D'Alembert's paradox
![Jean le Rond d'Alembert (1717-1783)](/Images/godic/202501/19/Alembert2546.jpg")
![From experiments it is known that there is always – except in case of superfluidity – a drag force for a body placed in a steady fluid onflow. The figure shows the drag coefficient Cd for a sphere as a function of Reynolds number Re, as obtained from laboratory experiments. The solid line is for a sphere with a smooth surface, while the dashed line is for the case of a rough surface. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:
•2: attached flow (Stokes flow) and steady separated flow,
•3: separated unsteady flow, having a laminar flow boundary layer upstream of the separation, and producing a vortex street,
•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic turbulent wake,
•5: post-critical separated flow, with a turbulent boundary layer.](/Images/godic/202501/19/Drag_sphere_nasa.svg2546.png")
![Steady and separated incompressible potential flow around a plate in two dimensions,[16] with a constant pressure along the two free streamlines separating from the plate edges.](/Images/godic/202501/19/Cavity_flow_plate.svg2546.png")
![Pressure distribution for the flow around a circular cylinder. The dashed blue line is the pressure distribution according to potential flow theory, resulting in d'Alembert's paradox. The solid blue line is the mean pressure distribution as found in experiments at high Reynolds numbers. The pressure is the radial distance from the cylinder surface; a positive pressure (overpressure) is inside the cylinder, towards the centre, while a negative pressure (underpressure) is drawn outside the cylinder.](/Images/godic/202501/19/Flow_pressure_with_friction.svg2546.png")
In fluid dynamics, d'Alembert's paradox (or the hydrodynamic paradox) is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to the fluid. Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to fluids, such as air and water; especially at high velocities corresponding with high Reynolds numbers. It is a particular example of the reversibility paradox.