Fingering instabilities of a reactive micellar interface

We present an experimental study of the fingering patterns in a Hele-Shaw cell occurring when a gel-like material forms at the interface between aqueous solutions of a cationic surfactant (cetyltrimethylammonium bromide) and an organic salt (salicylic acid), two solutions known to form a highly elastic wormlike micellar fluid when mixed homogeneously. A variety of fingering instabilities are observed, depending on the velocity of the front (the injection rate), and on which fluid is injected into which. We have found a regime of nonconfined stationary or wavy fingers for which width selection seems to occur without the presence of bounding walls, unlike the Saffman-Taylor experiment. Qualitatively, some of our observations share common mechanisms with instabilities of cooling lava flows or growing biofilms.

Figure 1

Onset of the fingering instability for an expanding circular front driven by an injection rate of $150\mathrm{ml}∕\mathrm{h}$, $30\mathrm{mM}$ CTAB solution injected into $30\mathrm{mM}$ NaSal solution: $t=\left(\mathrm{a}\right)$ 10.0, (b) 20.0, (c) 28.8, and (d) $64.7\mathrm{s}$. Scale: picture width is $8\mathrm{cm}$.

Figure 2

Fingering instability under the same conditions as Fig. 1, but with the opposite arrangement of fluids—$30\mathrm{mM}$ NaSal solution injected into $30\mathrm{mM}$ CTAB solution: $t=\left(\mathrm{a}\right)$ 8.2, (b) 11.0, (c) 16.3, and (d) $54.8\mathrm{s}$. Scale: picture width is $8\mathrm{cm}$

Figure 3

Fingering patterns for different flow rates after the same total volume $\left(4\mathrm{ml}\right)$ of $50\mathrm{mM}$ CTAB has been injected into $50\mathrm{mM}$ NaSal: (a) 20, (b) 50, (c) 100, and (d) $200\mathrm{ml}∕\mathrm{h}$. The dark borders are the gelled interfaces, which bound the interior, flowing conduit of fresh fluid that drives the further growth of the pattern. Scale: picture width is $11\mathrm{cm}$.

Figure 4

Phase diagram for the injection of $50\mathrm{mM}$ CTAB into $50\mathrm{mM}$ NaSal: Time at which successive regimes are triggered vs flow rate. (●) Onset of mushroom growth, (▵) onset of fans, and (엯) onset of tentacle growth. Dashed lines are guidelines.

Figure 5

A regular pattern of fingers in the fan state obtained when injecting $35\mathrm{mM}$ CTAB into $35\mathrm{mM}$ NaSal, $Q=600\mathrm{ml}∕\mathrm{h}$. Scale: picture width is $10\mathrm{cm}$.

Figure 6

Critical radius of breakdown of axial symmetry vs flow rate for three different CTAB and NaSal concentrations.

Figure 7

Growth of the micellar gel pattern in the tentacle state for the injection of a $50\mathrm{mM}$ CTAB solution into a $50\mathrm{mM}$ NaSal solution at a flow rate $Q=20\mathrm{ml}∕\mathrm{h}$. The width of each image is $11\mathrm{cm}$.

Figure 8

Total number of actively growing fingers $N_f$ in the tentacle growth state as a function of time ($50\mathrm{mM}$ CTAB solution injected into $50\mathrm{mM}$ NaSal solution at $20\mathrm{ml}∕\mathrm{h}$).

Figure 9

Linear finger growth regimes. Left to right: evanescent finger (concentration $c=30\mathrm{mM}$, flow rate $Q=1\mathrm{ml}∕\mathrm{h}$), steady finger ($c=40\mathrm{mM}$, $Q=2.5\mathrm{ml}∕\mathrm{h}$), wavy or oscillating finger growth ($c=30\mathrm{mM}$, $Q=1.5\mathrm{ml}∕\mathrm{h}$). Each picture is $2\mathrm{cm}$ wide, and the time interval between wavy finger pictures is $150\mathrm{s}$.