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\section{Preliminary Introduction}

Continuum Mechanics:
\begin{itemize}
    \item Fluid dynamics (liquids, gases)
    \item Solid mechanics (solids, elasticity,
        fracture)
    \item Other (complex fluids, soft matter,
        biomechanics)
\end{itemize}
Average over irrelevant molecular detail to get a
continuum description in terms of fields. For
example
\begin{itemize}
    \item \emph{velocity} $\bf{u}(\bf{x}, t)$
    \item \emph{pressure} $p(\bf{x}, t)$
    \item \emph{density} $\rho(\bf{x}, t)$
\end{itemize}
$\bf{x} = \bf{U}(t)$ is uniform flow, $\bf{u} =
\bf{U}(t) + \bf{R}(t) \times \bf{x}$ solid body
motion. $\bf{u}(\bf{x})$ steady, $\bf{u}(\bf{x},
t)$ unsteady.

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Simple physics - mass, momentum, Newton's Laws,
vector calculus, maths methods. Combine these to
find the velocity $\bf{u}$ and the accompanying
forces.

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Kinematics: velocities and trajectories

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Dynamic: forces and equations of motion.

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Forces: gravity, pressure, viscous stress
(internal friction).

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We will consider \emph{inviscid flow} good
approximation for water / air (except on very
small scales). We will also consider simple
viscous flow.

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More to come in Part II Fluids \& Waves and lots
of courses in Part III.

\subsubsection*{Applications}

Fluids are everywhere! Application for the
environment: atmosphere / polar ice caps (climate
change), pollution, ventilation, wind power. For
biology: breathing, blood, flying, swimming,
cells. Aerosols to astrophysics.

\newpage
\section{Kinematics}

\subsection{Pathlines \& Streamlines}

There are two natural perspectives on a flow
$\bf{u}(\bf{x}, t)$:
\begin{enumerate}[(1)]
    \item A stationary observer watching the flow
        go past (Eulerian picture)
    \item A moving observer, travelling along with
        (some bit of) the flow.
\end{enumerate}

\begin{definition*}[Streamlines]
    \emph{Streamlines} are curves that are
    everywhere parallel to the flow at a given
    instant. Given parametrically as $\bf{x} =
    \bf{x}(s; \bf{x}_0, t_0)$ from
    \[ \dfrac{\bf{x}}{s} = \bf{u}(\bf{x}, t_0) \]
    with $\bf{x} = \bf{x}_0$ at $s = 0$.
    (See later for stream functions). Similar to
    characteristics.
\end{definition*}

\noindent
The set of streamlines shows the direction of flow
at that instant -- all particles at one time. For
example $\bf{u} = (t, t)$
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/2a5f56a897e611ed.png}
\end{center}
\[ x = x_0 + s, y = y_0 + t_0 s \implies y = y_0 +
t_0(x - x_0) \]

\begin{definition*}[Pathline]
    A \emph{pathline} (particle path) is the
    trajectory of a fluid `particle' ($\equiv$ a
    very small blob of fluid). The pathline
    $\bf{x} = \bf{x}(t; \bf{x}_0)$ of the particle
    at $\bf{x}_0$ at $t = 0$ found from
    \[ \boxed{\dfrac{\bf{x}}{t} = \bf{u}(\bf{x},
    t) \quad \text{with} \quad \bf{x} = \bf{x}_0
    \quad \text{at $t = 0$}} \]
    one particle at many times.
\end{definition*}

For example, $\bf{u} = (1, t) \implies x = x_0 +
t, y = y_0 + \half t^2 \implies y = y_0 + \half(x
- x_0)^2$
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/09db413497e711ed.png}
\end{center}
Can consider lots of particles, for example at
$\bf{x}_0$ in a given region, to see how the shape
and position of a dyed patch of fluid evolves --
useful for thinking about transport (pollutant,
aerosols) and mixing problems. For a \emph{steady
flow} (only), streamlines and pathlines are the
same.

\subsection{The Material Derivative}

Rate of change moving along with the fluid
(material). For example:
\begin{center}
    \includegraphics[width=0.6\linewidth]
    {images/bf10daaa97e711ed.png}
\end{center}
$F$ varies along the flow.

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For any quantity $F(\bf{x}, t)$, the rate of
change (with time) seen by an observer moving with
the fluid, $\dfrac{F}{t}$ is found from
\begin{align*}
    \delta F
    &= F(x + \delta x, t + \delta t) - F(x, t) \\
    &= \delta \bf{x} \cdot \nabla F + \delta t
    \pfrac{F}{t} + \text{higher order terms}
\end{align*}
Displacement of observer moving with the fluid
$\delta \bf{x} = \bf{u}(\bf{x}, t) \delta t +
\text{h.o.t.}$.
\[ \Dfrac{F}{t} = \pfrac{F}{t} + \bf{u} \cdot
\nabla F \]
where $\Dfrac{F}{t}$ is the material / Lagrangian
derivative, $\pfrac{F}{t}$ is the local / Eulerian
derivative and $\bf{u} \cdot \nabla F$ is the
convectional advective derivative.
