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The primary aim of this dissertation is to construct and study exact <I>S</I>-matrices for (1+1)-dimensional integrable field theories with quantum affine symmetry. We introduce a general scheme of how to use trigonometric <I>R</I>-matrices for the construction of exact <I>S</I>-matrices. This scheme is then applied to a specific class of relativistic field theories defined in (1+1)-dimensional Minkowski space, the so-called affine Toda field theories with imaginary coupling constant. The most important feature of these theories is the fact that their classical equations of motion admit soliton solutions. As a step towards the consistent quantisation of these theories we attempt to construct <I>S-</I>matrices for the quantum scattering of affine Toda solitons. Apart from the solitons there are also bound states of solitons in affine Toda field theories. By using the bootstrap principle we derive the <I>S</I>-matrices for the scattering of bound states. We focus in particular on the scalar bound states which are the analogues of the breathers in Sine-Gordon theory, and show that the <I>S</I>-matrices for the lowest breathers in the theory are identical to the <I>S-</I>matrices for the fundamental quantum particles. We also provide evidence for the consistency of the conjectured <I>S</I>-matrices through a detailed examination of their pole structure. We find that a large number of poles can be explained in terms of higher order diagrams, many of which involve a generalised Coleman-Thun mechanism. The layout of this thesis is as follows. In the introduction the axioms of analytic <I>S-</I>matrix theory are reviewed and some of the main features of affine Toda field theories are introduced. In chapter 2 we provide an introduction to the theory of quantised universal enveloping algebras and their <I>R-</I>matrices, where there are trigonometric solutions of the Yang-Baxter equation. We also describe the main features of quantum affine symmetries in two-dimensional field theories. Chapters 3, 4 and 5 deal with the detailed discussion of soliton <I>S-</I>matrices and bound states in <I>a<SUB>n</SUB></I><SUP>(1)</SUP>, <I>d<SUB>n</SUB></I><SUP>(2)</SUP><I><SUB>+</SUB></I><SUB>1, </SUB><I>b<SUB>n</SUB></I><SUP>(1) </SUP>and <I>a</I><SUB>2n</SUB><SUP>(2) </SUP>affine Toda field theories. Special attention is given to the two cases of <I>a</I><SUB>2</SUB><SUP>(1) </SUP>and <I>d</I><SUB>3</SUB><SUP>(2)</SUP>, and the pole structures of their proposed <I>S</I>-matrices are examined in great detail. We also provide a conjecture of the complete quantum spectrum of these theories. The dissertation concludes with a summary of results and some remarks about open questions and unsolved problems. In the appendices we provide detailed proofs and calculations omitted from the main part of the thesis. We also attempt to construct integral representations of <I>S-</I>matrix scalar factors and give tables containing relevant data for affine Lie algebras.

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This thesis is concerned with the numerical prediction of the flow of liquids that display both elastic and viscous properties. Such liquids are termed visco-elastic liquids and are of considerable industrial importance. The properties of liquids are defined through a set of constitutive equations, which define a relationship between the rate of strain and extra-stress tensors. The derivation of constitutive equations that predict visco-elastic properties and are valid under all conditions of flow is a complex task that usually results in a relationship in the form of a differential or integral equation. To obtain a solution for the flow field the constitutive equation must be solved in conjunction with the equations of momentum and continuity. Because of the complexity of the equations and the flow geometries considered in this thesis a numerical method must be used. To represent the fluid an upper convected Maxwell model is taken as the constitutive relation. For the numerical method the finite element method is used with velocities, pressure, and extra-stresses as dependant variables. When using the finite element method care must be taken in the choice of shape functions for each dependant variable. An analysis is given that shows certain combinations of shape functions can give rise to a non-unique solution. The 'online' measurement of the elastic properties of a fluid is of industrial importance. The hole pressure technique for the 'on-line' measurement of the first normal stress difference is modelled. The effects of elasticity and inertia on the theoretical basis for the instrument is investigated. Flow through a 'U' shaped vessel with stirrers in the bottom corners of the vessel is modelled. The effect of elasticity and inertia on the flow field for different strengths of the stirring action is investigated.

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