Model of heap formation in vibrated gravitational suspensions

In vertically vibrated dense suspensions, several localized structures have been discovered, such as heaps, stable holes, expanding holes, and replicating holes. Because an inclined free fluid surface is difficult to maintain because of gravitational pressure, the mechanism of those structures is not understood intuitively. In this paper, as a candidate for the driving mechanism, we focus on the boundary condition on a solid wall: the slip-nonslip switching boundary condition in synchronization with vertical vibration. By applying the lubrication approximation, we derived the time evolution equation of the fluid thickness from the Oldroyd-B fluid model. In our model we show that the initially flat fluid layer becomes unstable in a subcritical manner, and heaps and convectional flow appear. The obtained results are consistent with those observed experimentally. We also find that heaps climb a slope when the bottom is slightly inclined. We show that viscoelasticity enhances heap formation and climbing of a heap on the slope.

Figure 1

(a) Schematic illustration of the system. (b, c) Schematic illustration of the boundary condition. (b) When $g\left(t\right)>$ 0, granules slip on the bottom. (c) When $g\left(t\right)<$ 0, granules stick on the bottom.

Figure 2

(a) Heap in steady state (blue line) and time-averaged velocity field (red arrows). (b, c) Spatiotemporal plot of heaps. Horizontal and vertical axes are time and position, respectively. Gray scale represents the height of $h(x,y)$. A brighter region indicates greater height (see color bar). (b) $\mathrm{\Gamma }=2.2$. (c) $\mathrm{\Gamma }=2.41$. (a–c) $\eta =100$ St, $\sigma =72$ dyne/cm, $f=10$ Hz, $h_0={\beta }_0\eta /{\eta }_w=0.5$ cm. We use a disjoining pressure. See Appendix pp3. (a) $B=10$ 000 Pa, $h_p=0.005$ cm. (b, c) $B=10\mathrm{Pa},h_p=0.005$ cm. (a–c) Initial condition is $h(x,y)=h_0+{\xi }_0{\sum }_{n=1}^{n=m}sin\left(2\pi nx/L+{\varphi }_n\right)$. $L$ is the system size. ${\varphi }_n$ is a random number uniformly distributed from 0 to $2\pi$. (a) ${\xi }_0=0.01$ cm, $L=10$ cm, $m=5$. (b, c) ${\xi }_0=0.05$ cm, $L=100$ cm, $m=10$.

Figure 3

(a) Dependence of $\gamma$ on $k$ calculated from Eq. (D15). Dashed black line is calculated from Eq. (83). Blue circles: $\eta =$ 100 St, and $\sigma =$ 21 dyne/cm. Green triangles: $\eta =$ 50 St, and $\sigma =$ 72 dyne/cm. Red squares: $\eta =$ 100 St, and $\sigma =$ 72 dyne/cm. (b) Blue circles indicate unstable branch of Eq. (60) with periodic boundary condition in two dimensions. ${\xi }_0$ is the initial deformation, $h=h_0+{\xi }_{0}sin\left(k_{m}x\right)$. Above the symbol, the fluid layer ends in rupture. Below the symbol, the fluid layer relaxes to become flat. Red solid line is unstable branch calculated from Eq. (78). $\eta =$ 50 St, and $\sigma =$ 72 dyne/cm. (a, b) Parameter values are $k_m=2\pi /5 {\mathrm{cm}}^{-1},f=10$ Hz, $h_0={\beta }_0\eta /{\eta }_w=0.5$ cm, and $\rho =1 {\mathrm{g}/\mathrm{cm}}^3$.

