Faraday wave instability analog in vibrated gas-fluidized granular particles

Granular materials are critical to many natural and industrial processes, yet the chaotic flow behavior makes granular dynamics difficult to understand, model, and control, causing difficulties for natural disaster mitigation as well as scale-up and optimization of industrial devices. Hydrodynamic instabilities in externally excited grains often resemble those in fluids, but have different underlying mechanisms, and these instabilities provide a pathway to understand geological flow patterns and control granular flows in industry. Granular particles subject to vibration have been shown to exhibit Faraday waves analogous to those in fluids; however, waves can only form at high vibration strengths and in shallow layers. Here, we demonstrate that combined gas flow and vibration induces granular waves without these limitations to enable structured, controllable granular flows at larger scale with lower energy consumption for potential industrial processes. Continuum simulations reveal that drag force under gas flow creates more coordinated particle motions to allow waves in taller layers as seen in liquids, bridging the gap between waves produced in conventional fluids and granular particles subject to vibration alone.

Figure 1

Images of Faraday waves in granular particles in a pseudo-2D system (a)–(c): (a) effect of $H$ on waves, (b) periodic repetition of the waves at $f/2$, and (c) effect of $f$ on waves. (d) Different waves patterns formed in a 3D system. (e), (f) Effect of $f$ on 3D wavelength with (e) showing the square patterns and (f) showing the stripe patterns. (g) shows 3D waves at different $H$. Particles: (a), (b) glass beads with $d_p=200–300$µm and ${\rho }_p=2500\mathrm{kg}/{\mathrm{m}}^3$; (c) silica-alumina beads with $d_p=53–106$µm and ${\rho }_p=1530\mathrm{kg}/{\mathrm{m}}^3$; (d)–(g) ceramic beads with $d_p=200–300$µm and ${\rho }_p=6060\mathrm{kg}/{\mathrm{m}}^3$.

Figure 2

Regime maps of $\mathrm{\Gamma }$ vs $f^{*}$ showing different wave patterns formed in a 3D bed for different $H/d_p$: (a) 8, (b) 24, (c) 40, and (d) 80 as well as different $U/U_{mf}$ (different columns). Particles: ceramic beads with $d_p=200–300$µm and ${\rho }_p=6060\mathrm{kg}/{\mathrm{m}}^3$.

Figure 3

Onset of Faraday waves in a 3D system for different gas velocities based on $f^{*}=f\sqrt{H/g}$ and (a) $\mathrm{\Gamma }$ and (b) $u_{\mathrm{tot}}^{*}=\left(2\pi Af+U-U_{mf}\right)/U_{mf}$. Particles: ceramic beads with $d_p=200–300$µm and ${\rho }_p=6060\mathrm{kg}/{\mathrm{m}}^3$. $H=2\mathrm{mm}$.

Figure 4

(a) Wavelength vs $f^{-2}$ and (b) wavelength vs $H$ for pseudo-2D and 3D systems with different patterns formed, different layer heights, and different particle properties. Lines in (a) show linear regressions through the data.

Figure 5

Continuum simulation results of patterns formed in grains with different layer heights ($H/d_p=24$ and 40) under constant gas flow conditions $\left(U/U_{mf}=1.1\right)$ with $\mathrm{\Gamma }=1.2$, no gas flow with $\mathrm{\Gamma }=1.8$, and pulsed gas flow conditions [26]. Particles: $d_p=250$µm and ${\rho }_p=2500\mathrm{kg}/{\mathrm{m}}^3$.

Figure 6

Time series of vertical slice images from continuum simulations for the cases with $H/d_p=40$ with and without constant gas flow from Fig. 5. Solids volume fraction ($\varphi$), solids pressure ($p_s$), vertical solids velocity relative to plate velocity ($v_{V,s}-v_p$), horizontal solids velocity ($v_{H,s}$), and normalized difference between drag force and solids weight $\left[(F_d-w_s\right)/w_s$] are shown.