# Neil Balmforth

#### Relevant Degree Programs

#### Affiliations to Research Centres, Institutes & Clusters

## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

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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##### Master's Student Supervision (2010 - 2020)

Double diffusive convection is a naturally occurring phenomenon playing important roles in geophysical, astrophysical, and oceanographic events alike. Herein, it is the transfer of heat by fluid movement driven by the differing rates of diffusion of temperature and salinity, developing into one of 2 regimes: diffusive convection or salt fingering. We consider this problem in a porous medium, relevant in situations regarding permafrost, magma, and soils amongst others. We begin by performing and comparing linear and nonlinear stability analyses near the onset of instability, as in existing work. We ultimately find that the two methods result in the same bounds for the onset of instability for salt fingering and steady diffusive convection, and so we conclude there are no subcritical cases. This is further confirmed in the third section, wherein we conduct a weakly nonlinear stability analysis using asymptotic expansions. In both the diffusive convection and the salt fingering cases, the amplitude equations obtained indicate that supercritical instabilities occur. In the case of oscillatory diffusive convection, the regime of criticality depends on the relative size of the density ratio to the Lewis number. Extending previous work by considering a porous medium, we consider modes creating the fastest growing fingers, resulting in fingers with a smaller horizontal/vertical aspect ratio. These fingers are studied first in a vertically unbounded domain, then in a bounded one. We find evolution equations for both cases, and plot the resulting steady-state solution of the temperature amplitude of the latter. We find that in the zero limit of the horizontal/vertical aspect ratio, the temperature amplitudes of the steady solutions converge in both the salt-heat and sugar-salt configurations.

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The purpose of this thesis is to study the problem when a microorganismswims very close to a shaped boundary. In this problem, we model the swimmerto be a two-dimensional, infinite periodic waving sheet. For simplicity,we only consider the case where the fluid between the swimmer and thewashboard is Newtonian and incompressible. We assume that the swimmerpropagates waves along its body and propels itself in the opposite direction.We consider two cases in our swimming sheet problem and the lubricationapproximation is applied for both cases. In the first case, the swimmer has aknown fixed shape. Various values of wavenumber, amplitude of the restoringforce and amplitude of the topography were considered. We found theinstantaneous swimming speed behaved quite differently as the wavenumberwas varied. The direction of the swimmer was also found to depend on theamplitude of the restoring force. We also found some impact of the topographicamplitude on the relationship between average swimming speed andthe wavenumber. We extended the cosine wave shaped washboard to be amore general shape and observed how it affected the swimming behaviour.In the second case, the swimmer is assumed to be elastic. We were interestedto see how different values of wavenumber, stiffness and amplitude ofthe restoring force could change the swimming behaviour. With normalized stiffness and wavenumber, we found the swimmer remained in a periodicstate with small forcing amplitude. While the swimmer reached a steadystate with unit swimming speed for high forcing amplitude. However, forother values of stiffness and wavenumber, we found the swimmer's swimmingbehaviour was very different.

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We have studied the propagation of 2D unit block of viscoplastic fluid of Bingham type over a horizontal plane, underneath another Newtonian fluid. We numerically simulate the dynamics of a two-layer fluid in a rectangle domain, using the volume-of-fluid method to deal with the evolution of the interface, and regularization scheme of the constitutive law, which replaces unyielded plugs with very viscous flow. We explore the final shape of the flow for varying yield stress, comparing the numerical results with the predictions of the asymptotic theory, a plasticity model based on slipline theory, and other past results.Numerical difficulties with the moving contact lines are encountered during the numerical simulation. A slip boundary condition is used to address this issue, the validity of which should be further investigated.

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Parametric subharmonic instability (PSI) is a nonlinear interaction between a resonanttriad of waves, in which energy is transferred from low wavenumber, highfrequency modes to high wavenumber, low frequency modes. In the ocean, PSI isthought to be one of the mechanisms responsible for transferring energy from M₂internal tides (internal gravity waves with diurnal tidal frequency) to near-inertialwaves (internal gravity waves with frequency equal to the local Coriolis frequency)near the latitude of 28.9 degrees. Due to their small vertical scale, near-inertial wavesgenerate large vertical shear and are much more efficient than M₂ internal tides atgenerating shear instability needed for vertical mixing, which is required to maintainocean stratification.The earlier estimate of the time-scale for the instability is an order of magnitudelarger than the time-scale observed in a recent numerical simulation (MacKinnon and Winters) (MW). An analytical model was developed by (Young et al. 2008) (YTB), and their findings agreed with the MW estimation; however as YTB assumed a constant Coriolis force, the model cannot explain the intensificaiton of PSI near 28.9 degrees as observed in the model of MW; in addition, thenear-ineartial waves can propagate a significant distance away from the latitude of28.9 degrees.This thesis extends the YTB model by allowing for a linearlyvarying Coriolis parameter (β-effect) as well as eddy diffusion. A linear stability analysis shows that the near-inertial wave field isunstable to perturbations. We show that the β-effect results in a shortening inwave length as the near-inertial waves propagate south; horizontal eddy diffusionis therefore enhanced to the south, and limits the meridional extent of PSI. Thehorizontal diffusion also affects the growth rate of the instability. A surprisingresult is that as the horizontal diffusion vanishes, the system becomes stable; thiscan be demonstrated both analytically and numerically.Mathematically, the β-effect renders the spatial differential operator nonnormal, which is characterized with the aid of pseudo-spectra. Our results suggest the possibility of large amplitude transient growth in near-inertialwaves in regimes that are asymptotically stable to perturbations.

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