Self-excited oscillations in a collapsible-tube flow driven by fixed upstream flux have been observed by numerical and laboratory experiments. In this thesis we attempt to understand the mechanism of onset of these oscillations by focusing on a reduced physical model. We consider flow in a finite-length planar channel, where a segment of one wall is replaced by a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. We describe the system using a one-dimensional model that accounts for viscous and fluid inertial effects. We perform linear stability analysis and weakly nonlinear analysis on the one-dimensional model, the resulting predictions are tested against two-dimensional Navier–Stokes numerical simulation. When the membrane has similar length to the rigid segment of channel downstream of the membrane, we find that in a narrow parameter regime we consider “mode-2” oscillations (i.e. membrane displacements with two extrema) are largely independent of the downstream segment but are driven by divergent instabilities of two non-uniform steady configurations of the membrane. When the downstream segment is much longer than the membrane, our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system. In this case, our analysis shows some agreements with experimental observations, namely that increasing the length of the downstream tube reduces the frequency of oscillations but has little effect on the conditions for onset. We also use linear stability analysis to show that steady highly-collapsed solutions, constructed by utilizing matched asymptotic expansions, are very unstable, which allows the possibility that they are a precursor to slamming motion whereby the membrane becomes transiently constricted very close to the opposite rigid wall before rapidly recovering.

Every time we breathe in we inhale thousands of particles, some of which may become trapped in the liquid lining of the airway wall. In this thesis we use theoretical fluid dynamics to model various aspects of the dynamics of these particles after their initial deposition on the airway wall. In Chapter 2 we consider the behaviour of an inhaled particle trapped in an alveolar corner, modelled as a two-dimensional cylinder partially immersed in a liquid pool in the corner of a rigid-walled wedge. We balance quasistatic capillary forces acting on the particle with viscous forces, modelled using lubrication theory, acting in a small gap between the particle and the wall. The direction of particle motion is non-intuitive and we obtain predictions for the fate of a particle dependent on the wedge angle, liquid volume and the size and deposition site of the particle. In practice, surface forces have been shown to pull particles into the airway liquid lining with sufficient force to depress underlying epithelial cells. In Chapters 3 and 4, using elastohydrodynamics, we consider the unsteady motion of a particle close to a deformable surface and the effect of wall deformation on the particle's behaviour. We model this initially as a two-dimensional cylinder moving in fluid perpendicularly and transversely close to a spring-backed plate, using simulations and asymptotic analysis based on lubrication theory. Viscous forces cause a transient overshoot of the force acting on the particle following a prescribed perpendicular displacement. Transverse motion of the particle causes the formation of a `corner' in the wall, which is particularly sharp immediately following the particle's initial displacement. In addition we consider the extension of the model into three dimensions and examine a sphere moving close to a deformable plane in a fluid environment. In Chapter 5 we consider the motion of a particle trapped in a mucus layer which is propelled by cilia acting within an underlying serous layer. We model this as a cylindrical disk moving within a viscous sheet, with a uniformly distributed body force in the lower layer representing the cilia. We predict the speed of the particle as a function of disk shape, ciliary activity and other material parameters.

We consider wave dynamics on networks of beams/plates coupled along 1D joints. This set-up can be mapped onto the wave dynamics on graphs and is introduced here as an extension to generic wave graph systems such as studied in quantum graph theory. In particular, we consider the elastic case which entails different mode-types (bending, longitudinal and shear waves) which propagate at different wave speeds and can mix at interfaces. The bending modes are described in terms of 4th order equations introducing an always evanescent wave component into the system. The scattering matrices describing reflection/transmission at interfaces thus contain both propagating (open) and evanescent (closed) channels. As a result, the scattering matrices and the transfer operator are no longer unitary; the consequences of this non-unitaritiness on secular equations and the Weyl law will be discussed. The findings are of relevance to describing complex engineering structures. We note that existing methods used to solve wave propagation problems often provide average solutions. As well as the aforementioned extension of quantum graphs to the elastic case, we consider fluctuations about this mean solution. This is done by propagating correlation functions on graphs; it turns out that this provides a suitable wave analogue of ray methods. This approach allows us to investigate response statistics and distributions; these properties are of real significance in, for example, the automotive industry.

