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Thesis: introduction

Most of us became familiar very early in life with the behaviour of mud. When it is thin and runny, it will splash like water; when it is slightly thicker, it will spread and pour in a sticky and reluctant manner; when it is thicker still, one can mould it into solid shapes; and when it dries out completely, it can be broken up into a light, fine dust.

Although we may not have been aware of it at the time, the properties which made mud so fascinating to us at the age of five are, by and large, the same properties which make it of great interest to the physical scientist: the variety of behaviour which it exhibits, the diverse situations in which it occurs, and the range of uses to which it can be put. These are also the reasons why most of this thesis is concerned with it.

Mud — a suspension of very fine clay particles in water — is found in a wide range of marine, fluvial and terrestrial environments, wherever a sufficient supply of sediment exists. It forms a substantial fraction of the material transported by some of the world's great rivers, and of its most fertile soils; it is a key component of natural hazards such as lahars and mudslides; and muddy flows may be involved in the formation of many important sedimentary deposits such as aquifers and oil reservoirs.

Although human beings have been making use of the peculiar physical properties of mud since the first pottery was invented, these properties are still not fully understood. The electrostatic interaction between clay particles can affect the bulk properties of muds — whether in their solid form or when flowing as a highly viscous fluid — in unexpected ways, and there is still no generally accepted constitutive law for muddy fluids. Much more experimental work is still required.

Despite these uncertainties, there is also a role for the theoretical modelling of mud and muddy flows. As in many other areas of fluid dynamics, a simple model may give physical insight into a system which seems at first prohibitively complex. Thus, we can sometimes gain a broad understanding of a physical problem even if we are unable to resolve in detail some of its components.

It is with this kind of modelling that this thesis is concerned. We consider two sets of problems which involve the flow and transport of mud, and in each case we apply mathematical tools to a relatively simple physical model of the system, with the aim of determining what behaviour is predicted by the model, and of relating this, where possible, to phenomena observed in the real system.

In the first part of the thesis, we deal with the morphological evolution of muddy coasts, such as the mudflats which fringe many parts of the coast of north-western Europe and provide some of our most effective defences against flooding and sea level rise. We aim to describe the changing profiles of these coasts over periods of decades and centuries, with the aim of understanding the processes which govern this change and thus of learning how better to control or predict the long-term evolution.

The second part is concerned with the dynamics of flows which propagate under a balance between viscous and gravitational forces: these may represent, for example, sewage spreading from an outfall pipe, mud injected into a fracture in an oil reservoir to keep it open, or the motion of a particle-laden current on the continental slope. We approach these problems through a related but better-developed topic, which is the buoyancy-driven flow of liquid through a porous medium — as in, for example, the intrusion of salt water into an aquifer. The models which may be developed for such systems share several features with the models of muddy flow, essentially because they are also controlled by viscous forces (acting within the pores of the rock), and by developing new models and solutions for them we gain some insight into the more general class of problems which includes muddy flows.

We now give a brief overview of the results presented in this thesis.

Part I: the morphodynamics of intertidal mudflats

The first part of the thesis deals with the morphodynamics of tidally-dominated intertidal mudflats. The modelling of muddy coasts is poorly developed compared to that of sandy coasts, and there is currently something of a gap between empirical descriptions and theoretical modelling work: one of the main aims of this thesis is to help bridge this gap.

In chapter 1, we describe the current state of knowledge of muddy intertidal systems. In particular, we discuss the idea of morphodynamic equilibrium, which is often used to relate physical process-based models to empirical typologies. The assumption which underlies the empirical classification of morphodynamical systems is that, in general, they are close to a state of "equilibrium" with forcing factors such as the local climate, mineralogy and hydrodynamical regime, and so, knowing these factors, we can predict the morphology and long-term behaviour of the system. A useful role for process-based modelling, therefore, is to characterise such equilibrium states, and to investigate how they develop and are maintained.

This is the aim of the next three chapters. Chapter 2 describes the construction of a cross-shore morphodynamical model, which incorporates tidal currents, sediment transport and morphological evolution. This model is more schematic than those which have previously been employed to simulate estuarine or coastal sediment transport processes, but it is more complete than any model which has been used to investigate long-term morphodynamics.

In chapter 3, we describe the cross-shore sediment transport processes which our model represents. On flats with an idealised cross-shore profile, we are able to obtain both analytical and numerical results for the suspended sediment concentration field. These provide insight into the mechanism of settling lag by which tidal currents transport sediment landwards, and they also allow us to obtain a far-field concentration as a function of the tidal regime, the sediment properties, and the large-scale morphology of the flat. This in turn allows us to obtain estimates for the gradient of an equilibrium flat as a function of the sediment properties and sediment supply, and we examine these estimates in some detail. In particular, we investigate their sensitivity to changes in the formulation of the model.

