Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

Abstract

An analytical solution is derived for the flow generated by a self-propelling twodimensional Marangoni boat driven by reactive insoluble surfactant on a deep layer of fluid of viscosity µ at zero Reynolds number, capillary number and surface Péclet number. In the model, surfactant emitted from the edges of the boat causes a surface tension disparity across the boat. Once emitted, the surfactant diffuses along the interface and sublimates to the upper gas phase. A linear equation of state relates the surface tension to the surfactant concentration. The propulsion speed of the boat is shown to bewhere Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, ∆σ is the surface tension disparity between the front and rear of the boat and K 0 is the order-zero modified Bessel function. Explicit expressions for the streamfunction associated with the Stokes flow beneath the boat are found facilitating ready examination of the Marangoni-induced streamlines. An integral formula, derived using the reciprocal theorem, is also given for the propulsion speed of the boat in response to a more general Marangoni stress distribution.

Hydrodynamic efficiency limit on a Marangoni surfer

“…In the following we derive a theorem for a lower bound on the dissipation by a Marangoni surfer moving with velocity V A . We follow the approach that led to a minimum dissipation theorem for surface-propelled swimmers with external dissipation (Nasouri et al 2021) and later with combined external and internal dissipation (Daddi-Moussa-Ider et al 2023a). Using this theorem, the flow field of an optimal nearly spherical swimmer with external dissipation has been obtained using a perturbative analytical method (Daddi-Moussa-Ider et al 2021a).…”

“…The derivation of the minimum dissipation theorem follows in the same way as for surface-driven swimmers (Nasouri et al 2021;Daddi-Moussa-Ider et al 2023a). We apply the inequality (2.5) to the superposition:…”

“…The derivation is adapted from the proof of the original Helmholtz minimum dissipation theorem as formulated in the textbook by Guazzelli & Morris (2009) and follows a similar line as our previous generalisations (Nasouri et al 2021;Daddi-Moussa-Ider et al 2023a).…”

“…For certain distributions of surface tension, the propulsion of a thin disk-shaped Marangoni surfer has been exactly solved (Lauga & Davis 2012;Elfring, Leal & Squires 2016;Crowdy 2021). Using the Lorentz reciprocal theorem (Masoud & Stone 2014), the propulsion velocity of a surfer with any shape can in principle be determined provided the solution of the passive particle pulled by an external force is known.…”

“…Acknowledgements. The authors thank the anonymous reviewer of Daddi-Moussa-Ider et al (2023a) for the suggestion to derive a minimum dissipation theorem for Marangoni surfers.…”