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Dense granular flows can spontaneously self-channelise by forming a pair of parallel-sided static levees on either side of a central flowing channel. This process prevents lateral spreading and maintains the flow thickness, and hence mobility, enabling the grains to run out considerably further than a spreading flow on shallow slopes. Since levees commonly form in hazardous geophysical mass flows, such as snow avalanches, debris flows, lahars and pyroclastic flows, this has important implications for risk management in mountainous and volcanic regions. In this paper an avalanche model that incorporates frictional hysteresis, as well as depth-averaged viscous terms derived from the
-rheology, is used to quantitatively model self-channelisation and levee formation. The viscous terms are crucial for determining a smoothly varying steady-state velocity profile across the flowing channel, which has the important property that it does not exert any shear stresses at the levee–channel interfaces. For a fixed mass flux, the resulting boundary value problem for the velocity profile also uniquely determines the width and height of the channel, and the predictions are in very good agreement with existing experimental data for both spherical and angular particles. It is also shown that in the absence of viscous (second-order gradient) terms, the problem degenerates, to produce plug flow in the channel with two frictionless contact discontinuities at the levee–channel margins. Such solutions are not observed in experiments. Moreover, the steady-state inviscid problem lacks a thickness or width selection mechanism and consequently there is no unique solution. The viscous theory is therefore a significant step forward. Fully time-dependent numerical simulations to the viscous model are able to quantitatively capture the process in which the flow self-channelises and show how the levees are initially emplaced behind the flow head. Both experiments and numerical simulations show that the height and width of the channel are not necessarily fixed by these initial values, but respond to changes in the supplied mass flux, allowing narrowing and widening of the channel long after the initial front has passed by. In addition, below a critical mass flux the steady-state solutions become unstable and time-dependent numerical simulations are able to capture the transition to periodic erosion–deposition waves observed in experiments.
Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness
on a rough slope inclined at an angle
to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle
, or the thickness is increased to
. This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to
. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness
) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness
it produces a deposit close to
for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to
for all initial thicknesses. This insensitivity to the initial conditions implies that
is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the
curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine
is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth
Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the
-rheology, which postulates that the bulk friction coefficient
(i.e. the ratio of the shear stress to the pressure) and the solids volume fraction
are functions of the inertial number
only. Although the
-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the
-rheology that does not suffer from such defects is proposed. In the framework of compressible
-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established
relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.
When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwards et al. (J. Fluid. Mech., vol. 823, 2017, pp. 278–315, J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.
Absolute K-shell ionization cross sections were measured for Ti, Co, Ge, Rb, and Sn for incident oxygen ions from 16-44 MeV. The x-rays were measured with a high resolution Si(Li) detector (166 eV at 5.9 keV). All of the data represents cross section measurements for thin targets. The measured cross sections for these elements are compared to the theoretical predictions of the Binary Encounter Approximation (BEA). Kα/Kβ ratios and energy shifts were also extracted from the data. The experimental data are compared to measured cross sections for other elements to give an overview of the systematics for oxygen ion induced x-ray production cross sections in this energy range. Some comment will also be given in regard to the use of oxygen ions to measure the parameters associated with ion implanted semiconductors.
Rapid shallow granular flows over inclined planes are often seen in nature in the form of avalanches, landslides and pyroclastic flows. In these situations the flow develops an inversely graded (large at the top) particle-size distribution perpendicular to the plane. As the surface velocity of such flows is larger than the mean velocity, the larger material is transported to the flow front. This causes size segregation in the downstream direction, resulting in a flow front composed of large particles. Since the large particles are often more frictional than the small, the mobility of the flow front is reduced, resulting in a so-called bulbous head. This study focuses on the formation and evolution of this bulbous head, which we show to emerge in both a depth-averaged continuum framework and discrete particle simulations. Furthermore, our numerical solutions of the continuum model converge to a travelling wave solution, which allows for a very efficient computation of the long-time behaviour of the flow. We use small-scale periodic discrete particle simulations to calibrate (close) our continuum framework, and validate the simple one-dimensional (1-D) model with full-scale 3-D discrete particle simulations. The comparison shows that there are conditions under which the model works surprisingly well given the strong approximations made; for example, instantaneous vertical segregation.
