\documentclass[final,pdf,HeilHazel,noaccumulate]{prosper}
\usepackage{epsfig}
\usepackage{amsbsy,amsmath}
\usepackage{rotating}
\usepackage{graphicx}
\usepackage[dvips]{color}
\usepackage{amsbsy}
\usepackage{epsfig}
\usepackage{natbib}

\setlength{\unitlength}{1cm}


\newcommand{\ee}{\end{equation}} 
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\newcommand{\ec}{\end{center}} 
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\newcommand{\bea}{\begin{eqnarray}} 
%\newcommand{\sumprime}{\mathop{{\sum}'}} 
\newcommand{\bd}{\begin{description}} 
\newcommand{\ed}{\end{description}} 
\newcommand{\bi}{\begin{itemize}} 
\newcommand{\ei}{\end{itemize}} 
\newcommand{\pa}{\partial} 
\newcommand{\hor}{\left(\frac{h}{R_0}\right)} 
 
\newcommand{\bis}{\begin{itemstep}} 
\newcommand{\eis}{\end{itemstep}} 

\title{Example slides from a few {\tt Prosper}-based seminar presentations}
\subtitle{ \ ...with lots of graphics}
\author{Matthias Heil \& Andrew L. Hazel}
\institution{Department of Mathematics \\ University of Manchester}
\email{. \\ M.Heil@maths.man.ac.uk \ \ \  \& \ \ \ ahazel@maths.man.ac.uk \\ 
http://www.maths.man.ac.uk/$\tilde{\ }$mheil}
\slideCaption{Matthias Heil \& Andrew L. Hazel, Department of Mathematics, University of
Manchester, UK \hspace{0.5cm} {\tt http://www.maths.man.ac.uk/$\tilde{\
}$mheil} }

