Figure 1

Top: sketch of a top-side view of the radial Hele-Shaw cell with rotation where the inner fluid (named fluid 2) and the outer fluid (named fluid 1) are indicated with different gray levels. At the bottom, a top view of the cell with characteristic initial condition with dumbbell shapes.Reuse & Permissions

Figure 2

Evolution of an initial Gaussian interface which is set to zero and with zero derivative in the interval $\left(x=\pm 5,\pm \infty \right)$. We plot the initial condition and two evolved interfaces corresponding to the cases with rotation $\left(D=2,B=1\right)$ and without rotation $\left(D=0,B=1\right)$ at respective times related by scaling (14). They are taken after $t=0.2$ and $t=0.0865$, respectively. The interface obtained at $t=0.2$ with rotation is reproduced exactly by scaling the interface evolved without rotation obtained at $t=0.0865$. This scaling involves stretching and reducing the height using an exponential factor as indicated in the text.Reuse & Permissions

Figure 3

The minimum value of the filament is compared for the lubrication approximation and the exact evolution in four different cases. Two runs with $D/B=4$ and two runs with $D/B=10$ using different initial conditions [different $H_{\text{min}}\left(t=0\right)=H_0$].Reuse & Permissions

Figure 4

Basin of attraction of five different attractors in the space generated by our uniparametric family of solutions (defined by $H_0$) and $D/B$. Region I is the basin of attraction of a centered stable droplet without pinch-off. Region II corresponds to a thin filament with a droplet at the top which reaches infinity. Regions IIIa, b, and c have in common that the minimum of the interface diminishes very rapidly at some moment of the evolution. The differences among them are explained in the text. Black dots and circles, together with gray diamonds, indicate the points where transitions among the different attractors have actually been computed. The thick line at the bottom left indicates the set of initial conditions for which the argument based on the nonuniform time rescaling (see Sec. 5B) would predict also finite-time pinch-off in the presence of rotation.Reuse & Permissions

Figure 5

Evolution of the global shape of the interfaces. Top: initial condition going to centered mass; corresponds to region I in Fig. 4. Bottom: example of region II where the initial condition goes to a drop reaching infinity without apparent finite-time pinch-off.Reuse & Permissions

Figure 6

Evolution of the global shape of the interfaces. Top: initial condition where the drops goes toward the center of the cell while the minimum decreases sharply (inset) and later rebounds slightly making finite-time pinch-off inconclusive (region IIIc). Middle: interface reaching finite-time pinch-off (region IIIa). Bottom: the drop evolves toward infinite and the minimum of the interface initially has a sharp decay but changes behavior suddenly toward infinite-time pinch-off (region IIIb) with the corresponding drop advancing but with a filament thickness which has reached extremely low values $\left(H_{\text{min}}={10}^{-8}\right)$.Reuse & Permissions

Figure 7

The minimum of the interface as a function of dimensionless time for two different initial conditions inside area IIIa.Reuse & Permissions

Figure 8

The minimum of the interface as a function of dimensionless time for two different initial conditions inside area IIIb.Reuse & Permissions

Figure 9

The minimum of the interface as a function of dimensionless time for five initial conditions all of them using $D/B=6$ belonging to area IIIc.Reuse & Permissions

Figure 10

Effect of boundary conditions imposed by the matching between the lubrication region and the droplet. The three interfaces in each plot represent the actual evolution with rotation up to the pinch-off time (black), the interface evolved without rotation with the correspondingly scaled time [dashed interface with $D=0$ at $\tau =\frac{B}{5D}\left(1-e^{-5Dt}\right)$], and the latter with the scaling defined by Eq. (14) (dotted). See discussion in the text.Reuse & Permissions

Figure 11

Evolution of an interface at $D/B=100$ with $H_0=0.15$ Rotation is stopped when the droplet reaches $l=10$ (interface A in the bottom graph). The initial exponential decay stops when rotation is switched off (top graph) and the droplet starts receding toward the center of the cell (interface B) at the same time that the thickness of the filament decays slowly. Once the droplet reaches the center of the cell (interface C), this behavior changes abruptly reaching finite-time pinch-off.Reuse & Permissions