Mathematical models of bubble evolution in a Hele-Shaw Cell

This thesis concerns the mathematical model of moving fluid interfaces in a Hele-Shaw cell: an experimental device in which fluid flow is studied by sandwiching the fluid between two closely separated plates. Analytic and numerical methods are developed to gain new insights into interfacial stability and bubble evolution, and the influence of different boundary effects is examined. In particular, the properties of the velocity-dependent kinetic undercooling boundary condition are analysed, with regard to the selection of only discrete possible shapes of travelling fingers of fluid, the formation of corners on the interface, and the interaction of kinetic undercooling with the better known effect of surface tension. Explicit solutions to the problem of an expanding or contracting ring of fluid are also developed.

Electrochemically induced actuation of liquid metal marbles

Controlled actuation of soft objects with functional surfaces in aqueous environments presents opportunities for liquid phase electronics, novel assembled super-structures and unusual mechanical properties. We show the extraordinary electrochemically induced actuation of liquid metal droplets coated with nanoparticles, so-called “liquid metal marbles”. We demonstrate that nanoparticle coatings of these marbles offer an extra dimension for affecting the bipolar electrochemically induced actuation. The nanoparticles can readily migrate along the surface of liquid metals, upon the application of electric fields, altering the capacitive behaviour and surface tension in a highly asymmetric fashion. Surprising actuation behaviours are observed illustrating that nanoparticle coatings can have a strong effect on the movement of these marbles. This significant novel phenomenon, combined with unique properties of liquid metal marbles, represents an exciting platform for enabling diverse applications that cannot be achieved using rigid metal beads.

Globalising the body : globalisation and reproductive rights

Globalisation is a phenomenon of the contemporary world. Everywhere around us there seem to be signs of the power of the forces of globalisation: in our media and popular culture; in our international linkages across continents through international travel and telecommunications; in our globalised trade; and with the global movement of people, a process which itself ranges from the movement of international tourists to the international movement of refugees and other displaced persons. The processes of globalisation seem to simultaneously unify and divide us. There is no doubt that we live in a globalised world and that we are connected to others in previously unimaginable ways by transportation, telecommunications and economics. Yet, while this global context increasingly links us to others, there is also a very real sense in which separation, difference and the local have also gained a new significance; we are locked in a tension between the universal and the particular that has come to typify contemporary society. This article explores the meanings of globalisation and this dynamic – or tension – between the universal and the particular in terms of its implications for the body and, in particular, its significance for women and their reproductive rights.

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

Including nonequilibrium interface kinetics in a continuum model for melting nanoscaled particles

The melting temperature of a nanoscaled particle is known to decrease as the curvature of the solid-melt interface increases. This relationship is most often modelled by a Gibbs--Thomson law, with the decrease in melting temperature proposed to be a product of the curvature of the solid-melt interface and the surface tension. Such a law must break down for sufficiently small particles, since the curvature becomes singular in the limit that the particle radius vanishes. Furthermore, the use of this law as a boundary condition for a Stefan-type continuum model is problematic because it leads to a physically unrealistic form of mathematical blow-up at a finite particle radius. By numerical simulation, we show that the inclusion of nonequilibrium interface kinetics in the Gibbs--Thomson law regularises the continuum model, so that the mathematical blow up is suppressed. As a result, the solution continues until complete melting, and the corresponding melting temperature remains finite for all time. The results of the adjusted model are consistent with experimental findings of abrupt melting of nanoscaled particles. This small-particle regime appears to be closely related to the problem of melting a superheated particle.

Stefan problems for melting nanoscaled particles

Under certain conditions, the mathematical models governing the melting of nano-sized particles predict unphysical results, which suggests these models are incomplete. This thesis studies the addition of different physical effects to these models, using analytic and numerical techniques to obtain realistic and meaningful results. In particular, the mathematical "blow-up" of solutions to ill-posed Stefan problems is examined, and the regularisation of this blow-up via kinetic undercooling. Other effects such as surface tension, density change and size-dependent latent heat of fusion are also analysed.