This thesis is devoted to the study of multiphase jet fragmentation using Smoothed Particle Hydrodynamics (SPH). The theoretical aspects of three hydrodynamic instabilities, namely the Kelvin-Helmholtz instability (KHI), Rayleigh-Taylor instability (RTI), and Rayleigh Plateau instability (RPI) are reviewed. The linear growth rate of the combined KHI and RTI are derived by means of linear perturbation in chapter 2. The linear growth rate of the multiphase RPI is presented in chapter 7. An overview of the Smoothed Particle Hydrodynamics is given in chapter 3. A pseudo-consistent SPH scheme is presented for the simulation of multiphase flow problems. Additionally, two interface stabilisation models are presented: quasi-buoyancy model and gas-repulsion model. When used in combination with the pseudo-consistent SPH scheme, these models are found to be superior than those presented in the weakly-compressible SPH literature and allows for the simulations for density ratio up to three-magnitudes. The development of an idealised KHI and a KHI subjected to constant gravitational acceleration (stratified shear instability) is examined in chapter 5. The extracted linear growth rate are compared with the theoretical growth rate presented both in the literature and in chapter 2 for the purpose of validation. The development of a single- and multi-mode RTI are studied by means of SPH in chapter 6. Chapter 7 presents the results for the three-dimensional RPI occurring between two fluids. Based on the knowledge acquired in chapter 5-7, the multiphase jet fragmentation driven by the previously mentioned hydrodynamic instabilities are presented in chapter 8. Finally, the major research findings and recommendations are summarised in chapter 9.

In this project, we consider a variety of ac-driven, inhomogeneous sine-Gordon equations describing an infinitely long Josephson junctions with phase shifts, driven by a microwave field. First, the case of a small driving amplitude and a driving frequency close to the natural (defect) frequency is considered. We construct a perturbative expansion for the breathing mode to obtain equations for the slow time evolution of the oscillation amplitude. We show that, in the absence of an ac-drive, a breathing mode oscillation decays with a rate of at least \mathcal{O}(t^{-1/4}) and \mathcal{O}(t^{-1/2}) for 0-\pi-0 and 0-\kappa junctions, respectively. Multiple scale expansions are used to determine whether, e.g., an external drive can excite the defect mode of a junction (a breathing mode), to switch the junction into a resistive state. Next, we extend the study to the case of large oscillation amplitude with a high frequency drive. Considering the external driving force to be rapidly oscillating, we apply an asymptotic procedure to derive an averaged nonlinear equation, which describes the slowly varying dynamics of the sine-Gordon field. We discuss the threshold distance of 0-\pi-0 junctions and the critical bias current in $0-\kappa$ junctions in the presence of ac drives. Then, we consider a spatially inhomogeneous sine-Gordon equation with two regions in which there is a \pi-phase shift, and a time periodic drive, modelling 0-\pi-0-\pi-0 long Josephson junctions. We discuss the interactions of symmetric and antisymmetric defect modes in long Josephson junctions. We show that the amplitude of the modes decay in time. In particular, exciting the two modes at the same time will increase the decay rate. The decay is due to the energy transfer from the discrete to the continuous spectrum. For a small drive amplitude, there is an energy balance between the energy input given by the external drive and the energy output due to radiative damping experience by the coupled mode. Finally, we consider spatially inhomogeneous coupled sine-Gordon equations with a time periodic drive, modelling stacked long Josephson junctions with a phase shift. We derive coupled amplitude equations considering weak coupling and strong coupling in the absence of ac-drive. Next, by considering the strong coupling with time periodic drive, we expect that the amplitude of oscillation tends to constant for long times.

