Irrotational flow, superposition of sources and sinks, and the Joukowski airfoil
Potential flow theory describes fluid motion that is both irrotational and inviscid. When the vorticity is everywhere zero (ω = ∇ × v = 0), the velocity field can be written as the gradient of a scalar potential:
where φ is the velocity potential. For incompressible flow, conservation of mass (∇ · v = 0) then requires that φ satisfy Laplace's equation: ∇²φ = 0. This is a linear equation, which means solutions can be superimposed — we can build complex flows from simple building blocks.
In two dimensions, we also define the stream function ψ, where the velocity components are u = ∂ψ/∂y and v = −∂ψ/∂x. Lines of constant ψ are streamlines, and lines of constant φ are equipotential lines. These two families of curves are always orthogonal to each other, forming a beautiful mathematical structure.
Every potential flow can be decomposed into a few canonical building blocks. Explore each one below: streamlines (blue, solid) show the direction of flow, while equipotential lines (orange, dashed) are perpendicular to them. Notice how the source and vortex are complementary — the streamlines of one are the equipotentials of the other.
Because Laplace's equation is linear, the velocity at any point is simply the vector sum of contributions from every element. Place sources, sinks, and vortices on the canvas and watch the combined streamline pattern emerge in real time. Try building a Rankine half-body (source + uniform flow) or a Rankine oval (source + sink + uniform flow).
A uniform stream plus a doublet produces flow around a circular cylinder. Adding circulation (a free vortex) breaks the symmetry and creates a net lift force — this is the Magnus effect, the same physics that makes a spinning baseball curve. The yellow dots mark the stagnation points, where the fluid velocity is zero.
The Joukowski transform w = z + c²/z is a conformal mapping that turns the simple cylinder flow into flow around a realistic airfoil shape. By shifting the circle center off the origin, you control the airfoil's thickness (x offset) and camber (y offset). The Kutta condition fixes the circulation so the flow leaves the trailing edge smoothly, giving a unique lift coefficient for each configuration.
Joukowski transform: w = z + c²/z maps the circle in the z-plane to an airfoil shape in the w-plane.
X offset controls thickness. Y offset controls camber. The Kutta condition sets circulation so the trailing edge stagnation point is at the cusp.
A remarkable result of potential flow theory is D'Alembert's paradox: a body moving at constant velocity through an inviscid, irrotational fluid experiences zero drag. The pressure distribution is perfectly symmetric fore and aft, so the net force in the direction of motion vanishes.
Of course, real fluids have viscosity. Even a tiny amount of viscosity creates a thin boundary layer near the surface where vorticity is generated, leading to pressure asymmetry and drag. Potential flow gives the correct pressure field far from the body and the correct lift (via the Kutta-Joukowski theorem), but it cannot predict drag — for that, you need to account for viscous effects in the boundary layer.
Nikolai Joukowski published his airfoil theory in 1910, providing the first rigorous mathematical explanation of how wings generate lift. His conformal mapping technique gave early aeronautical engineers a practical design tool — by adjusting the circle offset parameters, they could systematically explore airfoil shapes and predict their lift characteristics before building wind tunnel models.
While modern computational fluid dynamics has largely superseded these analytical methods for detailed design, the Joukowski airfoil remains a cornerstone of aerospace education. It illustrates how deep mathematical ideas — complex analysis, conformal mappings, and the theory of analytic functions — can solve profoundly practical engineering problems.