Hydrodynamic equations for a granular mixture from kinetic theory - fundamental considerations

In this review, a theoretical description is provided for the solid (granular) phase of the gas-solid flows that are the focus of this book. Emphasis is placed on the fundamental concepts involved in deriving a macroscopic hydrodynamic description for the granular material in terms of the hydrodynamic fields (species densities, flow velocity, and the granular temperature) from a prescribed “microscopic” interaction among the grains. To this end, the role of the interstitial gas phase, body forces such as gravity, and other coupling to the environment are suppressed and retained only via a possible nonconservative external force and implicit boundary conditions. The general notion of a kinetic equation is introduced to obtain macroscopic balance equations for the fields. Constitutive equations for the fluxes in these balance equations are obtained from special “normal” solutions to the kinetic equation, resulting in a closed set of hydrodynamic equations. This general constructive procedure is illustrated for the Boltzmann-Enskog kinetic equation describing a system of smooth, inelastic hard spheres. For weakly inhomogeneous fluid states the granular Navier-Stokes hydrodynamic equations are obtained, including exact integral equations for the transport coefficients. A method to obtain practical solutions to these integral equations is described. Finally, a brief discussion is given for hydrodynamics beyond the NavierStokes limitations. INTRODUCTION Activated granular materials occur ubiquitously in nature and practical realizations in industry. Many of the phenomena occur on length scales that are large compared to the size of constituent particles (grains) and time scales long compared to the time between collisions among the grains. In this case a description of the system in terms of the values for hydrodynamic fields in cells containing many particles, analogous to molecular fluids, can apply for granular fluids. The hydrodynamic fields for molecular fluids are the densities associated with the globally conserved quantities. In the absence of reactions, these are the species densities, total momentum density, and the energy density. More commonly the momentum density is replaced by a corresponding flow field, and the energy density is replaced by a related temperature. The time dependence of such a macroscopic description (hydrodynamics) follows from the exact conservation equations for these fields, supplemented by “constitutive equations” providing a closed description in terms of the fields alone. The key difference between granular and molecular fluids is that the former involves collisions between macroscopic grains. These collisions conserve momentum but dissipate energy since part of the kinetic energy of the grains goes into micro-deformations of the surface and exciting other internal modes of the grains. Even so, a hydrodynamic description for a fluid of grains can be given under appropriate conditions, following closely the approach developed in the context of molecular fluids, starting from the exact “balance equations” for the densities of interest. The objective of this chapter is to provide an overview of how general constitutive equations can be obtained from a fundamental basis in kinetic theory. The discussion does not make specific reference to a particular fluid state or kinetic theory. This overview is followed by a practical illustration for the special case of Navier-Stokes hydrodynamics for weakly non-uniform states, derived from the generalized Enskog kinetic theory (van Beijeren & Ernst 1973, 1979) extended to granular systems (Brey, Dufty & Santos 1997; see also Appendix A of Garzo, Dufty & Hrenya 2007). Extensive references to previous work on Navier-Stokes constitutive equations from Boltzmann and Enskog kinetic theories can be found in the review of Goldhirsch 2003, the text of Brilliantov & Poschel 2004, and in the recent articles Garzo, Dufty, & Hrenya 2007 and Garzo, Hrenya & Dufty 2007. The balance equations are local identities expressing the change in hydrodynamic fields of a cell due to their fluxes through the boundaries of that cell and local sources within the cell. The central problem is to represent the fluxes and sources in terms of these hydrodynamic fields and their gradients. In many cases the form of these constitutive equations is known from experiments (e.g., Fick’s diffusion law, Newton’s viscosity law). An important advantage of kinetic theory as the basis for constitutive equations, in contrast to such phenomenology generalized from experiment, is that both quantitative and qualitative predictions follow as mathematical consequences of the theory. Thus the form of the hydrodynamic equations, the values of their parameters, and the validity conditions for applications are provided as one unit. In practice, most applications to granular fluids have focused on low density conditions and moderate densities at low dissipation (e.g., the Boltzmann and Enskog kinetic theories) (Jenkins & Mancini 1989, Jenkins 1998, Lun 1991). However, the approach emphasized here is more general and provides a means to describe quite general complex fluid states such as those that occur more generally for granular fluids. The aim of this chapter is to provide a pedagogical overview of the basis for hydrodynamics as arising from kinetic theories (for a similar analysis based on the low density Boltzmann equation see Dufty & Brey 2005). With this goal in mind, attention is restricted to