Universal stability curve for pattern formation in pulsed gas-solid fluidized beds of sandlike particles

A granular layer can form regular patterns, such as squares, stripes, and hexagons, when it is fluidized with a pulsating gas flow. These structures are reminiscent of the well-known patterns found in granular layers excited through vibration, but, contrarily to them, they have been hardly explored since they were first discovered. In this work, we investigate experimentally the conditions leading to pattern formation in pulsed fluidized beds and the dimensionless numbers governing the phenomenon. We show that the onset to the instability is universal for Geldart B (sandlike) particles and governed by the hydrodynamical parameters $\mathrm{\Gamma }=u_{\mathrm{a}}/\left(u_{\mathrm{t}}\overline{\varphi }\right)$ and $f/f_{\mathrm{n}}$, where $u_{\mathrm{a}}$ and $f$ are the amplitude and frequency of the gas velocity, respectively, $u_{\mathrm{t}}$ is the terminal velocity of the particles, $\overline{\varphi }$ is the average solids fraction, and $f_{\mathrm{n}}$ is the natural frequency of the bed. These findings suggest that patterns emerge as a result of a parametric resonance between the kinematic waves originating from the oscillating gas flow and the bulk dynamics. Particle friction plays virtually no role in the onset to pattern formation, but it is fundamental for pattern selection and stabilization.

Figure 1

Example of regular patterns in pulsed beds of Glass 240 particles: (a) square pattern in shallow 3D geometry, $h=3\mathrm{mm},u=6.15+3.3\left[1+sin\left(2\pi 17t\right)\right]\mathrm{cm}/\mathrm{s}$, and (b) triangular bubble tessellation in deep quasi-2D geometry [16], $h=15$ cm, bed thickness $=1$ cm, $u=1.4+8.3\left[1+sin\left(2\pi 5t\right)\right]\mathrm{cm}/\mathrm{s}$.

Figure 2

Influence of the layer height $h$ on the threshold for pattern formation for Glass 240 particles at $u_{\min }=4\mathrm{cm}/\mathrm{s}$. $▽h=7\mathrm{mm},▴h=5\mathrm{mm},◯h=4\mathrm{mm},\blacksquare h=3\mathrm{mm}$. (Inset) The curves collapse when $f$ is normalized by the natural frequency, $f_{\mathrm{n}}=\frac{1}{\pi }\sqrt{\frac{g}{h}}$, of the layer.

Figure 3

Threshold for pattern formation for Glass 130 (red), Glass 240 (white), Zinc 130 (blue), Steel 130 (black), and Polystyrene 600 (asterisks). $u_{\min }=4\mathrm{cm}/\mathrm{s}$ for all cases and $h=2–8\mathrm{mm}$.

Figure 4

Influence of $u_{\min }$ on the threshold for pattern formation for Glass 240 (stars) and Glass 130 (circles). Black, gray, and white symbols represent $u_{\min }=3$, 4, and 6 cm/s, respectively.

Figure 5

Universal stability curve for pattern formation in pulsed fluidized beds of Geldart B particles. Plots include Glass 130 at $u_{\min }=3\mathrm{or}4\mathrm{cm}/\mathrm{s}$ and $h=2–7\mathrm{mm}$ (white symbols), Glass 240 at $u_{\min }=3\mathrm{or}6\mathrm{cm}/\mathrm{s}$ and $h=3–7\mathrm{mm}$ (red symbols), and Polysterene 600 at $u_{\min }=11.5\mathrm{cm}/\mathrm{s}$ and $h=10\mathrm{mm}$ (stars). For the sake of clarity, curves for Zinc 130 and Steel 130 are not plotted because they overlap with Glass 240 (see Fig. 3).

Figure 6

Stability curves when the driving parameter is defined as (a) $\mathrm{\Gamma }=2\pi f{u}_{\mathrm{a}}/g$ and (b) $\mathrm{\Gamma }=u_{\mathrm{a}}/u_{\mathrm{mf}}$. Symbols correspond to Fig. 5.