Convective and inertial Marangoni surfing

Document Type

Article

Publication Date

1-2-2026

Abstract

We study the surfing motion of an active particle along a planar interface, separating asemi-infinite layer of gas from a deep layer of liquid. The interface-trapped particle self-propels, thanks to an uneven distribution of surface tension in its immediate vicinity, whichitself results from a non-uniform release of an active agent from the particle’s surface.We use the reciprocal theorem in conjunction with singular perturbation expansions tocalculate the leading-order contributions to the propulsion speed of the surfer due tothe advective transport of mass and momentum when the Péclet and Reynolds numbers(denoted by Pe and Re, respectively) are small but finite. Assuming that the surface tensionvaries linearly with the concentration of the agent with a slope of negative α, we show,perhaps unexpectedly, that the normalised speed for a purely translating (but otherwisearbitrarily shaped) particle, independent of the agent discharge mechanism, can beexpressed as U = 1 + A(2Pe ln Pe + Re ln Re) + O(Pe) + O(Re), where the prefactor Ais positive for negative α and vice versa. For reference, the self-propulsion speed ofautophoretic Janus spheres varies with Pe as U = 1 + B Pe + · · ·, where B is positivewhen the mobility coefficient of the particle is negative and vice versa. Also, the speedof spherical squirmers changes with Re as U = 1 + C Re + O(Re)2 , with C being positivefor pushers and negative for pullers. Our asymptotic formula reveals that the speed ofa Marangoni surfer is a non-monotonic function of the Péclet and Reynolds numbers,hinting at the existence of optimal values for both Pe and Re. The information containedwithin the multiplier A also offers guidance for customising the shape of the surfer, as wellas the release rate and configuration of the agent, to enhance the self-surfing performance.Our general theoretical analysis is complemented by detailed numerical simulations for arepresentative spherical surfer. These simulations confirm our theoretical predictions and shed light on the effects of intermediate and large values of Pe and Re on the performance of Marangoni surfers.

Publication Title

Journal of Fluid Mechanics

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