The BAD project: data mining, database and prediction of protein adsorption on surfaces

Protein adsorption at solid-liquid interfaces is critical to many applications, including biomaterials, protein microarrays and lab-on-a-chip devices. Despite this general interest, and a large amount of research in the last half a century, protein adsorption cannot be predicted with an engineering level, design-orientated accuracy. Here we describe a Biomolecular Adsorption Database (BAD), freely available online, which archives the published protein adsorption data. Piecewise linear regression with breakpoint applied to the data in the BAD suggests that the input variables to protein adsorption, i.e., protein concentration in solution; protein descriptors derived from primary structure (number of residues, global protein hydrophobicity and range of amino acid hydrophobicity, isoelectric point); surface descriptors (contact angle); and fluid environment descriptors (pH, ionic strength), correlate well with the output variable-the protein concentration on the surface. Furthermore, neural network analysis revealed that the size of the BAD makes it sufficiently representative, with a neural network-based predictive error of 5% or less. Interestingly, a consistently better fit is obtained if the BAD is divided in two separate sub-sets representing protein adsorption on hydrophilic and hydrophobic surfaces, respectively. Based on these findings, selected entries from the BAD have been used to construct neural network-based estimation routines, which predict the amount of adsorbed protein, the thickness of the adsorbed layer and the surface tension of the protein-covered surface. While the BAD is of general interest, the prediction of the thickness and the surface tension of the protein-covered layers are of particular relevance to the design of microfluidics devices.

Saffman-Taylor fingers with kinetic undercooling

The mathematical model of a steadily propagating Saffman-Taylor finger in a Hele-Shaw channel has applications to two-dimensional interacting streamer discharges which are aligned in a periodic array. In the streamer context, the relevant regularisation on the interface is not provided by surface tension, but instead has been postulated to involve a mechanism equivalent to kinetic undercooling, which acts to penalise high velocities and prevent blow-up of the unregularised solution. Previous asymptotic results for the Hele-Shaw finger problem with kinetic undercooling suggest that for a given value of the kinetic undercooling parameter, there is a discrete set of possible finger shapes, each analytic at the nose and occupying a different fraction of the channel width. In the limit in which the kinetic undercooling parameter vanishes, the fraction for each family approaches 1/2, suggesting that this selection of 1/2 by kinetic undercooling is qualitatively similar to the well-known analogue with surface tension. We treat the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analogue with surface tension, since kinetic undercooling permits finger shapes which are corner-free but not analytic. We provide numerical evidence for the selection mechanism by setting up a problem with both kinetic undercooling and surface tension, and numerically taking the limit that the surface tension vanishes.

Influences of biodiesel chemical compositions and physical properties on engine exhaust particle emissions

This thesis presents a comprehensive study on the influences of biodiesel chemical composition and physical properties on diesel engine exhaust particle emissions. It examines biodiesels from several feedstocks having wide variations in their chemical composition (carbon chain length, unsaturation and oxygen content) and physical properties (density, viscosity, surface tension, boiling point etc.), and evaluates their influence on exhaust particle emissions. The outcome of this thesis is significant since it reveals the importance of regulating biodiesels chemical composition in order to ensure lowest possible emissions with better overall performance.

Discrete families of Saffman–Taylor fingers with exotic shapes

The mathematical problem of determining the shape of a steadily propagating Saffman–Taylor finger in a rectangular Hele-Shaw cell is known to have a countably infinite number of solutions for each fixed surface tension value. For sufficiently large surface tension values, we find that fingers on higher solution branches are non-convex. The tips of the fingers have increasingly exotic shapes as the branch number increases.

Two-dimensional bow and stern flows in a finite-depth fluid with constant vorticity

This thesis is concerned with two-dimensional free surface flows past semi-infinite surface-piercing bodies in a fluid of finite-depth. Throughout the study, it is assumed that the fluid in question is incompressible, and that the effects of viscosity and surface tension are negligible. The problems considered are physically important, since they can be used to model the flow of water near the bow or stern of a wide, blunt ship. Alternatively, the solutions can be interpreted as describing the flow into, or out of, a horizontal slot. In the past, all research conducted on this topic has been dedicated to the situation where the flow is irrotational. The results from such studies are extended here, by allowing the fluid to have constant vorticity throughout the flow domain. In addition, new results for irrotational flow are also presented. When studying the flow of a fluid past a surface-piercing body, it is important to stipulate in advance the nature of the free surface as it intersects the body. Three different possibilities are considered in this thesis. In the first of these possibilities, it is assumed that the free surface rises up and meets the body at a stagnation point. For this configuration, the nonlinear problem is solved numerically with the use of a boundary integral method in the physical plane. Here the semi-infinite body is assumed to be rectangular in shape, with a rounded corner. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterised by a train of waves upstream. In the limit that the height of the body above the horizontal bottom vanishes, the flow approaches that due to a submerged line sink in a $90^\circ$ corner. This limiting problem is also examined as a special case. The second configuration considered in this thesis involves the free surface attaching smoothly to the front face of the rectangular shaped body. For this configuration, nonlinear solutions are computed using a similar numerical scheme to that used in the stagnant attachment case. It is found that these solution exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a corner is also considered. Finally, the flow of a fluid emerging from beneath a semi-infinite flat plate is examined. Here the free surface is assumed to detach from the trailing edge of the plate horizontally. A linear problem is formulated under the assumption that the elevation of the plate is close to the undisturbed free surface level. This problem is solved exactly using the Wiener-Hopf technique, and subcritical solutions are found which are characterised by a train of sinusoidal waves in the far field. The nonlinear problem is also considered. Exact relations between certain parameters for supercritical flow are derived using conservation of mass and momentum arguments, and these are confirmed numerically. Nonlinear subcritical solutions are computed, and the results are compared to those predicted by the linear theory.

