Stream Function

Steady, incompressible, plane, two-dimensional flow represents one of the simplest types of flow of practical importance. By plane, two-dimensional flow we mean that there are only two velocity components, such as u and when the flow is considered to be in the $x$–$y$ plane.

The continuity equation for an incompressible fluid in 3D is expressed as:

$$\nabla \cdot \mathbf{V}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$

For 2D flows, this simplifies to:

$$\begin{array}{rl}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}& =0\\ \frac{\partial u}{\partial x}& =\frac{\partial (-v)}{\partial y}\end{array}$$

This possible to relate $u$ and $v$ using a single function $\psi$?

If $\psi (x,y)$ has continuous second-order derivatives:

$$\frac{{\partial }^2\omega }{\partial x\partial y}=\frac{{\partial }^2\omega }{\partial y\partial x}⟹u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}$$

This definition automatically satisfies the continuity equation.

What exactly is $\psi (x,y)$?

$$d\psi =\frac{\partial \psi }{\partial x}dx+\frac{\partial \psi }{\partial y}dy=-vdx+udy$$

On a curve on which $\psi =\text{const}$, we would have

$$d\psi =-vdx+udy=0⟹\frac{dy}{dx}=\frac{v}{u}$$

Thus, $\psi =\text{const}$ defines a streamline.

What is the meaning of the value of the value of $\psi$?

Consider the difference between difference in the stream function values between two streamlines:

$$dq=udy-vdx$$

Substituting the definitions of $u$ and $v$:

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

Integrating from one streamline to another:

$$q={\int }_{{\psi }_1}^{{\psi }_2}d\psi ={\psi }_2-{\psi }_1$$

The difference in $\psi$ values is equal to the flow rate between those two streamlines. This is a powerful result because it allows us to compute mass flow rates just by knowing the stream function!