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During the last fifteen years there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the $\mu (I)$-rheology (where $\mu$ is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This paper details theoretical and numerical frameworks (based on OpenFOAM) which unify these currently disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible $\mu (I)$-rheology, which is coupled to the gravity-driven segregation theory of Gray & Ancey (J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. Further inclusion of an excess air phase, which segregates away from the granular material, then allows the complex evolution of the free surface to be captured simultaneously. Three primary coupling mechanisms are identified: (i) advection of the particle concentrations by the bulk velocity, (ii) feedback of the particle-size and/or frictional properties on the bulk flow field and (iii) influence of the shear rate, pressure, gravity, particle size and particle-size ratio on the locally evolving segregation and diffusion rates. The numerical method is extensively tested in one-way coupled computations, before the fully coupled model is compared with the discrete element method simulations of Tripathi & Khakhar (Phys. Fluids, vol. 23, 2011, 113302) and used to compute the petal-like segregation pattern that spontaneously develops in a square rotating drum.
Emerging evidence suggests that parents’ nutritional status before and at the time of conception influences the lifelong physical and mental health of their child. Yet little is known about the relationship between diet in adolescence and the health of the next generation at birth. This study examined data from Norwegian cohorts to assess the relationship between dietary patterns in adolescence and neonatal outcomes. Data from adolescents who participated in the Nord-Trøndelag Health Study (Young-HUNT) were merged with birth data for their offspring through the Medical Birth Registry of Norway. Young-HUNT1 collected data from 8980 adolescents between 1995 and 1997. Linear regression was used to assess associations between adolescents’ diet and later neonatal outcomes of their offspring adjusting for sociodemographic factors. Analyses were replicated with data from the Young-HUNT3 cohort (dietary data collected from 2006 to 2008) and combined with Young-HUNT1 for pooled analyses. In Young-HUNT1, there was evidence of associations between dietary choices, meal patterns, and neonatal outcomes, these were similar in the pooled analyses but were attenuated to the point of nonsignificance in the smaller Young-HUNT3 cohort. Overall, energy-dense food products were associated with a small detrimental impact on some neonatal outcomes, whereas healthier food choices appeared protective. Our study suggests that there are causal links between consumption of healthy and unhealthy food and meal patterns in adolescence with neonatal outcomes for offspring some years later. The effects seen are small and will require even larger studies with more state-of-the-art dietary assessment to estimate these robustly.
Despite efforts to improve maternal and child nutrition, undernutrition remains a major public health challenge in Ghana. The current study explored community perceptions of undernutrition and context-specific interventions that could improve maternal and child nutrition in rural Northern Ghana.
Design:
This exploratory qualitative study used ten focus group discussions to gather primary data. The discussions were recorded, transcribed and coded into themes using Nvivo 12 software to aid thematic analysis.
Setting:
The study was conducted in rural Kassena-Nankana Districts of Northern Ghana.
Study participants:
Thirty-three men and fifty-one women aged 18–50 years were randomly selected from the community.
Results:
Most participants reported poverty, lack of irrigated agricultural land and poor harvests as the main barriers to optimal nutrition. To improve maternal and child nutrition, study participants suggested that the construction of dams at the community level would facilitate all year round farming including rearing of animals. Participants perceived that the provision of agricultural materials such as high yield seedlings, pesticides and fertiliser would help boost agricultural productivity. They also recommended community-based nutrition education by trained health volunteers, focused on types of locally produced foods and appropriate ways to prepare them to help improve maternal and child nutrition.
Conclusion:
Drawing on these findings and existing literature, we argue that supporting community initiated nutrition interventions such as improved irrigation for dry season farming, provision of agricultural inputs and community education could improve maternal and child nutrition.
Recently published diagnostic criteria for mild cognitive impairment with Lewy bodies (MCI-LB) include five neuropsychiatric supportive features (non-visual hallucinations, systematised delusions, apathy, anxiety and depression). We have previously demonstrated that the presence of two or more of these symptoms differentiates MCI-LB from MCI due to Alzheimer's disease (MCI-AD) with a likelihood ratio >4. The aim of this study was to replicate the findings in an independent cohort.
