Persistent Holes in a Fluid

We observe stable holes in a vertically oscillated 0.5 cm deep aqueous suspension of cornstarch. Holes appear only if a finite perturbation is applied to the layer for accelerations $a$ above $10g$. Holes are circular and approximately 0.5 cm wide, and can persist for more than ${10}^6$ cycles. Above $a\simeq 17g$ the rim of the hole becomes unstable, producing fingerlike protrusions or hole division. At higher acceleration, the hole delocalizes, growing to cover the entire surface with erratic undulations. We find similar behavior in an aqueous suspension of glass microspheres.

Figure 1

Top view of a vibrated layer of an aqueous suspension of (a)–(c) cornstarch and of (d)–(f) glass microspheres (image diameter, 9.4 cm). White corresponds to the highest points, and black to depressions that reach near the container bottom. Holes without Faraday waves: (a) $a=12g$, $f=150\mathrm{Hz}$; (d) $30g$, 100 Hz. Holes with Faraday waves: (b) $12g$, 60 Hz; (e) $27.3g$, 92 Hz. Delocalized state: (c) $30g$, 120 Hz; (f) $30g$, 60 Hz.

Figure 2

Phase diagram for a vibrated aqueous cornstarch suspension as a function of acceleration and frequency. In either the metastable or stable region, a surface perturbation forms a persistent hole. In the metastable regime these holes always collapse within ${10}^5$ container oscillations; in the stable regime some holes last for more than ${10}^5$ oscillations. In the areas marked unstable, holes collapse in less than ${10}^4$ cycles. In the delocalized regime the surface is highly irregular, as shown in Fig. 1c. Faraday waves appear with increasing acceleration at ▲, and disappear with decreasing acceleration at ▼.

Figure 3

Time evolution of the radii of six holes formed with a 2 s long puff of air. Holes grow during the forcing and then rapidly shrink after the air is cut off. Inset: hole diameter oscillates synchronously with the forcing; note time scale ($a=15g$, $f=150\mathrm{Hz}$).

Figure 4

(a) Vertical cross sections of holes at $a=14.5g$ (dashed and dotted lines) and $a=17g$ (solid and dot-dashed lines) for $f=150\mathrm{Hz}$. The two profiles for each acceleration show the extremes in a hole’s shape, separated by one-half cycle. (b) The tangent of the phase lag versus frequency at $a=15g$. The solid line is a fit to $\alpha /[1-(f/f_0{)}^2]$, where $\alpha$ and $f_0$ are fitting parameters.

Figure 5

Side view of the first steps toward the delocalized state in cornstarch. These photographs were taken every 0.9 s; time increases from left to right and top to bottom. An initial hump on the rim begins growing upward, reaches a maximum height, and then topples outward, enlarging the area of fluid motion. This process repeats until the entire surface of the liquid is active in the creation and destruction of vertical structures and voids [see Fig. 1c] ($a=25g$, $f=80\mathrm{Hz}$).

Figure 6

Apparent viscosity as a function of the maximum shear rate $\dot{\gamma }$ for two cornstarch suspensions prepared in the same manner as in Fig. 1. Measurements were made with a plate-plate rheometer (Paar Physica TEK 150) with radius $R=2.5\mathrm{cm}$ and gap 0.1 cm. The apparent viscosity was calculated as for a Newtonian fluid: $\eta =2\tau /(\pi \dot{\gamma }{R}^3)$ where $\tau$ is the torque. Since the shear rate increases with the radius, our measurement of the viscosity is a convolution of the geometry and the response of the fluid. The dashed extension of the curve represents measured values that are not reliable due to an instability in the flow (temperature, $27.6\pm 0.2°\mathrm{C}$).