Figure 1

(Color online) Snapshots of a gas-fluidized bed of a binary mixture of identical size but different density particles, undergoing spontaneous demixing, shown at times equally separated by $\mathrm{\Delta }t=100$. Light-colored (yellow) particles are twice as dense as the dark-colored (red) ones ($e=0.9$; $\mu =0.1$; $k_n=2.0\times {10}^5$; $U_s=0.41$; $d_p=100\mathrm{\mu }\mathrm{m}$; ${\rho }_s=0.90\mathrm{g}∕{\mathrm{cm}}^3$).Reuse & Permissions

Figure 2

(Color online) (Left panel) Snapshot of the bed at $t=150$ in Fig. 1. (Right panel) Time evolution of inverse CDFs (ICDFs) of the distribution of lighter particles (dark-colored; red in color online). Each line is separated by $\mathrm{\Delta }t=50$, started from a fully mixed, fully fluidized state.Reuse & Permissions

Figure 3

(Color online) An illustration of coarse projective integration for demixing. Starting from a coarse-grained initial condition (the first circle, representing an instantaneous ICDF), (1) the detailed model is appropriately initialized (through lifting) and (2) integrated for a short time, during which (3) a series of ICDFs are recorded (through restriction). By making use of temporal smoothness of ICDF evolution, (4) a local temporal slope is estimated (solid straight line), and an ICDF at a significantly later future time is projected (dashed line); the whole procedure is repeated in time. The solid line represents ICDF evolution from a full-time detailed MD-based simulation.Reuse & Permissions

Figure 4

(Color online) Direct, full integration (thin lines) and coarse projective integration (patches of thick lines), using an even discretization of the particle ICDFs as the coarse observables. (a) Slow demixing at $U_s=0.19$, where the bed hardly expands and no 1D-TWs form. Projective steps of $\mathrm{\Delta }t=80$ were taken through forward Euler, after direct integration for $\delta t=20$ (the latter half of data were used to estimate the local slope). (b) In the presence of 1D-TWs $(U_s=0.41)$, $\delta t=30$ and $\mathrm{\Delta }t=2T$, where $T$ is the average period estimated during short bursts of direct integrations; the last two periods of the locally oscillating data were used to estimate the coarse slope.Reuse & Permissions

Figure 5

(Color online) Evolving inverse CDFs (ICDFs) for the same case as in Fig. 1, averaged over 10 realizations: (A) ICDF of particles physically located in the upper half of the bed and (B) ICDF of the lighter particles, which are shown at $t=150$. Solid lines with negative slopes are snapshots of the difference between (A) and (B) at different times; compare with the functional fit shown as dashed lines; see Eq. (10).Reuse & Permissions

Figure 6

(Color online) Comparison between direct, full integrations (solid lines) and coarse projective integrations (groups of dots), using the four coarse observables in Eq. (10).Reuse & Permissions