Figure 4

Bistable states of heaps in viscoelastic fluid. Blue curve indicates time-averaged surface, and red arrows indicate time-averaged velocity field. Relaxation time ${\lambda }_1$ is set to exceed a critical value ${\lambda }_c$. (a) Small initial surface deformation. Initial condition is $h(x,y)=h_0-0.05cos\left(4\pi nx/L\right)$. $L=10$ cm is the system size. (b) Large initial surface deformation. Initial condition is $h(x,y)=h_0-0.3cos\left(2\pi nx/L\right)$. Parameter values are ${\lambda }_1=0.15$ s, $\mathrm{\Gamma }=2.37,\eta =100$ St, $\sigma =72$ dyne/cm, $h_0=\beta \eta /{\eta }_w=0.5$ cm, $\rho =1 {\mathrm{g}/\mathrm{cm}}^3$, and ${\lambda }_2=0.01$ s. We use a disjoining pressure. See Appendix pp3. $B=10$ Pa, $h_p=0.005$ cm.

Figure 5

(a) Dependence of ${\mathrm{\Lambda }}_k$ on $k$ at the critical acceleration ${\mathrm{\Gamma }}_c$. ${\lambda }_2=0.01$ s. Red solid line: viscous fluid. Green dashed line: ${\lambda }_1=1.135 \times {10}^{-1}$ s. Blue squares: ${\lambda }_1=1.150 \times {10}^{-1}$ s. Magenta circles: ${\lambda }_1=1.155 \times {10}^{-1}$ s. (b) Dependence of ${\mathrm{\Gamma }}_c$ on ${\lambda }_1$. Red circles: ${\lambda }_2=0.001$ s. Green triangles: ${\lambda }_2=0.01$ s. Blue squares: ${\lambda }_2=0.02$ s. (c) Dependence of $k_c$ on ${\lambda }_1$. Red circles: ${\lambda }_2=0.001$ s. Green triangles: ${\lambda }_2=0.01$ s. Blue squares: ${\lambda }_2=0.02$ s. (a–c) Parameter values are $\eta =100$ St, $\sigma =72$ dyne/cm, $h_0=\beta \eta /{\eta }_w=0.5$ cm, and $\rho =1 {\mathrm{g}/\mathrm{cm}}^3$.

Figure 6

(a) Dependence of ${\lambda }_c$ on parameter values. Horizontal and vertical axes are ${\lambda }_c$ calculated from Eq. (92) and ${\lambda }_c$ directly computed from Eq. (85), respectively. Blue circles: $\sigma =72$ dyne/cm, $\eta =100$ St. $h_0$ is varied from 0.1 to 1 cm. Green squares: $\sigma =72$ dyne/cm, $h_0=0.5$ cm. $\eta$ is varied from 10 St to 100 St. Red crosses: $\eta =100$ St, $h_0=0.5$ cm. $\sigma$ is varied from 10 dyne/cm to 100 dyne/cm. (b) Dependence of critical wave number $k_c$ on ${\lambda }_1$ around ${\lambda }_c$. Black dashed line is calculated from Eq. (93). Blue squares: $h_0=1$ cm, $\sigma =72$ dyne/cm. Green triangles: $h_0=0.5$ cm, $\sigma =21$ dyne/cm. Red circles: $h_0=0.5$ cm, $\sigma =72$ dyne/cm. $\eta$ is fixed at 100 St. (c) Dependence of ${\lambda }_c$ on frequency and ${\lambda }_2$. Red squares: ${\lambda }_2=0.000$ 1 s. Green crosses: ${\lambda }_2=0.001$ s. Blue circles: ${\lambda }_2=0.01$ s. Parameter values are $h_0=0.5$ cm, $\sigma =72$ dyne/cm, $\eta =100$ St. (a, b, c) We set ${\beta }_0\eta /{\eta }_w=h_0$.

