Rheology of dense polydisperse granular fluids under shear

The solution of the Enskog equation for the one-body velocity distribution of a moderately dense arbitrary mixture of inelastic hard spheres undergoing planar shear flow is described. A generalization of the Grad moment method, implemented by means of a novel generating function technique, is used so as to avoid any assumptions concerning the size of the shear rate. The result is illustrated by using it to calculate the pressure, normal stresses, and shear viscosity of a model polydisperse granular fluid in which grain size, mass, and coefficient of restitution vary among the grains. The results are compared to a numerical solution of the Enskog equation as well as molecular-dynamics simulations. Most bulk properties are well described by the Enskog theory and it is shown that the generalized moment method is more accurate than the simple (Grad) moment method. However, the description of the distribution of temperatures in the mixture predicted by Enskog theory does not compare well to simulation, even at relatively modest densities.

Figure 1

The absolute values of the kinetic part of the stress tensor (i.e., the second moments) normalized to $n{k}_{B}T$ for three densities as determined by the GME (solid lines), SME (open symbols), and DSMC (filled symbols). Note that the $xy$ moments are actually negative.

Figure 2

Same as Fig. 1, but showing results from the GME (solid lines) and MD simulations (symbols).

Figure 3

The diagonal components of the collisional contribution to the stress tensor as a function of ${\alpha }_0$ normalized to their equilibrium $({\alpha }_0=1)$ values. The lines are GME, the filled symbols DSMC, and the open symbols MD.

Figure 4

The reduced shear rate $a^{*}$ as a function of ${\alpha }_0$ as determined from the GME (lines), DSMC (filled symbols), and MD (open symbols).

Figure 5

Same as Fig. 4 but showing the reduced pressure $p^{*}=p∕n{k}_{B}T$.

Figure 6

The reduced shear viscosity and viscometric functions, as defined in Eqs. (49, 50) as functions of ${\alpha }_0$. The lines are from the GME and the symbols from MD.

Figure 7

The reduced temperature distribution $T^{*}\left(\sigma \right)=T\left(\sigma \right)∕T$ as a function of grain size, $\sigma$. The line is from the GME, the filled symbols from DSMC, the dashed line is the SME, and the dotted line is from the Boltzmann equation (also in the GME approximation).

Figure 8

The reduced temperature distribution $T^{*}\left(\sigma \right)=T\left(\sigma \right)∕T$ as a function of grain size, $\sigma$. The full line is from the GME, the open symbols are from MD, and the dashed line is from the Boltzmann equation (also in the GME approximation).