Potential flow and the Laplace equation

If a flow is irrotational, \(\nabla\times\mathbf{u} = 0\), then on a simply connected region \(\mathbf{u} = \nabla\phi\) for a velocity potential \(\phi\). Incompressibility then says \[ \Delta\phi = 0. \] The nonlinear Euler equations have become the linear Laplace equation for \(\phi\), with the Neumann condition \(\partial\phi/\partial n = 0\) on fixed walls. Solutions can be superposed. The pressure, the only place the nonlinearity survives, is recovered afterwards from Bernoulli's equation. By Kelvin's theorem, a flow that starts from rest in an ideal fluid stays irrotational, so this is a large and natural class.

In two dimensions both \(\phi\) and the stream function \(\psi\) are harmonic and satisfy the Cauchy-Riemann equations \(\phi_x = \psi_y\), \(\phi_y = -\psi_x\). So \(w(z) = \phi + i\psi\) is an analytic function of \(z = x+iy\), the complex potential, and \(dw/dz = u - iv\). The building blocks are \(w = Uz\) (uniform stream), \(w = \frac{m}{2\pi}\log z\) (source), \(w = -\frac{i\Gamma}{2\pi}\log z\) (point vortex) and \(w = \frac{\mu}{z}\) (doublet). The doublet alone has \(\phi = \mu x/(x^2+y^2)\), whose Laplacian is zero away from the origin.

A uniform stream plus a doublet gives flow past a circular cylinder of radius \(a\): \[ w = U\left(z + \frac{a^2}{z}\right), \qquad \phi = U\left(r + \frac{a^2}{r}\right)\cos\theta, \qquad \psi = U\left(r - \frac{a^2}{r}\right)\sin\theta. \] On \(r = a\), \(\psi = 0\), so the circle is a streamline, and the radial velocity \(U(1 - a^2/r^2)\cos\theta\) vanishes there. The surface speed is \(|u_\theta| = 2U|\sin\theta|\), twice the free stream at the top and bottom, and Bernoulli gives the pressure coefficient \(C_p = 1 - 4\sin^2\theta\).

That pressure distribution is symmetric front to back, so its integrated force in the stream direction is zero: \(\int_0^{2\pi} C_p\cos\theta\,d\theta = 0\). This is d'Alembert's paradox (1752): a steady potential flow exerts no drag on any body, while every real body in a stream is pushed downstream. Adding a vortex \(-\frac{i\Gamma}{2\pi}\log z\) breaks the top-bottom symmetry and produces a lift per unit length \(\rho U\Gamma\) (the Kutta-Joukowski theorem, 1902 to 1906), but still no drag. The paradox is resolved by viscosity: however small, it creates a thin boundary layer that can separate and leave a wake, as Prandtl explained in 1904.

Picture it: the streamlines part smoothly round the cylinder and close up behind it, a mirror image front and back, which is exactly why the pressure pushes equally both ways. Think it: potential flow is the theory of harmonic functions with Neumann data; on a simply connected region uniqueness up to a constant follows from \(\int|\nabla(\phi_1-\phi_2)|^2 = 0\) (outside a cylinder the circulation must be fixed as well), and the whole of 2D ideal irrotational flow is a chapter of complex analysis.