The spatial-intensity profile of light reflected during the interaction of an intense laser pulse with a microstructured target is investigated experimentally and the potential to apply this as a diagnostic of the interaction physics is explored numerically. Diffraction and speckle patterns are measured in the specularly reflected light in the cases of targets with regular groove and needle-like structures, respectively, highlighting the potential to use this as a diagnostic of the evolving plasma surface. It is shown, via ray-tracing and numerical modelling, that for a laser focal spot diameter smaller than the periodicity of the target structure, the reflected light patterns can potentially be used to diagnose the degree of plasma expansion, and by extension the local plasma temperature, at the focus of the intense laser light. The reflected patterns could also be used to diagnose the size of the laser focal spot during a high-intensity interaction when using a regular structure with known spacing.
Small perturbations to a steady uniform granular chute flow can grow as the material moves downslope and develop into a series of surface waves that travel faster than the bulk flow. This roll wave instability has important implications for the mitigation of hazards due to geophysical mass flows, such as snow avalanches, debris flows and landslides, because the resulting waves tend to merge and become much deeper and more destructive than the uniform flow from which they form. Natural flows are usually highly polydisperse and their dynamics is significantly complicated by the particle size segregation that occurs within them. This study investigates the kinematics of such flows theoretically and through small-scale experiments that use a mixture of large and small glass spheres. It is shown that large particles, which segregate to the surface of the flow, are always concentrated near the crests of roll waves. There are different mechanisms for this depending on the relative speed of the waves, compared to the speed of particles at the free surface, as well as on the particle concentration. If all particles at the surface travel more slowly than the waves, the large particles become concentrated as the shock-like wavefronts pass them. This is due to a concertina-like effect in the frame of the moving wave, in which large particles move slowly backwards through the crest, but travel quickly in the troughs between the crests. If, instead, some particles on the surface travel more quickly than the wave and some move slower, then, at low concentrations, large particles can move towards the wave crest from both the forward and rearward sides. This results in isolated regions of large particles that are trapped at the crest of each wave, separated by regions where the flow is thinner and free of large particles. There is also a third regime arising when all surface particles travel faster than the waves, which has large particles present everywhere but with a sharp increase in their concentration towards the wave fronts. In all cases, the significantly enhanced large particle concentration at wave crests means that such flows in nature can be especially destructive and thus particularly hazardous.
The full theory of polarized SiO maser emission from the near-circumstellar environment of Asymptotic Giant Branch stars has been the subject of debate, with theories ranging from classical Zeeman origins to predominantly non-Zeeman anisotropic excitation or propagation effects. Features with an internal electric vector position angle (EVPA) rotation of ∼π/2 offer unique constraints on theoretical models. In this work, results are presented for one such feature that persisted across five epochs of SiO ν = 1, J = 1 − 0 VLBA observations of TX Cam. We examine the fit to the predicted dependence of linear polarization and EVPA on angle (θ) between the line of sight and the magnetic field against theoretical models. We also present results on the dependence of mc on θ and their theoretical implications. Finally, we discuss potential causes of the observed differences, and continuing work.
In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the
-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number
. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. In this paper changes to the functional form of the
curve are considered, in order to maximise the range of well-posed inertial numbers, while preserving the overall structure of the equations. It is found that when the inertial number is low there exist curves for which the equations are guaranteed to be well posed. However when the inertial number is very large the equations are found to be ill posed regardless of the functional dependence of
. A new
curve, which is inspired by the analysis of the governing equations and by experimental data, is proposed here. In order to test this regularised rheology, transient granular flows on inclined planes are studied. It is found that simulations of flows, which show signs of ill posedness with unregularised models, are numerically stable and match key experimental observations when the regularised model is used. This paper details two-dimensional transient computations of decelerating flows where the inertial number tends to zero, high-speed flows that have large inertial numbers, and flows which develop into granular rollwaves. This is the first time that granular rollwaves have been simulated in two dimensions, which represents a major step towards the simulation of other complex granular flows.
