The exact potential-flow solution around a circular cylinder predicts a nonuniform pressure distribution, yet its net drag is zero. This conflict between a mathematically consistent inviscid solution and the nonzero drag observed on real bodies is a classic example of D'Alembert's paradox.

Pressure force on the cylinder

For a cylinder of radius $a$, the potential-flow pressure coefficient is

$$C_p=1-4\sin^2\theta.$$

Thus

$$p-p_\infty =\frac12\rho U^2\left(1-4\sin^2\theta\right).$$

Pressure acts normal to the surface. For a surface element per unit cylinder span,

$$dA=a,d\theta,$$

and the pressure force on the body is

$$d\mathbf F=-p\mathbf n,dA.$$

The $x$ component is

$$dF_x=-p\cos\theta,a,d\theta.$$

A uniform ambient pressure makes no net contribution around the closed cylinder, so the drag can be computed from gauge pressure:

$$D=-a\int_0^{2\pi}(p-p_\infty)\cos\theta,d\theta.$$

Substituting the ideal pressure distribution,

$$D=-\frac12\rho U^2a \int_0^{2\pi} \left(1-4\sin^2\theta\right)\cos\theta,d\theta.$$

Both terms integrate to zero over a full period, so

$$\boxed{D=0}.$$

The same fore-aft and top-bottom symmetries also give zero lift for the noncirculating cylinder.

Why the ideal pressure forces cancel

Potential flow around the noncirculating cylinder is symmetric about both the horizontal and vertical axes.

The front stagnation region has elevated pressure, but the ideal solution predicts an equivalent pressure recovery on the rear half of the cylinder. The rear therefore pushes forward just enough to cancel the backward pressure force on the front.

In the inviscid model there is also no viscous wall shear, because viscosity has been removed from the governing equations.

Thus both possible contributions to drag vanish:

• no skin-friction drag;
• no net pressure drag.

The paradox

Real cylinders at finite Reynolds number generally experience drag. The contradiction is not an algebra error in the potential-flow solution. It reveals that a seemingly small physical effect—viscosity—can qualitatively change the global force by altering the flow near the wall.

No slip creates a boundary layer in which vorticity and shear are concentrated. Under an adverse pressure gradient, that boundary layer can separate from the surface. Separation produces a wake and destroys the ideal fore-aft pressure recovery.

The resulting asymmetric pressure field creates pressure drag, often much larger than the direct viscous shear contribution on a bluff body.

Singular-perturbation character

At high Reynolds number, viscosity may be negligible through most of the outer flow while remaining essential in a thin near-wall region.

This means that taking

$$\mu\to0$$

is not always equivalent to setting

$$\mu=0$$

from the beginning.

If viscosity is set identically to zero, the no-slip condition disappears and the model cannot form the viscous boundary layer and separated wake responsible for real drag.

The high-Reynolds-number flow can therefore contain

• an outer region well approximated by inviscid flow;
• thin viscous boundary layers and shear layers that determine separation and drag.

A small coefficient multiplying the highest spatial derivatives can have an order-one effect on the final force. This is the hallmark of a singular perturbation.

Worked force comparison

Suppose a real cylinder in a fluid has reference area per unit span

$$A_{ref}=D\times1$$

and measured drag coefficient

$$C_D=1.0.$$

For

$$\rho=1.2,\mathrm{kg/m^3},$$

$$U=10,\mathrm{m/s},$$

and cylinder diameter

$$D=0.10,\mathrm m,$$

the measured drag per unit span would be

$$D'=\frac12\rho U^2C_DD.$$

Thus

$$D' =\frac12(1.2)(10^2)(1.0)(0.10) =\boxed{6.0,\mathrm{N/m}}.$$

The corresponding ideal potential-flow prediction is

$$\boxed{D'_{ideal}=0}.$$

The difference is not explained by a tiny correction to the symmetric potential-flow pressure field. It reflects a different flow structure containing viscous boundary layers and a wake.

What potential flow is still good for

D'Alembert's paradox does not make potential flow useless. The inviscid outer-flow solution can still predict

• approximate pressure and velocity away from viscous regions;
• stagnation-point structure;
• the forcing seen by a thin boundary layer;
• circulation-related lift when the appropriate circulation is supplied.

The lesson is instead about model scope: a model can satisfy its own equations exactly while omitting the mechanism that controls the observable of interest.

D'Alembert's paradox is therefore a central bridge between ideal flow and boundary-layer theory. It explains why viscosity can be asymptotically small in much of a flow while remaining indispensable for predicting drag.