Department of Physics, Technische Universität Dortmund, 44221, Dortmund, Germany.
Eur Phys J E Soft Matter. 2021 Feb 12;44(1):4. doi: 10.1140/epje/s10189-021-00034-9.
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.
我们提出了一种低雷诺数下对称半球形 Marangoni 船（肥皂船或樟脑船）自推进的理论。推进是通过释放（扩散排放或溶解）水溶性表面活性剂分子来实现的，这些分子调节气液界面张力。对于完全对称的游泳者，推进要么需要不对称释放，要么需要通过与平流耦合来自发打破对称。我们以 Stokes 流为模型，对球体的扩散-平流问题进行了分析和数值研究，同时考虑了恒定浓度和恒定通量边界条件。我们针对恒定通量边界条件下的浓度分布和努塞尔数（总发射通量与扩散通量的无量纲比）得出了新颖的结果。基于这些结果，我们分析了小 Marangoni 推进（低 Peclet 数）下的 Marangoni 船，结果表明存在两种游泳模式，即低速度下的扩散模式和高游泳者速度下的平流主导模式。我们描述了大 Marangoni 推进（高 Peclet 数）的极限以及蒸发的影响，通过近似解析理论来实现。游泳速度由力平衡决定，我们得到了 Marangoni 力的一般表达式，其中包括沿气-水-游泳者接触线的表面张力梯度产生的直接 Marangoni 力和 Marangoni 流力。我们揭示了 Marangoni 流力在推进过程中是施加向前还是向后的力。我们的主要结果是 Peclet 数和游泳速度之间的关系。对于完全对称的游泳者，当 Peclet 数超过一个临界值时，会发生自发的对称破缺，从而发生游泳，而对于较大的系统尺寸，这个临界值会变得很小。我们发现了对称游泳者的超临界游泳分岔和存在不对称时的避免分岔。