In this thesis we study four problems with potential biological and industrial applications which rely on fluid mixing and transport. The problem of simultaneous ultrafiltration, diffusion and osmosis across a membrane separating two fluids is studied, numerically and asymptotically, as a model for an artificial kidney dialyser. Couplings between the different transport mechanisms prove significant in determining overall transport rates. Our model appears to be the first to treat the three transport mechanisms in a spatially structured framework, and shows that previous, spatially averaged models can overestimate transport rates. Our results can be used to optimise dialyser geometry and to profile dialysis sessions. The remainder of this thesis concerns some fundamentals of fluid mixing and mixer design. Techniques for assessing the quality of fluid mixing are reviewed, and applied to a two-dimensional laminar chaotic flow. We find no outright optimum mixing method across the range of measures, suggesting that `sieving' a collection of mixing methods according to increasingly complicated mixing measures may fail to identify a global optimum. `Topological chaos' appears to allow good mixing stretch rate to be built-in to batch mixer design, avoiding the need to tune the mixer parameters, provided a correct flow topology is created. We show that the theoretical stretch rate predictions are achieved quite tightly, in practice in a significant fraction of the flow domain; we investigate the practicalities of topologically chaotic mixers. Finally, we discuss whether topological chaos may also apply to three-dimensional static mixer design, in a braided pipe mixer, in which pipe flow is mixed around carefully designed twisted inner pipes. We expect such a device to mix well if the inner pipes have appropriate topology. However, we demonstrate how three-dimensional flow features can undermine mixing performance.

Gas-liquid pipe flows are extremely important in many industries, one of which is the oil/gas industry which is where the motivation for this work comes from. In subsea natural gas pipelines the gas is compressed before being pumped through the pipe at high pressure. As it flows through the pipe some of the gas condenses into a low density mixture of hydrocarbon liquids. When gas and liquid flow together there are several possible flow regimes that can occur depending on the velocity of the gas and liquid, one of which is slug flow where the liquid forms a series of plugs (slugs) separated by relatively large gas pockets. The occurrence of slug flow is a major concern in the oil and gas industry due to the difficulty of dealing with large changes in the oil and gas flow rates at the exit of the pipe. We develop a hydraulic theory to describe the occurrence and structure of slugging in two-layer-gas-liquid flow generated by prescribed, constant, upstream flow rates in each layer. We will investigate how small-amplitude disturbances affect the flow in order to study the stability of spatially uniform solutions. We will then consider the existence of periodic travelling wave solutions numerically in order to investigate the influencing factors that may lead to a transition from stratified flow to slug flow. We then solve the governing equations numerically as an initial value problem in order to improve our understanding of how and why slugs form and are able to compare our solutions to those predicted by the periodic travelling wave theory. Finally, we investigate the effects of non-horizontal channels with small, slowly varying inclination on the development of slug flow by re-writing our equations in terms of a curvilinear co-ordinate system. From this we find that the height of the layer of liquid increases with the angle of the channel and our solutions are significantly different to those in the horizontal case.

This thesis provides a depth-averaged analytical and numerical approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to gravity-driven flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio e. Approximations are made assuming a Reynolds number, Re ~O (1/e) and a depth-averaged approach used to simplify the governing Navier-Stokes equations. A classical, parallel, Stokes flow is expected in the absence of any wave disturbance based on a local quadratic profile; in this work a generalised approach, which includes inertial effects, is solved. Flow structures are identified and compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are constructed to study film profile evolution subject to shear at the free surface. An evaluation of film profiles is given from a paramet- ric study for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken which identifies new profiles not previously predicted when inertial effects are neglected. A revised integral boundary layer model incorporating a more general cubic velocity profile is also introduced, to better capture fluid re- circulation associated with a capillary region, and is developed to provide a better understanding of the internal flow dynamics within the thin-film layer. Notably, the wavelength and amplitude of the capillary ripples are analysed. The effect of the boundary conditions between the fluid and the plane is undertaken to simulate slip properties of various substrates over which the fluid may flow. A Navier slip condition is proposed at this boundary and its effect on the wave structure is examined both with and without the inclusion of inertia. The corresponding film dynamics are analysed with increased slip at the fluid-plane boundary and the effect on the wave structures formed are discussed. In a subsequent chapter solitary wave structures are investigated through a study of gravity-driven flow structures as associated with an oscillating inlet. The effects of increasing the film Reynolds number of these flows is evaluated together with an investigation of the stability characteristics relevant to inlet frequency and inertial effects. The effect of surface shear on solitary waves is examined, both as a stabilising and a destabilising factor on perturbations introduced at the inlet. A final section provides an overview of the outcomes from this study.

