The main objective of the present study is the derivation of exact analytical expressions for the orientation and trajectory of a spherical microswimmer submitted to general linear flows and to an external (gravitational or magnetic) force field, a problem known as gyrotaxis or magnetotaxis. We consider linear shear, hyperbolic, solid-rotation, and stagnation flows. The evolution equations of the swimmer orientation and its position are nonlinear and analytical results are the exception rather than the rule. Most available results for cell orientation and trajectories are obtained numerically. The solution for the swimmer orientation is inspired from a method due to Bretherton, initially developed for a different nonlinear equation. We show here that this method can be generalized to our evolution equation. We see that the swimmer under flow exhibits both run (a motion where the orientation angle is kept constant with time) and tumble (the orientation angle is cyclic with time) regimes, and a variety of cell trajectories are extracted analytically, such as parabolic, elliptic, and helical. This study offers a framework to generalize the results to other types of flows.

中文翻译：

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流动和外场作用下球形微型游泳者的精确轨迹解

本研究的主要目标是推导受一般线性流动和外部（重力或磁）力场影响的球形微型游泳者的方向和轨迹的精确解析表达式，这个问题称为回旋或趋磁。我们考虑线性剪切流、双曲线流、固体旋转流和停滞流。游泳者方向及其位置的演化方程是非线性的，解析结果是例外而不是规则。大多数可用的细胞方向和轨迹结果都是通过数值获得的。游泳者方向的解决方案受到 Bretherton 方法的启发，该方法最初是为不同的非线性方程开发的。我们在这里展示了这种方法可以推广到我们的进化方程。运行（方向角随时间保持恒定的运动）和翻滚（方向角随时间循环）机制，并分析提取各种细胞轨迹，例如抛物线、椭圆和螺旋。本研究提供了一个框架，可以将结果推广到其他类型的流量。