Patterns in Flowing Sand: Understanding the Physics of Granular Flow

Dense granular flows are often unstable and form inhomogeneous structures. Although significant advances have been recently made in understanding simple flows, instabilities of such flows are often not understood. We present experimental and numerical results that show the formation of longitudinal stripes that arise from instability of the uniform flowing state of granular media on a rough inclined plane. The form of the stripes depends critically on the mean density of the flow with a robust form of stripes at high density that consists of fast sliding pluglike regions (stripes) on top of highly agitated boiling material—a configuration reminiscent of the Leidenfrost effect when a droplet of liquid lifted by its vapor is hovering above a hot surface.

Figure 1

Illustration of the experimental setup.

Figure 2

(a) The phase diagram of the system presenting the mean flow density ${\rho }_r$ as a function of $\tan \theta /\tan {\theta }_r$ and $\tilde{H}=H/d$ based on data taken for sand, glass beads, and various copper samples. Various flow regimes are indicated; the bullet in the dense stripe domain corresponds to the simulation data presented. Illustration of the flow structure in (b) dense and (c) dilute regimes, gray levels indicate local density.

Figure 3

Images of the flow taken in reflected light (illumination from the right) and normalized downstream surface velocity ${\tilde{u}}^s=u^s/\sqrt{gd}$ as a function of the normalized transverse coordinate $\tilde{y}=y/d$ for sand with $d=0.2\mathrm{mm}$ and downstream distance from the outlet $x=1.55\mathrm{m}$ at plane inclinations (a) $\theta =42.6°$; (b) 48.5° and (c) 52.2°, corresponding to $\tan \theta /\tan {\theta }_r=1.56$, 1.92, and 2.19, respectively. Data obtained at $x=2.13\mathrm{m}$ for sand with $d=0.4\mathrm{mm}$ at $\theta =41.3°$: (d) height profiles $h(y)$ taken at various hopper openings $H$, (e) laserline intensity (exposure time 4 ms), (f) transmitted light intensity, and (g) dimensionless wavelength $\lambda /h_{\mathrm{av}}$ of the pattern as a function of the normalized mean flow thickness ${\tilde{h}}_{\mathrm{av}}=h_{\mathrm{av}}/d$. To adjust $h_{\mathrm{av}}$, the hopper opening $H$ was varied.

Figure 4

Results of the DEM simulations: (a) cross section of the downstream velocity $\tilde{u}$, (b) speed in the $yz$ plane ($\tilde{s}=\sqrt{{\tilde{v}}^2+{\tilde{w}}^2}$) with streamlines, (c) relative density ${\rho }_r=\eta /0.6$ or packing fraction $\eta$, (d) the inertial number $I$, (e) dependence of relative density on $I$ and best quadratic fit, and (f) dependence of effective friction $\mu$ on the inertial number $I$. Solid line is best cubic fit. Dashed line is best fit to Pouliquen model (Eq. 2 in 7).