Air Flowing Around a Cylinder

Smoke in a wind tunnel traces streamlines that split around a pole and rejoin behind it. Each streamline solves an exact equation: it is a level curve of one function, the stream function. The surface below is that function; its contours are the streamlines.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Exact equations

In one sentence

$M\,dx + N\,dy = 0$ is exact if there is a $\psi$ with $\psi_x = M$ and $\psi_y = N$ (on a simply connected region, exactly when $M_y = N_x$ ); its solutions are then given implicitly by $\psi(x, y) = C$ .

General formula

{\htmlData{sym=4}{\frac{\partial\psi}{\partial x}}}\,dx + \frac{\partial{\htmlData{sym=0}{\psi}}}{\partial y}\,dy = 0 \quad\Rightarrow\quad {\htmlData{sym=0}{\psi}}(x, y) = {\htmlData{sym=1}{U}}\,y\left(1 - \frac{{\htmlData{sym=2}{a}}^2}{x^2 + y^2}\right) = {\htmlData{sym=3}{C}}

Along a solution curve $d\psi = \psi_x\,dx + \psi_y\,dy = 0$ , so $\psi$ never changes: each solution lives on one contour of the surface.

What the symbols mean

$\psi$
Stream function
A height map whose contour lines are the paths the air follows.
$U$
Wind speed
How fast the air moves far from the cylinder.
$a$
Cylinder radius
The size of the pole the air flows around.
$C$
Streamline label
Each value of C picks out one streamline.
$\frac{\partial\psi}{\partial x}$
x-slope of ψ
Plays the role of M in M dx + N dy = 0.

An equation $M\,dx + N\,dy = 0$ is exact when $M$ and $N$ are the partial derivatives of one function $\psi$ . Its solutions are the level curves $\psi = C$ .
For air moving past a cylinder, $\psi$ is the stream function: air moves along its level curves, never across them.
Far from the cylinder $\psi \approx Uy$ and the streamlines are straight; near it, the $a^2/(x^2 + y^2)$ term bends them around.
Check exactness with $M_y = N_x$ : both equal $\psi_{xy}$ .

Learn the math

Exact EquationsDiff Eq · Lesson 6, First Order Differential Equations