Bifurcation from stable holes to replicating holes in vibrated dense suspensions

In vertically vibrated starch suspensions, we observe bifurcations from stable holes to replicating holes. Above a certain acceleration, finite-amplitude deformations of the vibrated surface continue to grow until void penetrates fluid layers, and a hole forms. We studied experimentally and theoretically the parameter dependence of the holes and their stabilities. In suspensions of small dispersed particles, the circular shapes of the holes are stable. However, we find that larger particles or lower surface tension of water destabilize the circular shapes; this indicates the importance of capillary forces acting on the dispersed particles. Around the critical acceleration for bifurcation, holes show intermittent large deformations as a precursor to hole replication. We applied a phenomenological model for deformable domains, which is used in reaction-diffusion systems. The model can explain the basic dynamics of the holes, such as intermittent behavior, probability distribution functions of deformation, and time intervals of replication. Results from the phenomenological model match the linear growth rate below criticality that was estimated from experimental data.

Figure 1

Snapshot of holes formed in a vertically vibrated suspension. (a) Stable hole, $\Gamma$ $=$ 196 m/s${}^2$. (b) Replicating hole, $\Gamma$ $=$ 222 m/s${}^2$. (a,b) The parameters were $f$ $=$ 90 Hz and $\varphi$ $=$ 0.34, and the mean diameter of dispersed particles was 29.3 $\mu$m.

Figure 2

Phase diagram of holes. Blue circles: stable holes. Green triangles: replicating holes. Black crosses: holes whose lifetimes were shorter than 1 min. (a) We changed the mean diameter of potato starch by sieving. We used SP1-8 and their mixtures as dispersed particles (see Table ). The parameters are $f$ $=$ 90 Hz and $\varphi$ $=$ 0.34. (b) We changed the surface tension of water by adding SDS. We used hollow glass beads as the dispersed particles with $f$ $=$ 90 Hz and $\varphi$ $=$ 0.64.

Figure 3

(a) Time evolution of the normalized amplitude of mode $n=2$. Red circles indicate peaks of impulses. (b) Time evolution of the velocity of the centroid $V$. MD $=$ 26.7 $\mu$m, $\varphi$ $=$ 0.34, $f$ $=$ 90 Hz. (a1,b1) $\Gamma$ $=$ 199.0 m/s${}^2$. (a2,b2) $\Gamma$ $=$ 211.3 m/s${}^2$. (a3,b3) $\Gamma$ $=$ 219.5 m/s${}^2$. (a4,b4) $\Gamma$ $=$ 228.1 m/s${}^2$.

Figure 4

Time collapse images of the hole around the impulse in Fig. 3. $f$ $=$ 90 Hz, MD $=$ 26.7 $\mu$m, $\varphi$ $=$ 0.34, $\Gamma$ $=$ 234.5 m/s${}^2$.

Figure 5

(a) Dependence of the frequency of impulses on acceleration. Blue circles: MD $=$ 24.1 $\mu$m. Red squares: MD $=$ 26.7 $\mu$m. Green triangles: MD $=$ 29.3 $\mu$m. (b) Histogram of ${\varphi }_2-{\varphi }_v$. To make the histogram, we eliminated data in which $V$ was less than $0.03$ cm/s. MD $=$ 26.7 $\mu$m, $\Gamma$ $=$ 234.5 m/s${}^2$. (a,b) $\varphi$ $=$ 0.34, $f$ $=$ 90 Hz.

Figure 6

Phase diagram of Eqs. (11) and (12) with ${\mu }_{12}={\mu }_{21}=0$. Deep blue: supercritical pitchfork bifurcation. Red: subcritical pitchfork bifurcation. Solid red line: bifurcation line of $(V,A_2)=(0,0)$. Solid deep blue line: bifurcation line of $(V,A_2)=(0,0)$. Dashed red line: bifurcation line of $(V,A_2)=(0,A_{20})$. Dashed deep blue line: bifurcation line of $(V,A_2)=(0,A_{20})$. The region surrounded by the red line and the black dotted line is the hysteresis region.

