Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow

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 μ . 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 μ(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 μ . In this paper it is shown that the μ(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

A stochastic flow rule for granular materials

There have been many attempts to derive continuum models for dense granular flow, but a general theory is still lacking. Here, we start with Mohr-Coulomb plasticity for quasi-2D granular materials to calculate (average) stresses and slip planes, but we propose a "stochastic flow rule" (SFR) to replace the principle of coaxiality in classical plasticity. The SFR takes into account two crucial features of granular materials - discreteness and randomness - via diffusing "spots" of local fluidization, which act as carriers of plasticity. We postulate that spots perform random walks biased along slip-lines with a drift direction determined by the stress imbalance upon a local switch from static to dynamic friction. In the continuum limit (based on a Fokker-Planck equation for the spot concentration), this simple model is able to predict a variety of granular flow profiles in flat-bottom silos, annular Couette cells, flowing heaps, and plate-dragging experiments -- with essentially no fitting parameters -- although it is only expected to function where material is at incipient failure and slip-lines are inadmissible. For special cases of admissible slip-lines, such as plate dragging under a heavy load or flow down an inclined plane, we postulate a transition to rate-dependent Bagnold rheology, where flow occurs by sliding shear planes. With different yield criteria, the SFR provides a general framework for multiscale modeling of plasticity in amorphous materials, cycling between continuum limit-state stress calculations, meso-scale spot random walks, and microscopic particle relaxation

Constitutive relations for compressible granular flow in the inertial regime

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 μ(I),Φ(I)-rheology, which postulates that the bulk friction coefficient μ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction ϕ are functions of the inertial number I only. Although the μ(I),Φ(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 μ(I),Φ(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 μ(I) and Φ(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

Wide shear zones and the spot model: Implications from the split-bottom geometry

The spot model has been developed by Bazant and co-workers to describe quasistatic granular flows. It assumes that granular flow is caused by the opposing flow of so-called spots of excess free volume, with spots moving along the slip lines of Mohr-Coulomb plasticity. The model is two-dimensional and has been successfully applied to a number of different geometries. In this paper we investigate whether the spot model in its simplest form can describe the wide shear zones observed in experiments and simulations of a Couette cell with split bottom. We give a general argument that is independent of the particular description of the stresses, but which shows that the present formulation of the spot model in which diffusion and drift terms are postulated to balance on length scales of order of the spot diameter, i.e. of order 3-5 grain diameters, is difficult to reconcile with the observed wide shear zones. We also discuss the implications for the spot model of co-axiality of the stress and strain rate tensors found in these wide shear flows, and point to possible extensions of the model that might allow one to account for the existence of wide shear zones.Comment: 6 pages, 6 figures, to be published in EPJ

Toksikokinetika prometrina u mozgu miševa

Prometryne is a methylthio-s-triazine herbicide. Signifi cant trace amounts are found in the environment, mainly in water, soil, and food plants. The aim of this study was to establish brain and blood prometryne levels after single oral dose (1 g kg-1) in adult male and female mice. Prometryne was measured using the GC/MS assay at 1, 2, 4, 8, and 24 h after prometryne administration. Peak brain and blood prometryne values were observed 1 h after administration and they decreased in a time-dependent manner. Male mice had consistently higher brain and blood prometryne levels than female mice. The observed prometryne kinetics was similar to that reported for the structurally related herbicide atrazine.Prometrin je metiltio-s-triazinski herbicid. Značajne količine prometrina zaostaju u tragovima u okolišu, poglavito u vodi, tlu i biljkama koje rabimo za prehranu. Cilj je rada izmjeriti količinu prometrina koja se apsorbira u mozgu i krvi nakon primijenjene akutne oralne doze (1 g kg-1 tjelesne mase) u odraslih miševa obaju spolova. Razine prometrina u mozgu i krvi izmjerene su GC/MS-om tijekom 1., 2., 4., 8. i 24. sata nakon izlaganja. Utvrđeno je da je udio prometrina koji se zadržava u živčanom tkivu relativno nizak ali detektabilan u odnosu na koncentraciju u krvi i koncentraciju primijenjene doze. Najviše koncentracije u krvi i maseni udjeli u mozgu zabilježeni su tijekom 1. sata nakon izlaganja, a s vremenom izmjerene vrijednosti značajno opadaju. Uočena je značajna razlika između mužjaka i ženki pri čemu mužjaci imaju značajno više razine prometrina u mozgu i krvi nego ženke. Opisana toksikokinetika prometrina pokazuje sličnosti s otprije opisanom i poznatom toksikokinetikom strukturalno sličnog herbicida atrazina

Non-local rheology in dense granular flows -- Revisiting the concept of fluidity

Granular materials belong to the class of amorphous athermal systems, like foams, emulsion or suspension they can resist shear like a solid, but flow like a liquid under a sufficiently large applied shear stress. They exhibit a dynamical phase transition between static and flowing states, as for phase transitions of thermodynamic systems, this rigidity transition exhibits a diverging length scales quantifying the degree of cooperatively. Several experiments have shown that the rheology of granular materials and emulsion is non-local, namely that the stress at a given location does not depend only on the shear rate at this location but also on the degree of mobility in the surrounding region. Several constitutive relations have recently been proposed and tested successfully against numerical and experimental results. Here we use discrete elements simulation of 2D shear flows to shed light on the dynamical mechanism underlying non-locality in dense granular flows

Effect of Acinetobacter sp on Metalaxyl Degradation and Metabolite Profile of Potato Seedlings (Solanum tuberosum L.) Alpha Variety

One of the most serious diseases in potato cultivars is caused by the pathogen Phytophthora infestans, which affects leaves, stems and tubers. Metalaxyl is a fungicide that protects potato plants from Phytophthora infestans. In Mexico, farmers apply metalaxyl 35 times during the cycle of potato production and the last application is typically 15 days before harvest. There are no records related to the presence of metalaxyl in potato tubers in Mexico. In the present study, we evaluated the effect of Acinetobacter sp on metalaxyl degradation in potato seedlings. The effect of bacteria and metalaxyl on the growth of potato seedlings was also evaluated. A metabolite profile analysis was conducted to determine potential molecular biomarkers produced by potato seedlings in the presence of Acinetobacter sp and metalaxyl. Metalaxyl did not affect the growth of potato seedlings. However, Acinetobacter sp strongly affected the growth of inoculated seedlings, as confirmed by plant length and plant fresh weights which were lower in inoculated potato seedlings (40% and 27%, respectively) compared to the controls. Acinetobacter sp also affected root formation. Inoculated potato seedlings showed a decrease in root formation compared to the controls. LC-MS/MS analysis of metalaxyl residues in potato seedlings suggests that Acinetobacter sp did not degrade metalaxyl. GC–TOF–MS platform was used in metabolic profiling studies. Statistical data analysis and metabolic pathway analysis allowed suggesting the alteration of metabolic pathways by both Acinetobacter sp infection and metalaxyl treatment. Several hundred metabolites were detected, 137 metabolites were identified and 15 metabolic markers were suggested based on statistical change significance found with PLS-DA analysis. These results are important for better understanding the interactions of putative endophytic bacteria and pesticides on plants and their possible effects on plant metabolism