Pattern formation and interface pinch-off in rotating Hele-Shaw flows: A phase-field approach

Viscous fingering dynamics driven by centrifugal forcing is studied for arbitrary viscosity contrast. Theoretical methods, including exact solutions, and numerics based on a phase-field approach are used. Both confirm that pinch-off singularities in patterns originated from the centrifugally driven instability may occur spontaneously and be inherent to the two-dimensional Hele-Shaw dynamics. They are systematically more frequent for lower viscosity contrasts consistently with experimental evidence. The analytical insights provide an interpretation of this fact in terms of the asymptotic matching of the different regions of the fingering patterns. The phase-field numerical scheme is shown to be particularly adequate to elucidate the existence of finite-time singularities through the dependence of the singularity time on the interface thickness, in particular for varying viscosity contrast.

Figure 1

Pattern evolution for $A=1$, $b=0.5\text{mm}$, $\Omega =120\text{rev}/\text{min}$, and $R_0=50\text{mm}$ (a,b), and $A\simeq 0.45$, $\Omega =180\text{rev}/\text{min}$, and $R_0=38\text{mm}$ (c,d). Snapshots of size $3R_0\times 3R_0$ are shown at times 16.5 s (a), 22.5 s (b), 122 s (c), and 158 s (d) after the cell was set in rotation.

Figure 2

Asymptotic shapes of different regions of the $n$-fold symmetric pattern for $n=3$ and $n=18$: central region, starfish (stationary) solutions; dashed lines, infinitely thin filaments (lubrication region); circles, outer solution tending to circular droplets moving infinitely apart with an exponential velocity. Arrows indicate the direction of the arclength coordinate.

Figure 3

Sketch of a filament of fluid with a droplet at its tip.

Figure 4

Pattern evolution for a random initial condition (see text) and $B=1.03\times {10}^{-3}$, $ϵ=0.005$, $dx=ϵ$, $\widetilde{ϵ}=1$, and $A=1$ (a,b), $A=0.5$ (c,d), $A=0$ (e,f). Snapshots of size $3R_0\times 3R_0$ are shown at $t=1.32$ (left) and $t=1.74$ (right).

Figure 5

Filament thinning. Top: initial condition $\left(n=2,q=0.05\right)$. Middle and bottom: one later interface for $A=1$ and $A=0$. The other parameters are $B=0.09$, $ϵ=0.008$, and $dx=ϵ/2$.

Figure 6

Filament thinning. Width at rotation axis (in log scale) versus time. Dashed, solid, and dotted line(s): $dx=ϵ$, $dx=ϵ/2$, and $dx=ϵ/4$, respectively. Upper curves $\left(B=0.09\right)$: $ϵ=0.008$ (runs in Fig. 5, $q=0.05$). Lower curves $\left(B=0.01,q=0.2\right)$: three curves at $ϵ=0.02$, 0.01, and 0.005 for $dx=ϵ$ and three more at $ϵ=0.02$, 0.01, and 0.0067 for $dx=ϵ/2$ shown for each value of $A$. For $A=0$, lower curves correspond to lower values of $ϵ$. The only $dx=ϵ/4$ curve is for $A=1$, $ϵ=0.02$. The dashed (solid) vertical lines indicate the interval during which the droplet at the tip pinches off for $dx=ϵ\left(ϵ/2\right)$.

Figure 7

Pinch-off time as a function of interface thickness $ϵ$ in the phase-field model $\left(dx=ϵ/2,\widetilde{ϵ}=0.5\right)$. Black (gray) symbols represent $A=0\left(A=1\right)$. Squares correspond to dumbbells ($n=2$, and $B=0.01$, $q=0.2$) and diamonds to $n=4$ (four fingers, $B=0.0213$, $q=0.05$). In all cases, pinch-off occurs in the neck close to the advancing drop (as seen in Fig. 5) except for $A=1$, $n=2$, where it happens far away from the drop at the center of the cell. In this last case, a continuous straight line indicates the prediction explained in the text (with the theoretical slope).

Figure 8

Radial position (in log scale) of the center-of-mass of various droplets vs time for $A=1$ (thick) and $A=0$ (thinner curves). Solid, dashed, and dotted lines correspond to droplets indicated with an “x” in Figs. 4a, 4b, 4e, 4f, to those in Fig. 5, and to an off-center circular droplet $\left(B=0.32,ϵ=0.02,dx=ϵ,\widetilde{ϵ}=0.1,dt=0.25\widetilde{ϵ}d{x}^2\right)$, respectively. Curves are shown until the end of the respective runs, except for the isolated droplet, which is shown only up to times when finite-size effects of the simulation box are noticeable. Inset: linear regime of the isolated droplet continued in the main plot after translating the (dotted) curves to earlier times for comparison with the others. The inset preserves slopes with respect to the main plot.