When a fluid is both incompressible ((
abla cdot mathbf{u} = 0)) and irrotational ((
abla imes mathbf{u} = mathbf{0})), the non-linear Navier-Stokes equations undergo a miraculous mathematical reduction: they transform into Laplace’s equation, opening the gates to the immensely powerful machinery of complex analysis, harmonic function theory, and conformal mapping.
1. The Velocity Potential and Stream Function
For an irrotational flow, vector calculus guarantees that the velocity field can be expressed as the spatial gradient of a single scalar velocity potential (phi(mathbf{x}, t)):
[
abla imes mathbf{u} = mathbf{0} implies mathbf{u} =
abla phi
]
Substituting this representation into the incompressibility condition ((
abla cdot mathbf{u} = 0)) produces Laplace’s equation:
[
abla^2 phi = 0
]
In two dimensions ((x, y)), incompressibility also guarantees the existence of a scalar stream function (psi(x, y)) satisfying:
[
u = rac{partial phi}{partial x} = rac{partial psi}{partial y}, quad v = rac{partial phi}{partial y} = -rac{partial psi}{partial x}
]
These equations are precisely the Cauchy-Riemann equations of complex variable theory. Curves of constant (phi) (equipotential lines) and curves of constant (psi) (streamlines along which particles travel) form an orthogonal curvilinear mesh throughout the flow field.
2. The Complex Potential and Complex Velocity
By defining the complex coordinate (z = x + i y), the velocity potential and stream function combine into a holomorphic (analytic) complex potential (w(z)):
[
w(z) = phi(x, y) + i psi(x, y)
]
The derivative of (w(z)) with respect to (z) defines the complex velocity:
[
rac{dw}{dz} = rac{partial phi}{partial x} + i rac{partial psi}{partial x} = u – i v
]
The local physical flow speed is given by the modulus: (|mathbf{u}| = sqrt{u^2 + v^2} = |dw/dz|).
3. Elementary Flow Superposition
Because Laplace’s equation is strictly linear, complex potentials can be linearly superposed to synthesize intricate flow topographies:
- Uniform Flow at angle (alpha): (w(z) = U_infty e^{-ialpha} z).
- Source / Sink of strength (m): (w(z) = rac{m}{2pi} ln(z – z_0)).
- Point Vortex of circulation (Gamma): (w(z) = -rac{iGamma}{2pi} ln(z – z_0)).
- Doublet (Dipole) of strength (mu_d): (w(z) = rac{mu_d}{2pi z}).
Flow Past a Cylinder with Circulation
Superposing a uniform flow (U_infty), a doublet of strength (mu_d = 2pi U_infty R^2), and a clockwise vortex of circulation (Gamma) yields the canonical complex potential for flow past a rotating cylinder of radius (R):
[
w(z) = U_infty left( z + rac{R^2}{z}
ight) + rac{iGamma}{2pi} ln left( rac{z}{R}
ight)
]
On the cylinder surface ((|z| = R)), (psi = 0), confirming that the circle forms an impermeable boundary streamline. When (Gamma > 0), the stagnation points shift downward, accelerating fluid over the top surface and creating asymmetric pressure distribution that generates aerodynamic lift.
4. Conformal Mapping: The Joukowski Transformation
Conformal mapping utilizes analytic functions to transform simple geometries into complex physical bodies while preserving local angles. The celebrated Joukowski transformation:
[
zeta = z + rac{c^2}{z}
]
maps a circle centered slightly offset from the origin in the (z)-plane into an aerodynamically realistic airfoil with a sharp trailing edge in the (zeta)-plane. Applying the Kutta condition (fixing circulation (Gamma) such that the flow leaves the sharp trailing edge smoothly without infinite velocity) allows analytical computation of aerodynamic lift.
5. D’Alembert’s Paradox
Integrating surface pressure around any closed body in steady, unseparated potential flow using Bernoulli’s principle ((p + rac{1}{2}
ho |mathbf{u}|^2 = ext{const})) yields a devastating conclusion: the net hydrodynamic drag force is identically zero:
[
F_{ ext{drag}} = oint_{partial Omega} p , n_x , dA quiv 0
]
This is d’Alembert’s Paradox (1752). In inviscid theory, complete pressure recovery occurs at the rear stagnation point. In real fluids, boundary layer viscosity inevitably breaks this symmetry, inducing flow separation and generating wake pressure drag.
