An example chapter

chapter.tex

%!TEX root = ../thesis.tex
%!TEX TS-program = pdflatex
%!TEX encoding = UTF-8 Unicode

\chapter{Results} % (fold)
\label{cha:isothermal_results}

Isothermal flow solutions were calculated for 1666 flow configurations: these had 49 aspect ratios between 1/5$\;\leq\beta\leq\;$25, with 34 values of the Bingham number per aspect ratio. I only performed actual calculations for flows with $\beta \geq\;$2, since cases with $\beta<2$ can be obtained from these results via the symmetry of the flow configuration (i.e. by swapping $H$ and $W$ and rescaling other values). The Bingham number varied between zero for Newtonian flows, to the critical value $B=B^{\star}(\beta)$ when the weight of the fluid is completely supported by its yield strength (discussed in \S\ref{sub:viscoplastic_rheology}). I outline how the critical Bingham number $B^{\star}$ can be obtained analytically for rectangular channel flows in \S\ref{sec:critical_bingham_numbers}. The number of iterations required for convergence of the Lagrangian optimization algorithm ranged between 1 for Newtonian flows (with $B=0$) to around 1000 iterations for Bingham flows with $B=0.95B^{\star}$, but with a median of order 100 iterations for most configurations. Computations were run in parallel on an 8-core Mac Pro.

\section{Velocity and strain fields} % (fold)
\label{sec:velocity_fields}

\begin{figure}[p] 
	\begin{centering}
    \includegraphics[width=\figurewidth]{graphics/2011JB008550-p04} 
    \caption[Contour plots of the velocity (left column) and strain rate magnitude (right column) for a channel with aspect ratio of $\beta=2$, and Bingham numbers of $B = 0$, $0.274$ and $0.496$.]{Contour plots of the velocity (left column) and strain rate magnitude (right column) for a channel with aspect ratio of $\beta=\;$2, and Bingham numbers of $B =\;$0, 0.274 and 0.496. The definitions of Bingham number and aspect ratio are given in equation (\ref{eq:bingham_number_def}--\ref{eq:aspect_ratio_def}). The critical Bingham number for this aspect ratio is approximately 0.530 (see ({\ref{eq:crit_bingham_aspect_2}})). The yield surface is shown as a dashed line and the plug regions are shaded. Only half of the channel is shown as these distributions are symmetrical about the centre line of the channel.} 
    \label{fig:velocity_field_var_bingham}     
	\end{centering}
\end{figure}\addtocounter{table}{1}
\begin{figure}[p] 
	\begin{centering}
    \includegraphics[width=0.92\figurewidth]{graphics/2011JB008550-p05}
    \caption[Contour plots of the velocity and strain rate magnitude for a channel flow with a Bingham number of $B =\;$0.5, and aspect ratios of $\beta=\;$2 and 4.]{Contour plots of the velocity and strain rate magnitude for a channel flow with a Bingham number of $B =\;$0.5, and aspect ratios of $\beta=\;$2 and 4. The critical aspect ratio for this Bingham number is approximately 1.785 (see ({\ref{eq:crit_bingham_aspect_3}})). The yield surface is shown as a dashed line and the plug regions are shaded.} 
    \label{fig:velocity_field_var_aspect}   
	\end{centering}
\end{figure}\addtocounter{table}{1}

I first examine the behaviour of the flow for fixed aspect ratio $\beta$ and varying Bingham number $B$. Figure~\ref{fig:velocity_field_var_bingham} shows plots of the down-stream velocity and strain rate tensor magnitude fields for three flows with $\beta = 2$ and three different values of $B$. In this case the critical Bingham number is 
\begin{equation}
	B^{\star} = \frac{2}{2+\sqrt{\pi}} \approx 0.530; 
	\label{eq:crit_bingham_aspect_2} 
\end{equation} 
all the plotted flows have $ B < B^{\star}$.

For the two Bingham cases, dotted black lines delineate the yield surfaces, and the shading denotes the plug regions. In the Newtonian case the entire fluid yields, however the two plug regions grow with the development of a yield strength. One plug region forms in the corner of the channel, fixed to the channel walls with a concave circular arc forming the yield surface, while one forms around the channel centre, and moves at the maximum velocity in the channel. At low values of $B$ the central plug is almost circular, but as $B$ increases the sides of the centre plug flatten as they approach the walls, although the corners remain circular.

A layer of yielded fluid separates the two plug regions. The strain rate magnitudes in Figure~\ref{fig:velocity_field_var_bingham} show that the strain rate drops to zero at the plug boundaries. The strain rate magnitude reaches a maximum at the centre of the channel floor and the top of the channel walls, with a saddle point in the centre of the yielded layer between the plug regions. This layer is widest on the diagonal of the domain between the two plug regions, and narrows as it approaches the flow surface or centre line.

As the Bingham number increases (moving from top to bottom), the size of the central and corner plug regions increases until they meet at the critical Bingham number and flow ceases (not shown). The central plug grows at a faster rate than the corner plug. The width of the yielded layer between the plug regions narrows and the velocity of the central plug decreases, such that the yielded layer thickness goes to zero everywhere upon reaching the critical Bingham number.