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\section{Kinematics of Fluid Motion -the Eulerian picture} 
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Consider  two neighboring stations  (not two fluid particles) $\vec x$ and $\vec x'$ at the same instant $t$, where $\delta \vx=\vx'-\vx$ is  small. The fluid velocity at the two stations are related by 
\be \vec q(\vec x',t) = \vec q(\vec x,t) + (\vec x'-\vec x)\cdot \nabla \vec q(\vec x,t) + O(\vec x' - \vec x)^2 \ee
Hence 
\be \delta \vec q(\vec x,t) = \vec q(\vx',t) - \vec q(\vx,t) = 
\delta \vx\cdot \nabla \vec q(\vec x,t) +O(\delta\vec x)^2\label{delta} \ee
Let us introduce the index notation: 
 \be q_1=u, ~~q_2=v,~~q_3=w; \quad x_1=x,~~x_2=y,~~x_3=z\ee
and Einstein's convention: Repeated indices are summed over the range from 1 to 3, and the summation symbol is omitted but implied. For example,
\[ \sum_{i=1}^3q_iq_i = q_i q_i=q_1^2+q_2^2+q_3^2= \vq\cdot \vq\] 
Thus we may write (\ref{delta}) as
\be \delta q_i  = \delta x_j\f{\p q_i}{\p x_j}, \quad  i = 1,2,3.\label{eq:4.4}\ee
Now \be  \f{\p q_i}{\p x_j} = \f{1}{2} \lp \f{\p q_i}{\p x_j}+\f{\p q_j}{\p x_i}\rp + \f{1}{2}\lp \f{\p q_i}{\p x_j}-\f{\p q_j}{\p x_i}\rp \ee

 Define the rate-of -strain tensor by
 \be e_{ij}=\f{1}{2} \lp \f{\p q_i}{\p x_j}+\f{\p q_j}{\p x_i}\rp\ee
and the vorticity tensor by
\be \Omega_{ij} = \f{1}{2}\lp \f{\p q_i}{\p x_j}-\f{\p q_j}{\p x_i}\rp\ee
Note that 
\be e_{ij}=e_{ji}, \quad \Omega_{ij} = -\Omega_{ji}\ee
and (\ref{eq:4.4}) becomes
\be
\delta q_i = \delta x_j e_{ij} + \delta x_j \Omega_{ij} \ee
Let us examine the physics of these terms.

\subsection{Rate-of-strain tensor}
In matrix  form, the rate-of -strain tensor is :
\begin{eqnarray}
 \{e_{ij}\}&=&  \lp\begin{array}{ccc}
e_{11}&e_{12}&e_{13}\\ 
e_{21}&e_{22}&e_{23}\\
e_{31}&e_{32}&e_{33}\end{array}\rp   \nonumber\\
&=&
  \lp\begin{array}{ccc}
  \f{\p q_1}{\p x_1} &
\f{1}{2} \lp \f{\p q_1}{\p x_2}+\f{\p q_2}{\p x_1}\rp &
\f{1}{2} \lp \f{\p q_1}{\p x_3}+\f{\p q_3}{\p x_1}\rp \\
\f{1}{2} \lp \f{\p q_2}{\p x_1}+\f{\p q_1}{\p x_2}\rp &
 \f{\p q_2}{\p x_2} &
\f{1}{2} \lp \f{\p q_2}{\p x_3}+\f{\p q_3}{\p x_2}\rp\\
\f{1}{2} \lp \f{\p q_3}{\p x_1}+\f{\p q_1}{\p x_3}\rp &
\f{1}{2} \lp \f{\p q_3}{\p x_2}+\f{\p q_2}{\p x_3}\rp &
  \f{\p q_3}{\p x_3} \end{array}\rp\\ &=&
  \lp\begin{array}{ccc}
  \f{\p u}{\p x} &
\f{1}{2} \lp \f{\p u}{\p y}+\f{\p v}{\p x }\rp &
\f{1}{2} \lp \f{\p u}{\p z}+\f{\p w}{\p x}\rp \\
\f{1}{2} \lp \f{\p v}{\p x}+\f{\p u}{\p y}\rp &
 \f{\p v}{\p y} &
\f{1}{2} \lp \f{\p v}{\p z}+\f{\p w}{\p y}\rp\\
\f{1}{2} \lp \f{\p w}{\p x}+\f{\p u}{\p z}\rp &
\f{1}{2} \lp \f{\p w}{\p y}+\f{\p v}{\p z}\rp &
  \f{\p w}{\p z} \end{array}\rp\nonumber 
\end{eqnarray}
First, the diagonal terms.  It is easy to see that $e_{11}=  {\p u}/{\p x} $ is the rate of stretching per unit length  in the direction of $x$, $e_{22}=  {\p v}/{\p y} $ is the rate of stretching  per unit length in the direction of $y$, and 
$e_{33}=  {\p w}/{\p z} $ is the rate of stretching    per unit length in the direction of $z$.  They are the normal components of the rate of strain tensor. 