Figure 7

(a) Time series of $A_2$. (b) Time series of $V$. (c) Frequency of impulses as a function of ${\kappa }_2$. Red circles: ${\kappa }_1=-2$. Green triangles: ${\kappa }_1=-1.75$. Blue squares: ${\kappa }_1=-1.54$. (a)–(c) $c_1=33.8$, $c_2=5.21$, ${\mu }_{11}=-0.91$, ${\mu }_{22}=-200$, ${\mu }_{12}={\mu }_{21}=0$, ${\sigma }_v=0.035$, ${\sigma }_2=0.005$. (a,b) ${\kappa }_2$ $=$ 0.1.

Figure 8

(a1) ${\kappa }_1$ (dashed line) and ${\kappa }_2$ (solid line) estimated from experimental data, (a2) ${\sigma }_v$ (dashed line) and ${\sigma }_2$ (solid line) estimated from experimental data, (b1) ${\kappa }_1$ (red circle) and ${\kappa }_2$ (blue square) estimated from simulation results, (b2) ${\sigma }_v$ (red circle) and ${\sigma }_2$ (blue square) estimated from simulation results. (a1,a2) Green triangle: MD $=$ 24.3 $\mu$m. Blue square: MD $=$ 26.7 $\mu$m. Red circle: MD $=$ 29.3 $\mu$m. $\varphi =0.34$, $f=90$ Hz. (b1,b2) Parameters are the same as for Figs. 7a and 7(b). Dashed black lines are true values.

Figure 9

Probability distribution functions of $a_2$. (a) Simulation result. Supercritical bifurcation occurs at ${\kappa }_2=0$. ${\mu }_{22}=-200$. (b) Simulation result. Subcritical bifurcation occurs at ${\kappa }_2=0$. ${\mu }_{22}$ $=$ 20. (c) Experimental data. (a,b) Magenta line: ${\kappa }_2=-0.6$. Green line: ${\kappa }_2=-0.2$. Deep blue line: ${\kappa }_2=0.2$. Other parameters are the same as for Fig. 7. (c) Magenta line: $\Gamma =190.0$ m/s${}^2$. Green line: $\Gamma =211.3$ m/s${}^2$. Deep blue line: $\Gamma =228.1$ m/s${}^2$. MD $=$ 26.7 $\mu$m, $\varphi$ $=$ 0.34, $f$ $=$ 90 Hz.

Figure 10

(a) Simulation result: probability distribution function (PDF) for time intervals between impulses. Magenta line: ${\kappa }_2=0.2$. Green line: ${\kappa }_2=0.3$. Deep blue line: ${\kappa }_2=0.4$. Other parameters are the same as for Fig. 7a. (b) Experimental result: PDF for time intervals between replications. Blue circle: original potato starch, $\varphi$ $=$ 0.38, $f$ $=$ 100 Hz, $\Gamma$ $=$ 180 m/s${}^2$. Purple square: original potato starch, $\varphi$ $=$ 0.37, $f$ $=$ 80 Hz, $\Gamma$ $=$ 157 m/s${}^2$. Dashed line is fit to the simulation of Eqs. (3) and (4). Magenta dashed line: ${\kappa }_1=-0.123$, $c_1=5.01$, ${\mu }_{11}=-11.4$, ${\kappa }_2=2.40\times {10}^{-2}$, $c_2=65.1$, ${\mu }_{22}=-16.0$, ${\sigma }_v=4.20\times {10}^{-2}$, ${\sigma }_2=6.00\times {10}^{-3}$. Green dashed line: ${\kappa }_1=-0.177$, $c_1=2.37$, ${\mu }_{11}=-18.2$, ${\kappa }_2=1.50\times {10}^{-2}$, $c_2=43.4$, ${\mu }_{22}=-10.0$, ${\sigma }_v=4.20\times {10}^{-2}$, ${\sigma }_2=6.00\times {10}^{-3}$.