Methods
Participants ⩾60 years old with MCI were recruited. Each participant had a detailed clinical, cognitive and imaging assessment including FP-CIT SPECT and cardiac MIBG. The presence of neuropsychiatric supportive symptoms was determined using the Neuropsychiatric Inventory (NPI). Participants were classified as MCI-AD, possible MCI-LB and probable MCI-LB based on current diagnostic criteria. Participants with possible MCI-LB were excluded from further analysis.
Results
Probable MCI-LB (n = 28) had higher NPI total and distress scores than MCI-AD (n = 30). In total, 59% of MCI-LB had two or more neuropsychiatric supportive symptoms compared with 9% of MCI-AD (likelihood ratio 6.5, p < 0.001). MCI-LB participants also had a significantly greater delayed recall and a lower Trails A:Trails B ratio than MCI-AD.
Conclusions
MCI-LB is associated with significantly greater neuropsychiatric symptoms than MCI-AD. The presence of two or more neuropsychiatric supportive symptoms as defined by MCI-LB diagnostic criteria is highly specific and moderately sensitive for a diagnosis of MCI-LB. The cognitive profile of MCI-LB differs from MCI-AD, with greater executive and lesser memory impairment, but these differences are not sufficient to differentiate MCI-LB from MCI-AD.
To disrupt cycles of health inequity, traceable to dietary inequities in the earliest stages of life, public health interventions should target improving nutritional wellbeing in preconception/pregnancy environments. This requires a deep engagement with pregnant/postpartum people (PPP) and their communities (including their health and social care providers, HSCP). We sought to understand the factors that influence diet during pregnancy from the perspectives of PPP and HSCP, and to outline intervention priorities.
Design:
We carried out thematic network analyses of transcripts from ten focus group discussions (FGD) and one stakeholder engagement meeting with PPP and HSCP in a Canadian city. Identified themes were developed into conceptual maps, highlighting local priorities for pregnancy nutrition and intervention development.
Setting:
FGD and the stakeholder meeting were run in predominantly lower socioeconomic position (SEP) neighbourhoods in the sociodemographically diverse city of Hamilton, Canada.
Participants:
All local, comprising twenty-two lower SEP PPP and forty-three HSCP.
Results:
Salient themes were resilience, resources, relationships and the embodied experience of pregnancy. Both PPP and HSCP underscored that socioeconomic-political forces operating at multiple levels largely determined the availability of individual and relational resources constraining diet during pregnancy. Intervention proposals focused on cultivating individual and community resilience to improve early-life nutritional environments. Participants called for better-integrated services, greater income supports and strengthened support programmes.
Conclusions:
Hamilton stakeholders foregrounded social determinants of inequity as main factors influencing pregnancy diet. They further indicated a need to develop interventions that build resilience and redistribute resources at multiple levels, from the household to the state.
Lewy body dementia, consisting of both dementia with Lewy bodies (DLB) and Parkinson's disease dementia (PDD), is considerably under-recognised clinically compared with its frequency in autopsy series.
Aims
This study investigated the clinical diagnostic pathways of patients with Lewy body dementia to assess if difficulties in diagnosis may be contributing to these differences.
Method
We reviewed the medical notes of 74 people with DLB and 72 with non-DLB dementia matched for age, gender and cognitive performance, together with 38 people with PDD and 35 with Parkinson's disease, matched for age and gender, from two geographically distinct UK regions.
Results
The cases of individuals with DLB took longer to reach a final diagnosis (1.2 v. 0.6 years, P = 0.017), underwent more scans (1.7 v. 1.2, P = 0.002) and had more alternative prior diagnoses (0.8 v. 0.4, P = 0.002), than the cases of those with non-DLB dementia. Individuals diagnosed in one region of the UK had significantly more core features (2.1 v. 1.5, P = 0.007) than those in the other region, and were less likely to have dopamine transporter imaging (P < 0.001). For patients with PDD, more than 1.4 years prior to receiving a dementia diagnosis: 46% (12 of 26) had documented impaired activities of daily living because of cognitive impairment, 57% (16 of 28) had cognitive impairment in multiple domains, with 38% (6 of 16) having both, and 39% (9 of 23) already receiving anti-dementia drugs.
