Well-posed continuum equations for granular flow with compressibility and
            <i>μ</i>
            (
            <i>I</i>
            )-rheology

Barker, T.; Schaeffer, David G.; Shearer, Michael; Gray, J.O.

doi:10.1098/rspa.2016.0846

Cited by 63 publications

(100 citation statements)

References 36 publications

(99 reference statements)

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“…Well-posedness ensures that the solution is stable, as opposed to ill-posedness that could lead to instabilities that depend for instance on the grid size spacing. This led Barker et al (2017) to propose a modified friction coefficient:…”

Section: Inertial Rheologymentioning

confidence: 99%

“… μ d is the friction coefficient that μ ( I ) tends to as I increases, and I o is a dimensionless parameter that depends on material property. To ensure well‐posedness of the constitutive equations derived from equation , the upper formula is only valid across a small range of Inertial number (e.g., ~0.001–0.3 for monodisperse glass beads; Barker et al, ). Well‐posedness ensures that the solution is stable, as opposed to ill‐posedness that could lead to instabilities that depend for instance on the grid size spacing.…”

Section: Introductionmentioning

confidence: 99%

“…Well‐posedness ensures that the solution is stable, as opposed to ill‐posedness that could lead to instabilities that depend for instance on the grid size spacing. This led Barker et al () to propose a modified friction coefficient:

μ ((), I) = ((), \begin{matrix} \sqrt{\frac{α}{ln (\frac{A_{1}}{I})}} I \leq I_{1}^{N} \\ \frac{μ_{static} I_{0} + μ_{d} I + μ_{\infty} I^{2}}{I_{0} + I} I > I_{1}^{N} \end{matrix})

A_{1} = I_{1}^{N} exp ((), \frac{α {((), I_{0} + I_{1}^{N})}^{2}}{{(μ_{static} I_{0} + μ_{d} I_{1}^{N} + μ_{\infty} {((), I_{1}^{N})}^{2})}^{2}})

where I 0 (same as in equation (4)), μ ∞ , and α are constants.

I_{1}^{N}

is the lower end of I where equation is well‐posed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Continuum Modeling of Pressure‐Balanced and Fluidized Granular Flows in 2‐D: Comparison With Glass Bead Experiments and Implications for Concentrated Pyroclastic Density Currents

2019

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Granular flows are found across multiple geophysical environments and include pyroclastic density currents, debris flows, and avalanches, among others. The key to describing transport of these hazardous flows is the rheology of these complex multiphase mixtures. Here we use the multiphase model MFIX in 2‐D for concentrated currents to examine the implications of rheological assumptions and validate this approach through comparison to experiments of both frictional and fluidized flows made of glass beads (Sauter mean grain‐size of 75 μm). Because the rheology of highly polydisperse, highly angular, polydensity granular mixtures is poorly known, we focus on simplified monodisperse or bidisperse mixtures described by the frictional flow theory of Schaeffer (1987, https://doi.org/10.1016/0022-0396(87)90038-6) and Srivastava‐Sundaresan, often referred to as the Princeton model (Srivastava & Sundaresan, 2003, https://doi.org/10.1016/S0032-5910(02)00132-8). We show that simulations including the latter model replicate well the flow shape, kinematics, and pore fluid pressure that match well‐constrained dam‐break experiments of initially fluidized or pressure‐balanced granular flows. Simulations reveal that pore fluid pressure is intrinsically modulated by dilation and compaction of the flow and hence can be generated in concentrated pyroclastic density currents. We use these simulations to interpret basal pore pressure signals from local flow properties (mixture density, solid velocity, and pore fluid pressure). Rheological changes retard these simulated flows considerably near the predicted runout, but most continuum models cannot inherently predict a zero velocity. We suggest an inertial number parameter that can be used to approximate deposition, and this approach could be a valuable tool used to validate simulations against natural pyroclastic current examples.

