Spatial correlations in sheared isothermal liquids: From elastic particles to granular particles

Spatial correlations in sheared isothermal liquids for both elastic and granular cases are theoretically investigated. Using the generalized fluctuating hydrodynamics, correlation functions for both the microscopic scale and the macroscopic scale are obtained. We find the existence of long-range correlations obeying power laws. The validity of our theoretical predictions has been verified from molecular-dynamics simulation.

Figure 1

The density correlation function ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.185$ with $L=89\sigma$ and $e=0.83$ as a function of the distance $r$. The solid line represents $g_0(r,e)$ obtained by Lutsko [6].

Figure 2

The density correlation function ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.37$ with $L=72\sigma$ and $e=0.90$ as a function of the distance $r$. The solid line expresses $g_0(r,e)$ obtained by Lutsko [6].

Figure 3

The density correlation function ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.50$ with $L=32.5\sigma$ and $e=0.90$ as a function of the distance $r$. The solid line expresses $g_0(r,e)$ obtained by Lutsko [6].

Figure 4

The double-log plot of ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.185$ with $L=89\sigma$, and $e=0.83$ as a function of the distance $r$. Here, the solid line is that for the homogeneous case obtained by Lutsko [6], and the plotted data are obtained in the case of $\dot{\gamma }=0.92{\tau }_0^{-1}$ and $1.84{\tau }_0^{-1}$.

Figure 5

The double-log plot of ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.37$ with $L=72\sigma$, $e=0.90$ as a function of the distance $r$. Here, the solid line is that for the homogeneous case obtained by Lutsko [6].

Figure 6

The double-log plot of ${\overline{C}}_{nn}\left(r\right)$ for the volume fraction $\varphi =0.50$ with $L=32.5\sigma$, $e=0.90$ as a function of the distance $r$. Here, the solid line is that for the homogeneous case obtained by Lutsko [6].

Figure 7

The double-log plot of ${\overline{C}}_{pp}\left(r\right)$ for the volume fraction $\varphi =0.37$ with $L∕\sigma =36$, 72 and $e=0.90$ as a function of the distance $r$. Here, $a$ is a fitting parameter, which is 0.4 for $L=36\sigma$ or 1.0 for $L=72\sigma$.

Figure 8

The double-log plot of ${\overline{C}}_{pp}\left(r\right)$ for the volume fraction $\varphi =0.50$ with $L=32.5\sigma$ and $e=0.90$ as a function of the distance $r$.

Figure 9

The double-log plot of ${\overline{C}}_{pp}\left(r\right)$ for the volume fraction $\varphi =0.093$ with $L=112\sigma$ and $e=1.0$ at $t=40{\tau }_0$ as a function of the distance $r$.

Figure 10

The double-log plot of ${\overline{C}}_{pp}\left(r\right)$ for the volume fraction $\varphi =0.37$ with $L=72\sigma$ and $e=1.0$ at $t=30{\tau }_0$ as a function of the distance $r$.

Figure 11

The profile of the velocity $u\left(y\right)$ along the flow direction as a function of the position $y$ for $\varphi =0.37$, $e=0.90$, and $L=32\sigma$ with $t=20{\tau }_0$.

Figure 12

The time dependence of ${\overline{C}}_{pp}\left(r\right)$ for the volume fraction $\varphi =0.37$ at $t=20{\tau }_0,40{\tau }_0,60{\tau }_0$ for $L=36\sigma$ and $72\sigma$ in the case of $e=0.90$.