Hydrodynamic response of a surfactant-laden interface to a radial flow

We study the features of a radial Stokes flow due to a submerged jet directed toward a liquid-air interface. The presence of surface-active impurities confers to the interface an in-plane elasticity that resists the incident flow. Both analytical and numerical calculations show that a minute amount of surfactants is enough to profoundly alter the morphology of the flow. The hydrodynamic response of the interface is affected as well, shifting from slip to no-slip boundary condition as the surface compressibility decreases. We argue that the competition between the divergent outward flow and the elastic response of the interface may actually be used as a practical way to detect and quantify a small amount of impurities.

Figure 1

Schematic representation of the system. A submerged jet is directed toward an interface covered with insoluble surface-active impurities (represented as surfactants). Impurities are then swept away toward the cell boundaries, which induces a Marangoni counterflow. The system is invariant by rotation around the $z$ axis.

Figure 2

Streamlines of the Landau-Squire jet perpendicular to a clean interface [see Eq. (10)]. The point source is marked by a purple dot.

Figure 3

Streamlines of the total velocity field $\mathbf{v}={\mathbf{v}}^{\left(0\right)}+{\mathbf{v}}^{\left(1\right)}$ in the low-compressibility regime $0<\beta <1$. The point source is marked by a purple dot. The orange dots indicate the position of the centerline of the vortex ring.

Figure 4

Evolution of the surfactant concentration $\mathrm{\Gamma }\left(r\right)$ with the compressibility $\beta =\eta V_0/E_0$. The curves correspond to Eq. (19) for $0\le \beta \le 1$; for $\beta >1$, the concentration is obtained by numerical inversion of its Hankel transform [Eqs. (15) and (22)]. Note the crossover between the low- and high-compressibility regimes that occurs when $\beta =1$.

Figure 5

Interfacial velocity field $v_r(r,0)$ in the high-compressibility regime $\beta >1$. The limit $\beta \to \infty$ corresponds to the clean interface limit.

Figure 6

(a) Flow streamlines for a clean interface ($x=0$). Small cell ($R=72a$), gap $H=8a$. (b) Radial position of the torus centerline for a clean interface ($x=0$).

Figure 7

(a) Flow streamlines for a surfactant-laden interface ($x=0.152$). Small cell ($R=72a$), gap $H=8a$. (b) Evolution of the position of the torus centerline. Large cell ($R=144a$) and small cell ($R=72a$), gap $H=8a$. The shaded area corresponds to the high-compressibility regime.

Figure 8

(a) Evolution of the concentration of surfactant for different surface coverages. Large cell ($R=144a$), gap $H=8a$. (b) Same as (a) but the graph is zoomed in to better visualize the concentration profiles for the largest values of $x$.

Figure 9

Evolution of the interfacial radial velocity profiles for different surface coverages. Large cell ($R=144a$), gap $H=8a$. (a) High-compressibility regime ($x<x^{*}$). (b) Low-compressibility regime ($x>x^{*}$). Notice that the velocity scales are different between the two graphs.

Figure 10

Normalized interfacial velocity $v_r(r,0)/[V_{\mathrm{inj}}\times h(H/a)]$ as a function of the distance to the origin. The full line corresponds to Eq. (C2). Inset: same data presented over a wider range.