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

An analytical solution is derived for the flow generated by a self-propelling two-dimensional 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 be U0=Δσ(2πμ)−1e√DaK0(√Da) where Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, Δσ is the surface tension disparity between the front and rear of the boat, and K0 is the order-zero modified Bessel function. Explicit expressions for the stream function 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