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.

We made heat-capacity measurements of two dimensional (2D) 3 He adsorbed on graphite preplated with monolayer 4 He in a wide temperature range (0.1 ≤ T ≤ 80 mK) at densities higher than that for the 4/7 phase ( = 6.8 nm −2 ). In the density range of 6.8 ≤ ρ ≤ 8.1 nm −2 , the 4/7 phase is stable against additional 3 He atoms up to 20% and they are promoted into the third layer. We found evidence that such promoted atoms form a self-bound 2D Fermi liquid with an approximate density of 1 nm −2 from the measured density dependence of the γ-coefficient of heat capacity. We also show evidence for the first-order transition between the commensurate 4/7 phase and the ferromagnetic incommensurate phase in the second layer in the density range of 8.1 ≤ ρ ≤ 9.5 nm −2 .

Nonlinear relaxation dynamics of a vertically vibrated granular pile is experimentally studied. In the experiment, the flux and slope on the relaxing pile are measured by using a high-speed laser profiler. The relation of these quantities can be modeled by the nonlinear transport law assuming the uniform vibrofluidization of an entire pile. The fitting parameter in this model is only the relaxation efficiency, which characterizes the energy conversion rate from vertical vibration into horizontal transport. We demonstrate that this value is a constant independent of experimental conditions. The actual relaxation is successfully reproduced by the continuity equation with the proposed model. Finally, its specific applicability toward an astrophysical phenomenon is shown.

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