Elastic fingering in two-dimensional Stokes flow

We investigate the linear and weakly nonlinear behaviors of a gel-like elastic interface separating two viscous fluids moving in two dimensions (2D). Induced by fluid suction or injection, a 2D Stokes flow takes place and the interface deforms due to the interplay between viscous and elastic forces. By modeling the elastic boundary as having a curvature-dependent bending rigidity and by employing a perturbative mode-coupling approach, we study the linear stability of the interface and the formation of nonlinear patterned structures. In addition to capturing a number of interesting linear properties, we have identified the emergence of suction-driven, nonlinear polygonal-like patterns presenting either convex or concave edges that can develop peculiar near-cusp fingers.

Figure 1

A schematic of the elastic fingering 2D Stokes flow problem with injection ($Q>0$) or suction ($Q<0$), where $Q$ is the areal rate. The outer (inner) fluid has viscosity ${\mu }_2$ (${\mu }_1$). The interface separating the fluids is elastic, and has a curvature-dependent bending rigidity $\nu =\nu \left(\kappa \right)$. The time-dependent unperturbed interface radius is denoted by $R\left(t\right)$ and the interface perturbation is represented by $\zeta (\theta ,t)$, where $\theta$ is the azimuthal angle. The deformed elastic interface is represented as $\mathcal{R}(\theta ,t)=R\left(t\right)+\zeta (\theta ,t)$ [Eq. (12)].

Figure 2

Plot of $\nu \left(\kappa \right)/{\nu }_0$ [as given by Eq. (21)] as a function of $\kappa$, for two values of the bending rigidity fraction ($C=0.2$ and $C=0.5$) and two values of characteristic elastic length scale ($\lambda =0.01$ cm e $\lambda =0.05$ cm).

Figure 3

Behavior of the functions $A_1(C,\eta )$ [Eq. (26)] and $A_2(C,\eta )$ [Eq. (27)] as $\eta$ is varied, for a few values of the bending rigidity fraction $C$: 0, 0.3, 0.5, and 0.56. It is clear that within the interval $0\le C\le 0.56$, the function $A_1<0$, whereas the function $A_2>0$, ensuring that $\mathrm{\Lambda }\left(n\right)$ is bounded for all $\eta$.

Figure 4

The mode of maximum growth rate $n_{\max }$ as a function of $\eta$, for a few increasing values of $C$: 0, 0.4, 0.5, 0.55, and 0.56. Note that as $C$ gets closer to $C_{\mathrm{critical}}=0.56...$ [Eq. (30)], the curve for $n_{\max }$ presents a higher and sharper peak and arbitrarily large values of $n_{\max }$ are possible.

Figure 5

Linear stability diagram on the ($C$, $\eta$) plane showing neutral stability curves of the system for some values of the mode $n$. Here $Q=-9{\mathrm{cm}}^2/\mathrm{s}$, ${\nu }_0=8.0\times {10}^{-1}\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$, and $\lambda =1$ cm.

Figure 6

Representative weakly nonlinear time evolution of the elastic interface for 2D Stokes flow with suction, at times $t_i\le t\le t_f$, where $t_i=46$ s and $t_f=48$ s, at equal intervals of 0.2 s. The patterns are obtained by considering the coupling of 30 participating modes ($N=30$), for $n=8$. [(a)–(c)] $A=1$, [(d)–(f)] $A=0$, and [(g)–(i)] $A=-1$. In addition, for panels (a), (d), and (g) ${\nu }_0={10}^{-1}\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$; for panels (b), (e), and (h) ${\nu }_0={10}^{-2}\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$; while for panels (c), (f), and (i) ${\nu }_0={10}^{-5}\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$. The innermost interfacial pattern (taken at final time $t=t_f=48$ s) is highlighted, represented by a ticker curve. The horizontal bar in panel (a) indicates 1 cm. Here $Q=-9{\mathrm{cm}}^2/\mathrm{s}$, $C=0.56$, and $\lambda ={10}^{-2}$ cm.

Figure 7

Representative weakly nonlinear time evolution of the elastic interface for 2D Stokes flow with suction, at times $t_i\le t\le t_f$, where $t_i=45.4$ s and $t_f=48.2$ s, at equal intervals of 0.28 s. The patterns are obtained by considering the coupling of 30 participating modes ($N=30$) for $n=4$. Three values of the viscosity contrast are considered: (a) $A=1$, (b) $A=0$, and (c) $A=-1$. Here $Q=-9{\mathrm{cm}}^2/\mathrm{s}$, $\lambda =0.525$ cm, ${\nu }_0={10}^{-3}\mathrm{g}{\mathrm{cm}}^2/{\mathrm{s}}^2$, and $C=0.56$. In panels (a)–(c), the innermost interfacial pattern (taken at final time $t=t_f=48.2$ s) is highlighted, being represented by a ticker curve. The horizontal bar in (a) indicates 1 cm. [(d)–(f)] Time evolution of the dimensionless bending rigidity field $\nu \left(\kappa \right)/{\nu }_0$ for the evolving interfaces illustrated in panels (a)–(c).