D'Alembert's Paradox: Why a Perfect Fluid Has Zero Drag

Push a hand through water and you feel resistance. Yet in 1752 the French mathematician Jean le Rond d’Alembert proved that, for an incompressible, inviscid potential flow, a body moving at constant velocity feels zero drag D'Alembert's paradox, Wikipedia^[1]. He called it “a singular paradox which I leave to future Geometers to elucidate” D'Alembert's paradox, Wikipedia^[1]. The story of how the paradox arose and was worked out is traced in detail by Grimberg, Pauls and Frisch Grimberg, Pauls and Frisch 2008^[2]. It is the idea behind our Dalembertia zero-drag sticker.

What is d’Alembert’s paradox?

The paradox is a clash between a clean theory and every experiment. The theory assumes the fluid is incompressible, has no viscosity and has no swirl (it is irrotational), and that the flow is steady D'Alembert's paradox, Wikipedia^[1]. Under those assumptions the net force along the direction of motion is exactly zero. Real air and water always produce drag, especially at high Reynolds numbers, the regime of fast or large bodies D'Alembert's paradox, Wikipedia^[1]. The mismatch split the subject for a long time into hydraulics, which observed what it could not explain, and theoretical fluid mechanics, which explained what could not be observed D'Alembert's paradox, Wikipedia^[1].

How does a perfect fluid flow around a cylinder?

The simplest worked example is a long cylinder of radius R (in metres) in a uniform stream of speed U (in m s⁻¹). Because the flow is irrotational, the velocity is the gradient of a potential φ, and because it is incompressible, φ obeys Laplace’s equation. The solution in polar coordinates r and θ, with θ measured from the downstream direction, is Potential flow around a circular cylinder, Wikipedia^[3]

φ = Ur(1 + R²/r²) cos θ

Here φ has units of m² s⁻¹. Bernoulli’s equation then gives the pressure p (in pascals) from the local flow speed V Potential flow around a circular cylinder, Wikipedia^[3]:

p = p_∞ + ½ρ(U² − V²)

where ρ is the fluid density in kg m⁻³ and p_∞ is the pressure far away. On the cylinder surface, the dimensionless pressure coefficient is Potential flow around a circular cylinder, Wikipedia^[3]

C_p = (p − p_∞)/(½ρU²) = 1 − 4 sin²θ

The flow stops at the front and back stagnation points, where C_p = 1, and is fastest on the sides, where V = 2U and C_p = −3 Potential flow around a circular cylinder, Wikipedia^[3].

Why is the drag exactly zero?

Look at the formula: sin²θ takes the same value at θ and at π − θ. So the pressure on the back of the cylinder is a perfect mirror image of the pressure on the front. The high pressure that pushes the front backward is matched by an equally high pressure pushing the back forward, and the suction on the sides is symmetric too. Add up the pressure around the surface and the forces cancel: the net drag is zero Potential flow around a circular cylinder, Wikipedia^[3]. The same holds for any body in steady potential flow, not just cylinders D'Alembert's paradox, Wikipedia^[1].

The flow does have inertia, though. Accelerating a body through a perfect fluid takes extra force, as if the body were heavier. This is called added mass Added mass, Wikipedia^[4]:

F = (m + m_a)a, m_a = ⅔πR³ρ

where F is the force in newtons, m the body’s own mass, a its acceleration and m_a the added mass, which for a sphere of radius R is half the mass of the fluid it displaces Added mass, Wikipedia^[4]. Bessel introduced the idea in 1828 to explain why a pendulum swings more slowly in a fluid Added mass, Wikipedia^[4]. So the paradox is about steady motion only.

How was d’Alembert’s paradox resolved?

The missing ingredient is viscosity D'Alembert's paradox, Wikipedia^[1]. Saint-Venant began modelling viscous friction in the 1840s, and Stokes calculated the drag on a slowly moving sphere in 1851 D'Alembert's paradox, Wikipedia^[1]. But that was not enough. As viscosity becomes small, the equations seem to approach the inviscid ones, which still give zero drag D'Alembert's paradox, Wikipedia^[1].

Ludwig Prandtl’s answer in 1904 was the boundary layer. However small the viscosity, the fluid in contact with a surface does not slip, so a thin layer forms in which the speed rises from zero to the outer flow speed D'Alembert's paradox, Wikipedia^[1] Boundary layer, Wikipedia^[5]. In that layer viscosity always matters. It creates friction drag on streamlined bodies. On blunt bodies the layer peels away from the surface, leaving a low-pressure wake behind the body, so the front-to-back pressure symmetry is broken and form drag appears D'Alembert's paradox, Wikipedia^[1]. That wake can roll up into a vortex street D'Alembert's paradox, Wikipedia^[1], as on our Vortessa vortex-street sticker and in our guide to vortex streets and hydraulic jumps.

Most fluid dynamicists regard the paradox as solved in practice along Prandtl’s lines, but a formal mathematical proof is still lacking, as for so many problems involving the Navier–Stokes equations D'Alembert's paradox, Wikipedia^[1]. Those equations are the subject of our Navia fluid-problem tee.

Quick FAQ

Is there any fluid with truly zero drag?

Experiments always show drag, except in the case of superfluidity D'Alembert's paradox, Wikipedia^[1]. Read more in our guide to superfluid helium.

What is the drag crisis?

Real drag has surprises of its own. In the drag crisis, the drag coefficient of a sphere drops suddenly from about 0.5 to 0.2 at Reynolds numbers of around 300,000, as the flow pattern changes and leaves a narrower turbulent wake Drag crisis, Wikipedia^[6]. On a rough ball, such as a dimpled golf ball, it happens at lower Reynolds numbers Drag crisis, Wikipedia^[6]. More fluid designs are in our physics stickers.

References

1. Wikipedia, “D'Alembert's paradox”. https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox
2. G. Grimberg, W. Pauls and U. Frisch (2008), “Genesis of d’Alembert’s paradox and analytical elaboration of the drag problem,” Physica D: Nonlinear Phenomena 237, 1878–1886 (DOI 10.1016/j.physd.2008.01.015; metadata verified via Crossref). https://doi.org/10.1016/j.physd.2008.01.015
3. Wikipedia, “Potential flow around a circular cylinder”. https://en.wikipedia.org/wiki/Potential_flow_around_a_circular_cylinder
4. Wikipedia, “Added mass”. https://en.wikipedia.org/wiki/Added_mass
5. Wikipedia, “Boundary layer”. https://en.wikipedia.org/wiki/Boundary_layer
6. Wikipedia, “Drag crisis”. https://en.wikipedia.org/wiki/Drag_crisis

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.