Figure 7

Climbing of a heap. (a–c) The bottom wall is inclined at an angle ${\theta }_b$ from the $x$ axis. The heap (blue line) shows directional motion. The dashed line is the initial condition. ${\theta }_b=-2^{\circ }$. (a) $\mathrm{\Gamma }=2.6$. (b) $\mathrm{\Gamma }=3.05$. (c) $\mathrm{\Gamma }=3.2$. (d) Time series of $V_g$. (e) Velocity of the centroid of a heap. The bottom plate is slightly inclined. A positive velocity indicates that a heap climbs a slope. (f) Dependence of $C_0/sin{\theta }_b$ on $\mathrm{\Gamma }$. (g) Dependence of $S_0$ on $\mathrm{\Gamma }$. (h) Dependence of $V_g/sin{\theta }_b$ on $\mathrm{\Gamma }$. (e–h) Purple crosses: ${\theta }_b=0.5^{\circ }$. Blue squares: ${\theta }_b=1.0^{\circ }$. Green triangles: ${\theta }_b=1.5^{\circ }$. Red circles: ${\theta }_b=2.0^{\circ }$. (a–h) $f=10$ Hz, $\rho =1 {\mathrm{g}/\mathrm{cm}}^3,\eta =100$ St, $\sigma =72$ dyne/cm, $h_0=\beta \eta /{\eta }_w=0.5$ cm, $B=10$ Pa, $h_p=0.005$ cm.

Figure 8

Effect of elasticity on climbing. (a) Velocity of the centroid of a heap. A positive velocity indicates that a heap climbs a slope. Blue squares: velocity of a heap with elasticity. ${\lambda }_1=0.1$ s, ${\lambda }_2=0.01$ s. Red circles: velocity of a heap without elasticity (Newtonian fluid). (b) Steady shape of the heap in faster branch. (c) Steady shape of the heap in slower branch. (d) Blue circles indicate onset acceleration at which heap starts to climb. ${\lambda }_2=0.01$ s. Red square indicates data point of Newtonian fluid. (a–d) $f=10$ Hz, $\rho =1 {\mathrm{g}/\mathrm{cm}}^3,\eta =100$ St, $\sigma =72$ dyne/cm, $h_0=\beta \eta /{\eta }_w=0.5$ cm, $B=10$ Pa, $h_p=0.005$ cm, ${\theta }_b=2^{\circ }$.

Figure 9

(a) Drift velocity of a heap driven by horizontal vibration. Blue squares: ${\mathrm{\Gamma }}_h=0.035$. Green triangles: ${\mathrm{\Gamma }}_h=0.07$. Red circles: ${\mathrm{\Gamma }}_h=0.14$. (b) Shape of a heap under large horizontal acceleration. ${\mathrm{\Gamma }}_h=1.00,{\phi }_h=0$. Blue solid line: shape of a heap at $t=32$ s. Red dashed line: initial shape of a heap. Black dashed arrow indicates a position of rear contact line at $t=32$ s. (a, b) $f=10$ Hz, $\rho =1 {\mathrm{g}/\mathrm{cm}}^3,\eta =100$ St, $\sigma =72$ dyne/cm, $h_0=\beta \eta /{\eta }_w=0.5$ cm, $B=100$ Pa, $h_p=0.005$ cm.

Figure 10

Drift of a heap in quasi-2D container. Dashed line indicates the position of the contact line. (a) ${\mathrm{\Gamma }}_h=1.04,{\phi }_h=3.62$ rad, $f=70$ Hz. (b) ${\mathrm{\Gamma }}_h=0.21,{\phi }_h=0.78$ rad, $f=70$ Hz. (c) ${\mathrm{\Gamma }}_h=1.26,{\phi }_h=5.13$ rad, $f=80$ Hz. (a–c) $\mathrm{\Gamma }=10.1$. The total volume of the suspension is $1.66 {\mathrm{cm}}^3$. The packing fraction of the titanium beads is 58%.

Figure 11

Instantaneous flow field under downward and upward gravity. Blue curve and red arrows indicate surface of the fluid and flow field, respectively. (a) When $g\left(t\right)<$ 0, velocity is 0 on the bottom. (b) When $g\left(t\right)>$ 0, velocity is nonzero on the bottom.

Figure 12

Touching time $T_1$ as a function of $\alpha =k_b/\left({\gamma }_b\omega \right)$. Black dashed line indicates $\pi -T_0$. Red circles: $\mathrm{\Gamma }=1.5$. Green triangles: $\mathrm{\Gamma }=3.0$. Blue squares: $\mathrm{\Gamma }=9.0$.