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

2017 ◽  
Vol 830 ◽  
pp. 553-568 ◽  
Author(s):  
J. Heyman ◽  
R. Delannay ◽  
H. Tabuteau ◽  
A. Valance
Keyword(s):  
Friction Coefficient ◽  
Acoustic Waves ◽  
Compressible Flows ◽  
Granular Flows ◽  
Potential Candidate ◽  
Volume Changes ◽  
Ill Posed ◽  
The Stability ◽  
Well Posed ◽  
Dense Granular Flows

The $\unicode[STIX]{x1D707}(I)$-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., J. Fluid Mech., vol. 779, 2015, pp. 794–818). In this rapid communication, we extend the stability analysis of Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818) to compressible flows. We show that compressibility regularizes 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 $\unicode[STIX]{x1D707}$, we introduce a bulk friction coefficient $\unicode[STIX]{x1D707}_{b}$, associated with volume changes and show that the problem is well-posed if $\unicode[STIX]{x1D707}_{b}>1-7\unicode[STIX]{x1D707}/6$. Moreover, we show that the ill-posed domain defined by Barker et al. (J. Fluid Mech., vol. 779, 2015, pp. 794–818) 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.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Well-posed and ill-posed behaviour of the -rheology for granular flow

2015 ◽  
Vol 779 ◽  
pp. 794-818 ◽  
Author(s):  
T. Barker ◽  
D. G. Schaeffer ◽  
P. Bohorquez ◽  
J. M. N. T. Gray
Keyword(s):  
Friction Coefficient ◽  
Blow Up ◽  
Stokes Equations ◽  
Momentum Balance ◽  
Navier Stokes ◽  
Major Step ◽  
Navier Stokes Equations ◽  
Newtonian Flows ◽  
Ill Posed ◽  
Well Posed

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.

Finite element simulation of complex dense granular flows using a well-posed regularization of the μ(I)-rheology

2019 ◽  
Vol 188 ◽  
pp. 102-113 ◽  
Author(s):  
Linda Gesenhues ◽  
José J. Camata ◽  
Adriano M.A. Côrtes ◽  
Fernando A. Rochinha ◽  
Alvaro L.G.A. Coutinho
Keyword(s):  
Finite Element ◽  
Finite Element Simulation ◽  
Granular Flows ◽  
Element Simulation ◽  
Well Posed ◽  
Dense Granular Flows

scholarly journals Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow

2017 ◽  
Vol 828 ◽  
pp. 5-32 ◽  
Author(s):  
T. Barker ◽  
J. M. N. T. Gray
Keyword(s):  
High Speed ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Functional Dependence ◽  
Granular Flows ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Major Step ◽  
Ill Posed ◽  
Well Posed

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.

scholarly journals Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mohammed Hamaidi ◽  
Ahmed Naji ◽  
Fatima Ghafrani ◽  
Mostafa Jourhmane
Keyword(s):  
Boundary Condition ◽  
Heat Conduction Problem ◽  
Space Time ◽  
Optimization Scheme ◽  
Ill Posed ◽  
The Stability ◽  
Iterative Procedures ◽  
Conventional Optimization ◽  
Inverse Boundary Value Problems ◽  
Well Posed

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

scholarly journals Influence of granular temperature and grain rotation on the wall friction coefficient in confined shear granular flows

2021 ◽  
Vol 249 ◽  
pp. 03026
Author(s):  
Cheng-Chuan Lin ◽  
Riccardo Artoni ◽  
Fu-Ling Yang ◽  
Patrick Richard
Keyword(s):  
Friction Coefficient ◽  
Granular Matter ◽  
Scaling Law ◽  
Three Dimensional ◽  
Granular Flows ◽  
Force Sensor ◽  
Wall Friction ◽  
Granular Temperature ◽  
Research Perspective ◽  
Dense Granular Flows

A depth-weakening wall friction coefficient, µw, has been reported from three-dimensional numerical simulations of steady and transient dense granular flows. To understand the degradation mechanisms, a scaling law for µw/ f and χ has been proposed where f is the intrinsic particle-wall friction and χ is the ratio of slip velocity to square root of granular temperature (Artoni & Richard, Phys. Rev. Lett., vol. 115 (15), 2015, 158001). Independently, a friction degradation model has been derived which describes a monotonically diminishing friction depends on a ratio of grain angular and slip velocities, Ω (Yang & Huang, Granular Matter, vol. 18 (4), 2016, 77). In search of experimental evidence for how these two parameters degrade the µw, an annular shear cell experiment was performed to estimate the bulk granular temperature, angular and slip velocities at sidewall through image-processing. Meanwhile, µw was measured by a force sensor to confirm the weakening towards the creep zone. The measured µw/ f − χ and µw/ f − Ω were both well-fitted to the corresponding models showing that both granular temperature and angular velocity are significant mechanisms to degrade the µw which broadens the research perspective on modeling the boundary condition of dense granular flows.

scholarly journals Segregation-induced finger formation in granular free-surface flows

2016 ◽  
Vol 809 ◽  
pp. 168-212 ◽  
Author(s):  
J. L. Baker ◽  
C. G. Johnson ◽  
J. M. N. T. Gray
Keyword(s):  
Granular Flows ◽  
Free Surface Flows ◽  
Pyroclastic Flows ◽  
Coarse Grained ◽  
Momentum Balance ◽  
Large Particles ◽  
Averaged Model ◽  
Well Posed ◽  
Dense Granular Flows ◽  
New System

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 $\unicode[STIX]{x1D707}(I)$-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.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

scholarly journals Self-diffusion scalings in dense granular flows

Soft Matter ◽  
2021 ◽  
Author(s):  
Riccardo Artoni ◽  
Michele Larcher ◽  
James T. Jenkins ◽  
Patrick Richard
Keyword(s):  
Shear Rate ◽  
Solid Fraction ◽  
Granular Flows ◽  
The Self ◽  
Granular Temperature ◽  
Restitution Coefficient ◽  
Self Diffusion ◽  
Diffusivity Tensor ◽  
Dense Granular Flows

The self-diffusivity tensor in homogeneously sheared dense granular flows is anisotropic. We show how its components depend on solid fraction, restitution coefficient, shear rate, and granular temperature.

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