The Chapman–Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions – to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the ‘temperature anisotropy’, characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier–Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to ε∇ log n, where ε is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to ε∇n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman–Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.

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The kinetics of a collection of inelastically colliding smooth disks in a plane, in a state of constant shear rate, is studied by performing an analysis of the pertinent Boltzmann equation. The fact that the granular temperature T satisfies T∝γ2ℓ2/ε, where ℓ is the mean free path, γ is the shear rate and ε≡1−e2, where e is the coefficient of normal restitution, leads to the observation that when γ∝√ε the limit ε→0 of the above problem corresponds to a system of elastically colliding particles in equilibrium (at temperature T). This observation enables the construction of a systematic perturbative expansion (for the single particle distribution function) in powers of √ε , in which the equilibrium (Maxwellian) distribution function serves as zeroth order. The limitations of this expansion are discussed alongside possible generalizations. Explicit expressions for the single particle distribution function to O(ε) and expressions for the corresponding stress tensor are obtained. The phenomenon of normal stress difference is shown to be of O(ε), i.e. of second (Burnett) order in the shear-rate and its calculated magnitude compares well with results of numerical simulations. A comparison of the present theory with that of Jenkins and Richman is presented as well.

The question whether one-dimensional granular systems can be described by hydrodynamic equations is the main theme of the present work. Numerical simulations are used to create a database with which theory is compared. The system investigated in the numerical work is that of a one-dimensional collection of point particles colliding inelastically. The dependence of the dynamical properties on both the degree of inelasticity and the number of particles is investigated. A hydrodynamic theory which describes the large-scale motion of such systems has been developed. It is shown that the standard set of hydrodynamic fields (density, velocity, and granular temperature) is insufficient for this purpose and that an additional hydrodynamic field corresponding to the third moment of the fluctuating velocity field must be added to that set. The results of a linear stability analysis of the derived hydrodynamic equations are in a close agreement with those of the numerical simulations. The question of the effects of velocity correlations on the hydrodynamics is addressed as well. It is shown that these correlations, though not negligible, do not affect the hydrodynamic equations. The form of the single particle initial distribution function is shown to slightly affect the form of the hydrodynamic equations for transient times. Except for this minor effect the hydrodynamic equations possess a universal form. Possible implications for higher dimensional systems are mentioned.

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