Finally, in chapter 4, we describe a series of numerical experiments in which the full morphodynamical system was integrated forwards in time (using a numerical method which is described in appendix A) until an equilibrium state was reached. The tidal range and sediment supply were varied independently, and the behaviour of the flats was found to be qualitatively the same in each case. The flats prograde over decadal timescales, and the equilibrium cross-shore profile can be characterised by a velocity $U_{\mathrm{crit}}$, which corresponds to the maximum current speed attained throughout a tidal cycle, and is roughly constant across the flat. The variation of $U_{\mathrm{crit}}$ with tidal range and sediment supply was found to be described to within 10% by the estimates derived in chapter 3, and the sediment transport processes were also qualitatively very similar to those identified over the more idealised flats.

We have also carried out experiments to determine how the morphology is affected by an asymmetrical tidal signal, and by a spring-neap variation in tidal range. An asymmetrical tide produces a steeper flat, the gradient of which is predicted well by the theory of chapter 3; a spring-neap cycle leads to a morphology which is very similar to that for a fixed tidal range, but rather steeper than an estimate based on the median tide would predict.

Part II: gravity-driven flows at low Reynolds number

The second part of the thesis deals with models of gravity-driven flows at low Reynolds number. In chapter 5, we consider slowly draining flows through a layered porous medium; in chapter 6, we consider particle-driven viscous flows over a rigid horizontal surface; and in chapter 7 we lay the foundations for an extension of the models of chapter 6 to non-Newtonian fluids. A brief overview of appropriate mud rheologies is given in appendix B.

The models of flow in a porous medium have applications to the contamination of aquifers by salt water or other pollutants, and to the extraction of oil from layered sandstone reservoirs, while the work on viscous flow described in chapters 5 and 6 has applications to mud flow in various contexts, including possibly to the large-scale submarine mass flows which are believed to be responsible for forming many important sedimentary deposits.

When fluid spreads at low Reynolds number on a horizontal surface, its propagation is governed by a balance between gravitational and viscous forces. In a porous medium, the viscous forces are respresented by Darcy's law, and for spreading under an ambient fluid, they can be modelled using lubrication theory. The resulting models for porous and viscous flows are very similar, and each consists of a non-linear diffusion equation which admits similarity solutions.

In chapters 5 and 6 we extend existing models for these flows by incorporating a mass loss term. For currents in a porous medium, this represents the drainage of fluid into an underlying layer of lower permeability, driven by hydrostatic pressure or by flow in the ambient. For viscous currents whose buoyancy is due to the presence of suspended particles, it represents mass loss by the settling of these particles. The modelling approach, at least for currents in a porous medium, is supported by the results of analogue laboratory experiments.

We have obtained analytical solutions to several of these equations. These solutions are not similarity solutions, but share their properties to some extent. In particular, the shape of the current, when scaled by its depth and horizontal extent, is unchanging in time, and the solutions are attracting for flows with non-ideal initial conditions. We have investigated numerically the adjustment towards these flows from a range of initial conditions.

For the rest of these problems, we have been unable to obtain exact solutions. Instead, we have developed perturbation expansions, and have investigated various means of extending the first-order perturbation approximations to describe almost the entire evolution of the current to reasonable accuracy. These approximate solutions are compared with numerical integration of the governing equations.

A complication arises when we try to extend the modelling approach of chapter 6 to non-Newtonian, shear-thinning fluids, because in a shear-thinning fluid the settling velocity of a particle depends on the local shear rate of the surrounding fluid. Chapter 7 investigates in more detail the consequences of this effect.

We consider particles settling through a power-law fluid which is flowing downhill under gravity: this provides a simple prototype for a gravity-driven flow of mixed mud and sand. The effect of shear-dependent settling is that an initially uniform suspension of particles becomes stratified, and that in a suspension containing a range of particle sizes, the various species become segregated.

For a dilute sand suspension, in which the bulk density and viscosity of the flow are not affected by the suspended sand fraction, we investigate in some detail the deposits formed from monodisperse and polydisperse suspensions. In particular, we find that the deposit may contain inversely graded regions. We also investigate the dynamics of less dilute flow, in which the sand particles affect the buoyancy or viscosity of the current, and phenomena such as "shocks" in the concentration profile may result. These models may provide a basis for further work to develop more realistic models of particle-laden muddy flows.