Stability of Freely Falling Granular Streams

A freely falling stream of weakly cohesive granular particles is modeled and analyzed with the help of event driven simulations and continuum hydrodynamics. The former show a breakup of the stream into droplets, whose size is measured as a function of cohesive energy. Extensional flow is an exact solution of the one-dimensional Navier-Stokes equation, corresponding to a strain rate, decaying like $t^{-1}$ from its initial value, ${\dot{\gamma }}_0$. Expanding around this basic state, we show that the flow is stable for short times, ${\dot{\gamma }}_{0}t\ll 1$, whereas for long times, ${\dot{\gamma }}_{0}t\gg 1$, perturbations of all wavelengths grow. The growth rate of a given wavelength depends on the instant of time when the fluctuation occurs, so that the observable patterns can vary considerably.

Figure 1

Snapshots of the system for different times; colors (gray scales) indicate relative velocities $\Delta v$ of cohesively interacting particles; small droplets of size $N_{\mathrm{drop}}<100$ are ignored for better visibility. See Ref. 19 for a movie.

Figure 2

Mean droplet size $⟨N_{\mathrm{drop}}⟩$ as a function of $w=W_{\mathrm{coh}}/W_{\mathrm{coh},\exp }$; data points are results from the simulation and the solid line is a power law fit; inset: mean droplet size $⟨N_{\mathrm{drop}}⟩$ as a function of time for $w=8$ (high final value) and $w=1$ (low final value); at the grey vertical line all systems have reached a steady state, which is used for the main plot.

Figure 3

Length vs width of individual clusters with $N_{\mathrm{drop}}\ge 1000$; $w=1$ (disk), $w=3$ (triangle), and $w=8$ (square); gray lines correspond to aspect ratios of 1 and 3; insets: time ${\tau }_c={\dot{\gamma }}_0{t}_c$ for an imposed perturbation of wave number $k$ (in units of particle diameter $d$) to grow beyond a given value $A_c$ (1.7, 1.5, 1.3, 1.1 from top to bottom); left inset: simulation with $r_0=10d$; right inset: analytical theory

Figure 4

Real part of the unstable eigenvalue as a function of wave number and time. $\tilde{\Gamma }$ has the experimental value ${\tilde{\Gamma }}_{\exp }$ and $\nu =0$.

Figure 5

Example for growth of perturbations, obtained from integrating the linearized equations; inset: most unstable wave number and time of growth to increase by 50% in dependence on $w=\Gamma /{\Gamma }_{\exp }$.