Hydrodynamic efficiency limit on a Marangoni surfer

Abstract

A Marangoni surfer is an object embedded in a gas–liquid interface, propelled by gradients in surface tension. We derive an analytical theorem for the lower bound on the viscous dissipation by a Marangoni surfer in the limit of small Reynolds and capillary numbers. The minimum dissipation can be expressed with the reciprocal difference between drag coefficients of two passive bodies of the same shape as the Marangoni surfer, one in a force-free interface and the other in an interface with surface incompressibility. The distribution of surface tension that gives the optimal propulsion is given by the surface tension of the solution for the incompressible surface and the flow is a superposition of both solutions. For a surfer taking the form of a thin circular disk, the minimum dissipation is $16\mu a V^2$ , giving a Lighthill efficiency of $1/3$ . This places the Marangoni surfers among the hydrodynamically most efficient microswimmers.

Hydrodynamics of a disk in a thin film of weakly nematic fluid subject to linear friction

“…The problem with finite subphase depth was subsequently addressed by Stone and Ajdari [8], partially through numerical methods. For infinite depth and vanishing surface viscosity (Bq → 0), the translational drag coefficient was found to be 50% larger than the drag coefficient for a fluid with a free surface [8,9]. More recently, the translational motion of a disk embedded in a nearly inviscid Langmuir film, marking the limit Bq ≪ 1, was revisited [10].…”