# Defining stream function and velocity potential function

I'm struggling with the definition of the stream function. Please check if my following understanding correct:

First we start with an undefined analytical function that we are going to call "complex potential": $$W=\phi+i\psi$$. From the properties of analytic functions, we have $$\frac{dW}{dz}=\phi_x-i\phi_y$$. Now, we are going to define $$\phi$$: $$\nabla{\phi}=(\phi_x,\phi_y):=(u,v)$$, where $$(u,v)$$ is the velocity field of our given 2-dimensional flow. We call $$\phi$$ the velocity potential function. Now, we are going to define the stream function as the imaginary part $$\psi$$ of the complex potential $$W$$. The uniqueness follows from analyticity of $$W$$.

We can now obtain the physical meaning of $$\psi$$ by noticing that from the Cauchy-Riemann equations it follows that $$(\phi_x,\phi_y)=(\psi_y,-\psi_x)$$. Since $$(\psi_x,\psi_y)\cdot(\psi_y,-\psi_x)=0$$, tangent curves of $$\phi$$ are orthogonal to the tangent curves of $$\psi$$. Now i refer to the theorem that states: "Theorem: If the function f is differentiable, the gradient of f at a point is either zero, or perpendicular to the level set of f at that point.". From this theorem and $$\nabla{\phi}=(\phi_x,\phi_y):=(u,v)$$ it follows that level curves of $$\phi$$ are orthogonal to the velocity field $$(u,v)$$. And since tangent curves of $$\phi$$ are orthogonal to the tangent curves of $$\psi$$, i conclude that level curves of $$\psi$$ are parallel to the velocity field $$(u,v)$$, in other words, the fluid flow follows the lines $$\psi=const$$.

Is this sound or not? When i started explaining the stream function like i wrote above, the professor said that my question was about the stream function and not about the complex potential. And my answer wasn't accepted.

## 1 Answer

Your statement is correct, but it isn't fully general. This is because your version only works for potential flows—you can also define streamfunctions for flows that are not potential flows (and therefore cannot be defined by a complex potential).

Usually, you define the streamfunction as a scalar $$\psi$$ whose isocontours represent flow lines, and then determine the mathematical form of the streamfunctions for potential flows using a complex potential followed by your argument.