the simplest case of smooth grains interacting through pair-wise additive short ranged interactions. Other important effects such as those due to the interstitial fluid phase, gravity etc. are included only at the level of an external body force acting on the grains, and are addressed briefly in the next section. The layout of the chapter is as follows. Section 2 provides an overview of role of kinetic theory and hydrodynamics in the context of gas-solid flows, highlighting the advantages and limitations of each. In the next section the notion of a kinetic theory as a “mesoscopic” theory is introduced in its most general form. Next, the balance equations for the hydrodynamic fields are obtained from the kinetic theory with explicit expressions for the fluxes in terms of the solution to the kinetic equation. Finally, the notion of constitutive equations is introduced for special “normal” solutions to the kinetic equation. Together, the balance equations supplemented with the constitutive equations yield the closed hydrodynamic description of the fluid in terms of the local fields. These general considerations are formally exact and provide the basis for specialization to particular applications and practical approximations. The remainder of this chapter is then focused on the important case of states with small spatial and temporal variations, for which the Navier-Stokes hydrodynamic equations are obtained. The force law for particle-particle collisions in the kinetic theory is idealized to that of smooth, inelastic hard spheres, and the collision operator is specialized to a practical form (revised Enskog kinetic equation) appropriate for a wide range of space and time scales, and densities. As an important illustration, the normal solution is described for weakly inhomogeneous states as an expansion in the small spatial and temporal gradients, leading to the explicit constitutive equations and expressions for the associated transport coefficients. Finally, the need to go beyond Navier-Stokes hydrodynamics for many granular states is discussed. Applications of the Enskog kinetic theory to uniform shear flow is noted as an important example. The scope of topics covered is quite broad and a complete citation of all the important literature is not practical. Instead, in many cases reference will be given to reviews in which extensive bibliographies appear. Generally, it is hoped that the material presented is self-contained in the sense that the logical presentation can be followed even though the full details of calculations are left implicit. CONTEXT: GAS-SOLID FLOWS Before embarking on the details of a kinetic theory and its basis for hydrodynamics, it is useful to review the context of each in the description of the complex flows encountered in gas-solid systems (e.g. gas-fluidized beds). The complexity arises from a number of sources, e.g. gravitational field, geometry (boundary conditions), and formation of heterogeneous structures (macroscopic bubbles or high density clusters) (van Swaaij 1990, Gidaspow 1994). Since most features of interest occur at the laboratory scale, hydrodynamics has been a primary tool in attempts to model gas-solid flows (Jackson 2000, Kuipers, Hoomans & van Swaaij 1998). On this large scale, both gas and particle subsystems are described by Navier-Stokes continuum equations with sources coupling the two self-consistently (the Two Fluid Model, Anderson & Jackson 1967). However, the parameters of these equations such as the particle-gas force and transport properties must be supplied phenomenologically and their detailed forms can make a significant difference in the flows predicted. To overcome this limitation, a more detailed description on smaller length scales is required. One approach is to describe the particle dynamics by numerical simulation of the associated Newton’s equations of motion, while retaining a hydrodynamic description for the gas. This is the Discrete Particle Model (also referred to as the Discrete Element Method) (Hoomans, Kuipers, Briels, van Swaaij 1996; Deen, van Sint Annaland, van der Hoef & Kuipers 2007, Zhu, Zhou, Yang & Yu 2007 ). This provides a detailed and accurate description of the particle trajectories for given forces. In addition to gravity, pressure gradients, particle-particle and particle-wall forces, an essential force in Newton’s equation is that of the gas on the fluid particle. Implementation of the DPM can lead to different results depending on the nature and treatment of the forces on the particle (Feng & Yu 2004, Leboriero, Joseph & Hrenya 2008, van Wachem et al., 2007). Although this drag force on the particle is localized at the surface of the particle, it can depend on the details of the gas fluid state including the indirect influence of other particles on this state. For example, it can be different for dense monodisperse and polydisperse gassolid systems (van der Hoef, Beetstra & Kuipers 2005). Recently, accurate modeling of the gas-particle drag force has been possible using lattice gas Boltzmann methods that allow an accurate simulation of the gas on a lattice smaller than the particle size, accounting for both the momentum transfer to the particles and incorporating details of the boundary conditions. In this way lattice gas Boltzmann simulations on the smallest scale provide the needed input for DPM on the mesoscopic scale (Hill, Koch & Ladd 2001; Benyahia, Syamlal & O'Brien 2006; Yin & Sundaresan 2009; van der Hoef, van Sint Annaland & Kuipers 2004). A kinetic theory description for the particles provides an alternative to the