Note that 
\be  e_{11} + e_{22} + e_{33} = e_{kk} = \f{\p u}{\p x} + \f{\p v}{\p y} + \f{\p w}{\p z} = \nabla\cdot \vq \ee
is the rate of volume dilatation due to fluid motion. For a proof,  let us consider a cube with sides $(x,  x+\Delta x)$, $(y,  y+\Delta y)$ and  $(z,  z+\Delta z)$. After $\delta t$, the side along $x$ will lengthen from $\Delta x$ to  $  \Delta x+  \Delta x\f{\p u}{\p x}\delta  t =\Delta x \lp 1+ \f{\p u}{\p x} \delta t\rp$.  Similarly, the side  along $y$ will lengthen from $\Delta y$ to $\Delta y\lp 1+\f{\p v}{\p y} \delta t\rp$,  and the side along $z$ lengthens from  $\Delta z$ to $\Delta z\lp 1+\f{\p w}{\p z} \delta t\rp$. Consequently the   volume  $V(t)= \Delta x\Delta y\Delta z $ will change to  
\[ V(t+\delta  t)=  \Delta x \lp 1+ \f{\p u}{\p x} \delta t\rp \Delta y \lp 1+ \f{\p v}{\p y}\delta t\rp\Delta z \lp 1+ \f{\p w}{\p z}\delta t\rp \]
\[ =V(t)\lb 1+ \lp \f{\p u}{\p x}+\f{\p v}{\p y}+\f{\p w}{\p z}\rp\delta t+ O(\delta t)^2\rb\]
Hence, the rate of volume dilatation is
\be\lim_{\delta t = 0} \f{1}{V}\f{V(t+\delta t) - V(t)}{\delta t}=\f{1}{V}\f{dV}{dt}= \lp \f{\p u}{\p x}+\f{\p v}{\p y}+\f{\p w}{\p z}\rp= \nabla\cdot \vq\ee
 
\begin{figure} 
\includegraphics[scale=1]{f1-2-1.eps}
\caption{Rate of strain tensor components}
\label{fig:rateofshearstrain}\end{figure}

 Next, the off-diagonal terms. Referring to Figure 1.2.1, consider a plane flow in which $\f{\p u}{\p y}$ and $\f{\p v}{\p x} $ do not vanish. In the time interval $\delta t$ the 
side $\Delta x$ rotates counterclockwise for an angle $\delta \theta_1=\f{\Delta v \delta t}{\Delta x}= \f{\p v}{\p x}\delta t$. The  side $\Delta y$ rotates  counterclockwise for an angle $\delta \theta_2=-\f{\Delta u \delta t}{\Delta y}= -\f{\p u}{\p y}\delta t$. The total rate of  angular deformation is 
\be  \f{\delta \theta_1}{\delta t}-\f{\delta \theta_2}{\delta t} = \f{\p v}{\p x}+\f{\p u}{\p y}\ee
Thus $e_{12}=e_{xy}$  is a rate of angular deformation, called the rate of shear strain. Other components $e_{13}$ and $e_{23}$ can be   interpreted similarly. 

