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 $\mu$ 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 $U_0=\mathrm{\Delta }\sigma {\left(2\pi \mu \right)}^{-1}{e}^{\sqrt{\text{Da}}}{K}_0\left(\sqrt{\text{Da}}\right)$ where Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, $\mathrm{\Delta }\sigma$ 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.

Figure 1

A Marangoni boat of length $2L$ in an $(x,y)$ plane with surfactant levels ${\mathrm{\Gamma }}^{\pm }$ imposed at its front and rear moves with speed $U_0$ atop a fluid of viscosity $\mu$ due to a surface tension disparity $\mathrm{\Delta }\sigma$ caused by the surfactant. The surfactant sublimates to the upper gas phase and diffuses along the interface with the relative strength of these two effects characterized by a Damköhler number Da.

Figure 2

The mathematical problem for the lower analytic function $f\left(z\right)$ to which the physical problem in Fig. 1 is reduced. It is a mixed-type boundary value problem for a lower-analytic function $f\left(z\right)$. There are square root branch points in $f\left(z\right)$ at $z=\pm L$. The solution can be found using conformal mapping methods as shown in Appendix pp2.

Figure 3

Graph of $U_0\left(2\pi \mu \right)/\mathrm{\Delta }\sigma$ against Da. The dashed line is the asymptote $\sqrt{\pi /2}{\left(\text{Da}\right)}^{-1/4}$.

Figure 4

Fore-aft symmetric eddy structure beneath the boat when $\text{Da}=1$ and $r=1$ so that equal amounts of surfactant are ejected both fore and aft. The flow induced by the Marangoni stress is symmetric about the center of the boat which consequently remains stationary: $U_0=0$.

Figure 5

Streamlines in the fixed frame (left column) and the cotraveling frame (right column) for $\text{Da}=1$ and $r=0.75$ (top row), $0,5,0.25$, and 0 (bottom row). The case $r=0$ corresponds to actuation at the rear only. The boat propulsion speed $U_0$ is identical in all cases shown although the streamline topology changes dramatically with $r$.

Figure 6

Streamlines in the fixed frame (left column) and the cotraveling frame (right column) for $\text{Da}=10$ and $r=0.75$ (top row), $0,5,0.25$, and 0 (bottom row). The case $r=0$ corresponds to actuation at the rear only. The boat propulsion speed $U_0$ is identical in all cases shown.

Figure 7

Conformal mapping (B1) from the lower-half unit disk in a complex $\zeta$ plane to the fluid region $y<0$ below the Marangoni boat. The point $\zeta =-i$ is the preimage of the point at infinity in the lower-half $z$ plane $y\to -\infty$. The preimage of the boat is the real diameter $\zeta \in (-1,1)$. Tracing arrows around the boundaries in each plane shows the correspondences under (B1).