Snow avalanches are typically initiated on marginally stable slopes with a surface layer of fresh snow that may easily be incorporated into them. The erosion of snow at the front is fundamental to the dynamics and growth of snow avalanches and they may rapidly bulk up, making them much more destructive than the initial release. Snow may also deposit at the rear, base and sides of the flow and the net balance of erosion and deposition determines whether an avalanche grows or decays. In this paper, small-scale analogue experiments are performed on a rough inclined plane with a static erodible layer of carborundum grains. The static layer is prepared by slowly closing down a flow from a hopper at the top of the slope. This leaves behind a uniform-depth layer of thickness
at a given slope inclination. Due to the hysteresis of the rough bed friction law, this layer can then be inclined to higher angles provided that the thickness does not exceed
, which is the maximum depth that can be held static on a rough bed. An avalanche is then initiated on top of the static layer by releasing a fixed volume of carborundum grains. Dependent on the slope inclination and the depth of the static layer three different behaviours are observed. For initial deposit depths above
, the avalanche rapidly grows in size by progressively entraining more and more grains at the front and sides, and depositing relatively few particles at the base and tail. This leaves behind a trough eroded to a depth below the initial deposit surface and whose maximal areal extent has a triangular shape. Conversely, a release on a shallower slope, with a deposit of thickness
, leads to net deposition. This time the avalanche leaves behind a levee-flanked channel, the floor of which lies above the level of the initial deposit and narrows downstream. It is also possible to generate avalanches that have a perfect balance between net erosion and deposition. These avalanches propagate perfectly steadily downslope, leaving a constant-width trail with levees flanking a shallow trough cut slightly lower than the initial deposit surface. The cross-section of the trail therefore represents an exact redistribution of the mass reworked from the initial static layer. Granular flow problems involving erosion and deposition are notoriously difficult, because there is no accepted method of modelling the phase transition between static and moving particles. Remarkably, it is shown in this paper that by combining Pouliquen & Forterre’s (J. Fluid Mech., vol. 453, 2002, pp. 133–151) extended friction law with the depth-averaged
-rheology of Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–544) it is possible to develop a two-dimensional shallow-water-like avalanche model that qualitatively captures all of the experimentally observed behaviour. Furthermore, the computed wavespeed, wave peak height and stationary layer thickness, as well as the distance travelled by decaying avalanches, are all in good quantitative agreement with the experiments. This model is therefore likely to have important practical implications for modelling the initiation, growth and decay of snow avalanches for hazard assessment and risk mitigation.
Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.
Geophysical granular flows, such as landslides, pyroclastic flows and snow avalanches, consist of particles with varying surface roughnesses or shapes that have a tendency to segregate during flow due to size differences. Such segregation leads to the formation of regions with different frictional properties, which in turn can feed back on the bulk flow. This paper introduces a well-posed depth-averaged model for these segregation-mobility feedback effects. The full segregation equation for dense granular flows is integrated through the avalanche thickness by assuming inversely graded layers with large particles above fines, and a Bagnold shear profile. The resulting large particle transport equation is then coupled to depth-averaged equations for conservation of mass and momentum, with the feedback arising through a basal friction law that is composition dependent, implying greater friction where there are more large particles. The new system of equations includes viscous terms in the momentum balance, which are derived from the
-rheology for dense granular flows and represent a singular perturbation to previous models. Linear stability calculations of the steady uniform base state demonstrate the significance of these higher-order terms, which ensure that, unlike the inviscid equations, the growth rates remain bounded everywhere. The new system is therefore mathematically well posed. Two-dimensional simulations of bidisperse material propagating down an inclined plane show the development of an unstable large-rich flow front, which subsequently breaks into a series of finger-like structures, each bounded by coarse-grained lateral levees. The key properties of the fingers are independent of the grid resolution and are controlled by the physical viscosity. This process of segregation-induced finger formation is observed in laboratory experiments, and numerical computations are in qualitative agreement.
The movements of fluid–fluid interfaces and the common curve are an important aspect of two-fluid-phase flow through porous media. The focus of this work is to develop, apply and evaluate methods to simulate two-fluid-phase flow in porous medium systems at the microscale and to demonstrate how these results can be used to support evolving macroscale models. Of particular concern is the problem of spurious velocities that confound the accurate representation of interfacial dynamics in such systems. To circumvent this problem, a combined level-set and lattice-Boltzmann method is advanced to simulate and track the dynamics of the fluid–fluid interface and of the common curve during simulations of two-fluid-phase flow in porous media. We demonstrate that the interface and common curve velocities can be determined accurately, even when spurious currents are generated in the vicinity of interfaces. Static and dynamic contact angles are computed and shown to agree with existing slip models. A resolution study is presented for dynamic drainage and imbibition in a sphere pack, demonstrating the sensitivity of averaged quantities to resolution.
