# Rotational, Potential Flow

The short answer is "by definition, potential flow is irrotational", but please hear me out.

I was working through "Fundamentals of Aerodynamics" by Anderson, and I noticed the following when he introduced vortex flow as an elementary flow type during his discussion of potential flow modeling. In this section he starts with the definition of circulation as:

$$\Gamma=-\oint_c \vec{v}\cdot d\vec{s}$$

And, by corollary:

$$\Gamma = -\iint_S(\nabla\times\vec{v})\cdot d\vec{s}$$

Since, for vortex flow, you can solve for circulation as a circumference multiplied by a tangential velocity, $$v_\theta$$, this means that your circulation is $$\Gamma=-v_\theta 2\pi r$$.

As a result, it would appear that the vorticity of the flow surrounding a point vortex is nonzero, and thus the flow is rotational, which violates the assumption of potential flow. How is this not the case?

The vorticity is zero everywhere except at the point vortex where (for a vortex at the origin) $$\omega=\nabla\times v = \Gamma \delta(x)\delta(y)$$. Then $$v_\theta= \frac{\Gamma}{2\pi r}, \quad v_r=0.$$ Now in polar coordinates the gradient operator is $$\nabla = \hat {\bf e}_r\frac{\partial}{\partial r}+\hat{\bf e}_\theta\frac 1 r \frac{\partial}{\partial \theta}$$ so this velocity is the gradient of $$\Gamma \theta(x,y)/2\pi$$, where $$\theta(x,y)= \tan^{-1}(y/x)$$ is the polar angle. Of course $$\theta(x,y)$$ is not an actual function of $$x,y$$ in the whole $${\mathbb R}^2$$ because it is not single-valued. Its derivative is single valued though, and $$\theta$$ is unambiguously defined in any domain that does not wrap around the origin. Because the velocity is he gradient of a locally defined function, its curl vanishes everywhere that the local function exists, i.e in$${\mathbb R}^2$$ minus the origin.
In cartesian coordinates the velocity field is $$v_x(x,y) = \Gamma \frac{-y}{(x^2+y^2)}\\ v_y(x,y) = \Gamma\frac {+x}{(x^2+y^2)}$$ so $$\omega_z(x,y)=\frac{\partial v_y}{\partial x}- \frac{\partial v_x}{\partial y}$$ which is easily computed to be zero except at the origin where the derivatives do not exist.