Universality of shear-banding instability and crystallization in sheared granular fluid

Abstract

The linear stability analysis of an uniform shear flow of granular materials is revisited using several cases of a Navier-Stokes'-level constitutive model in which we incorporate the global equation of states for pressure and thermal conductivity (which are accurate up-to the maximum packing density ν m ) and the shear viscosity is allowed to diverge at a density ν µ (< ν m ), with all other transport coefficients diverging at ν m . It is shown that the emergence of shear-banding instabilities (for perturbations having no variation along the streamwise direction), that lead to shear-band formation along the gradient direction, depends crucially on the choice of the constitutive model. In the framework of a dense constitutive model that incorporates only collisional transport mechanism, it is shown that an accurate global equation of state for pressure or a viscosity divergence at a lower density or a stronger viscosity divergence (with other transport coefficients being given by respective Enskog values that diverge at ν m ) can induce shear-banding instabilities, even though the original dense Enskog model is stable to such shear-banding instabilities. For any constitutive model, the onset of this shear-banding instability is tied to a universal criterion in terms of constitutive relations for viscosity and pressure, and the sheared † M. Alam, P. Shukla and S. Luding granular flow evolves toward a state of lower "dynamic" friction, leading to the shear-induced band formation, as it cannot sustain increasing dynamic friction with increasing density to stay in the homogeneous state. A similar criterion of a lower viscosity or a lower viscous-dissipation is responsible for the shear-banding state in many complex fluids. M. Alam, P. Shukla and S. Ludingiting cases of these models. One important finding is that a global equation of state (without viscosity divergence at a lower density) leads to a shear-banding instability in the framework of a "dense" model that incorporates only collisional transport mechanism. Even with the local equation of state, if we use a constitutive relation for viscosity that has a stronger divergence (at the same maximum packing density) than other transport coefficients, we recover shear-banding instabilities in the framework of the dense-model of Haff (1983). This brings us to a crossroad: is there any connection among the results of Alam & Nott, Khain & Meerson, and the present work? Is there any universal criterion for the onset of the shear-banding instability in granular shear flow? Such an universality for the shear-banding instability indeed exists, solely in terms of the constitutive relations, as we show in this paper. Possible connections of the present criterion of a lower dynamic friction for the shear-banding state to explain the onset of shear-banding in many complex fluids as well as in an elastic hard-sphere fluid are discussed.

Dense Granular Poiseuille Flow

“…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.…”

Steady base states for non-Newtonian granular hydrodynamics

“…This may have implications to segregation conditions for a granular impurity (Jenkins & Yoon 2002;. Moreover, a complete determination of the steady base states is convenient for studies of instabilities (Hopkins & Louge 1991;Wang, Jackson & Sundaresan 1996;Alam & Nott 1998;Nott, Alam, Agrawal, Jackson & Sundaresan 1999;Khain & Meerson 2003;Alam, Shukla & Luding 2008). Furthermore, analogous temperature curvature properties are observed for the same geometry in moderately dense granular gases, except that for higher densities a region with temperature curvature inflections grows from the boundaries (Lun 1996;Alam & Nott 1998).…”

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].…”

“…The most recent results, to our knowledge, involve shear bands with dilute zones with high shear-rate and static, solid-like plugs without internal shear [52,54,100,103], but no effect of elasticity. Different equations of state at different densities and dissipation strengths were examined [131], but no systematic study of the elasticity was performed yet, to our knowledge.…”