Shear-induced crystallization of a dense rapid granular flow: Hydrodynamics beyond the melting point

Abstract

We investigate shear-induced crystallization in a very dense flow of mono-disperse inelastic hard spheres. We consider a steady plane Couette flow under constant pressure and neglect gravity. We assume that the granular density is greater than the melting point of the equilibrium phase diagram of elastic hard spheres. We employ a Navier-Stokes hydrodynamics with constitutive relations all of which (except the shear viscosity) diverge at the crystal packing density, while the shear viscosity diverges at a smaller density. The phase diagram of the steady flow is described by three parameters: an effective Mach number, a scaled energy loss parameter, and an integer number m: the number of half-oscillations in a mechanical analogy that appears in this problem. In a steady shear flow the viscous heating is balanced by energy dissipation via inelastic collisions. This balance can have different forms, producing either a uniform shear flow or a variety of more complicated, nonlinear density, velocity and temperature profiles. In particular, the model predicts a variety of multi-layer two-phase steady shear flows with sharp interphase boundaries. Such a flow may include a few zeroshear (solid-like) layers, each of which moving as a whole, separated by fluid-like regions. As we are dealing with a hard sphere model, the granulate is fluidized within the "solid" layers: the granular temperature is non-zero there, and there is energy flow through the boundaries of the "solid" layers. A linear stability analysis of the uniform steady shear flow is performed, and a plausible bifurcation diagram of the system, for a fixed m, is suggested. The problem of selection of m remains open.

Dense Granular Poiseuille Flow

“…Importantly, the particles inside the cluster are fluidized, and the granular temperature is larger than zero. Similar fluid-solid coexistence was observed in dense granular Couette flows [20,18,22]. Particles inside the plug form an ordered crystalline structure, though the density is smaller than n max .…”

“…This is a two-phase flow, where a dense "solid-like" cluster is surrounded by two fluid layers, similarly to what was recently found in the Couette geometry [18,20,21,22]. The velocity does not vary across the cluster; the cluster moves as a solid block.…”

Towards dense, realistic granular media in 2D

“…5.1, see also [51,68,100,103,[151][152][153][154], which can lead to shear-banding [52,54,131,155], or even to horizontal heat flux [156], i.e., a higher order phenomenon, possibly related to anisotropy. This anisotropy 10 is special about granular systems not only in vibrated systems [157,158], but also under shear [77], where a finite first normal stress difference shows up even for low densities [78], and interestingly changes sign when density is increased [73].…”

“…Of special interest is structure formation during free dissipative cooling of a granular gas [25,[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. Starting from a homogeneous system, structures evolve and a dilute granular gas coexists with denser, possibly much denser and even solid-like clusters -in non-equilibrium.…”

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

“…A satisfactory description of granular gases at moderate densities (at a fraction of the close-packing density) is provided by a Navier-Stokes granular hydrodynamics with the equation of state of Carnahan and Starling [44] and semi-empiric transport coefficients derived from kinetic theory in the spirit of Enskog approach [45]. A hydrodynamic description at even higher densities is even more challenging, although some particular flow regimes at very high densities have been successfully described by using empirical constitutive relations [12,[46][47][48] and, when necessary, by taking into account a multi-phase character of the flow [49,50]. term in the transformed energy equation.…”