Granular gases
The granular material behaves like a gas if the volume fraction is much less than, where inelastic collisions dissipate the kinetic energy so that granular particles exhibit clustering instabilities. However, granular gases can flow when they are driven by external forces, where the external supply of energy is balanced with the dissipation of energy, i.e. the system reaches a non-equilibrium steady state. Therefore, there are still many challenges of describing the flows of granular gases.
Approach: To understand the most basic flow property of granular gases, we study their simple shear flows [1]. First, we simulate granular shear flows by molecular dynamics (MD) simulations (Fig. 1(a)), where granular particles lose their kinetic energy by inelastic collisions. Initially, the granular particles homogeneously distribute in space. However, small clusters are generated and merge with each other to make a large cluster in the bulk. As time goes on, the cluster is slowly elongated along the sheared direction and the system reaches a non-equilibrium steady state (Fig. 2(a)), which is clearly different from a usual clustering process without external forces. To describe such a shear-induced clustering instability, we numerically solve hydrodynamic equations of granular gases, where transport coefficients are derived from kinetic theory based on the inelastic Boltzmann equation [1]. By using a bumpy boundary condition for the numerical calculations, we have confirmed a good agreement between MD simulations and the hydrodynamic equations. Then, we carry out a linear stability analysis to study the dependence of the instability on control parameters, i.e. the restitution coefficient of granular particles, e, and shear rate applied to the system [5], where we find that the instability is caused by too much dissipation, or too less supply of energy. Here, we have analytically solved an eigenvalue problem for the growth rate of disturbance by using perturbation theory, where our result is validated by a numerical calculation without any fitting parameters (Fig. 2(c) left).
We further investigate granular shear flows by a weakly non-linear analysis, where the time-dependent Ginzburg-Landau (TDGL) equation for the disturbance amplitude is derived from the granular hydrodynamic equations [5]. Figure 2(c) (right) shows our results of the coefficients for the 3rd and 5th order terms in the TDGL equation, where both the supercritical and subcritical bifurcations can be seen as the onset of the instability. In addition, our TDGL equation with time dependent diffusion coefficients (Fig. 2(b)) well reproduces the clustering process observed in our MD simulations (Fig. 2(a)) [6, 13].
Method overview: We used MD simulations of frictional granular particles, where the force between granular particles has both the normal and tangential components to model a microscopic friction. Newton's equation and Euler's equation are numerically integrated for the translational and rotational motions, respectively. The Lees-Edwards boundary condition and a bumpy boundary condition are used to apply simple shear deformations to the system. To validate our perturbative calculations for the linear stability analysis, we numerically solved eigenvalue problems by using the LAPACK sub-routines. The applicant made all the numerical programs and did all the analytical calculations.
The underlying assumption of kinetic theory is violated once granular particles are aggregated, which is unavoidable for cohesive granular particles because of their attractive interactions. In our MD simulations of cohesive granular particles [9], we have observed many kinds of clustering processes which cannot be explained by the method described in the previous subject 1, where gas and liquid phases coexist under shear.
Approach: To describe such a new clustering instability, we modify a diffuse interface model for multi-phase fluids to include the effect of inelastic collisions, where the constitutive relations have higher order gradients of the density field to represent the interface between two different phases [15]. In our knowledge, this is the first attempt to model multi-phase flows of granular materials. Numerically solving the hydrodynamic equations (Fig. 2(d)), we find that all the clustering processes observed in MD simulations are well reproduced. Then, we carry out the linear stability analysis and confirm that the neutral curve well explains our numerical simulations without any fitting parameters (the solid line in Fig. 2(e)). Interestingly, the clustering instability is triggered if the system is thermodynamically unstable, where the shear-induced instability (subject 1) competes with the thermodynamic instability if the system size is infinitely large.
Method overview: Hydrodynamic equations of cohesive granular gases, which are compressive viscous fluids with the higher order gradients and the bulk dissipation, are numerically integrated by the explicit MacCormack scheme. The applicant made all the numerical codes (except for the MD simulations) and did all the analytical calculations. The 3D-isosurface (Fig. 2(d)) is drawn by the applicant with the OpenGL libraries.