% vim: tw=50
% 07/02/2023 10AM

\subsubsection{The vorticity equation}

Start with
\[ \rho \left( \pfrac{\bf{u}}{t} + (\bf{u} \cdot
\nabla) \bf{u} \right) = -\nabla p + \bf{f} \]
Assume $\rho$ is constant and $\bf{f}$ is
conservative (i.e. $\bf{f} = -\nabla \chi$).
Earlier vector identity $(\bf{u} \cdot
\nabla)\bf{u} = \nabla \left( \half u^2 \right) -
\bf{u} \times \bf{w}$. Take curl:
\[ \rho \left( \pfrac{\bf{w}}{t} = \nabla \times
(\bf{u} \times \bf{w}) \right) = 0 \]
Second vector identity:
\begin{align*}
    (\nabla \times (\bf{u} \times \bf{w}))_i
    &= \eps_{ijk} \pfrac{}{x_j} (\eps_{klm} u_L
    w_m) \\
    &= (\delta_{il} \delta_{jm} - \delta_{im}
    \delta_{jl}) \left( \pfrac{u_l}{x_j} w_m + u_l
    \pfrac{w_m}{x_j} \right) \\
    &= (\bf{w} \cdot \nabla) \bf{u} -
    \bf{w}\ub{\cancel{(\nabla \cdot
    \bf{u})}}_{\text{incompressible}} + \bf{u}
    \ub{\cancel{(\nabla \cdot \bf{w})}}_{\nabla
    \cdot \nabla \times = 0} - (\bf{w} \cdot
    \nabla) \bf{u}
\end{align*}
\[ \implies \boxed{
\Dfrac{\bf{w}}{t}
\equiv \pfrac{\bf{w}}{t} + (\bf{u} \cdot
\nabla) \bf{w} = (\bf{w} \cdot \nabla) \bf{u}} \]
\emph{Vorticity equation}. We call the
$\pfrac{\bf{w}}{t}$ term the local derivative, the
$(\bf{u} \cdot \nabla)\bf{w}$ term the advection
of $\bf{w}$ and $\bf{w} \cdot \nabla)\bf{u}$ the
``vortex stretching''.

\noindent
i.e. moving with the fluid, $\bf{w}$ changes if
$\bf{u}$ changes in the direction of $\bf{w}$.

\subsubsection{Vortex stretching and the ``ballerina effect''}

Compare
\[ \dfrac{}{t} \bf{\delta l} = (\bf{\delta l}
\cdot \nabla) \bf{u} \qquad \text{and} \qquad
\Dfrac{\bf{w}}{t} = (\bf{w} \cdot \nabla)\bf{u} \]
Moving with the fluid, $\bf{w}$ changes just like
a material line element $\bf{\delta l}$ initially
aligned with $\bf{w}$. In particular, if
$\bf{\delta l}$ gets longer (stretching) then
$\bf{w}$ gets bigger -- this is just conservation
of angular momentum! For example, consinder a
uniformly rotating fluid cylinder:
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/ff94cb46a6d211ed.png}
\end{center}
\begin{itemize}
    \item Mass conservation: $a_1^2 l_1 = a_2^2
        l_2$
    \item Angular momentum: $a_1^4 l_1 w_1 = a_2^4
        l_2 w_2$
\end{itemize}
Hence $w_2 = w_1 \frac{l_2}{l_1}$, i.e. $\bf{w}$
increases as fluid stretches in the direction of
existing $\bf{w}$ -- called \emph{vortex
stretching} or the ``ballerina effect''.

\begin{example*}
    Amplification of bathtub vortex:
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/630df728a6d411ed.png}
    \end{center}
\end{example*}

\begin{example*}
    Hurricane / tornado:
    \begin{center}
        \includegraphics[width=0.6\linewidth]
        {images/ad1673f4a6d411ed.png}
    \end{center}
\end{example*}

\begin{example*}
    \begin{align*}
        \bf{u}
        &= \left( -\half \beta x, -\half \beta y,
        \beta z \right) + \Omega(t) (y, -x, 0) \\
        \implies \bf{w}
        &= (0, 0, 2\Omega) \\
        \pfrac{\bf{w}}{t} + (\bf{u} \cdot
        \nabla)\bf{w}
        &= (\bf{w} \cdot \nabla)\bf{u}
    \end{align*}
    $z$-component $2\dot{\Omega}+ 0 =
    (2\Omega)\beta$ implies $\Omega \propto
    e^{\beta t}$.
\end{example*}

\subsubsection*{Circulation}

The \emph{circulation around a closed curve}
$\Gamma$ is defined by $C(\Gamma) = \oint_\Gamma
\bf{u} \cdot \dd \bf{l} = \int_S \bf{w} \cdot \dd
\bf{S}$
\begin{center}
    \includegraphics[width=0.3\linewidth]
    {images/fc202760a6d411ed.png}
\end{center}
For example a line vortex $\bf{u} = \left( 0,
\frac{\kappa}{2\pi r}, w \right)$ has circulation
$\kappa$ for any circle $r = a$.

\begin{note*}[Non-examinable]
    Kelvin's Circulation Theorem. If $\Gamma(t)$
    is material curve, i.e. one moving with the
    fluid, then
    \[ \dfrac{}{t} [C(\Gamma)] = \oint_\Gamma
    \left( \Dfrac{\bf{u}}{t} \cdot \dd \bf{l} +
    \bf{u} \cdot \ub{(\bf{u} \cdot \nabla)
    \bf{\delta l}}_{\frac{}{t} \bf{\delta l}} \right)
    \stackrel{\text{exercise}}{=} 0 \]
\end{note*}

For example $C = \pi a^2 (2\Omega)$ above is
constant for a material circle $a(t) = a(0)
e^{-\half \beta t}$.

\newpage
\section{Introduction to Viscous Flow}

Section 2 was about \emph{inviscid} flow.
\begin{itemize}
    \item We neglected friction (viscous stresses) between
        layers of fluid or boundaries.
    \item Inviscid fluid
        exerts only a normal stress $-p \bf{n}$.
    \item Forces $-\nabla p + \bf{f}$ were balanced by
        inertia $\rho \Dfrac{\bf{u}}{t}$. 
    \item If $\bf{f} = -\nabla \chi$ then energy and
        angular momentum are conserved.
\end{itemize}

\myskip
With viscous flow, we will find:
\begin{itemize}
    \item Velocity gradients give rise to visous
        stresses (friction)
    \item Fluids also exert tangential (shear)
        stresses on boundaries
    \item New term in equation of motion
    \item Dissipation of energy (non-examinabl)
        and diffusion of vorticity.
\end{itemize}
We will focus on simple case of \emph{2D parallel
viscous flow}
\[ \bf{u} = (u(y, t), 0, 0) \]
-- the full treatment of viscosity is in Part II.