Freezing colloidal suspensions: Periodic ice lenses and compaction

AbstractRecent directional solidification experiments with aqueous suspensions of alumina particles (Anderson & Worster, Langmuir, vol. 28 (48), 2012, pp. 16512–16523) motivate a model for freezing colloidal suspensions that builds upon a theoretical framework developed by Rempel et al. (J. Fluid Mech., vol. 498, 2004, pp. 227–244) in the context of freezing soils. Ice segregates from the suspension at slow freezing rates into discrete horizontal layers of particle-free ice, known as ice lenses. A portion of the particles is trapped between ice lenses, while the remainder are pushed ahead, forming a layer of fully compacted particles separated from the bulk suspension by a sharp compaction front. By dynamically modelling the compaction front, the growth kinetics of the ice lenses are fully coupled to the viscous flow through the evolving compacted layer. We examine the periodic states that develop at fixed freezing rates in a constant, uniform temperature gradient, and compare the results against experimental observations. Congruent with the experiments, three periodic regimes are identified. At low freezing rates, a regular periodic sequence of ice lenses is obtained; predictions for the compacted layer thickness and ice-lens characteristics as a function of freezing rate are consistent with experiments. At intermediate freezing rates, multiple modes of periodic ice lenses occur with a significantly diminished compacted layer. When the cohesion between the compacted particles is sufficiently strong, a sequence of mode-doubling bifurcations lead to chaos, which may explain the disordered ice lenses observed experimentally. Finally, beyond a critical freezing rate, the regime for sustained ice-lens growth breaks down. This breakdown is consistent with the emergence of a distinct regime of ice segregation found experimentally, which exhibits a periodic, banded structure that is qualitatively distinct from ice lenses.This is the author's accepted manuscript and will be under embargo until the 14th of April 2015. The final version is published by CUP in the Journal of Fluid Mechanics here:http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9382757&fileId=S002211201400500X

Lubricated viscous gravity currents

AbstractWe present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.We are grateful to Dr Mark Hallworth for his valuable help wit h our experiments and the technicians of the G.K. Batchelor Laboratory for assist ance with the setup of our experiments. KNK is supported by a NERC PhD studentship.This is the author accepted manuscript. The final version is available via CUP at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9553100&fileId=S0022112015000300

An experimental and theoretical study of the dynamics of grounding lines

AbstractWe present an experimental and theoretical study of a thin, viscous fluid layer that flows radially under gravity from a point source into a denser inviscid fluid layer of uniform depth above a rigid horizontal surface. Near the source, the viscous layer lies in full contact with the surface, forming a vertical-shear-dominated viscous gravity current. At a certain distance from the source, the layer detaches from the surface to form a floating current whose dynamics are controlled by the viscous stresses due to longitudinal extension. We describe the dynamics of the grounded and floating components using distinct thin-layer theories. Separating the grounded and floating regions is the freely moving line of detachment, or grounding line, whose evolution we model by balancing the horizontal forces between the two regions. Using numerical and asymptotic analysis, we calculate the evolution of the system from a self-similar form at early times towards a steady state at late times. We use our solutions to illustrate how three-dimensional stresses within marine ice sheets, such as that of West Antarctica, can lead to stabilization of the grounding line. To assess the validity of the assumptions underlying our model, we compare its predictions with data from a series of laboratory experiments.This work was supported by the Engineering and Physical Sciences Research Council.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2013.26

Freeze fracturing of elastic porous media: a mathematical model.

We present a mathematical model of the fracturing of water-saturated rocks and other porous materials in cold climates. Ice growing inside porous rocks causes large pressures to develop that can significantly damage the rock. We study the growth of ice inside a penny-shaped cavity in a water-saturated porous rock and the consequent fracturing of the medium. Premelting of the ice against the rock, which results in thin films of unfrozen water forming between the ice and the rock, is one of the dominant processes of rock fracturing. We find that the fracture toughness of the rock, the size of pre-existing faults and the undercooling of the environment are the main parameters determining the susceptibility of a medium to fracturing. We also explore the dependence of the growth rates on the permeability and elasticity of the medium. Thin and fast-fracturing cracks are found for many types of rocks. We consider how the growth rate can be limited by the existence of pore ice, which decreases the permeability of a medium, and propose an expression for the effective 'frozen' permeability.This research was supported by a PhD studentship from the EPSRC.This is the final version. It first appeared at http://dx.doi.org/10.1098/rspa.2014.074

The formation of grounding zone wedges: Theory and experiments

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Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear

A novel viscous fingering instability, involving a less viscous fluid intruding underneath a current of more viscous fluid, was recently observed in the experiments of Kowal & Worster (J. Fluid Mech., vol. 766, 2015, pp. 626-655). We examine the origin of the instability by asking whether the instability is an internal instability, arising from internal dynamics, or a frontal instability, arising from viscous intrusion. We find it is the latter and characterise the instability criterion in terms of viscosity difference or, equivalently, the jump in hydrostatic pressure gradient at the intrusion front. The mechanism of this instability is similar to, but contrasts with, the Saffman-Taylor instability, which occurs as a result of a jump in dynamic pressure gradient across the intrusion front. We focus on the limit in which the two viscous fluids are of equal density, in which a frontal singularity, arising at the intrusion, or lubrication, front, becomes a jump discontinuity, and perform a local analysis in an inner region near the lubrication front, which we match asymptotically to the far field. We also investigate the large-wavenumber stabilisation by transverse shear stresses in two dynamical regimes: A regime in which the wavelength of the perturbations is much smaller than the thickness of both layers of fluid, in which case the flow of the perturbations is resisted dominantly by horizontal shear stresses; and an intermediate regime, in which both vertical and horizontal shear stresses are important.K.N.K. acknowledges the support of a NERC PhD studentship

