Drift instability in the motion of a fluid droplet with a chemically reactive surface driven by Marangoni flow

We theoretically derive the amplitude equations for a self-propelled droplet driven by Marangoni flow. As advective flow driven by surface tension gradient is enhanced, the stationary state becomes unstable and the droplet starts to move. The velocity of the droplet is determined from a cubic nonlinear term in the amplitude equations. The obtained critical point and the characteristic velocity are well supported by numerical simulations.

Figure 1

Schematic illustration of the system in this study. Surfactants dissolve in the outer fluid and some are adsorbed at the interface between the inner and the outer fluids. These surfactants reduce the surface tension of the droplet. (a) No droplet motion occurs for an isotropic distribution of surfactants. (b) When the surfactant distribution becomes asymmetric, the flow (thin red arrows) occurs and the droplet starts to move in the direction of the thick black arrow. The flux of surfactants are shown in broken arrows. The background gradation represents surfactant concentration.

Figure 2

Distribution of bulk concentration field. (a) Isotropic distribution when $\lambda =0.5$, $\alpha =0.01$, and $u_1/D=0$. (b) Anisotropic distribution with $u_1/D=2.0$. The blue (dark gray) line shows $c(r,\theta =0)$ (front) and the red (light gray) line shows $c(r,\theta =\pi )$ (rear). The uniform distribution of (a) is shown in (b) as a dashed line.

Figure 3

Bifurcation diagram for spontaneous motion. The bifurcation parameters are chosen to be $u_1$ (a) and $R$ (b).

Figure 4

Normalized velocity of a droplet without (a) and with (b) surface advection in Eq. (24). The slope of the line is 0.5.