Abstract
We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.Graphic abstract
Highlights
Swimming on the microscale is governed by low Reynolds numbers and requires special propulsion mechanisms which are effective in the presence of dominating viscous forces
These results for Marangoni forces as a function of Ufor a critical Peclet number Pe > Pec; if Marangoni flows forces are included, we find Pec 1 and the symmetry-breaking bifurcation takes place within are inserted into the force balance or swimming condition this regime. – U < 1 and 1 < Pe < Sc: All fluid flows are still at low Reynolds numbers, but Marangoni flows are rel
At low Reynolds numbers, we developed a complete theory for Marangoni boat propulsion for a completely symmetric, half-spherical, surfactant emitting swimmer
Summary
Swimming on the microscale is governed by low Reynolds numbers and requires special propulsion mechanisms which are effective in the presence of dominating viscous forces. Marangoni flow forces drag and decrease the direct driving force (FM,fl < 0) This result will change as we (i) consider 3D diffusion and (ii) as symmetry breaking is only caused by advection, which can focus the concentration field and lead to higher Legendre components becoming relevant in c(ρ). The highest total force is obtained if a long-range − cos θ-component is by advection and described by a linear response in diffusion–advection (iii) with respect to Uand Pe. Only the linear response in Uis relevant for sympresent in the concentration profile, as we will find metry breaking; the Marangoni flow can for small swimmer velocities; Marangoni flows increase the direct Marangoni driving force. The remaining total Marangoni force mainly comes from the net forward motion in the horizontal vortex pairs but will be weaker than the direct force
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