Flow instabilities due to the interfacial formation of surfactant–fatty acid material in a Hele-Shaw cell

We present an experimental study of pattern formation during the penetration of an aqueous surfactant solution into a liquid fatty acid in a Hele-Shaw cell. When a solution of the cationic surfactant cetylpyridinium chloride is injected into oleic acid, a wide variety of fingering patterns are observed as a function of surfactant concentration and flow rate, which are strikingly different than the classic Saffman-Taylor (ST) instability. We observe evidence of interfacial material forming between the two liquids, causing these instabilities. Moreover, the number of fingers decreases with increasing flow rate $Q$, while the average finger width increases with $Q$, both trends opposite to the ST case. Bulk rheology on related mixtures indicates a gel-like state. Comparison of experiments using other oils indicates the importance of pH and the carboxylic head group in the formation of the surfactant–fatty acid material.

Figure 1

(a) Instability pattern of a 400 mM CPCl solution injected into oleic acid at $Q=100$ ml/h $\left(t=10\text{s}\right)$; (b) close up of the CPCl-oleic acid fingers taken about 10 min after finishing the test, showing clearly the material at the interface. Scale bar represents 5 mm.

Figure 2

The classic Saffman-Taylor instability is observed when only one of the required components is used: (a) pure water–oleic acid, (b) 425 mM CPCl solution–silicone oil $(Q=100$ ml/h, $t=15\text{s})$.

Figure 3

Viscoelastic behavior of bulk homogeneous surfactant–fatty acid material: Gel 1 (red), and 2 (blue), see text. Closed and open circles show elastic and loss moduli, respectively.

Figure 4

Fingering pattern made with 50 mM CPCl–oleic acid at time $t=800/Q$: (a) $Q=150$ ml/h, (b) $Q=250$ ml/h, (c) $Q=350$ ml/h, (d) $Q=400$ ml/h.

Figure 5

Quantitative aspects of the finger patterns at 50 mM CPCl, taken at time $t=800/Q$: (a) the effective number of fingers vs flow rate; (b) the effective width of fingers as a function flow rate. The inset plots show average width and number of fingers measured at varying radii $r$ from the center (see text). The inset in (a) shows the binary picture of fingering pattern in Fig. 4. The blue circle shows the radius where $N=N_{max}$, and the red circles show where $N$ reaches $0.75N_{max}$.

Figure 6

Fingering pattern made in CPCl–oleic acid at fixed $Q=100$ ml/h and $t=8$ s for CPCl concentration of (a) $C=0.5$ mM, (b) $C=25$ mM, (c) $C=50$ mM, (d) $C=100$ mM, and (e) $C=200$ mM.

Figure 7

(a) $\tilde{N}$, and (b) $\tilde{w}\left(mm\right)$ versus concentration, calculated similar to Fig. 5. Triangle shows $C=0$, i.e., ST pattern.

Figure 8

Pattern formed by injecting a 400 mM CPCl solution at $Q=100$ ml/h into caprylic acid (taken at $t=8$ s).