DPM on the same scale of the particle positions and velocities. There are two main advantages of kinetic theory. First, it does not have the practical limitations of discrete particle simulations to small (compared to laboratory) systems of particles. Second, as described below, the transition to hydrodynamics and identification of its parameters is straightforward from kinetic theory, but very much less so for DPM. It requires the same input forces, both for collisions and for coupling to the gas phase (still described by Navier-Stokes equations), and so can benefit from the recent developments for DPM. On the other hand, for very dense clusters and glassy structures the form of the particle-particle collisions in kinetic theory is only known semiphenomenologically at this point. Applications of kinetic theory to gas-solid flows are mainly in the context of providing the form and parameters of the two fluid model (see however Minier & Peirano 2001). Early examples of this approach include Sinclair & Jackson 1989, and Koch (1990); for a review and references see Gidaspow, Jung, & Singh 2004. Recent improvements in the kinetic theory and its systematic application for the normal solution have led to a more accurate solid phase hydrodynamics, as described below. The coupled sets of continuum equations in the two fluid model then constitute a problem in computational fluid dynamics, often including additional assumptions for the gas phase to describe turbulent conditions, as described elsewhere in this book. The derivation of hydrodynamics (specifically, constitutive equations) generally entails necessary conditions, e.g., sufficiently small Knudsen numbers for Navier-Stokes hydrodynamics. As emphasized below, the term “hydrodynamics” includes more general fluid states with correspondingly more complex constitutive equations. In any case, such closures have associated validity conditions that must be checked before application to a given problem. The complication arising from the particle-gas drag force can affect these validity conditions significantly. The kinetic equation remains valid more generally, just as DPM, but the possibility of a hydrodynamic description might be precluded. Under these conditions, the particle hydrodynamic description must be replaced with a more general solution to the kinetic equation itself. The objective of the current chapter is focused on the method of deriving systematically hydrodynamic equations from a given kinetic equation. The specific kinetic equation considered and the complexity of the resulting hydrodynamic description can depend on the flow conditions, geometry, degree of heterogeneity, etc. and such applications are the subject of other chapters in this book. The illustration of this method given here is for an ideal granular fluid of inelastic hard spheres described by the generalized Enskog equation. It is expected that this is a good compromise between accuracy and practical utility, not limited by conditions of Knudsen number, Reynolds number, density heterogeneity, or geometry. In most respects it has both the generality of DPM and the advantages of kinetic theory. The application of that kinetic theory to Navier-Stokes hydrodynamics described here, however, has the additional more severe limitations to small Knudsen numbers. While this is the most common hydrodynamics currently in use for the two fluid description of gas-solid flows, it is clear from the derivation here that failure should be expected for many conditions of interest (e.g., bubbles, plugs, rheology). Nevertheless, the systematic derivation of Navier-Stokes hydrodynamics provides accurate results under its validity conditions for both the form of those equations as well as quantitative values for the transport coefficients. For example, the physical mechanisms that govern polydisperse mixing and separation processes (Duran Rajchenbach and Clement 1993) are not well understood yet; the recent results described here have been applied to a controlled, quantitative means to study one of those mechanisms, thermal diffusion (Garzo 2009). KINETIC THEORY AS A BASIS FOR HYDRODYNAMICS Kinetic Theory Consider a mixture of s species of smooth spherical particles with masses { }. Their sizes and material composition can be suppressed at this point as they enter only through the force laws for the particle-particle and particle-gas interactions. These are taken to be short ranged (compared to relevant cell sizes for the macroscopic description), conserve momentum, but dissipate energy. The hydrodynamic fields of interest describe a few densities at each spatial point in the system. A more complete mesoscopic description is given by the distribution of particles in the six dimensional phase space defined by the points where is the position and is the velocity of a particle. For a system with different species, there is a set of distribution functions for all the species . If these functions are normalized to unity, then each s i mi .. 1 ; = , , v r r v s ( ) { } s i t f i ,..., 1 ; ; , = v r ( ) t f i ; , v r is the probability density of finding a particle of species i at position r with velocity at the time t . In the following, the normalization is chosen instead to be the species densities so the interpretation is that of a number density of species i at . v t , , v r The species densities , energy density ( ) { t ni , r } ( ) t e , r , and momentum density are defined in terms of the distribution functions by ( t , r p ) ( ) ( ) ∫ = = s i t f d t n i i ,..., 1 , ; , , v r v r (1) ( ) ( ) ( ) ( ∫ ∑∫ = − + = s