The Resolution of d'Alembert's Paradox and Thermodynamic Admissibility

The d'Alembert paradox, annunciated in 1752, was established after it was shown that the result of a net zero drag, obtained by applying potential theory to the incompressible flow past a sphere, was in contradiction with experimental results which showed a positive drag. Interpreting the result as a flaw in the theory, resulted in the declaration of the paradox. Following d'Alembert, we assume a potential motion, and proceed to analyze the consequences of this assumption using the global principles of continuum mechanics. We show that if the fluid is inviscid, the potential motion is thermodynamically admissible, the drag is zero, and the motion can persist indefinitely. Although no conventional fluid is available to either falsify (or validate), this result experimentally, in principle, the theory could be tested by using a superfluid, such as liquid Helium. If the fluid is viscous, we show that the potential irrotational motion is thermodynamically inadmissible, it is in violation of the second law of thermodynamics, and hence such a motion cannot persist. Indeed, observations show that when a rigid body is impulsively set into motion, an irrotational motion may exist initially but does not persist. Any flow property which is derived from a thermodynamically inadmissible motion cannot be expected to be in agreement with experimental data. As an illustration we show that the drag, calculated from the viscous potential solution of the flow past a sphere, is zero. We submit that since the theory of continuum mechanics predicts that no agreement between results obtained from viscous potential theory and experimental data can be expected, there is no room for a paradox once a contradiction is in fact observed. In nature, or under experimental conditions, the nonpersistence of the thermodynamically inadmissible motion proceeds in a breakup of the irrotational motion which transforms into a rotational and obviously admissible motion. We show that by selecting boundary conditions, required in the solution of the differential equations of motion, such that they satisfy the Clausius–Duhem jump conditions inequality, the thermodynamic admissibility of the solution is a priori assured. We also show the vorticity distribution at the wall associated with the particular choice of boundary condition.