During 1990 we surveyed the southern sky using a multi-beam receiver at frequencies of 4850 and 843 MHz. The half-power beamwidths were 4 and 25 arcmin respectively. The finished surveys cover the declination range between +10 and −90 degrees declination, essentially complete in right ascension, an area of 7.30 steradians. Preliminary analysis of the 4850 MHz data indicates that we will achieve a five sigma flux density limit of about 30 mJy. We estimate that we will find between 80 000 and 90 000 new sources above this limit. This is a revised version of the paper presented at the Regional Meeting by the first four authors; the surveys now have been completed.
The subsurface exploration of other planetary bodies can be used to unravel their geological history and assess their habitability. On Mars in particular, present-day habitable conditions may be restricted to the subsurface. Using a deep subsurface mine, we carried out a program of extraterrestrial analog research – MINe Analog Research (MINAR). MINAR aims to carry out the scientific study of the deep subsurface and test instrumentation designed for planetary surface exploration by investigating deep subsurface geology, whilst establishing the potential this technology has to be transferred into the mining industry. An integrated multi-instrument suite was used to investigate samples of representative evaporite minerals from a subsurface Permian evaporite sequence, in particular to assess mineral and elemental variations which provide small-scale regions of enhanced habitability. The instruments used were the Panoramic Camera emulator, Close-Up Imager, Raman spectrometer, Small Planetary Linear Impulse Tool, Ultrasonic drill and handheld X-ray diffraction (XRD). We present science results from the analog research and show that these instruments can be used to investigate in situ the geological context and mineralogical variations of a deep subsurface environment, and thus habitability, from millimetre to metre scales. We also show that these instruments are complementary. For example, the identification of primary evaporite minerals such as NaCl and KCl, which are difficult to detect by portable Raman spectrometers, can be accomplished with XRD. By contrast, Raman is highly effective at locating and detecting mineral inclusions in primary evaporite minerals. MINAR demonstrates the effective use of a deep subsurface environment for planetary instrument development, understanding the habitability of extreme deep subsurface environments on Earth and other planetary bodies, and advancing the use of space technology in economic mining.
Debris and pyroclastic flows often have bouldery flow fronts, which act as a natural dam resisting further advance. Counter intuitively, these resistive fronts can lead to enhanced run-out, because they can be shouldered aside to form static levees that self-channelise the flow. At the heart of this behaviour is the inherent process of size segregation, with different sized particles readily separating into distinct vertical layers through a combination of kinetic sieving and squeeze expulsion. The result is an upward coarsening of the size distribution with the largest grains collecting at the top of the flow, where the flow velocity is greatest, allowing them to be preferentially transported to the front. Here, the large grains may be overrun, resegregated towards the surface and recirculated before being shouldered aside into lateral levees. A key element of this recirculation mechanism is the formation of a breaking size-segregation wave, which allows large particles that have been overrun to rise up into the faster moving parts of the flow as small particles are sheared over the top. Observations from experiments and discrete particle simulations in a moving-bed flume indicate that, whilst most large particles recirculate quickly at the front, a few recirculate very slowly through regions of many small particles at the rear. This behaviour is modelled in this paper using asymmetric segregation flux functions. Exact non-diffuse solutions are derived for the steady wave structure using the method of characteristics with a cubic segregation flux. Three different structures emerge, dependent on the degree of asymmetry and the non-convexity of the segregation flux function. In particular, a novel ‘lens-tail’ solution is found for segregation fluxes that have a large amount of non-convexity, with an additional expansion fan and compression wave forming a ‘tail’ upstream of the ‘lens’ region. Analysis of exact solutions for the particle motion shows that the large particle motion through the ‘lens-tail’ is fundamentally different to the classical ‘lens’ solutions. A few large particles starting near the bottom of the breaking wave pass through the ‘tail’, where they travel in a region of many small particles with a very small vertical velocity, and take significantly longer to recirculate.
Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged
-rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional
-rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.
In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient
. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent
-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number
in the friction coefficient
. In this paper it is shown that the
-rheology is well-posed for intermediate values of
, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.