A liquid droplet, immersed into a Newtonian fluid, can be propelled solely by internal flow. In a simple model, this flow is generated by a collection of point forces, which represent externally actuated devices or model autonomous swimmers. We work out the general framework to compute the self-propulsion of the droplet as a function of the actuating forces and their positions within the droplet. A single point force, $\mathbf{F}$, with general orientation and position, ${\mathbf{r}}_0$, gives rise to both translational and rotational motion of the droplet. We show that the translational mobility is anisotropic and the rotational mobility can be nonmonotonic as a function of $|{\mathbf{r}}_0|$, depending on the viscosity contrast. Due to the linearity of the Stokes equation, superposition can be used to discuss more complex arrays of point forces. We analyze force dipoles, such as a stresslet, a simple model of a biflagellate swimmer and a rotlet, representing a helical swimmer, driven by an external magnetic field. For a general force distribution with arbitrary high multipole moments the propulsion properties of the droplet depend only on a few low order multipoles: up to the quadrupole for translational and up to a special octopole for rotational motion. The coupled motion of droplet and device is discussed for a few exemplary cases. We show in particular that a biflagellate swimmer, modeled as a stresslet, achieves a steady comoving state, where the position of the device relative to the droplet remains fixed. There are two fixed points, symmetric with respect to the center of the droplet. A tiny external force selects one of them and allows one to switch between forward and backward motion.

中文翻译：

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由内部有源设备驱动的液滴的动力学

浸入牛顿流体的液滴只能通过内部流动来推动。在一个简单的模型中，此流量是由一组点力生成的，这些点力代表外部驱动的设备或模型自主游泳者。我们制定了通用框架，根据驱动力及其在液滴中的位置来计算液滴的自推进力。单点力，$\mathbf{F}$，具有大致的方向和位置， ${\mathbf{[R}}_0$引起液滴的平移和旋转运动。我们证明平移迁移率是各向异性的，旋转迁移率可以是非单调的，取决于$|{\mathbf{[R}}_0|$，取决于粘度对比。由于斯托克斯方程的线性，可以使用叠加来讨论更复杂的点力阵列。我们分析了受力偶极子，例如应力小波，双鞭毛状游泳者的简单模型和代表螺旋游泳者的小子，它们是由外部磁场驱动的。对于具有任意高多极矩的一般力分布，液滴的推进特性仅取决于几个低阶多极：高达四极用于平移，高达八极用于旋转运动。针对一些示例性情况，讨论了液滴与装置的耦合运动。我们特别表明，双鞭毛游泳者（建模为Stresslet）实现了稳定的共同运动状态，其中设备相对于液滴的位置保持固定。有两个相对于液滴中心对称的固定点。很小的外力就可以选择其中之一，并且可以在向前和向后运动之间切换。