Control of radial fingering patterns: A weakly nonlinear approach

It is well known that the constant injection rate flow in radial Hele-Shaw cells leads to the formation of highly branched patterns, where finger tip-splitting events are plentiful. Different kinds of patterns arise in the lifting Hele-Shaw flow problem, where the cell’s gap width grows linearly with time. In this case, the morphology of the emerging structures is characterized by the strong competition among inward moving fingers. By employing a mode-coupling theory we find that both finger tip-splitting and finger competition can be restrained by properly adjusting the injection rate and the time-dependent gap width, respectively. Our theoretical model approaches the problem analytically and is capable of capturing these important controlling mechanisms already at weakly nonlinear stages of the dynamics.

Figure 1

Schematic setup for the injection-driven radial Hele-Shaw flow.

Figure 2

(Color online) (a) Behavior of the finger tip function $T\left(2n,n\right)$ as time is varied, considering the coupling of modes $n=5$ and $2n=10$; (b) time evolution of the cosine perturbation amplitude for the first harmonic mode $a_{2n}$. The solid curves describe the situation at constant injection rate, while the dashed curves are plotted by taking a time-varying injection rate $Q=f\left(n,A\right)t^{-1/3}$. The constant injection rate is $Q=61.5$ and $f\left(5,1\right)=76.8$.

Figure 3

Snapshots of the evolving interface, plotted at equal time intervals for the interaction of two cosine modes $n=5$ and $2n=10$ when (a) $Q\left(t\right)\sim \text{const}$ and (b) $Q\left(t\right)=f\left(n,A\right)t^{-1/3}$. Here, the constant injection rate is $Q=61.5$ and $f\left(5,1\right)=76.8$. The area in black represents the more viscous fluid. Finger tip splitting is clearly present in (a) and suppressed in (b).

Figure 4

Schematic setup for the time-dependent gap radial Hele-Shaw flow.

Figure 5

(Color online) (a) Behavior of the finger competition function $C\left(n\right)$ as time is varied; (b) time evolution of the cosine $\left(a_{n/2}\right)$ and sine $\left(b_{n/2}\right)$ perturbation amplitudes for the subharmonic mode. The solid curves describe the situation where $b\left(t\right)=1+\beta t$, with $\beta =0.03$. The dashed curves are plotted by assuming that $b\left(t\right)={\left[1+g\left(n,A,\delta \right)t\right]}^{-2/7}$, where $g\left(56,-1,200\right)=0.066$.

Figure 6

Snapshot of the fluid-fluid interface at time $t=4$ for the interaction of two modes $n=56$ and $n/2=28$ (both sine and cosine) when (a) $b\left(t\right)=1+\beta t$, with $\beta =0.03$, and (b) $b\left(t\right)={\left[1+g\left(n,A,\delta \right)t\right]}^{-2/7}$, where $g\left(56,-1,200\right)=0.066$. The area in black represents the more viscous fluid. Finger competition among inward moving fingers is detected in (a) and inhibited in (b).