Instability of extensional flows

The interface of a fluid that displaces another fluid in a quasi- two dimensional geometry can develop fingering patterns, which are common to a wide range of natural systems. It is believed that such interfaces remain stable, having planer or circular shapes, when the displacing fluid is more viscous (less mobile). However, some systems, such as ice sheets or squeezed pastes, develop fingering patterns in spite of having a more viscous displacing fluid.
We show that a more viscous displacing fluid can develop fingering patterns if that fluid is nonlinear and is discharged axisymmetrically along frictionless boundaries. Unlike the classical viscous fingering, the patterns we observe resemble tears and their characteristic wavelength progressively grows.

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Viscous flows on porous substrate

We study the propagation of viscous gravity currents over a thin porous substrate with finite capillary entry pressure. Near the origin, where the current is deep, propagation of the current coincides with leakage through the substrate. Near the nose of the current, where the current is thin and the fluid pressure is below the capillary entry pressure, drainage is absent. Consequently the flow can be characterised by the evolution of drainage and fluid fronts. We analyze this flow using numerical and analytical techniques combined with laboratory-scale experiments. At early times, we find that the position of both fronts evolve as $t^{1/2}$, similar to an axisymmetric gravity current on an impermeable substrate. At later times, the growing effect of drainage inhibits spreading, causing the drainage front to logarithmically approach a steady position. In contrast, the asymptotic propagation of the fluid front is quasi-self-similar, having identical structure to the solution of gravity currents on an impermeable substrate, only with slowly varying fluid flux. We benchmark these theoretical results with laboratory experiments that are consistent with our modeling approximation, but that also highlight the detailed dynamics of drainage inhibited by finite capillary pressure.

Related publications:

  • Sayag, R. and J. A. Neufeld, Propagation of viscous currents on a porous substrate with finite capillary entry pressure, J. Fluid Mech., 801, 65-90, 2016. PDF VIDEO

Floating extensional flows

Ice sheets spread like viscous gravity currents into the surrounding oceans. As they spread they thin and can detach from the bed, owing to the hydrostatic pressure of the ocean, and float as ice shelves. The detachment position is a contact line that separates a grounded, shear-dominated flow and a floating, extension-dominated flow. Differences between these two flow regimes can be intensified because of the complex rheology of ice. In particular, floating shelves can fracture and potentially shatter, which may ultimately affect the stability of ice sheets and lead to a catastrophic rise in sea level. We explore fundamental aspects of this problem using laboratory experiments. We model the flow of ice sheets as viscous gravity currents that propagate in a circular geometry, and focus on the influence of the fluid rheology on the flow pattern. In particular, we study the Newtonian limit using golden syrup, and the non-Newtonian response using an aqueous suspension of Xanthan gum.

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Axisymmetric gravity currents of power-law fluids

We analyse axisymmetric gravity currents of power-law fluids theoretically and experimentally. We use aqueous suspensions of Xanthan gum in laboratory experiments of constant-volume and constant-flux release to resolve the rheological parameters of the fluid, which we then compare with measurements made using a strain-controlled rheometer. We find that the constant-volume release of highly shear-thinning fluids involves an early-time evolution dominated by inertia, and non-convex free surfaces that make the application of similarity solutions of the late-time viscously dominated evolution inefficient at resolving material properties. In contrast, constant-flux release of the same fluids can be viscously dominated and consistent with the self-similar solution from early in the evolution, which makes it a more useful method for measuring rheological parameters.

Related publications:

  • Sayag, R., and M. G. Worster, Axisymmetric gravity currents of power-law fuids over a rigid horizontal surface , J. Fluid Mech., 716, R5, 2013. PDF sup VIDEO

Elastic response of a grounded ice sheet coupled to a floating ice shelf

An ice sheet that spreads into an ocean is forced to bend owing to its buoyancy, and detaches from the bedrock to form a floating ice shelf. The location of the transition between the grounded sheet and the floating shelf, defined as the grounding-line, behaves as a free boundary. We develop a model of an elastic grounded sheet resting on a deformable elastic bed and coupled to an elastic floating shelf. We find that the grounding line position is determined by the geometry of the bed and the bending-buoyancy lengthscale of the system. These two contributions depend on the reaction modulus of the bed in opposite ways. We show that the structure of the floating shelf depends on the bending-buoyancy lengthscale only, allowing us to calculate the bending stiffness of the elastic sheet independently of the properties of the bed. Relations between the structure of the floating shelf and the grounding line position are also developed. Our theoretical predictions agree with laboratory experiments made using thick elastic sheets and a dense salt solution. Our findings may provide new insights into the dynamics near grounding lines, as well as methods to infer the bending stiffness of ice sheets and the grounding line position from satellite altimetery that can be applied to elastic sheets in general.

