Elastic Convection in Vibrated Viscoplastic Fluids

We observe a new type of behavior in a shear-thinning yield stress fluid: freestanding convection rolls driven by vertical oscillation. The convection occurs without the constraint of container boundaries, yet the diameter of the rolls is spontaneously selected for a wide range of parameters. The transition to the convecting state occurs without hysteresis when the amplitude of the plate acceleration exceeds a critical value. We find that a nondimensional stress, the stress due to the inertia of the fluid normalized by the yield stress, governs the onset of the convective motion.

Figure 1

(a) Overhead view of the convecting rolls on a horizontal plate, vertically vibrated at $f=40\mathrm{Hz}$ and acceleration amplitude $\Gamma =11.1$. The volume of the gel was $V=6\mathrm{mL}$. Arrows represent the direction of the flow. (b) Roll pattern using a smaller volume, $V=3\mathrm{mL}$, at $f=40\mathrm{Hz}$, $\Gamma =12.0$. (c) Ring pattern obtained at higher volumes. The mean convecting flow occurs in the outer ring region, e.g., $f=40\mathrm{Hz}$, $\Gamma =15.0$, and $V=9\mathrm{mL}$. (d) Rolls with a branch can be formed with a different initial condition. $f=100\mathrm{Hz}$, $\Gamma =32.6$, and $V=5\mathrm{mL}$.

Figure 2

Side view of the oscillating motion of the rolls at various phases of the plate oscillation ($f=40\mathrm{Hz}$, $\Gamma =14.8$, and $V=5\mathrm{mL}$). Time proceeds from left to right at 5 ms intervals. The oscillation of the gel is locked to the plate frequency but out of phase [see Fig. 5c]. However, the mean convective flow occurs on a time scale much slower than that of the oscillation period.

Figure 3

State diagram as a function of acceleration and frequency. The dashed line shows the state boundary obtained by a linear fit. Below the line, the rolls do not rotate. Above the line, they can rotate if the roll shape has been clearly formed. Inset: The same plot for nondimensional stress defined by Eq. (1).

Figure 4

The mean angular velocity $\Omega$ as a function of the acceleration above onset, $\Gamma -{\Gamma }_c$. ${\Gamma }_c$ denotes the onset value for convection. Results for $f=40\mathrm{Hz}$ with $V=5.0\mathrm{mL}$ and 60 Hz with $V=2.2\mathrm{mL}$. The curve for J1 gel is also shown for $f=60\mathrm{Hz}$, $V=2.2\mathrm{mL}$. Each is fit by $\Omega =\alpha (\Gamma -{\Gamma }_c{)}^{\beta }$, where $\alpha$ and $\beta$ are fit parameters. This results in $\Omega =(0.82\pm 0.04)\times (\Gamma -{\Gamma }_c{)}^{1.50\pm 0.04}$ [${\mathrm{s}}^{-1}$] and $\Omega =(0.47\pm 0.02)\times (\Gamma -{\Gamma }_c{)}^{1.55\pm 0.03}$ [${\mathrm{s}}^{-1}$]. Inset: The same plot without subtracting ${\Gamma }_c$.

Figure 5

(a) Dots: Time evolution of the diameter of a single roll as observed from above, for $f=40\mathrm{Hz}$, $V=5\mathrm{mL}$, and $\Gamma =14.3$. dashed line: Position of the plate showing the phase difference. (b) Dependence of the roll diameters on acceleration $\Gamma$ for $f=40\mathrm{Hz}$, $V=5\mathrm{mL}$, ${\Gamma }_C=10.1$ (solid triangle) and $f=30\mathrm{Hz}$, $V=12\mathrm{mL}$, ${\Gamma }_c=7.6$ (unfilled triangle). The upper (lower) symbols show the maximum (minimum) value of the measured diameter. Note that there is a plateau region for the maximum of the diameters when the acceleration $\Gamma$ is well above the onset value ${\Gamma }_C$. (c) Size dependence of the roll on the volume of the material for $f=40\mathrm{Hz}$, $\Gamma =12.3$ and $f=30\mathrm{Hz}$, $\Gamma =9.2$. Symbols are a similar to (b).

Figure 6

Time evolution of the gel in a vertically vibrated cylindrical container at time intervals of 2 s ($f=50\mathrm{Hz}$, $\Gamma =27.3$, and $V=25\mathrm{mL}$). The gel eventually gathers into one area enclosed by rolls and the wall, vacating the other areas.