Convective and inertial Marangoni surfing

We study the surfing motion of an active particle along a planar interface, separating a semi-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, which itself 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 to calculate the leading-order contributions to the propulsion speed of the surfer due to the advective transport of mass and momentum when the Péclet and Reynolds numbers (denoted by $\textit{Pe}$ and $\textit{Re}$ , respectively) are small but finite. Assuming that the surface tension varies linearly with the concentration of the agent with a slope of negative $\alpha$ , we show, perhaps unexpectedly, that the normalised speed for a purely translating (but otherwise arbitrarily shaped) particle, independent of the agent discharge mechanism, can be expressed as $\mathscr{U} = 1 + \mathscr{A} ( 2 \textit{Pe} \ln \textit{Pe} + \textit{Re} \ln \textit{Re} ) + \mathscr{O}(\textit{Pe}) + \mathscr{O}(\textit{Re})$ , where the prefactor $\mathscr{A}$ is positive for negative $\alpha$ and vice versa . For reference, the self-propulsion speed of autophoretic Janus spheres varies with $\textit{Pe}$ as $\mathscr{U} = 1 + \mathscr{B} \, \textit{Pe} + {\cdots}$ , where $\mathscr{B}$ is positive when the mobility coefficient of the particle is negative and vice versa . Also, the speed of spherical squirmers changes with $\textit{Re}$ as $\mathscr{U} = 1 + \mathscr{C} \, \textit{Re} + \mathscr{O}(\textit{Re})^2$ , with $\mathscr{C}$ being positive for pushers and negative for pullers. Our asymptotic formula reveals that the speed of a Marangoni surfer is a non-monotonic function of the Péclet and Reynolds numbers, hinting at the existence of optimal values for both $\textit{Pe}$ and $\textit{Re}$ . The information contained within the multiplier $\mathscr{A}$ also offers guidance for customising the shape of the surfer, as well as 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 a representative spherical surfer. These simulations confirm our theoretical predictions and shed light on the effects of intermediate and large values of $\textit{Pe}$ and $\textit{Re}$ on the performance of Marangoni surfers.