Related publications:

  • Sayag, R., and M. G. Worster, Elastic response of a grounded ice sheet coupled to a floating ice shelf, Phys. Rev. E, 84, 3, 2011a PDF

  • Sayag, R., and M. G. Worster, Elastic Dynamics and tidal migration of ice-sheet grounding lines modify sub-glacial lubrication and melting, Geophys. Rev. Lett., 40, 2013, PDF

A mechanism for the spatiotemporal variability of ice streams.

Ice streams are regions of fast flowing glacier ice that transport a significant portion of the total ice flux from present ice sheets. The flow pattern of ice streams can vary both temporally and spatially, in particular, ice streams can become stagnant and change their path. We present a mechanism that accounts for such spatiotemporal variability of ice streams. The major element of this mechanism is a triple valued relation between the ice velocity and the shear stress at the base of the ice.
A triple valued sliding law was previously suggested to explain the surging behaviour of glaciers (Fowler and Johnson, 1996). We show that such a sliding law can be heuristically motivated by the transverse velocity profile of an ice-stream, and use stability analysis to explain the adjustment of the lateral shear margins to changes in the driving stress.
We then use a 2D numerical model to demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. Specifically we find stable and time oscillatory solutions of rapid stream-like regions within slow ice-sheet flow separated by narrow shear margins. The width of those ice streams is related to the dimensions of the mass source distribution in the onset regions if the bed is flat, or in the presence of a spatially uniform mass source, to the length scale imposed by bed topography at the onset regions. This spatiotemporal behaviour is sustained whether the ice rheology is Newtonian or power-law viscous. We present a quantitative relation between the parameters of the ice rheology and the width of the ice-streams shear-margins and show how these parameters can affect the minimum width of an ice stream before it shuts down. We then study the interaction of two ice streams that share a common mass source and find complex and asymmetric spatiotemporal patterns that resemble patterns that are observed in Siple coast ice streams, Antarctica.

Related publications:

  • Sayag, R., and E. Tziperman, Interaction and variability of ice streams under a triple-valued sliding law and non-Newtonian rheology, J. Geophys. Res., 116, F01009, doi:10.1029/2010JF001839., 2011 PDF

  • Sayag, R., and E. Tziperman, Spatiotemporal dynamics of ice streams due to a multivalued sliding law, Journal of Fluid Mechanics, 640, 483-505, 2009 [Movies] PDF

Ice Streams and shear flow instability

A significant portion of the ice discharge in ice sheets is drained through ice streams, with subglacial sediment (till) acting as a lubricant. The known importance of horizontal friction in shear margins to ice stream dynamics suggests a critical role of longitudinal stresses.
The effects of subglacial till and longitudinal stresses on the stability of an ice sheet flow are studied by linear stability analysis of an idealized ice-till model in two horizontal dimensions. A power law-viscous constitutive relation is used, explicitly including longitudinal shear stresses. The till, which has compressible-viscous rheology, affects the ice flow through bottom friction.
We examine the possibility that pure ice streams develop via a spontaneous instability of ice flow. We demonstrate that this model can be made intrinsically unstable for a seemingly relevant range of parameters, and that the wavelengths and growth rates that correspond to the most unstable modes are in rough agreement with observed pure ice streams. Instabilities occur due to basal friction and melt water production at the ice-till interface. The most unstable wavelength arise due to selective dissipation of both short and long perturbation scales. Longitudinal stress gradients stabilize short transverse wavelengths while Nye diffusion stabilizes long transverse
wavelengths. The selection of an intermediate unstable wave length occurs, however, only for certain parameter and perturbation structure choices. These results do not change qualitatively for a Newtonian ice flow law, indicating no significant role to shear thinning, although this may very well be due to the restrictive assumptions of the model and analysis.

Related publications:

  • Sayag, R., and E. Tziperman, Spontaneous generation of pure ice-streams via flow instability: Role of longitudinal shear stresses and subglacial till, J. Geophys. Res., 113, B05411, 2008 PDF

Last update: Jan, 2014