Forward, reverse, and no motion of Marangoni surfers under confinement

We examine the mobility of a chemically active particle straddling the interface between a liquid layer of finite depth and a semi-infinite layer of gas. A surface-active agent is released asymmetrically from the particle that locally lowers the interfacial surface tension. It is commonly presumed that the uneven distribution of the surface tension and the associated Marangoni flow lead to the propulsion of the active surfer opposite to the release direction, where the surface tension is higher. This is considered forward motion. However, our recent theoretical analysis—in the limits of negligible inertia and diffusion-dominated transport of the active agent—has shown that this trend may be reversed for certain shapes of the surfer and shallow enough liquid layers. Advancing beyond the Stokes regime, here, we study the Marangoni-driven motion of thin cylindrical disks and oblate spheroids for a wide range of release rates and diffusivity of the exuded chemical species, that control the effective Reynolds and Péclet numbers. We consider various degrees of confinement represented by the thickness of the liquid film, and show that indeed the surfers can undergo a forward, a backward, or an arrested motion. We also identify the links between these modes of mobility and the forces acting on the surfers as well as the flow structure in their vicinity. Rather unexpectedly, we discover that negative pressure is the primary contributor to the fluid force experienced by the surfer and that this suction force is mainly responsible for the reverse Marangoni propulsion. The reported results are based on closely corroborating numerical calculations and experimental measurements.

Figure 1

Schematics of (a) experimental and (b) computational setups depicting, respectively, a fully submerged hemispherical and a half-submerged spherical surfer at a water-air interface. The area of the hemisphere dip coated with a layer of Dawn soap is colored red in panel (a), similar to the active area of the sphere in panel (b). The color map and vector plots in panel (b) represent the concentration distribution and liquid velocity field in the vicinity of the surfer, respectively.

Figure 2

Comparison between the results of experimental measurements (filled symbols) and numerical calculations (empty symbols) for the normalized propulsion velocity $\mathcal{U}/{\mathcal{U}}_{\infty }$ vs the dimensionless minimum gap between the surfer and the confining wall $\delta /R$. The Reynolds numbers corresponding to the motion of the sphere or hemisphere (square symbols) and disk (diamond symbols) are $\mathrm{Re}\approx 23$ and 25, respectively. The Péclet number is of the order of $\mathrm{Pe}\sim \mathcal{O}\left({10}^5\right)$ in the experiments and is set to $\mathrm{Pe}\approx 1000$ in the simulations, which is sufficiently high to justify the comparison.

Figure 3

Flow field plots corresponding to the Marangoni propulsion of a hemisphere or sphere under various degrees of confinement. The results are produced from the PIV measurements (left panels) and numerical simulations (right panels), and are plotted with respect to a fixed frame of reference. The top four rows show the fluid flow in the $x\text{−}z$ plane bisecting the surfer, and the fifth and sixth rows illustrate the velocity field at the water-air interface for, respectively, the least and most confined cases (i.e., the first and fourth rows). In the right panels, the thick arrows highlight the direction of the flow and the color maps display the concentration distribution of the active agent, where the concentration is the highest at the active area of the surfer (colored red) and is the lowest at the far field (colored blue). The black arrows are scaled independently in each panel to facilitate flow visualization. The red and green arrows atop each panel show the direction of propulsion and the black circles in the last two rows represent the three-phase contact line.

Figure 4

Flow field plots corresponding to the Marangoni propulsion of a disk under various degrees of confinement. The results are produced from the PIV measurements (left panels) and numerical simulations (right panels), and are plotted with respect to a fixed frame of reference. The top four rows show the fluid flow in the $x\text{−}z$ plane bisecting the surfer, and the fifth and sixth rows illustrate the velocity field at the water-air interface for, respectively, the least and most confined cases (i.e., the first and fourth rows). In the right panels, the thick arrows highlight the direction of the flow and the color maps display the concentration distribution of the active agent, where the concentration is the highest at the active area of the surfer (colored red) and is the lowest at the far field (colored blue). The black arrows are scaled independently in each panel to facilitate flow visualization. The red and green arrows atop each panel show the direction of propulsion and the black circles in the last two rows represent the three-phase contact line.

Figure 5

Normalized propulsion velocity $\mathcal{U}/{\mathcal{U}}_{\infty }$ vs the dimensionless minimum gap between the surfer and the confining wall $\delta /R$. The results are shown for $\mathrm{Pe}\approx 1000$ and half-submerged oblate spheroidal surfers of aspect ratio $\varepsilon =1,0.5,\text{and}0.2$. Panels (a)–(d) correspond to $\mathrm{Re}\approx 0.1,1,10,\text{and}100$, respectively. The inset in panel (d) shows the replots of $\mathcal{U}/{\mathcal{U}}_{\infty }$ vs $\delta /R$ curves for $\varepsilon =0.2$ from panels (a)–(d).

Figure 6

Flow field plots corresponding to the Marangoni propulsion of a half-submerged oblate spheroid of aspect ratio $\varepsilon =0.5$. The results are produced from numerical simulations for $\mathrm{Pe}\approx 1000$ (see Fig. 5 and Supplemental Material [55]). The gap size is $\delta /R=5$ and 1 in the left and right columns, respectively. The Reynolds number increases in each successive row from $\mathrm{Re}\approx 1$ to 10 and then to $\mathrm{Re}\approx 100$. The black arrows are scaled independently in each panel to facilitate flow visualization. The green arrows atop each panel show the direction of propulsion and the purple ones highlight the flow pattern.

Figure 7

Flow field plots corresponding to the Marangoni propulsion of half-submerged oblate spheroids of various aspect ratios under extreme confinements ($\delta /R=0.1$). The results are produced from numerical simulations for $\mathrm{Pe}\approx 1000$ (see Fig. 5 and Supplemental Material [55]). The left, middle, and right columns illustrate the flow for $\varepsilon =1$, 0.5, and 0.2, respectively. The Reynolds number increases in each successive row from $\mathrm{Re}\approx 1$ to 10 and then to $\mathrm{Re}\approx 100$. The black arrows are scaled independently in each panel to facilitate flow visualization. The red and green arrows atop each panel show the direction of propulsion and the purple ones highlight the flow pattern.