Orientational Correlation and Velocity Distributions in Uniform Shear Flow of a Dilute Granular Gas

Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth ($\beta =-1$) and rough ($\beta =1$) limits for elastic collisions ($e=1$); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases. Away from these two limits, the orientational as well as spatial velocity correlations are responsible for the emergence of non-Gaussian high-velocity tails. The tails of both distribution functions follow stretched exponentials, with the exponents depending on normal ($e$) and tangential ($\beta$) restitution coefficients.

Figure 1

(a) Variation of $\Lambda (t)={cos}^2\Psi$ with time for different values of $e$ with $\beta =0$, $\varphi =0.01$, and $N=8000$. Left and right insets show the distributions of ${cos}^2\Psi$ and $\cos \Psi$, respectively. (b) Variation of $⟨{cos}^2\Psi ⟩$ with $\beta$ for different $e$. Larger symbols (triangle and hatched circle) at $\beta =0$ for each $e$ correspond to simulations with $N=4000$ and 16 000, respectively.

Figure 2

Probability distributions of mean density (main panel) and rotational temperature (right upper inset) for different values of $e$. Right lower inset shows the variation of temperature ratio, $\theta /(\theta +T)$, with $\beta$. Two left insets show projected particle snapshots in the $xy$ and $yz$ planes at steady states for $e=0.5$.

Figure 3

(a) Translational (main panel) and rotational (lower inset) VDFs for $\beta =-0.999$. (b) Rotational (main panel) and translational (lower inset) VDFs for $\beta =0$. The upper right inset in each panel shows the deviation of the low-velocity regions from a Gaussian: $\Delta P({\mathbf{C}}_x)$ in Fig. 3a and $\Delta P({\mathbf{\Omega }}_z)$ in Fig. 3b.

Figure 4

Variation of the exponent for the stretched exponential, ${\alpha }_i$, with $\beta$; translational (main panel, ${\alpha }_{\mathbf{C}}$) and rotational (inset, ${\alpha }_{\mathbf{\Omega }}$) velocity. Typical error bars are shown on few data points.

Figure 5

Spatial velocity correlation: $\beta =0$ (main panel) and $\beta =-0.999$ (inset).