Conclusions
Our results show the pathway to diagnosis of DLB is longer and more complex than for non-DLB dementia. There were also marked differences between regions in the thresholds clinicians adopt for diagnosing DLB and also in the use of dopamine transporter imaging. For PDD, a diagnosis of dementia was delayed well beyond symptom onset and even treatment.
Although neuroimaging studies suggest brain regional abnormalities in depressive disorders, it remains unclear whether abnormalities are present at illness onset or reflect disease progression.
Objectives
We hypothesized that cerebral variations were present in adolescents with subthreshold depression known to be at high risk for later full-blown depression.
Aims
We examined brain structural and diffusion-weighted magnetic resonance images of adolescents with subthreshold depression.
Methods
The participants were extracted from the European IMAGEN study cohort of healthy adolescents recruited at age 14. Subthreshold depression was defined as a distinct period of abnormally depressed or irritable mood, or loss of interest, plus two or more depressive symptoms but without diagnosis of Major Depressive Episode. Comparisons were performed between adolescents meeting these criteria and control adolescents within the T1-weighted imaging modality (118 and 475 adolescents respectively) using voxel-based morphometry and the diffusion tensor imaging modality (89 ad 422 adolescents respectively) using tract-based spatial statistics. Whole brain analyses were performed with a statistical threshold set to p< 0.05 corrected for multiple comparisons.
Results
Compared with controls, adolescents with subthreshold depression had smaller gray matter volume in caudate nuclei, medial frontal and cingulate cortices; smaller white matter volume in anterior limb of internal capsules, left forceps minor and right cingulum; and lower fractional anisotropy and higher radial diffusivity in the genu of corpus callosum.
Conclusions
The findings suggest that adolescents with subthreshold depression have volumetric and microstructural gray and white matter changes in the emotion regulation frontal-striatal-limbic network.
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 $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-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 $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ 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.
Potential routes to the formation of urea were investigated using electronic structure methods. The most likely pathways involve either the reaction of the formamide and amine radicals or involve protonated isocyanic acid as a starting point.
Evidence supporting the Developmental Origins of Health and Disease (DOHaD) hypothesis indicates that improving early life environments can reduce non-communicable disease risks and improve health over the lifecourse. A widespread understanding of this evidence may help to reshape structures, guidelines and individual behaviors to better the developmental conditions for the next generations. Yet, few efforts have yet been made to translate the DOHaD concept beyond the research community. To understand why, and to identify priorities for DOHaD Knowledge Translation (KT) programs, we review here a portion of published descriptions of DOHaD KT efforts and critiques thereof. We focus on KT targeting people equipped to apply DOHaD knowledge to their everyday home or work lives. We identified 17 reports of direct-to-public DOHaD KT that met our inclusion criteria. Relevant KT programs have been or are being initiated in nine countries, most focusing on secondary school students or care-workers-in-training; few target parents-to-be. Early indicators suggest that such programs can empower participants. Main critiques of DOHaD KT suggest it may overburden mothers with responsibility for children’s health and health environments, minimizing the roles of other people and institutions. Simultaneously, though, many mothers-to-be seek reliable guidance on prenatal health and nutrition, and would likely benefit from engagement with DOHaD KT. We thus recommend emphasizing solidarity, and bringing together people likely to one day become parents (youth), people planning pregnancies, expecting couples, care workers and policymakers into empowering conversation about DOHaD and about the importance and complexity of early life environments.
