Stream function

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

$\mathbf{F}=⟨{\psi }_y,-{\psi }_x⟩,$

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. They run in close analogy: “conservative” (zero curl, has a potential) on one side, “source-free” (zero divergence, has a stream function) on the other.

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 $\mathbf{F}$, the 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, which makes streamlines the 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. Both $\varphi$ and $\psi$ automatically satisfy Laplace’s equation ${\nabla }^2\varphi ={\nabla }^2\psi =0$, i.e. they’re harmonic. That’s why complex analysis handles 2D steady-state physics so well: 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 just 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.