Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth

The displacement of a viscous fluid by an air bubble in the narrow gap between two parallel plates can readily drive complex interfacial pattern formation known as viscous fingering. We focus on a modified system suggested recently by Zheng et al. [Phys. Rev. Lett. 115, 174501 (2015)], in which the onset of the fingering instability is delayed by introducing a time-dependent (power-law) plate separation. We perform a complete linear stability analysis of a depth-averaged theoretical model to show that the plate separation delays the onset of nonaxisymmetric instabilities, in qualitative agreement with the predictions obtained from a simplified analysis by Zheng et al. We then employ direct numerical simulations to show that in the parameter regime where the axisymmetrically expanding air bubble is unstable to nonaxisymmetric perturbations, the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed in constant-width Hele-Shaw cells.

Figure 1

Schematic diagram of a perturbed interface in the radial Hele-Shaw cell with time-varying gap width.

Figure 2

Plot of the stability criterion Eq. (11) for $J=10,50,$ and 100. The axisymmetric state is unstable to nonaxisymmetric perturbations with integer wave number $k$ if $S_k<0$. The predictions corresponding to Zheng et al.'s approach [17] are shown by dashed lines.

Figure 3

(a) Series of snapshots showing the evolution of the interface for $J=600$ [28]. The simulation was started at $t_0=0.00125$ when the axisymmetric interface (of radius $R_{\mathrm{init}}=0.05$) was subjected to an initial perturbation with wave number $k=k_{\max }=13$ and amplitude $\epsilon =5\times {10}^{-4}$. The innermost interface is shown for $t=0.011$ and the time interval between contours is $\mathrm{\Delta }t=0.083$. (b) Interface contours from panel (a) scaled by the radius of the finger tip, $R_{\mathrm{tip}}$. In panels (a) and (b), the first and the last interfaces in the sequence are highlighted with green and red colours, respectively.

Figure 4

Log-log plot of $R_{\mathrm{tip}}$ as a function of time for $J=220$, 600, and 1000 [30]. In each case, the interface is initially perturbed by the single, most-rapidly growing mode. The dotted line shows the radius of the axisymmetrically growing bubble, $\overline{R}\left(t\right)$.

Figure 5

Series of snapshots showing the evolution of the interface for $J=1000$ [28]. The simulations were started at $t_0=0.00125$ when the axisymmetric interface (of radius $R_{\mathrm{init}}=0.05$) was subjected to an initial perturbation with wave number (a) $k=k_{\max }=17$, (b) $k=7$, (c) $k=5$. The amplitude of the initial perturbation is $\epsilon =5\times {10}^{-4}$ for panels (a) and (b) and $1.25\times {10}^{-2}$ for panel (c). The time interval between contours is (a) $\mathrm{\Delta }t=0.069$, (b) $\mathrm{\Delta }t=0.078$, (c) $\mathrm{\Delta }t=0.08$. In each case the first contour is shown at $t=0.011$. Dashed lines indicate a sector within which a single finger develops at early times. The insets show the early evolution of the interface with successive contours chosen so that the average radius $\langle R\rangle =\frac{1}{2\pi }{\int }_0^{2\pi }Rd\theta$ increases by 0.022.

Figure 6

Instantaneous interface shapes with average radius $\langle R\rangle =0.05$ (black), 0.1 (red), 0.17 (cyan), 0.31 (dark green), and 0.6 (blue), respectively, computed for $J=600$ [28]. The arrow points to a region of the interface that undergoes tip-splitting and finger competition.

Figure 7

Normalized spectra of the instantaneous interface shapes shown in Fig. 6 for $J=600$. The mean radii are (a) $\langle R\rangle =0.05$, (b) 0.1, (c) 0.17, (d) 0.31, and (e) 0.6 [30]. The vertical dashed line indicates the value of $k_{\max }=13$ predicted by the linear stability analysis.

Figure 8

The variation of $K_{\max }$ with $J$ from direct numerical simulations (DNS) [30]. The error bars denote the standard deviation across 50 simulations. The lines show $k_{\max }$ from the results of the linear stability analysis, using our own predictions (solid) and those from Ref. [17] (dashed). $k_{\max }$ can only take integer values, which we connect by vertical line segments, resulting in steps in the curves.