Neil Balmforth

This research studies the impingement of a free surface Newtonian liquid jet on a dry solid moving wall using theoretical, numerical and experimental methods. The study focus mainly on the parameter range of jet Reynolds numbers 100
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In this thesis, we study baroclinic critical layers in rotating stratified shear flows. Baroclinic critical layers are characterized by strong amplitudes surrounding the singular points of the steady inviscid wave solution, and play crucial roles in the mixing and transition to turbulence in ocean, atmosphere and astrophysical disks. This thesis studies the baroclinic critical layers in strato-rotational instability and the forced baroclinic critical layers. The first problem we study is the baroclinic critical layer in strato-rotational instability. Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between internal gravity waves or Kelvin waves. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by baroclinic critical layers. The critical-level coupling removes the requirement for resonance near specific wavenumbers, resulting in an extensive continuous band of unstable modes. The second problem we study is the forced baroclinic critical layers. Linear theory predicts the baroclinic critical layer dynamics is characterized by the secular growth of flow perturbations over a region of decreasing width. Once it enters the nonlinear regime, the nonlinear dynamics filters out harmonics and the modification to the mean flow controls the evolution. At late times, we show that the vorticity begins to focus into yet smaller regions whose width decreases exponentially with time, and that the addition of dissipative effects can arrest this focussing to create a drifting coherent structure. In the last problem, we show that the mean-flow defect generated in the forced baroclinic critical layer can make the flow unstable, and we study this 'secondary instability'. The instability is a horizontal shear instability with a distinct phase velocity compared to that of the forced baroclinic critical layers, and thus will excite new baroclinic critical layers. A WKB solution for the exponential growth is derived, which indicates the secondary instability grows faster than a common normal mode due to the unsteadiness of the mean-flow defect. At the later stage, the short-wave harmonics grow at extraordinarily high speeds and will finally make the linear problem ill-posed.

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Dewatering of pulp fibre suspensions is a fundamental process in many unit operationsin the production of pulp and paper. A theoretical understanding of thedynamics of these networking fibre suspensions can prove valuable in the designof industrial equipment and supplement the general field of compressive rheology.This project aims to provide a general understanding of the consolidation, inparticular, how the network of fibres responds to the stresses experienced duringdewatering events. This begins with assessing the robustness of our previous modellingeffort, which extended the traditional multi-phase, deformable porous mediaframeworks, through the accommodation of a rate-dependent constitutive modelfor the solid effective stress (bulk viscosity). Robustness studies include: testinga large variety of pulp types, investigating low concentration dynamics, and usingindustrial and lab equipment. Results from these studies motivated extending studiesof the solid network’s response during high speed dewatering and shear stressapplication.For each study, a combined theoretical and experimental approach was undertakento formulate appropriate model equations, independently calibrate therequired material parameters, and collect experimental dynamic dewatering results.Model robustness for varying pulp suspensions at intermediate concentrationsutilized a Darcian flow cell to calibrate permeability, a uni-axial dewateringexperiment to determine their compressive yield stress, and dynamic dewateringexperiments at modest rates to characterize the suspension’s bulk viscosity. Thelow concentration investigation introduced experiments for calibrating permeabilityand compressive yield stress around the suspension’s gel point and utilizedgravitational drainage experiments to gauge bulk viscosity’s importance. In both investigations, the inclusion of a sizable bulk viscosity was necessary to effectivelyrepresent the dewatering behaviour. Dewatering dynamics in the Twin Roll press,collected at a pilot-scale facility, primarily highlighted the limitations of our previousmodelling efforts. Rebuilding of the uni-axial experiment and constitutivemodel for the solid effective stress was undertaken to capture the solid network’selastic response evident at elevated dewatering rates. Additionally, a unique apparatuswas developed to experimentally calibrate a pulp suspension’s significantshear yield stress at concentrations above traditional rheometer approaches.

