For a two-dimensional incompressible velocity field

$$\mathbf v=u(x,y)\hat{\mathbf x}+v(x,y)\hat{\mathbf y},$$

mass conservation requires

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.$$

A stream function $\psi(x,y)$ represents the velocity as

$$\boxed{u=\frac{\partial\psi}{\partial y},\qquad v=-\frac{\partial\psi}{\partial x}}.$$

With this sign convention, incompressibility is satisfied automatically because

$$\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} =\psi_{yx}-\psi_{xy}=0$$

for a sufficiently smooth $\psi$.

Thus the stream function does for two-dimensional incompressibility what a velocity potential does for irrotationality: it builds one governing constraint directly into the representation.

Constant $\psi$ curves are streamlines

A streamline is tangent to the velocity field. Along a curve,

$$d\psi =\frac{\partial\psi}{\partial x}dx +\frac{\partial\psi}{\partial y}dy.$$

Using the velocity relations,

$$\boxed{d\psi=-v,dx+u,dy}.$$

Along a streamline,

$$\frac{dy}{dx}=\frac{v}{u},$$

so

$$d\psi =-v,dx+u\left(\frac{v}{u}dx\right)=0.$$

Therefore

$$\boxed{\psi=\text{constant along a streamline}}.$$

Contours of the stream function directly draw the streamline pattern of a steady two-dimensional flow.

Difference in stream function equals volume flow per unit depth

Consider any curve from point $A$ to point $B$. The volume flow rate crossing that curve per unit depth perpendicular to the $xy$ plane is

$$q'_{A\to B}=\int_A^B(u,dy-v,dx).$$

But

$$u,dy-v,dx=d\psi,$$

so

$$\boxed{q'_{A\to B}=\psi_B-\psi_A}.$$

Thus the difference between two streamline values measures the volume flow per unit depth passing between them.

This gives $\psi$ the units

$$[\psi]=\mathrm{m^2/s}.$$

Relation to vorticity

The two-dimensional vorticity is

$$\omega_z =\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y}.$$

Substituting the stream-function definitions,

$$\omega_z =-\frac{\partial^2\psi}{\partial x^2} -\frac{\partial^2\psi}{\partial y^2}.$$

Therefore

$$\boxed{\omega_z=-\nabla^2\psi}.$$

A two-dimensional incompressible flow is irrotational precisely where

$$\boxed{\nabla^2\psi=0}.$$

In such a region, the stream function is harmonic.

Worked example: uniform flow

Consider uniform flow in the positive $x$ direction:

$$u=U,\qquad v=0.$$

The stream-function relations require

$$\frac{\partial\psi}{\partial y}=U,$$

and

$$\frac{\partial\psi}{\partial x}=0.$$

Therefore

$$\boxed{\psi=Uy+C}.$$

Contours of constant $\psi$ are

$$y=\text{constant},$$

which are horizontal straight streamlines, as expected.

Take two streamlines at $y_1$ and $y_2$. Their stream-function difference is

$$\psi_2-\psi_1=U(y_2-y_1).$$

This equals the volume flow per unit depth through a vertical segment of height $y_2-y_1$:

$$q'=U(y_2-y_1).$$

Worked contrast: simple shear

For

$$u=Gy,\qquad v=0,$$

we obtain

$$\psi=\frac12Gy^2+C.$$

Then

$$-\nabla^2\psi=-G,$$

which is exactly the nonzero vorticity of simple shear.

A stream function therefore does not require irrotational flow. It requires two-dimensional incompressibility. Only when the flow is also irrotational does $\psi$ satisfy Laplace's equation.

Stream function versus velocity potential

The two scalar representations encode different constraints:

• a velocity potential exists for a suitable irrotational velocity field and automatically gives zero curl;
• a stream function represents a two-dimensional incompressible flow and automatically gives zero divergence.

A two-dimensional flow that is both incompressible and irrotational can possess both representations. This combination is the foundation of two-dimensional potential-flow analysis.