Continuum description of finite-size particles advected by external flows: The effect of collisions

The equation of the density field of an assembly of macroscopic particles advected by a hydro-dynamic flow is derived from the microscopic description of the system. This equation allows one to recognize the role and the relative importance of the different microscopic processes implicit in the model: the driving of the external flow, the inertia of the particles, and the collisions among them. The validity of the density description is confirmed by comparisons of numerical studies of the continuum equation with direct simulation Monte Carlo simulations of hard disks advected by a chaotic flow. We show that the collisions have two competing roles: a dispersinglike effect and a clustering effect (even for elastic collisions). An unexpected feature is also observed in the system: the presence of collisions can reverse the effect of inertia, so that grains with lower inertia are more clusterized.

Figure 1

Distributions of the residual term (left, $x$ component; right, $y$ component) of Eq. (8) which is assumed to be a Gaussian white noise. The form of the distribution and the self-correlation (displayed in the insets) confirm this assumption. The flow used in this simulation is a sinusoidal shear, i.e., $u_x\mathbf{\left(}{\mathbf{r}}_i\right(t),t\mathbf{)}=U\cos \left[2\pi y_i\right(t)∕L]$, $u_y=0$, where ${\mathbf{r}}_i\left(t\right)=\left[x_i\right(t),y_i(t\left)\right]$. For the upper plot (large ${\tau }_p$) the time step used in the DSMC is $\Delta t=0.0001$, size of the DSMC cells is $0.6\times 0.6$, mean free time is $0.01$, mean free path $5$, length of the time period used to calculate $\eta$ is $\overline{\Delta t}=0.005$, $\widehat{{\tau }_p}=1$, $r=1$, $N=L^2=1000$, $\sigma =0.3$, and $U=\mathrm{10\hspace{0.17em}000}$ so that ${\tau }_p=U\widehat{{\tau }_p}∕L\sim 300$. For the bottom plot (small ${\tau }_p$) $\Delta t=0.01$, DSMC cells are $2\times 2$, mean free time is $0.6$, mean free path $18$, $\overline{\Delta t}=0.5$, $\widehat{{\tau }_p}=1$, $r=1$, $N=L^2=\mathrm{10\hspace{0.17em}000}$, $\sigma =0.6$, and $U=50$ so that ${\tau }_p=0.5$.

Figure 2

(Color online). Instantaneous density patterns, obtained with DSMC, for the $2d$ cellular flow, $N=\mathrm{500\hspace{0.17em}000}$ and $L=\sqrt{N}$. Panels (b) and (d) are without collisions. Frames (a) and (b): $U=30$, $B_0=1\omega =1({\tau }_f\sim 0.4)$, $\widehat{{\tau }_p}=10$, $\sigma =0.3$, $r=1$, ${\tau }_c\sim 0.02$ [only for (a)]. Panels (c) and (d): $U=110$, $B_0=1$, $\omega =10({\tau }_f\sim 0.3)$, ${\tau }_p=0.003$, $\sigma =0.5$, $r=1$, ${\tau }_c\sim 0.0002$ [for (c)]. come from DSMC. Note that a small part $(\sim 2\%)$ of the whole system is potrayed. Solid (red) lines represent streamlines of the flux with $B_0=0$. The inset in (b) shows these for the entire spatial domain.

Figure 3

$P_M\left(n\right)$ function for the distribution of particles obtained from DSMC (right panels) and for the density of particles from the continuum equation (16) (left) for the same parameter values. Upper right are the cases in Figs. 2a, 2b, that is, with high inertia, and the lower right corresponds to the Figs. 2c, 2d panels (low inertia).