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Section: Inertial Rheologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

μ ((), I) = ((), \begin{matrix} \sqrt{\frac{α}{ln (\frac{A_{1}}{I})}} I \leq I_{1}^{N} \\ \frac{μ_{static} I_{0} + μ_{d} I + μ_{\infty} I^{2}}{I_{0} + I} I > I_{1}^{N} \end{matrix})

A_{1} = I_{1}^{N} exp ((), \frac{α {((), I_{0} + I_{1}^{N})}^{2}}{{(μ_{static} I_{0} + μ_{d} I_{1}^{N} + μ_{\infty} {((), I_{1}^{N})}^{2})}^{2}})

where I 0 (same as in equation (4)), μ ∞ , and α are constants.

I_{1}^{N}

is the lower end of I where equation is well‐posed.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuum Modeling of Pressure‐Balanced and Fluidized Granular Flows in 2‐D: Comparison With Glass Bead Experiments and Implications for Concentrated Pyroclastic Density Currents

2019

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“…Nonetheless, (3.2) appears to be well-supported experimentally for steady chute flows (GDR MiDi 2004). Figure 4 of Barker et al (2017) shows that the corrections due to compressibility are small except at the free surface. In this article, we are interested in the dynamics at the base of the flow, away from the free surface, so for the analysis in §4 we assume that (3.2) holds.…”

Section: The µ(I) Rheologymentioning

confidence: 99%

The granular Blasius problem

2019

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We consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for $x<0$) but rough downstream (imposing a no-slip condition on $x>0$), with a sharp transition at $x=0$. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number $I$; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The $\unicode[STIX]{x1D707}(I)$ rheology (Jop et al., Nature, vol. 441 (7094), 2006, pp. 727–730) asserts that $\text{d}\unicode[STIX]{x1D707}/\text{d}I=O(1/I^{2})$ as $I\rightarrow \infty$, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the $\unicode[STIX]{x1D707}(I)$ rheology, with different dependencies of $\unicode[STIX]{x1D707}(I)$ on $I$, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.

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“…We acknowledge the fact that, by the time our paper was considered for publication in Journal of Fluid Mechanics, a similar study had been published in another journal ( Barker et al 2017). In this study, compressibility is introduced based on the critical state concept borrowed from soil mechanics while we preferred a fluid mechanics-like formulation based on a second viscosity (both formulations are shown to be equivalent in Appendix A).…”

Section: Acknowledgementmentioning

confidence: 99%

Compressibility regularizes the 𝜇(I)-rheology for dense granular flows

et al. 2017

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The µ(I)-rheology was recently proposed as a potential candidate to model the incompressible flow of frictional grains in the dense inertial regime. However, this rheology was shown to be ill-posed in the mathematical sense for a large range of parameters, notably in the low and large inertial number limits (Barker et al. 2015). In this rapid communication, we extend the stability analysis of Barker et al. (2015) to compressible flows. We show that compressibility regularizes mostly the equations, making the problem well-posed for all parameters, with the condition that sufficient dissipation be associated with volume changes. In addition to the usual Coulomb shear friction coefficient µ, we introduce a bulk friction coefficient µ b , associated with volume changes and show that the problem is well-posed if µ b > 1 − 7µ/6. Moreover, we show that the ill-posed domain defined by Barker et al. (2015) transforms into a domain where the flow is unstable but remains well-posed when compressibility is taken into account. These results suggest the importance of taking into account dynamic compressibility for the modelling of dense granular flows and open new perspectives to investigate the emission and propagation of acoustic waves inside these flows.

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Well-posed continuum equations for granular flow with compressibility and μ ( I )-rheology

Cited by 63 publications

References 36 publications

Continuum Modeling of Pressure‐Balanced and Fluidized Granular Flows in 2‐D: Comparison With Glass Bead Experiments and Implications for Concentrated Pyroclastic Density Currents

Continuum Modeling of Pressure‐Balanced and Fluidized Granular Flows in 2‐D: Comparison With Glass Bead Experiments and Implications for Concentrated Pyroclastic Density Currents

The granular Blasius problem

Compressibility regularizes the 𝜇(I)-rheology for dense granular flows

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