Objectives: Maintaining two active languages may increase cognitive and brain reserve among bilingual individuals. We explored whether such a neuroprotective effect was manifested in the performance of memory tests for participants with amnestic mild cognitive impairment (aMCI). Methods: We compared 42 bilinguals to 25 monolinguals on verbal and nonverbal memory tests. We used: (a) the Loewenstein-Acevedo Scales for Semantic Interference and Learning (LASSI-L), a sensitive test that taps into proactive, retroactive, and recovery from proactive semantic interference (verbal memory), and (b) the Benson Figure delayed recall (nonverbal memory). A subsample had volumetric MRI scans. Results: The bilingual group significantly outperformed the monolingual group on two LASSI-L cued recall measures (Cued A2 and Cued B2). A measure of maximum learning (Cued A2) showed a correlation with the volume of the left hippocampus in the bilingual group only. Cued B2 recall (sensitive to recovery from proactive semantic interference) was correlated with the volume of the hippocampus and the entorhinal cortex of both cerebral hemispheres in the bilingual group, as well as with the left and right hippocampus in the monolingual group. The memory advantage in bilinguals on these measures was associated with higher inhibitory control as measured by the Stroop Color-Word test. Conclusions: Our results demonstrated a superior performance of aMCI bilinguals over aMCI monolinguals on selected verbal memory tasks. This advantage was not observed in nonverbal memory. Superior memory performance of bilinguals over monolinguals suggests that bilinguals develop a different and perhaps more efficient semantic association system that influences verbal recall. (JINS, 2019, 25, 15–28)
A large body of research has explored opportunities to mitigate climate change in agricultural systems; however, less research has explored opportunities across the food system. Here we expand the existing research with a review of potential mitigation opportunities across the entire food system, including in pre-production, production, processing, transport, consumption and loss and waste. We detail and synthesize recent research on the topic, and explore the applicability of different climate mitigation strategies in varying country contexts with different economic and agricultural systems. Further, we highlight some potential adaptation co-benefits of food system mitigation strategies and explore the potential implications of such strategies on food systems as a whole. We suggest that a food systems research approach is greatly needed to capture such potential synergies, and highlight key areas of additional research including a greater focus on low- and middle-income countries in particular. We conclude by discussing the policy and finance opportunities needed to advance mitigation strategies in food systems.
Objectives: The aim of this study was to determine the presence and severity of potential cultural and language bias in widely used cognitive and other assessment instruments, using structural MRI measures of neurodegeneration as biomarkers of disease stage and severity. Methods: Hispanic (n=75) and White non-Hispanic (WNH) (n=90) subjects were classified as cognitively normal (CN), amnestic mild cognitive impairment (aMCI) and mild dementia. Performance on the culture-fair and educationally fair Fuld Object Memory Evaluation (FOME) and Clinical Dementia Rating Scale (CDR) between Hispanics and WNHs was equivalent, in each diagnostic group. Volumetric and visually rated measures of the hippocampus entorhinal cortex, and inferior lateral ventricles (ILV) were measured on structural MRI scans for all subjects. A series of analyses of covariance, controlling for age, depression, and education, were conducted to compare the level of neurodegeneration on these MRI measures between Hispanics and WNHs in each diagnostic group. Results: Among both Hispanics and WNH groups there was a progressive decrease in volume of the hippocampus and entorhinal cortex, and an increase in volume of the ILV (indicating increasing atrophy in the regions surrounding the ILV) from CN to aMCI to mild dementia. For equivalent levels of performance on the FOME and CDR, WNHs had greater levels of neurodegeneration than did Hispanic subjects. Conclusions: Atrophy in medial temporal regions was found to be greater among WNH than Hispanic diagnostic groups, despite the lack of statistical differences in cognitive performance between these two ethnic groups. Presumably, unmeasured factors result in better cognitive performance among WNH than Hispanics for a given level of neurodegeneration. (JINS, 2018, 24, 176–187)
In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the
$\unicode[STIX]{x1D707}(I)$
-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number
$I$
. 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
$\unicode[STIX]{x1D707}(I)$
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
$\unicode[STIX]{x1D707}$
on
$I$
. A new
$\unicode[STIX]{x1D707}(I)$
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.
A few studies have evaluated the impact of clinical trial results on practice in paediatric cardiology. The Infant Single Ventricle (ISV) Trial results published in 2010 did not support routine use of the angiotensin-converting enzyme inhibitor enalapril in infants with single-ventricle physiology. We sought to assess the influence of these findings on clinical practice.
