D'Alembert's paradox
The d'Alembert paradox is a statement in fluid mechanics that can be derived from the law of conservation of energy (see below). The paradox was first formulated in 1752 by Jean Baptiste le Rond d'Alembert .
It says that an impermeable body of any shape in an unlimited, smooth and steady parallel flow does not experience any resistance , i.e. no force in the direction of the flow, if the prerequisite is met that the flow disturbance caused by the body is at a great distance from the body subsides. Forces perpendicular to the direction of flow ( dynamic lift as with a wing) are not excluded, so that, for example, unlimited horizontal glider flight would be possible without using energy.
A source , i.e. a body from which liquid flows out, even experiences a propulsion, i.e. H. a force that opposes the flow. Conversely, a sink experiences a force in the direction of flow.
The statement of the paradox of d'Alembert is paradoxical in the sense that it contradicts our everyday experience, since in real flows in general neither the frictional forces nor the disturbance of the flow through the body can be completely neglected, so that the requirements of the statement are not met are.
The justification with the energy law goes as follows: If a force component in the direction of the uniform movement of the body were required to maintain the movement, then the force would constantly do mechanical work on the liquid (or the gas). However, the energy cannot be absorbed by the ideal liquid (or the gas) because there are neither frictional losses that would allow conversion into heat, nor changes in the kinetic energy of the liquid (because of the assumed stationarity, i.e. the constant flow velocities everywhere ). According to the law of conservation of energy, the force component in the direction of motion must therefore be zero.
Individual evidence
- ↑ Lev Landau , Jewgeni Lifschitz : Textbook of Theoretical Physics , Volume 6 Hydrodynamics , Akademie-Verlag, Berlin 1991, Chapter I § 11