Stream function

A stream function for a 2D vector field $\mathbf{F}=⟨P,Q⟩$ is a scalar function $\psi (x,y)$ such that

$\mathbf{F}=⟨{\psi }_y,-{\psi }_x⟩,$

i.e., $P=\partial \psi /\partial y$ and $Q=-\partial \psi /\partial x$.

A field admits a stream function iff it is source-free (zero divergence) on a simply connected domain. The stream function is the flux-side analog of the potential function for conservative fields.

The flux side of the parallel story

Vector calculus has two parallel stories — the circulation story and the flux story — strongly analogous between “conservative” (zero curl, has a potential) and “source-free” (zero divergence, has a stream function).

Property	Circulation story	Flux story
Local condition	$\nabla \times \mathbf{F}=0$	$\nabla \cdot \mathbf{F}=0$
Global condition	Zero closed-loop circulation	Zero closed-curve outward flux
Scalar function	Potential $\varphi$ with $\mathbf{F}=\nabla \varphi$	Stream function $\psi$ with $\mathbf{F}=⟨{\psi }_y,-{\psi }_x⟩$
Bridging theorem	FTLI, Green’s theorem (circ), Stokes’ theorem	Green’s theorem (flux), Divergence theorem

Streamlines

Level curves of $\psi$ are streamlines — the integral curves of $\mathbf{F}$, paths a particle would follow if carried by the field at every instant.

Proof: along a streamline, $d\psi /dt={\psi }_{x}dx/dt+{\psi }_{y}dy/dt={\psi }_x{P}_{\text{streamline}}+{\psi }_y{Q}_{\text{streamline}}$. But $P={\psi }_y$ and $Q=-{\psi }_x$, so the directional derivative is ${\psi }_x{\psi }_y+{\psi }_y(-{\psi }_x)=0$. So $\psi$ is constant along streamlines — streamlines are level sets of $\psi$.

Flux across a curve

$\psi (B)-\psi (A)$ equals the flux of $\mathbf{F}$ across any curve from $A$ to $B$ (with appropriate orientation). This is the flux-side analog of $\varphi (B)-\varphi (A)={\int }_C\mathbf{F}\cdot d\mathbf{r}$ for conservative fields.

Finding a stream function

Same algorithm as finding a potential, but using the stream-function equations $P={\psi }_y$, $-Q={\psi }_x$. Given source-free $\mathbf{F}=⟨P,Q⟩$:

1. Integrate ${\psi }_y=P$ w.r.t. $y$: $\psi =\int Pdy+g(x)$.
2. Differentiate w.r.t. $x$: ${\psi }_x=g^{\prime }(x)+\text{terms}$. Set ${\psi }_x=-Q$, solve for $g^{\prime }(x)$.
3. Integrate.

Doubly special: conservative AND source-free

A 2D field that is both conservative (has potential $\varphi$, zero curl) and source-free (has stream function $\psi$, zero divergence) satisfies

${\varphi }_x={\psi }_y,{\varphi }_y=-{\psi }_x.$

These are exactly the Cauchy-Riemann equations for $\varphi$ and $\psi$ to be the real and imaginary parts of an analytic function $f(z)=\varphi (x,y)+i\psi (x,y)$.

So doubly-special vector fields in 2D are real-imaginary pairs of analytic functions, in disguise. Both $\varphi$ and $\psi$ automatically satisfy Laplace’s equation ${\nabla }^2\varphi ={\nabla }^2\psi =0$ — they’re harmonic. This is why complex analysis is the natural language for 2D steady-state physics — fluid flow, electrostatics, heat conduction — anything governed by Laplace’s equation.

Worked example

$\mathbf{F}=⟨x,-y⟩$.

Divergence: $1-1=0$. Source-free.

Curl: $0-0=0$. Also conservative.

Find $\varphi$: ${\varphi }_x=x$ gives $\varphi =x^2/2+g(y)$. ${\varphi }_y=g^{\prime }(y)=-y$ gives $g=-y^2/2$. So $\varphi =(x^2-y^2)/2$.

Find $\psi$: ${\psi }_y=x$ gives $\psi =xy+h(x)$. ${\psi }_x=y+h^{\prime }(x)=-(-y)=y$ gives $h^{\prime }(x)=0$. So $\psi =xy$.

Combining: $f=\varphi +i\psi =(x^2-y^2)/2+ixy=\frac{1}{2}((x+iy{)}^2)=z^2/2$.

The doubly-special field $⟨x,-y⟩$ is secretly the analytic function $z^2/2$.

Caveat: domains

Existence of a stream function (or potential) requires the topology of the domain to cooperate. On non-simply-connected domains, zero divergence doesn’t guarantee a single-valued stream function — same issue that affects potentials in non-simply-connected domains.