Elastic fingering patterns in confined lifting flows

The elastic fingering phenomenon occurs when two confined fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes elastic. We study elastic fingering pattern formation in Newtonian fluids flowing in a lifting (time-dependent gap) Hele-Shaw cell. Using a mode-coupling approach, nonlinear effects induced by the interplay between viscous and elastic forces are investigated and the weakly nonlinear behavior of the fluid-fluid interfacial patterns is analyzed. Our results indicate that the existence of the elastic interface allows the development of unexpected morphological behaviors in such Newtonian fluid flow systems. More specifically, we show that depending on the values of the governing physical parameters, the observed elastic fingering structures are characterized by the occurrence of either finger tip splitting or side branching. The impact of the elastic interface on finger-competition events is also discussed.

Figure 1

Representative sketch of the flow in a lifting (time-dependent gap) Hele-Shaw cell presenting an elastic fluid-fluid interface.

Figure 2

Time evolution of the fluid-fluid interface position $\mathcal{R}$ as a function of the polar angle $\theta$ (left panels), illustrating the representative elastic fingering patterns that may emerge during lifting Hele-Shaw flows. The corresponding time evolution of the rescaled cosine amplitudes $a_n\left(t\right)/R\left(t\right)$ for modes $n, 2n$, and $3n$, where $n=34$, is depicted in the right panels. The values of the bending rigidity fraction parameter are (a) $C=0$, (b) $C=0.35$, and (c) $C=0.45$. In addition, $\mathrm{\Gamma }=0.1, q=300$, and $\lambda =0.85$. The final times used are (a) $\tau =0.240$, (b) $\tau =0.235$, and (c) $\tau =0.225$. Note that the polar angle $\theta$ is given in radians.

Figure 3

Time evolution of the fluid-fluid interface position $\mathcal{R}$ as a function of the polar angle $\theta$ (left panels), illustrating the typical elastic fingering patterns that may arise during lifting Hele-Shaw flows. The corresponding time evolution of the rescaled cosine amplitudes $a_n\left(t\right)/R\left(t\right)$ for modes $n, 2n$, and $3n$, where $n=33$, is depicted in the right panels. The values of the bending rigidity fraction parameter are (a) $C=0$, (b) $C=0.125$, and (c) $C=0.25$. In addition, $\mathrm{\Gamma }=0.1, q=300$, and $\lambda =0.7$. The final times used are (a) $\tau =0.240$, (b) $\tau =0.203$, and (c) $\tau =0.166$. The polar angle $\theta$ is given in radians.

Figure 4

Behavior of the finger-competition function $\mathcal{C}(n,t)$ as the bending rigidity fraction $C$ is varied for three values of the characteristic radius $\lambda$.