\subsection{Vorticity tensor}

The matrix form of $\Omega_{ij}$ is 
\begin{eqnarray}
 \{\Omega_{ij}\}&=&  \lp\begin{array}{ccc}
\Omega_{11}&\Omega_{12}&\Omega_{13}\\ 
\Omega_{21}&\Omega_{22}&\Omega_{23}\\
\Omega_{31}&\Omega_{32}&\Omega_{33}\end{array}\rp   \nonumber\\
&=&
  \lp\begin{array}{ccc}
0 &
\f{1}{2} \lp \f{\p q_1}{\p x_2}-\f{\p q_2}{\p x_1}\rp &
\f{1}{2} \lp \f{\p q_1}{\p x_3}-\f{\p q_3}{\p x_1}\rp \\
\f{1}{2} \lp \f{\p q_2}{\p x_1}-\f{\p q_1}{\p x_2}\rp &
0 &
\f{1}{2} \lp \f{\p q_2}{\p x_3}-\f{\p q_3}{\p x_2}\rp\\
\f{1}{2} \lp \f{\p q_3}{\p x_1}-\f{\p q_1}{\p x_3}\rp &
\f{1}{2} \lp \f{\p q_3}{\p x_2}-\f{\p q_2}{\p x_3}\rp &
 0 \end{array}\rp\\ &=&
  \lp\begin{array}{ccc}
0 &
\f{1}{2} \lp \f{\p u}{\p y}-\f{\p v}{\p x }\rp &
\f{1}{2} \lp \f{\p u}{\p z}-\f{\p w}{\p x}\rp \\
\f{1}{2} \lp \f{\p v}{\p x}-\f{\p u}{\p y}\rp &
0 &
\f{1}{2} \lp \f{\p v}{\p z}-\f{\p w}{\p y}\rp\\
\f{1}{2} \lp \f{\p w}{\p x}-\f{\p u}{\p z}\rp &
\f{1}{2} \lp \f{\p w}{\p y}-\f{\p v}{\p z}\rp &
 0 \end{array}\rp\nonumber 
\end{eqnarray}

Because of the anti-symmetry,   there are only three independent components, which can also be used to define   the vorticity vector $\vec \zeta$:
\begin{eqnarray} \vec \zeta &=& \nabla\times \vq = \left|\begin{array}{ccc}
\vec i&\vec j&\vec k\\
\f{\p}{\p x}& \f{\p}{\p y}& \f{\p}{\p z}\\ \nonumber
u&v&w\end{array}\right| \\
&=&\vec i\lp\f{\p w}{\p y}-\f{\p v}{\p z} \rp + \vec j\lp\f{\p u}{\p z}-\f{\p w}{\p x}\rp + \vec k \lp  \f{\p v}{\p x}-\f{\p u}{\p y}\rp \end{eqnarray}
Hence \be \{ \Omega_{ij} \}= \f{1}{2}\lp \begin{array}{ccc}
0&-\zeta_3& \zeta_2\\
\zeta_3 & 0 & -\zeta_1\\
-\zeta_2 & \zeta_1& 0 \end{array}
\rp \ee

What is the physical meaning of $\vec \zeta$\, ?
Consider a plane circular disc $A$ bounded by the circle $C$ of radius $a$, see Figure 1.2.2. By Stokes' theorem
\[ \2int_A (\nabla \times \vq ) \cdot \vec n \, dA = \oint_C \vq \cdot d\vec r\]
Now let $a\to 0$, then,
\[(\nabla \times \vq)_n \2int_A dA = \oint_C\vq \cdot d\vec r \]
or, 
\[ \f{1}{2}\zeta_n =\f{1}{2} (\nabla \times \vq)_n= \f{1}{a}\lb\f{1}{2\pi a}\oint_C\vq\cdot d\vec r\rb\]
The quantity \[\lb \f{1}{2\pi a}\oint_C\vq\cdot d\vec r\rb\]
is the average tangential velocity along the circle. Hence $\zeta_n/2$ is the average angular speed  of the fluid  circling along   $C$, i.e., the average rate of rotation. The line integral above is also known as the {\em circulation}.
 
\begin{figure}\vspace{1.6in}\hspace{2.2in}
\includegraphics[scale=1]{f1-2-2.eps}
\caption{Circulation along a closed circle}
\label{fig:circulation}\end{figure}
\end{document}