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 1-7vorti.tex 
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\section{Vorticity Theorem for a viscous fluid}
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Consider an incompressible fluid. 
 \be \nabla \cdot \vec q = 0 \label{mass}\ee
  \be \f{\p \vec{q}}{\p t} + \vec q \cdot \nabla  \vec{q} = \vec{f} - \f{\nabla p}{\rho} + \nu \nabla^2 \vec{q}  \label{Eq:(9.1)} \ee
where
\be \mu = \rho \nu,  \qquad \nu = \, \mbox{kinematic viscosity} . \label{Eq:(9.2)} \ee
In this section we shall assume that $\nu=$constant. 


Note  the following identity in index form: \begin{eqnarray*}
q_j \f{\p q_i}{\p x_j} & = & q_j \lp \f{\p q_i}{\p x_j} - \f{\p q_j}{\p x_i} \rp + q_j \f{\p q_j}{\p x_i} \\
& = & 2q_j\Omega_{ij} + \f{\p}{\p x_i} \lp \f{1}{2} q_jq_j \rp \\
& = & \lp \vec{\zeta} \, \times \, \vec{q} \rp_i + \lp \nabla \f{\vec{q}^2}{2} \rp_i . 
\end{eqnarray*} where $ \vec{\zeta} = \nabla \times \vec{q} $. Hence, in vector form, the convective inertia term can be written as 
\be \vec{q} \cdot \nabla \vec{q} = \nabla \f{\vec{q}^2}{2} + \vec{\zeta} \, \times \, \vec{q}  \label{Eq:(9.3)} \ee
We can now rewrite  the momentum equation (\ref{Eq:(9.1)})
by using   (\ref{Eq:(9.3)})
\be \f{\p \vec{q}}{\p t} + \nabla \f{\vec{q}^2}{2} + \vec{\zeta} \, \times \vec{q} = \vec{f} - \f{\nabla p}{\rho} + \nu \nabla^2 \vec{q} . \label{Eq:(9.4)} \ee
 
The   equation governing the transport of vorticity follows by taking   the curl of  (\ref{Eq:(9.4)}). 
For a conservative body force such as gravity, 
\be \nabla \times \vec{f} \equiv 0 \ee
Recall  the vector identities:  \be \nabla \times \nabla S = 0 \label{veciden1}\ee for any scalar $S$, 
 \be  \nabla \cdot \nabla \times \vec{T} = 0 \label{veciden2}\ee
for any vector $\vec{T}$, and 
\be   \nabla \times\lp \vec{\zeta} \, \times \, \vec{q} \rp = \vec{q} \cdot \nabla \zeta - \vec{\zeta} \cdot \nabla \vec{q} +\vec{ \zeta } \nabla\cdot  \vec{q} - \vec{q} \nabla \cdot \vec{\zeta} \label{veciden3}\ee
 for any cross  product of two vectors.  
In (\ref{veciden3}), let $\vec{q}$ be the fluid velocity and $\vec{\zeta}$ be the vorticity. Then the third term  on the right  vanishes on account of incompressibility, and  the last vanishes because $\vec{\zeta} = \nabla \times \vec{q}$. 

Finally, we get the vorticity transport equation:
\be 
  \f{\p \vec{\zeta}}{\p t} + \vec{q} \cdot \nabla \vec{\zeta} = \vec{\zeta} \cdot \nabla \vec{q} + \nu \nabla^2 \vec{\zeta} + \f{\nabla \rho \, \times \, \nabla p}{\rho^2}  \label{Eq:vort-trans}  \ee
\begin{figure}[h] \label{fig:Rayleigh}
	\begin{center}
		\includegraphics[scale=1]{f1-7-1}
	\end{center}
\caption{Two neighboring points on a vortex line}
	\label{fig:f1-7-1}
\end{figure}
 
We now examine the 
physical meaning   of each term  in (\ref{Eq:vort-trans}):
\begin{itemize}
\item{
$\f{D\vec{\zeta}}{Dt}$ is the rate of change of vorticity following the moving fluid. }

\item{$ \nu \nabla^2 \vec{\zeta} $ represents the diffusion of vorticity by viscosity.}

\item{ $\vec{\zeta} \cdot \nabla \vec{q} $: Let  us  rewrite Eq. (\ref{Eq:vort-trans}) 
\[ \f{1}{|\zeta|} \f{D\vec{\zeta}}{Dt} = \f{\vec{\zeta}}{|\zeta|} \cdot \nabla \vec{q} + \cdots . \]
Define a   vortex line as the curve  which   is tangent to all the local vorticity vectors.   Then  
\[\f{\vec{\zeta}}{|\zeta|}= \vec{e}_\zeta \]
is the local unit tangent to the vortex line.  Let $s$ be the arc-length along the vortex line:
\[ \f{\vec{\zeta}}{|\zeta|} \cdot \nabla \vec{q}=\vec{e}_{\zeta} \cdot \nabla \vec{q}= \f{d \vec{q}}{ds} =   \f{d\vec{q}_{\parallel}}{ds} + \f{d\vec{q}_{\perp}}{ds} , \]
where
$\vec{q}_{\parallel}$ denotes the fluid velocity along the vortex line while 
 $\vec{q}_{\perp} $ denotes the velocity normal to the vortex line. Hence 
$\f{d\vec{q}_{\parallel}}{ds}$ is the rate of stretching of the vortex line, while  
$\f{d\vec{q}_{\perp}}{ds}$  is the rate of rotation of the vortex line. 
It follows that 
$ \vec{\zeta} \cdot \nabla \vec{q} $ represents the rate of production of vorticity by stretching and turning of the vortex line. }

\item{Finally $\nabla \rho \times \nabla p/\rho^2 $  is the vorticity source due to density nonuniformity; this is important  in stratified fluids.}
\end{itemize}

In summary, following the fluid motion,  the increase in vorticity is caused by production due to stretching and turning of the vortex lines, diffusion by viscosity and density inhomogeneity. 









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