Instability of radially spreading extensional flows. Part 2. Theoretical analysis

The interface of a strain-rate-softening fluid that displaces a low-viscosity fluid in a circular geometry with negligible drag can develop finger-like patterns separated by regions in which the fluid appears to be torn apart. Such patterns were observed and explored experimentally in Part 1 using polymeric solutions. They do not occur when the viscosity of the displacing fluid is constant, or when the displacing fluid has no-slip conditions along its boundaries. We investigate theoretically the formation of tongues at the interface of an axisymmetric initial state. We show that finger-like patterns can emerge when circular interfaces of strain-rate-softening fluids displace low-viscosity fluids between stress-free boundaries. The instability, which is fundamentally different from the classical Saffman–Taylor viscous fingering, is driven by the tension that builds up along the circular front of the propagating fluid. That destabilising tension is a geometrical consequence and is present independently of the nonlinear properties of the fluid. Shear stresses stabilise the growth either along extended circumferential streamlines or through a street of vortices. However, such stabilising processes become weaker, thereby allowing the instability to develop, the more strain-rate-softening the fluid is. The theoretical model that we present predicts the main experimental observations made in Part 1. In particular, the patterns we predict using linear-stability theory are consistent with the strongly nonlinear experimental patterns. Our model depends on a single dimensionless number representing the power-law exponent, which implies that the instability we describe could arise in any extensional flow of strain-rate-softening material, ranging from suspensions that rupture in squeeze experiments to rifts formed in ice shelves.Israel Science Foundation (grant No. 1368/16

A physically based parameterization of gravity drainage for sea-ice modeling

We incorporate a physically derived parameterization of gravity drainage, in terms of a convective upwelling velocity, into a one-dimensional, thermodynamic sea- ice model of the kind currently used in coupled climate models. Our parameterization uses a local Rayleigh number to represent the important feedback between ice salinity, porosity, permeability and desalination rate. It allows us to determine salt fluxes from sea ice and the corresponding evolution of the bulk salinity of the ice, in contrast to older, established models that prescribe the ice salinity. This improves the predictive power of climate models in terms of buoyancy fluxes to the polar oceans, and also the thermal properties of sea ice, which depend on its salinity. We analyze the behaviour of exist- ing fixed-salinity models, elucidate the physics by which changing salinity affects ice growth and compare against our dynamic-salinity model, both in terms of laboratory experiments and also deep-ocean calculations. These comparisons explain why the direct effect of ice salinity on growth is relatively small (though not always negligible, and sometimes dif- ferent from previous studies), and also highlight substantial differences in the qualita- tive pattern and quantitative magnitude of salt fluxes into the polar oceans. Our study is particularly relevant to growing first-year ice, when gravity drainage is the dominant mechanism by which ice desalinates. We expect that our dynamic model, which respects the underlying physics of brine drainage, should be more robust to changes in polar cli- mate and more responsive to rapid changes in oceanic and atmospheric forcing.This is the accepted manuscript. The final version is available from Wiley/American Geophysical Union at http://onlinelibrary.wiley.com/doi/10.1002/2013JC009296/abstract

Dynamics of laterally confined marine ice sheets

We present an experimental and theoretical study of the dynamics of laterally confined marine ice sheets in the natural limit in which the long, narrow channel into which they flow is wider than the depth of the ice. A marine ice sheet comprises a grounded ice sheet in contact with bedrock that floats away from the bedrock at a ‘grounding line’ to form a floating ice shelf. We model the grounded ice sheet as a viscous gravity current resisted dominantly by vertical shear stresses owing to the no-slip boundary condition applied at the bedrock. We model the ice shelf as a floating viscous current resisted dominantly by horizontal shear stresses owing to no-slip boundary conditions applied at the sidewalls of the channel. The two shear-dominated regions are coupled by jump conditions relating force and fluid flux across a short transition region downstream of the grounding line. We find that the influence of the stresses within the transition region becomes negligible at long times and we model the transition region as a singular interface across which the ice thickness and mass flux can be discontinuous. The confined shelf buttresses the sheet, causing the grounding line to advance more than it would otherwise. In the case that the sheet flows on a base of uniform slope, we find asymptotically that the grounding line advances indefinitely as $t^{1/3}$, where $t$ is time. This contrasts with the two-dimensional counterpart, for which the shelf provides no buttressing and the grounding line reaches a steady state (Robison, J. Fluid Mech., vol. 648, 2010, pp. 363–380).We would like to thank Dr M. Hallworth for valuable help with running the experiments and the technicians of the DAMTP G. K. Batchelor Laboratory for help with the set-up of the experimental apparatus. K.N.K. is supported by a NERC PhD studentship. The experimental data are available as supplementary material.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.3

Permeability measurements using oscillatory flows

© 2020, Springer-Verlag GmbH Germany, part of Springer Nature. Abstract: We describe a versatile apparatus for measuring the permeability of porous materials using oscillatory flows. The permeabilities are measured by an original spectral analysis of the pressure and fluid-displacement signals. The measurements are shown to be in very good agreement with classical drainage experiments performed on the same device. Our apparatus and methodology will be useful if small fluid displacements are required, for example in reactive porous media. Graphic abstract: [Figure not available: see fulltext.].Royal Society University Research Fellowship British Antartic Survey Foundation Isaac Newton Trust