Recent experiments [P. Brunet, J. Eggers, and R. Deegan, Phys. Rev. Lett. 99, 114501 (2007)] have shown that a liquid droplet on an inclined plane can be made to move uphill by sufficiently strong, vertical oscillations. In order to investigate this counterintuitive phenomenon we will derive three different models that qualitatively reproduce the main features of the experiment. For the first model the liquid's inertia and viscosity are assumed negligible, so that the motion of the droplet is dominated by the applied acceleration due to the oscillation of the plate, gravity and surface tension and that the droplet is thin. We explain how the leading order motion of the droplet can be separated into a spreading mode and a swaying mode. For a linear contact line law, the maximum rise velocity occurs when the frequencies of oscillation of the two modes are in phase. We show that, both with and without contact angle hysteresis, the droplet can climb uphill and also that, for certain contact line laws, the motion of the droplet can produce footprints similar to experimental results. We show that if the two modes are out of phase when there is no contact angle hysteresis, the inclusion of hysteresis can force them into phase. This in turn increases the rise velocity of the droplet and can, in some cases, cause a sliding droplet to climb. For the second model we use a two-dimensional flow where the Reynolds number is assumed large enough for viscosity to be neglected. We show that the leading order motion of the droplet can be separated into the same two modes and the net motion of the droplet is an oscillatory function of the frequency. For increasingly non-wetting droplets we discover that the rise velocity begins to oscillate very rapidly as a function of the static contact angle. What we also discover is that the change in the free surface of the droplet is actually a wave travelling travelling across the droplet, and the amount of modes present coincide with the rapid change in the rise velocity. Using a cubic contact line law and contact angle hysteresis we observe a droplet that can climb uphill for parameter values similar to that of the experiment. With the addition of a time dependent term within the contact line law we show that it is possible to obtain a multi-valued relationship between the velocity of the contact line and the respective contact angles, reproducing experimental observations seen for unsteady, moving contact lines. For the third model we again assume that the liquid's viscosity is negligible, similar to model 2, only now for a three-dimensional, thin droplet. For very small amplitudes the motion of the droplet is a combination of a swaying mode and a spreading mode that interact causing a net motion of the droplet. This motion is found to be an oscillatory function of the driving frequency and the magnitude of the peak rise velocity is proportional to one over the frequency squared. By examining the velocity of the centre of the droplet and the displacement of the contact line we see that the absolute maximums of both of these, over one period of oscillation, contain natural frequencies, which are evenly spaced with respect to the square root of the frequency of the oscillation.

A general displacement-based shear and transverse normal deformable plate theory is reviewed. Shear and transverse normal deformable plate theories suitable for cylindrical bending problems have been deduced from the general plate theory by introducing certain general functions of the transverse coordinate into the displacement field approximation. This theory takes into account the transverse shear and normal deformation effects and unifies most of the classical and shear deformable theories available in the literature. A predictor-corrector method has been used for improving the accuracy of transverse stress analysis results and assessing the accuracy of composite plate/beam theories. In more detail, uniform shear deformable plate theory, parabolic shear deformable plate theory, general three-degree-of-freedom shear deformable plate theory, general four-degree- of-freedom transverse shear and normal deformable plate theory and general five-degree-of-freedom shear deformable plate theory are employed to improving their prediction performances of transverse shear and normal stresses. By means of the assessment of plate theories for simply supported beams, general three-degree-of-freedom shear deformable plate theory, general four-degree-of-freedom transverse shear and normal deformable plate theory are applied for other sets of boundary conditions of cross-ply laminates subject to mechanical loading. General five-degree-of-freedom shear deformable plate theory is applied for angle-ply laminates subject to mechanical loading. In addition, general four-degree-of-freedom transverse shear and normal deformable plate theory is employed for cross-ply laminates subject to thermal loading. The numerical results of the present studies are compared with the corresponding exact solution results available in the literature.