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Abstract: The dynamics of free surface flow of yield stress fluid under gravity has been an open problem, both in theory and in computation. The contribution of this thesis comes in three parts. First, we report the results of computations for two dimensional dambreaks of viscoplastic fluid, focusing on the phenomenology of the collapse, the mode of initial failure, and the final shape of the slump. The volume-of-fluid method (VOF) is used to evolve the surface of the viscoplastic fluid, and its rheology is captured by either regularizing the viscosity or using an augmented-Lagrangian scheme. The interface is tracked by the Piecewise-Linear-Interface-Calculation (PLIC) scheme, modified in order to avoid resolution issues associated with the over-ridden finger of ambient fluid that results from the no slip condition and the resulting inability to move the contact line. We establish that the regularized and augmented-Lagrangian methods yield comparable results. The numerical results are compared with asymptotic theories valid for relatively shallow or vertically slender flow, with a series of previously reported experiments, and with predictions based on plasticity theory. Second, we report computations of the axisymmetric slump of viscoplastic fluid, with the PLIC scheme improved for mass conservation. The critical yield stress for failure is computed and bounded analytically using plasticity methods. The simulations are compared with experiments either taken from existing literature or performed using Carbopol. The comparison is satisfying for lower yield stresses; discrepancies for larger yield stresses suggest that the mechanism of release may affect the experiments. Finally, we report asymptotic analyses and numerical computations for surges of viscoplastic fluid down an incline with low inertia. The asymptotic theory applies for relatively shallow gravity currents. The anatomy of the surge consists of an upstream region that converges to a uniform sheet flow, and over which a truly rigid plug sheaths the surge. The plug breaks further downstream due to the build up of the extensional stress acting upon it, leaving instead a weakly yielded superficial layer, or pseudo-plug. Finally, the surge ends in a steep flow front that lies beyond the validity of shallow asymptotics.

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The current thesis investigates the controlled spreading of droplets of complex fluids. This thesis makes four primary scientific contributions. Firstly, we provide detailed theoretical analysis on spreading of yield stress fluids. We employ lubrication theory, asymptotic solutions, and numerical simulations to explain the dynamics and final static shape of a viscoplastic droplet on a solid horizontal surface. We show that the final radius of the droplet becomes smaller with increasing the yield stress. Secondly, we provide experimental data to verify our theoretical solutions. In our experiments, we first provide a method to eliminate the apparent slip of the yield stress fluid. The method uses a chemical modification of glass surfaces to generate permanent positive charges, resulting in a no-slip boundary condition. We directly observe the slip and no-slip of the Carbopol droplets, using a visualization method based on confocal microscopy. We then perform shadowgraphy experiments to measure the final radius of the droplets under different conditions such as extruding and impacting droplets. We compare the theoretical and experimental results and discuss the similarities and differences. Briefly, the asymptotic solutions overestimates the experimental results (most likely due to the assumption of a shallow layer), while numerical solutions are much closer to the experimental outcomes. Thirdly, we provide a comprehensive rheological characterization of a particular thermo-responsive fluid, Pluronic F127. We show that the aqueous solution of the polymer undergoes a sol(Newtonian)-gel(yield stress) transition upon heating. We further characterize the properties of the gel in detail. Finally, we show one can thermally trigger a thermo-responsive droplet to externally control the final shape of the droplet on a surface. In short, the final radius of the droplet can be controlled by heating the surface; for a given concentration, the larger the surface temperature, the smaller the final shape of a droplet. In the same part of the thesis, we introduce a novel experimental method based on optical coherence tomography to identify the solidified region inside a droplet.

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Despite a century of study, the macroscopic behaviour of quasistatic granular materials remains poorly understood. In particular, we lack a fundamental system of continuum equations, comparable to the Navier-Stokes equations for a Newtonian fluid. In this thesis, we derive continuum models for two-dimensional granular materials directly from the grain scale, using tools of discrete calculus, which we develop.To make this objective precise, we pose the canonical isostatic problem: a marginally stable granular material in the plane has 4 components of the stress tensor σ, but only 3 continuum equations in Newton’s laws ∇ ‧σ = 0 and σ = σT. At isostaticity, there is a missing stress-geometry equation, arising from Newton’s laws at the grain scale, which is not present in their conventional continuum form.We first show that a discrete potential ψ can be defined such that the stress tensor is written as σ = ∇ × ∇ × ψ, where the derivatives are given an exact meaning at the grain scale, and converge to their continuum counterpart in an appropriate limit. The introduction of ψ allows us to understand how force and torque balance couple neighbouring grains, and thus to understand where the stress-geometry equation is hidden.Using this formulation, we derive the missing stress-geometry equation ∆(F^ : ∇∇ψ) = 0, introducing a fabric tensor F^ which characterizes the geometry. We show that the equation imposes granularity in a literal sense, and that on a homo- geneous fabric, the equation reduces to a particular form of anisotropic elasticity.We then discuss the deformation of rigid granular materials, and derive the mean-field phase diagram for quasistatic flow. We find that isostatic states are fluid states, existing between solid and gaseous phases. The appearance of iso- staticity is linked to the saturation of steric exclusion and Coulomb inequalities.Finally, we present a model for the fluctuations of contact forces using tools of statistical mechanics. We find that force chains, the filamentary networks of con- tact forces ubiquitously observed in experiments, arise from an entropic instability which favours localization of contact forces.

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