Inviscid Flow and Potential Flow Theory

TL;DR

You'll learn how dropping viscosity from Navier-Stokes gives the Euler equations, and how adding one more assumption — irrotational flow — turns the problem into solving Laplace's equation. You'll build flow fields by adding together simple building blocks (uniform flow, sources, vortices, doublets). You'll also see why this beautiful theory predicts zero drag on a cylinder, and why that's wrong.

1. The Mental Model

Real fluids have friction (viscosity), but far from walls, friction often barely matters. If you also assume the flow doesn't spin locally (no vorticity), the whole velocity field collapses into one scalar function that obeys the same equation as electric potentials and heat conduction. That's the trick: a hard, nonlinear fluid problem becomes an easy, linear one you can solve by addition. Potential flow theory works because "no spin" turns velocity into the gradient of a single number.

2. The Core Material

2.1 From Navier-Stokes to a Linear Problem

Start with the incompressible Navier-Stokes momentum equation:

$$\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot abla\mathbf{u}\right) = - abla p + \mu abla^2\mathbf{u} + \rho\mathbf{g}$$

Step 1 — drop viscosity. If the Reynolds number is huge, the viscous term $\mu
abla^2\mathbf{u}$ is tiny compared to the inertial terms almost everywhere (away from walls and wakes). Setting $\mu = 0$ gives the Euler equations:

$$\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot abla\mathbf{u}\right) = - abla p + \rho\mathbf{g}$$

This is still nonlinear — the $\mathbf{u}\cdot
abla\mathbf{u}$ term didn't go anywhere. Euler flow can still have vortices, shear layers, all sorts of complicated behavior.

Step 2 — assume irrotational flow. Define vorticity $\boldsymbol{\omega} =
abla \times \mathbf{u}$. If a fluid parcel starts with zero vorticity and there's no viscosity to generate any (Kelvin's circulation theorem guarantees circulation is conserved in inviscid, barotropic flow), then $\boldsymbol{\omega} = 0$ everywhere downstream too. A vector field with zero curl can always be written as the gradient of a scalar:

$$\mathbf{u} = abla \phi$$

$\phi$ is the velocity potential. This is the whole reason "potential flow" is worth learning: instead of tracking three velocity components, you track one scalar function.

Step 3 — apply continuity. For an incompressible fluid, $
abla \cdot \mathbf{u} = 0$. Substitute $\mathbf{u} =
abla\phi$:

$$ abla \cdot ( abla \phi) = abla^2 \phi = 0$$

That's Laplace's equation — the same equation governing electrostatics and steady heat conduction. It's linear, which means if $\phi_1$ and $\phi_2$ are both valid solutions, so is $\phi_1 + \phi_2$. You can build complicated flows by stacking simple ones on top of each other. That's the superposition trick used constantly below.

flowchart TD
    A["Navier-Stokes (viscous, nonlinear)"] -->|"set mu = 0"| B["Euler equations (inviscid, still nonlinear)"]
    B -->|"assume curl(u) = 0"| C["u = grad(phi), potential flow"]
    C -->|"apply div(u) = 0"| D["Laplace's equation: grad-squared(phi) = 0"]
    D -->|"linear PDE"| E["Superposition of elementary solutions"]

In 2D it's often easier to work with the stream function $\psi$, defined so continuity is satisfied automatically: $u = \partial\psi/\partial y$, $v = -\partial\psi/\partial x$. Lines of constant $\psi$ are streamlines. For irrotational flow, $\psi$ also satisfies Laplace's equation, and $\phi$ and $\psi$ are related by the Cauchy-Riemann conditions — which is why complex analysis works so well for 2D potential flow, though you don't need that machinery to follow this note.

2.2 The Building Blocks

Four elementary solutions, expressed in polar coordinates $(r,\theta)$, cover almost every classic problem:

Uniform flow at speed $U$ in the $x$-direction:
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