Dalembert-Euler Paradox

a proposition in hydrodynamics according to which there is zero resistant force acting on a body in uniform and rectilinear motion within an unbounded fluid that has no viscosity, eddying, and surface of velocity discontinuity. This principle was stated by the French scientist J. d’AIembert in 1744 and by the St. Petersburg academician L. Euler in 1745. The absence of resistance to the body’s motion by the fluid, under the above hydrodynamic assumptions, is rigorously proved mathematically for both incompressible and compressible fluids. Physically, the absence of resistance is explained by the fact that under these conditions the flow should close in behind the moving body, and thus the fluid exerts an action on the rear portion of the body which balances the action (which always occurs) in the front.

In reality a body moving in a liquid or gas always experiences resistance. The contradiction between reality and the substance of the d’Alembert-Euler paradox is explained by the fact that the assumptions on which the proof of the paradox is constructed are not fulfilled under actual conditions. When a body moves in a fluid there will always be a viscosity of the fluid, eddying (particularly behind the body), and the occurrence of surface of velocity discontinuity. All of these factors cause the fluid to resist the motion of a body.