Methods
A web-based survey was distributed via e-mail to over 2000 paediatric cardiologists, intensivists, cardiothoracic surgeons, and cardiac advance practice nurses during three distribution periods. The results were analysed using McNemar’s test for paired data and Fisher’s exact test.
Results
The response rate was 31.5% (69% cardiologists and 65% with >10 years of experience). Among respondents familiar with trial results, 74% reported current practice consistent with trial findings versus 48% before trial publication (p<0.001); 19% used angiotensin-converting enzyme inhibitor in this population “almost always” versus 36% in the past (p<0.001), and 72% reported a change in management or improved confidence in treatment decisions involving this therapy based on the trial results. Respondents familiar with trial results (78%) were marginally more likely to practise consistent with the trial results than those unfamiliar (74 versus 67%, p=0.16). Among all respondents, 28% reported less frequent use of angiotensin-converting enzyme inhibitor over the last 3 years.
Conclusions
Within 5 years of publication, the majority of respondents was familiar with the Infant Single Ventricle Trial results and reported less frequent use of angiotensin-converting enzyme inhibitor in single-ventricle infants; however, 28% reported not adjusting their clinical decisions based on the trial’s findings.
This study evaluated dosimetric parameters for cervical high-dose-rate (HDR) brachytherapy treatment using varying dose prescription methods.
Methods
This study includes 125 tandem-based cervical HDR brachytherapy treatment plans of 25 patients who received HDR brachytherapy. Delineation of high-risk clinical target volumes (HR-CTVs) and organ at risk were done on original computed tomographic images. The dose prescription point was defined as per International Commission in Radiation Units and Measurements Report Number 38 (ICRU-38), also redefined using American Brachytherapy Society (ABS) 2011 criteria. The coverage index (V100) for each HR-CTV was calculated using dose volume histogram parameters. A plot between HR-CTV and V100 was plotted using the best-fit linear regression line (least-square fit analysis).
Results
Mean prescribed dose to ICRU-38 Point A was 590·47±28·65 cGy, and to ABS Point A was 593·35±30·42 cGy. There was no statistically significant difference between planned ICRU-38 and calculated ABS Point A doses (p=0·23). The plot between HR-CTV and V100 is well defined by the best-fit linear regression line with a correlation coefficient of 0·9519.
Conclusion
For cervical HDR brachytherapy, dose prescription to an arbitrarily defined point (e.g., Point A) does not provide consistent coverage of HR-CTV. The difference in coverage between two dose prescription approaches increases with increasing CTV. Our ongoing work evaluates the dosimetric consequences of volumetric dose prescription approaches for these patients.
Recently, large-scale trials of behavioural interventions have failed to show improvements in pregnancy outcomes. They have, however, shown that lifestyle support improves maternal diet and physical activity during pregnancy, and can reduce weight gain. This suggests that pregnancy, and possibly the whole periconceptional period, represents a ‘teachable moment’ for changes in diet and lifestyle, an idea that was made much of in the recent report of the Chief Medical Officer for England. The greatest challenge with all trials of diet and lifestyle interventions is to engage people and to sustain this engagement. With this in mind, we propose a design of intervention that aims simultaneously to engage women through motivational conversations and to offer access to a digital platform that provides structured support for diet and lifestyle change. This intervention design therefore makes best use of learning from the trials described above and from recent advances in digital intervention design.
In this paper we consider PDE-constrained optimization problems which incorporate an H1 regularization control term. We focus on a time-dependent PDE, and consider both distributed and boundary control. The problems we consider include bound constraints on the state, and we use a Moreau-Yosida penalty function to handle this. We propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.
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
${\it\mu}(I)$
-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
${\it\mu}(I)$
-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
${\it\mu}$
. 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
${\it\mu}(I)$
-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
$I$
in the friction coefficient
${\it\mu}$
. In this paper it is shown that the
${\it\mu}(I)$
-rheology is well-posed for intermediate values of
$I$
, 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.