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\overlays{4}{%
\begin{slide}[R]{Lagrangian wall mechanics}
\vspace{-1cm}
\onlySlide*{1}{%
\begin{figure}[h!]
\begin{center}
\epsfig{file=pictures/Ring.eps,height=8cm}
\end{center}
\end{figure}
}
\fromSlide*{2}{%
\begin{figure}[h!]
\begin{center}
\epsfig{file=pictures/Ring.eps,height=3cm}
\end{center}
\end{figure}
\bi
\item Principle of virtual displacements for a linearly elastic ring of 
undeformed radius $R_0$ and thickness $h$, subject to load ${\bf f}$:
\begin{eqnarray*}
\hspace{-1cm}
\int_0^{2 \pi}   \left[ 
\gamma \  
\delta \gamma  +  
\frac{1}{12} \left(\frac{h}{R_0}\right)^2 \kappa \  
\delta \kappa  -  \frac{1}{12} \left(\frac{h}{R_0}\right)^3 \left( \left(\frac{R_0}{h}\right)  {\bf f} -
 \lambda_T^2 \frac{\partial^2 {\bf R}_w}{\partial t^2} \right) \cdot  
\delta {\bf R}_w 
 \right] \ d\zeta  = 0.
\end{eqnarray*}
\ei
}
\onlySlide*{3}{%
\bi
\item $\gamma$ and $\kappa$ represent the ring's mid-plane strain and
change of curvature, respectively.
\ei
}
\onlySlide*{4}{%
\bi
\item Load non-dimensionalised by bending stiffness $K$, i.e.
$$
{\bf f}^* = K \ {\bf f} \mbox{ \ \ \ \ \ \ \ where \ \ \ \ \ \ \ }
K = \frac{E}{12(1-\nu^2)}\left(\frac{h}{R_0}\right)^3.
$$
\ei
}
\end{slide}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\overlays{9}{%
\begin{slide}{Fluids: Weak form of equations}
\vspace*{-1cm}
\tiny Momentum:
\untilSlide*{2}{%
$$\iiint \left[-\frac{\partial p}{\partial x_{i}} 
  + \frac{\partial}{\partial x_{j}}
 \left(\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial
 u_{j}}{\partial x_{i}}\right)\right]\psi^{(F)} \text{ d}V = 0
$$}
\onlySlide*{2}{\red [...integrate by parts.]}
\fromSlide*{3}{%
$$  \iiint \left[p\frac{\partial
\psi^{(F)}}{\partial x_{i}} - \left(\frac{\partial u_{i}}{\partial
x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right)
\frac{\partial \psi^{(F)}}{\partial x_{j}} \right]\text{ d}V$$}
\onlySlide*{3}{%
$$+ 
\iint \left[-p n_{i} 
+ \left(\frac{\partial u_{i}}{\partial
x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right) n_{j} \right]
\psi^{(F)} \text{ d}S = 0$$
{\red [...split the surface integral into $\text{ d}S = \text{
d}S_{f} + \text{ d}S_{\backslash S_{f}}$]}}
\onlySlide*{4}{%
$$+ 
\iint \left[-p n_{i} 
+ \left(\frac{\partial u_{i}}{\partial
x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right) n_{j} \right]
\psi^{(F)} \text{ d}S_{\backslash S_{f}}$$
$$ +  \iint \left[-p n_{i} 
+ \left(\frac{\partial u_{i}}{\partial
x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right) n_{j} \right]
\psi^{(F)} \text{ d}S_{f}= 0$$
\red [Note: $\psi^{(F)} = 0$ on $S_{\backslash S_{f}}$]}
\onlySlide*{5}{%
$$ + \iint \left[-p n_{i} 
+ \left(\frac{\partial u_{i}}{\partial
x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}\right) n_{j} \right]
\psi^{(F)} \text{ d}S_{f} = 0$$
{\red [...use the traction boundary condition on the free surface $S_{f}$.]}}
\onlySlide*{6}{%
$$ - \iint \left[p_{b} + \frac{1}{\text{Ca}} \kappa
\right] \psi^{(F)} n_{i}\text{ d}S_{f} = 0$$
{\red [...apply Weatherburn's surface divergence theorem to the surface integral.]}}
\fromSlide*{7}{%
$$
{+ \frac{1}{\text{Ca}}\iint 
 \frac{1}{g} \left[\mbox{\boldmath$g$}_{1}\right]_{i} \left(g_{22}
\frac{\partial\psi^{(F)}}{\partial \zeta_{1}} - g_{12} 
 \frac{\partial\psi^{(F)}}{\partial \zeta_{2}}\right)
+ \frac{1}{g} \left[\mbox{\boldmath$g$}_{2}\right]_{i} \left(g_{11}
\frac{\partial\psi^{(F)}}{\partial \zeta_{2}} 
- g_{12} \frac{\partial\psi^{(F)}}{\partial \zeta_{1}}\right) \text{
 d}S_{f}}$$
$$- \iint p_{b} \psi^{(F)} n_{i}\text{ d}S_{f}
{-\frac{1}{\text{Ca}}\oint \psi^{(F)}m_{i} \text{ d} s}
= 0$$}
\onlySlide*{8}{%
\begin{picture}(12,6)
\put(1.5,1.5){\resizebox{8cm}{6cm}{\includegraphics{pictures/vols.eps}}}
\put(3.5,6){$\mbox{\boldmath$m$}$}
\put(8.8,5.2){$\mbox{\boldmath$m$}$}
\put(6.3,2){$\mbox{\boldmath$m$}$}
\put(6.5,4.5){\Large$S_{f}$}
\put(1.7,3.3){\Large $s$}
\put(4.5,4){$\mbox{\boldmath$g$}_{2}$}
\put(5.5,3.5){$\mbox{\boldmath$g$}_{1}$}
\end{picture}}
\onlySlide*{9}{%
Conservation of mass:
$$ \iiint \frac{\partial u_{i}}{\partial x_{i}}
\psi^{(P)} \text{ d}V = 0$$
Non-penetration on free surface:
$$\iint u_{i}n_{i} \psi^{(H)} \text{ d}S_{f} = 0$$
\bi
\item Discretise using Taylor--Hood elements 
\item Solve matrix equations directly by frontal method (HSL 2000)\ei
}
\end{slide}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\overlays{4}{%
\begin{slide}[Replace]{Results (displayed as a pseudo-animation)}
\vspace{-1cm}
Surface-tension-driven collapse of a liquid-lined elastic ring:
\vspace{-0.3cm}
\begin{figure}[h!]
\onlySlide*{1}{\epsfig{file=pictures/Sigma5.0Pic1.eps,height=7cm}}
\onlySlide*{2}{\epsfig{file=pictures/Sigma5.0Pic2.eps,height=7cm}}
\onlySlide*{3}{\epsfig{file=pictures/Sigma5.0Pic3.eps,height=7cm}}
\onlySlide*{4}{\epsfig{file=pictures/Sigma5.0Pic4.eps,height=7cm}}
\end{figure}
\vspace{-0.5cm}
\bi
\item Fully coupled discretisation of the free-surface Navier-Stokes
equations and the equations of large-displacement shell theory.
\item Solution by the Newton-Raphson method.
\ei
\end{slide}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\overlays{4}{%
\begin{slide}[Replace]{Results (displayed as real animation)}
\vspace{-1cm}
Self-excited oscillations during finite Reynolds number 
flow in a collapsible channel.
\begin{center}
\href{run:Stream.avi}{\red Start external animation}
\end{center}


\begin{center}
\fbox{
\parbox[b]{6cm}{
If this doesn't work, check that (i) the file Stream.avi
is located in the same directory as your pdf file when you display these slides
and (ii) that acroread knows how to display avi files (maybe download
a later version...)
}}


\vspace{3cm}
\Acrobatmenu{PrevPage}{\blue Back to the previous page?}

\end{center}

\end{slide}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
