Math 2.0

Fluid mechanics Ideal fluids Counterexample

d'Alembert's paradox

Builds on Potential flowInviscid flowPotential flow around a circular cylinderIncompressible flow

Counterexample

Let a rigid body move with constant velocity through an incompressible, inviscid fluid. The fluid fills the rest of three-dimensional space and is at rest at infinity. Suppose the flow is a steady potential flow in the frame of the body. Then the total force that the fluid exerts on the body is zero; in particular the drag is zero. In two dimensions the drag is also zero, although a lift proportional to the circulation can remain. Experiments show that a body moving steadily through a real fluid such as air or water feels a drag force. So the ideal-fluid model with potential flow fails to predict drag.

Remarks

The idea of the proof: in three dimensions the velocity potential of the disturbance decays like that of a dipole, so the disturbance velocity decays like r−3r^{-3} at distance rr. A momentum balance on a large sphere around the body expresses the force as a flux through the sphere, and this flux tends to zero as the radius grows. For the circular cylinder one can see it directly: the pressures on the front half and on the back half are mirror images. The resolution is viscosity: however small, it enforces the no-slip condition and creates a thin boundary layer at the surface. The boundary layer can separate from the body, and the resulting wake has a lower pressure than the front, which gives a nonzero drag coefficient.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Circulation, concept, no evidence levelCirculationconceptComplex potential, concept, no evidence levelComplex potentialconceptEuler equations, concept, no evidence levelEuler equationsconceptHolomorphic function, concept, no evidence levelHolomorphic functionconceptIncompressible flow, concept, no evidence levelIncompressible flowconceptInviscid flow, concept, no evidence levelInviscid flowconceptPotential flow, concept, no evidence levelPotential flowconceptStream function, concept, no evidence levelStream functionconceptStreamlines, streaklines and pathlines, concept, no evidence levelStreamlines, streakli…conceptVelocity potential, concept, no evidence levelVelocity potentialconceptVorticity, concept, no evidence levelVorticityconceptMilne-Thomson circle theorem, statement, no evidence levelMilne-Thomson circle …statementBernoulli's principle, statement, no evidence levelBernoulli's principlestatementPotential flow around a circular cylinder, example, no evidence levelPotential flow around…exampled'Alembert's paradox, counterexample, no evidence leveld'Alembert's paradoxcounterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circulation concept
  2. Complex potential concept
  3. Euler equations concept
  4. Holomorphic function concept
  5. Incompressible flow concept
  6. Inviscid flow concept
  7. Potential flow concept
  8. Stream function concept
  9. Streamlines, streaklines and pathlines concept
  10. Velocity potential concept
  11. Vorticity concept
  12. Milne-Thomson circle theorem statement
  13. Bernoulli's principle statement
  14. Potential flow around a circular cylinder example
  15. d'Alembert's paradox counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox
About this page
TagMJEX13ZQWR
URIm2://MJEX13ZQWR
Kindcounterexample
WritingW0 stub
MSC76G25, 76B99
UsesPotential flow, Inviscid flow, Potential flow around a circular cylinder, Incompressible flow
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Changed3 November 2026

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