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\begin{document}
\title{The Prediction of Flutter at Transonic Speeds}
\author{Brendan Perry}
% Faculty of Life Sciences people should comment the next line out
\school{Mathematics}
\faculty{Engineering and Physical Sciences}
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%\beforeabstract
%I developed a 2D transonic code that took for ever to debug, even longer to run
%properly and then wrote some matlab code that plotted pretty pictures. My
%research to date %has re-invented the wheel but instead of the usual circular
%configuration I have gone for a square. My wheel will do the same job as many
%other, older, wheels but is harder %to use, less efficient and less pretty to
%look at.
%I now hate wheels, and squares.
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% The next part is optional; however it is a good place to thank your
% supervisor and the people responsible for providing computer support ;-)
%\prefacesection{Acknowledgements}
%I would like to thank ...
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% Finally, you can start writing about all the new theorems you have proved
% and all the new results that you have discovered
%\chapter{Introduction}
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%This chapter covers the background material for my thesis and introduces
%the notation that I will use throughout.
\chapter{Small Disturbance Theory For Transonic Flow}
\chaptermark{Small Disturbance Theory}
\section{The System Model}
The strutural features of the system are represented by the \emph{typical section} airfoil model, which includes 2 degrees of freedom: pitch $\alpha(t^*)$ and plunge $\eta(t^*)^*$. The airfoil displacement is measured relative to the equilibrium configuration, shown by the dashed lines. The airfoil pitches about the flexural axis $x_f$ with the positive direction being defined as nose-up. The plunge displacement is defined as positive downwards. The pitch and plunge displacements are restrained by linear springs with stiffness $k_{\alpha}$ and $k_{\eta}$ respectively; the corresponding uncoupled pitch and plunge frequencies are $\omega_{\alpha}$ and $\omega_{\eta}$. The equilibrium position of the airfoil includes a steady angle of attack $\alpha_0$ about which the pitching displacement is measured. In what follows it is helpful to introduce the effective angle of attack $\theta(t^*)$, which is the sum of the steady and the pitching components.
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The flow at the domain boundaries is steady, although the airfoil is free to undergo unsteady pitching and plunging oscillations. This motion of the airfoil introduces time-dependent disturbances into the flowfield which cause the position, and strength, of embedded shocks to vary with time. In order that the flow remains approximately isentropic, these shocks must be weak. This effectively limits the theory to thin airfoils operating at small effective angles of attack:
\begin{equation}
\theta, \delta\ll 1 \label{small disturbance}
\end{equation}
where $\delta$ is the airfoil thickness to chord ratio. A natural consequence of this \emph{small disturbance approximation} is that flow variables can be represented as the sum of the freestream and small disturbance components; an asymptotic expansion can be used to represent each variable using $\delta$ as the small parameter.
\end{document}