Granular fountains: Convection cascade in a compartmentalized granular gas

This paper extends the two-compartment granular fountain [D. van der Meer, P. Reimann, K. van der Weele, and D. Lohse, Phys. Rev. Lett. 92, 184301 (2004)] to an arbitrary number of compartments: The tendency of a granular gas to form clusters is exploited to generate spontaneous convective currents, with particles going down in the well-filled compartments and going up in the diluted ones. We focus upon the bifurcation diagram of the general $K$-compartment system, which is constructed using a dynamical flux model and which proves to agree quantitatively with results from molecular dynamics simulations.

Figure 1

(Color online) (a) Granular fountain in $K=2$ compartments: the clustering effect induces a steady convective current in the system [13]. (b) One of the convection patterns for a cyclic array of $K=5$ compartments. (c), (d) The two convective states in a cyclic array of $K=3$ compartments. Both states are threefold degenerate, since the cluster (the region of low density) can be located in any of the three compartments.

Figure 2

(Color online) Bifurcation diagram for the two-compartment granular fountain, depicting the steady states of the system as a function of the driving parameter $B$ defined in Eq. (2) (from [13]). This diagram contains experimental measurements [indicated by the (black) error bars], data from molecular dynamics simulations using uniform (asterisks) and clustered (stars) initial conditions, and the theoretical prediction from our flux model described in Sec. 3 (gray lines) with $\lambda =0.018$. The stable states are indicated by a solid line, the unstable ones by a dashed line. The model parameter $\lambda$, representing the ratio of the fluxes through hole and slit in the limit of very strong shaking, will be defined more precisely in Sec. 3.

Figure 3

(Color online) Two typical time-evolution plots of the cyclic three-compartment granular fountain, obtained from particle simulations with 600 particles. The quantity $n_k$ along the vertical axis denotes the particle fraction in compartment $k$ with respect to the uniform distribution: $n_k=N_k∕200$, with $N_k$ the number of particles in compartment $k$. (a) starting from the uniform distribution, with 200 particles in each compartment, at $(af{)}^{-2}=1.11\times {10}^3(\mathrm{m}∕\mathrm{s}{)}^{-2}$; (b) starting from the clustered state in which one compartment contains all 600 particles, at $(af{)}^{-2}=1.89\times {10}^3(\mathrm{m}∕\mathrm{s}{)}^{-2}$. In both cases the system evolves to a steady two-cluster state in which two compartments share most of the particles and one compartment is nearly empty.

Figure 4

(Color online) Bifurcation diagram of the cyclic granular fountain with $K=3$ compartments, from particle simulations with 600 particles. Asterisks (blue) are steady states obtained from the uniform initial distribution with 200 particles in each compartment; stars (black) are reached from an initial one-cluster state; solid dots (red) are obtained from an initial two-cluster state. The gray curves (drawn as aids to the eye) outline the stable branches of the bifurcation diagram (cf. Fig. 9); the dashed line represents the interval where the uniform state is unstable.

Figure 5

(Color online) Bifurcation diagram, obtained from particle simulations with 1000 particles, showing the stable steady states of the cyclic five-compartment system. The various symbols denote the different initial states of the simulation: asterisks (blue) are obtained from a uniform initial distribution (200 particles in each compartment); stars (black) and open triangles (black) from a one-cluster state; solid dots (red) from a two-cluster state; squares (green) from a three-cluster state; triangles (magenta) from an initial four-cluster state. The gray lines help to distinguish the stable branches of the bifurcation diagram (cf. Fig. 11). Unlike in Fig. 4, here the fractions in the dilute compartments belonging to each of the multicluster states cannot be distinguished from each other. The shaded areas are explained in the main text.

Figure 6

(Color online) Top row: The outflux $H\left(n_k\right)$ from compartment $k$ as a function of its particle contents $n_k$ [(blue) solid curves], being the sum of the flux through the slit $F\left(n_k\right)$ [(green) dotted curves] and the flux through the hole $G\left(n_k\right)$ [(red) dashed curves], for three different values of the parameter $\lambda$ (related to the size of the hole). (a) The critical value ${\lambda }_c=e^{-2}\approx 0.1353$, $B=0.75$; (b) $\lambda =0.08$, $B=0.75$; (c) $\lambda =0.02$, $B=1.50$. Bottom row: The associated bifurcation diagrams for the two-compartment system. (d) For $\lambda \ge {\lambda }_c$ the hole is too large to allow for clustering, so the system always remains uniform; (e) at $\lambda =0.08$ there is an interval of $B$ values for which the uniform state is unstable (dashed) and the system goes into a stable convective clustered state; (f) for $\lambda =0.02$ we see again a new feature, namely, an interval $(5.0\lesssim B\lesssim 7.5)$ where the uniform and convective state are both stable; the corresponding size of the hole is close to its minimum value of one particle diameter.

