Sediment motion induced by Faraday waves in a Hele-Shaw cell

The interaction between the oscillatory boundary-layer flow induced by Faraday waves and a sedimentary granular layer was studied in a Hele-Shaw cell vertically vibrated. The experimental parameters were the vibration frequency $f$ and acceleration $a$ and the particle diameter $d_p$. At a critical value for the depth of the supernatant fluid layer $\mathrm{\Delta }{h}_c$, a transition between a flat motionless granular layer and a second regime in which the granular layer undulates and oscillates periodically was observed. For the smallest value of $d_p$ (for which the Stokes number was $\text{St}\ll 1$) the reduced acceleration $\mathrm{\Gamma }=a/g$ ($g$ is the acceleration of gravity) is independent of $\mathrm{\Delta }{h}_c$, while for the larger ones, $\mathrm{\Gamma }$ depends linearly on $\mathrm{\Delta }{h}_c$. Finally, it is shown that at the onset of grain motion, the wave velocity $V_w=h_{w}f/2$ ($h_w$ is the wave amplitude) depends linearly on $\mathrm{\Delta }{h}_c$ and is independent of $d_p$.

Figure 1

Sketch of the experimental setup.

Figure 2

Threshold of Faraday instability as a function of the reduced acceleration $\mathrm{\Gamma }$ and supernatant liquid depth $\mathrm{\Delta }h$ for $f=24\text{Hz}$. Open circles: only fluid. Solid circles: fluid plus particles.

Figure 3

Faraday waves patterns in half a cycle for $f=24\text{Hz}$, $\mathrm{\Gamma }=1.68$, and $\mathrm{\Delta }h=7.8\text{mm}$: (a) with the granular bottom and (b) with a solid bottom. See Supplemental Material [25] for a video of Faraday waves.

Figure 4

The dimensionless wave height $h_w/\mathrm{\Delta }h$ as a function of $f/\sqrt{a/\mathrm{\Delta }h}$ for measurements above the layer of motionless grains and the acrylic bottom of the cell. Equation (1) is plotted as a black dashed line for fitting coefficients $\alpha =1.21$ and $\beta =0.51$.

Figure 5

Wavelengths of the Faraday waves as a function of $\mathrm{\Delta }h$ at the onset of Faraday instability for $f=24\mathrm{Hz}$. The dashed line corresponds to the theoretical wavelength from the dispersion relation, Eq. (2), for a nonviscous fluid in an unbounded domain.

Figure 6

Bed profiles at the Faraday threshold for $f=24\text{Hz}$ and $\mathrm{\Delta }h/w=2.5$. See Supplemental Material [25] for a video of Faraday waves.

Figure 7

The amplitude of the sediment profile wave $h_p$ as a function of the supernatant liquid depth $\mathrm{\Delta }h$ for $f=24\text{Hz}$. The onset of sediment motion occurs at $\mathrm{\Delta }{h}_c\simeq 7.1\text{mm}$.

Figure 8

Gray level standard deviation map, showing active zones of moving grains in the granular bed: (a) the active layer of sedimented particles shown below the Faraday waves extends over several grain sizes ($\mathrm{\Delta }h=4.7\text{mm}$), and (b) below the threshold (for $\mathrm{\Delta }h>\mathrm{\Delta }{h}_c$) there are still waves, but no grain displacement is perceived ($\mathrm{\Delta }h=7.5\text{mm}$).

Figure 9

Phase diagram in the parameters plane $\mathrm{\Gamma }-\mathrm{\Delta }h$ for $f=24\text{Hz}$ and $d_p=40\mu \text{m}$.

Figure 10

Height of the sediment profile $h_p$ as a function of the supernatant fluid layer $\mathrm{\Delta }h$ for three values of the forcing frequency $f$ at the onset of Faraday waves.

Figure 11

Experimental relation between the depth of the supernatant liquid at the onset of grain motion $\mathrm{\Delta }{h}_c$ and the velocity scale based on the wave height, $V_w=h_{w}f/2$.

Figure 12

Parameter plane $\mathrm{\Gamma }-\mathrm{\Delta }h$ for (a) $d_p=140\mu \text{m}$ and (b) $d_p=250\mu \text{m}$.

Figure 13

Wave based velocity scale, $V_w=h_{w}f/2$, against the critical depth of the supernatant fluid $\mathrm{\Delta }{h}_c$ for different grain diameters $d_p$.

Figure 14

(a) The length ${\lambda }_w$ and (b) the amplitude $h_w$ of the Faraday waves, measured at the onset of grain motions, as a function of the forcing frequency $f$ and parameterized in $d_p$.