Surfactant-driven instability of a divergent flow

Extremely small amounts of surface-active contaminants are known to drastically modify the hydrodynamic response of the water-air interface. Surfactant concentrations as low as a few thousand molecules per square micron are sufficient to eventually induce complete stiffening. In order to probe the shear response of a water-air interface, we design a radial flow experiment that consists in an upward water jet directed to the interface. We observe that the standard no-slip effect is often circumvented by an azimuthal instability with the occurrence of a vortex pair. Supported by numerical simulations, we highlight that the instability occurs in the (inertia-less) Stokes regime and is driven by surfactant advection by the flow. The latter mechanism is suggested as a general feature in a wide variety of reported and yet unexplained observations.

Figure 1

(a) Sketch of the experimental setup. Water is injected near the interface and produces an axisymmetrical flow with a toroidal structure, as shown in (b) (top view) and (c) (side view). In (b), the green line marks the vertical laser sheet passing near the injection point (orange disk). The image (c) is the corresponding cross-sectional view. The streamlines are obtained by time-averaging video images. Parameters: $V_{\mathrm{inj}}=58\mathrm{mm}{\mathrm{s}}^{-1}$, $H=10.5$ mm, $C_{\mathrm{SDS}}=\mathrm{CMC}/8$.

Figure 2

(a) Top view of the dipolar flow. The $+$ (respectively, $-$) sign indicates clockwise (respectively, counterclockwise) rotation. The red arrows mark the flow direction along the symmetry axis of the dipole. The green dashed lines 1 and 2 indicate the orientations of the vertical laser sheets for bulk flow observation. (b) Cross-sectional view along line 1 in (a). (c) Cross-sectional view along line 2 in (a). Parameters: $V_{\mathrm{inj}}=55\mathrm{mm}{\mathrm{s}}^{-1}$, $H=4.9$ mm, $C_{\mathrm{SDS}}=\mathrm{CMC}/8$.

Figure 3

Photographs of the 3D structure of the dipolar flow as revealed by dye (fluorescein) injection in a tracer-free solution. The $+$ (respectively, $-$) red sign denotes clockwise (respectively, counterclockwise) vortex rotation. The gap is reported below each side view. The magenta line points out structural links between the views. TV: top view; SV: side view (along the symmetry plane of the dipole); BV: back view. Parameters: $V_{\mathrm{inj}}=5.5\mathrm{cm}{\mathrm{s}}^{-1}$, $C_{\mathrm{SDS}}=\text{CMC}/100$.

Figure 4

(a) Schematic illustration of the motion of surface (S) and subsurface (SS) tracers along the symmetry axis of the dipole [e.g., line 1 in Fig. 2]. ${\mathrm{SS}}^{\prime }$ denotes the mirror image of SS tracers through the water-air interface. [(b)–(d)] Dimensionless velocity of S (blue squares) and SS (black circles) tracers at fixed gap $H=2.1$ mm for different injection speeds. (b) $V_{\text{inj}}=11.8\mathrm{mm}{\mathrm{s}}^{-1}$, axisymmetric flow; (c) $V_{\text{inj}}=17.7\mathrm{mm}{\mathrm{s}}^{-1}$, intermediate state; (d) $V_{\text{inj}}=23.6\mathrm{mm}{\mathrm{s}}^{-1}$, dipolar flow. Concentration: $C_{\mathrm{SDS}}=\mathrm{CMC}/100$. Solid and dashed lines are guides to the eyes.

Figure 5

[(a)–(c)] Computed surface velocity field ${\mathbf{v}}_s$ (color maps) at steady state for a surface fraction $\phi ={10}^{-3}$ for several Péclet numbers. The white arrows (arbitrary lengths) indicate the flow direction. (a) $\text{Pe}=0.1$, (b) $\text{Pe}=10$, and (c) $\text{Pe}={10}^3$. In (b), the orange spot marks the location of a stagnant point. In (c), the dashed green lines indicate the main axes of the dipole. (d) Normalized surface velocity and concentration ($\mathrm{\Gamma }$) along line 1 in (c).

Figure 6

(a) Surface concentration $\mathrm{\Gamma }$ (color map) and surface velocity ${\mathbf{v}}_{\mathbf{s}}$ (arrows) for $\text{Pe}={10}^3$. The green dashed lines indicate the main axes of the dipole. (b) Surface concentration along the $x$ axis of the laboratory frame for different values of $\text{Pe}$. Parameter: $\phi ={10}^{-3}$.