Kinetic theoretical study of a simply sheared two‐dimensional granular gas to Burnett order

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∝γ2 l 2/ε, where l is the mean free path, γ is the shear rate and ε≡1−e 2, 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.