Impaction of spray droplets on leaves: Influence of formulation and leaf character on shatter, bounce and adhesion

This paper combines experimental data with simple mathematical models to investigate the influence of spray formulation type and leaf character (wettability) on shatter, bounce and adhesion of droplets impacting with cotton, rice and wheat leaves. Impaction criteria that allow for different angles of the leaf surface and the droplet impact trajectory are presented; their predictions are based on whether combinations of droplet size and velocity lie above or below bounce and shatter boundaries. In the experimental component, real leaves are used, with all their inherent natural variability. Further, commercial agricultural spray nozzles are employed, resulting in a range of droplet characteristics. Given this natural variability, there is broad agreement between the data and predictions. As predicted, the shatter of droplets was found to increase as droplet size and velocity increased, and the surface became harder to wet. Bouncing of droplets occurred most frequently on hard to wet surfaces with high surface tension mixtures. On the other hand, a number of small droplets with low impact velocity were observed to bounce when predicted to lie well within the adhering regime. We believe this discrepancy between the predictions and experimental data could be due to air layer effects that were not taken into account in the current bounce equations. Other discrepancies between experiment and theory are thought to be due to the current assumption of a dry impact surface, whereas, in practice, the leaf surfaces became increasingly covered with fluid throughout the spray test runs.

Particle emissions from biodiesels with different physical properties and chemical composition

Biodiesels produced from different feedstocks usually have wide variations in their fatty acid methyl ester (FAME) so that their physical properties and chemical composition are also different. The aim of this study is to investigate the effect of the physical properties and chemical composition of biodiesels on engine exhaust particle emissions. Alongside with neat diesel, four biodiesels with variations in carbon chain length and degree of unsaturation have been used at three blending ratios (B100, B50, B20) in a common rail engine. It is found that particle emission increased with the increase of carbon chain length. However, for similar carbon chain length, particle emissions from biodiesel having relatively high average unsaturation are found to be slightly less than that of low average unsaturation. Particle size is also found to be dependent on fuel type. The fuel or fuel mix responsible for higher particle mass (PM) and particle number (PN) emissions is also found responsible for larger particle median size. Particle emissions reduced consistently with fuel oxygen content regardless of the proportion of biodiesel in the blends, whereas it increased with fuel viscosity and surface tension only for higher diesel–biodiesel blend percentages (B100, B50). However, since fuel oxygen content increases with the decreasing carbon chain length, it is not clear which of these factors drives the lower particle emission. Overall, it is evident from the results presented here that chemical composition of biodiesel is more important than its physical properties in controlling exhaust particle emissions.

Ink-jet printing of graphene for flexible electronics: An environmentally-friendly approach

Mechanical flexibility is considered an asset in consumer electronics and next-generation electronic systems. Printed and flexible electronic devices could be embedded into clothing or other surfaces at home or office or in many products such as low-cost sensors integrated in transparent and flexible surfaces. In this context inks based on graphene and related two-dimensional materials (2DMs) are gaining increasing attention owing to their exceptional (opto)electronic, electrochemical and mechanical properties. The current limitation relies on the use of solvents, providing stable dispersions of graphene and 2DMs and fitting the proper fluidic requirements for printing, which are in general not environmentally benign, and with high boiling point. Non-toxic and low boiling point solvents do not possess the required rheological properties (i.e., surface tension, viscosity and density) for the solution processing of graphene and 2DMs. Such solvents (e.g., water, alcohols) require the addition of stabilizing agents such as polymers or surfactants for the dispersion of graphene and 2DMs, which however unavoidably corrupt their properties, thus preventing their use for the target application. Here, we demonstrate a viable strategy to tune the fluidic properties of water/ethanol mixtures (low-boiling point solvents) to first effectively exfoliate graphite and then disperse graphene flakes to formulate graphene-based inks. We demonstrate that such inks can be used to print conductive stripes (sheet resistance of ~13 kΩ/□) on flexible substrates (polyethylene terephthalate), moving a step forward towards the realization of graphene-based printed electronic devices.