PX285 - K1 - vorticity

definition

vorticity is the curl of a flow:

\vec{ω} = \vec{\nabla} \times \vec{u}

a flow is irrotational if $\vec{ω} = 0$ everywhere

circulation

the circulation is defined around an arbitrary closed path $(Γ) :$

K = \oint_{Γ} \vec{u} \cdot d \vec{l}

using stokes' theorem:

K = \oint_{Γ} \vec{u} \cdot d \vec{l} = \iint_{S} (\vec{\nabla} \times \vec{u}) \cdot d \vec{S}

where, $S$ is the surface bounded by the contour, $Γ$

in irrotational flows, $ω = \vec{\nabla} \times \vec{u} = 0 ⟹ K = 0$

irrotational rotation

considering a vortex:

\vec{u} = (\frac{A}{r}) \hat{θ}

where, $\hat{θ}$ is the polar angle in cylindrical coordinate system

\begin{aligned} \vec{ω} & = \vec{\nabla} \times \vec{u} \\ = | \begin{array}{c} \hat{r} & r \hat{θ} & \hat{z} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial z} \\ u_{r} & r u_{θ} & u_{z} \end{array} | \\ = | \begin{array}{c} \hat{r} & r \hat{θ} & \hat{z} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial θ} & \frac{\partial}{\partial z} \\ 0 & A & 0 \end{array} | \\ = 0 \end{aligned}

thus, the flow is irrotational