to the rear. This leads to d Alembert s paradox that the net force acting on the sphere is predicted to be zero. This paradox can only be resolved, and nonzero drag obtained, by accounting for the viscosity of the fluid. For in viscid flow, the surface velocity and pressure follow as... 

At infinite Reynolds number, potential flow theory may be used to investigate bubble motion. As a consequence of D Alembert s paradox (see, e.g., Ref. 10), potential flow theory predicts no drag on a steadily rising bubble. However, it provides a result for the force on an accelerating sphere ... 

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