Figure 7

(Color online) Flux function $\tilde{H}\left(z_k\right)$ as a function of the new variable $z_k=\sqrt{B}{n}_k$, for a fountain setup with a relatively small hole $(\lambda =0.02)$. The steady states of the system must satisfy the constant-flux condition represented by the dashed horizontal line [Eq. (13)]. For ${\tilde{H}}_{\min }<\tilde{H}<{\tilde{H}}_{\max }$ this condition has three solutions $z_{-}$, $z_0$, and $z_{+}$ (which may be combined in any order to construct a steady state; see Fig. 8), whereas for $\tilde{H}$ outside this interval there is only one solution, which means that in these regimes only a uniform steady state exists.

Figure 8

(Color online) (a) The ten sum functions $S_{i,j,k}$ for $K=3$ compartments, i.e., the sum of $i$ times $z_{-}$ (on the lower branch of the dashed gray S-shaped curve), $j$ times $z_0$ (on the middle branch), and $k$ times $z_{+}$ (on the upper branch), with $i+j+k=K=3$. The uniform state (with three equal components) is represented by the successive sum functions $S_{3,0,0}$, $S_{0,3,0}$, and $S_{0,0,3}$ (solid black lines); each transition from one sum function to the next signals a bifurcation and associated change of stability. The one-cluster state is represented by the sum functions $S_{2,1,0}$, $S_{2,0,1}$, and $S_{0,2,1}$, respectively. The two-cluster state is represented by $S_{1,2,0}$, $S_{1,0,2}$, and $S_{0,1,2}$. Finally, the central sum function $S_{1,1,1}$ (solid red line) corresponds to a state with a different fraction in each of the three compartments. (b) Graphical representation of the conservation condition $S_{i,j,k}\left(\tilde{H}\right)=K\sqrt{B}$ [Eq. (15)], from which (given any value of $B$, here $B=9$) the steady-state fractions $z_k$ are determined. These are the solid dots (blue) on the S-shaped curve (red), which—translated back to the fractions $n_k=z_k∕\sqrt{B}$—yield the bifurcation diagram of Fig. 9.

Figure 9

(Color online) Bifurcation diagram of the three-compartment system, evaluated from the flux model. It shows both the stable steady states (solid lines), which agree well with the ones found from numerical experiments in Fig. 4, and the unstable states (dashed lines). The S-shaped curve at $B\approx 4$ (red) corresponds to a steady state that is always unstable, consisting of three different fractions $n_k$ in the three compartments. The dashed line at $B=9$ is the counterpart of the horizontal dashed line in Fig. 8b; the three coexisting steady states at this value of $B$ (uniform state, unstable two-cluster state, and stable two-cluster state) are discussed in the text.

Figure 10

(Color online) Bifurcation structure of the three-compartment granular fountain, highlighted at six successive values of the parameter $B$. The triangular plots (a)–(f) depict the configuration space, as explained in the text; white regions indicate the basin of attraction of the uniform steady state (the fixed point in the center), light gray (yellow) regions correspond to the three one-cluster states (the fixed points toward the corners of the triangle), and dark gray (blue) regions correspond to the three two-cluster states (the fixed points close to the middle of the sides of the triangle). Beyond $B\approx 11$ the situation of plot (a) is recovered again.

Figure 11

(Color online) Bifurcation diagram of the five-compartment system, evaluated from the flux model with $\lambda =0.02$. The stable steady states (solid lines) agree well with the ones found from numerical experiments (Fig. 5) and are seen to be interconnected via a whole maze of unstable steady states (dashed lines, increasingly fragmented with increasing number of unstable eigenvalues). Among the unstable states are also six states with three different particle fractions [the S-shaped (red) lines]; these never become stable, but play an important role in stabilizing and destabilizing the regular fountain states with only two different fractions.

Figure 12

(Color online) Construction of a stable two-fraction fountain state, with the hot (dilute) compartments containing a fraction $z_{-}∕\sqrt{B}$ and the cold (dense) compartments $z_{+}∕\sqrt{B}$. (a) The flux function $H\left(z\right)$ (blue curve) and its derivative $H^{\prime }\left(z\right)$ (red curve) for $\lambda =0.02$. The solid dots are pairs of points that satisfy the flux balance $H\left(z_{+}\right)=H\left(z_{-}\right)$ and the derivative equality $H^{\prime }\left(z_{+}\right)=H^{\prime }\left(z_{-}\right)$, respectively. The arrows indicate the direction in which these pairs evolve as the shaking parameter $B$ is increased. (b) The curves of $z_{+}$ vs $z_{-}$ corresponding to the condition $H\left(z_{+}\right)=H\left(z_{-}\right)$ (blue) and to $H^{\prime }\left(z_{+}\right)=H^{\prime }\left(z_{-}\right)$ (red). These curves cross each other in one point: the corresponding two-fraction state is stable, as explained in the text.