Mathematical models for swimming in a viscoelastic fluid

在本项目中，我们将开发在粘弹流体中游泳的新数学模型。 该项目包括三个部分。第一种是将精子鞭毛的滑动细丝模型与线性粘弹性流体的麦克斯韦模型中的连续力理论联合收割机相结合，从而确定游动速度。 这个简单的模型将捕捉到许多功能，这将是目前在更复杂的，但现实的模型，并阐明了在粘弹性流体中游泳的基本原理。 第二部分的目的是推导粘弹性介质中细杆受力的细长体方程，类似于斯托克斯流中常见的细长体方程。 最后，第三部分的目的是将这些模型应用于精子在粘液中游泳的情况。例如，在受精过程中的某个时刻，精子细胞将其跳动模式改变为不对称的鞭子状“超激活”状态。我们将研究多动如何影响子宫和输卵管的粘弹性环境中的运动。这个项目的动机是，许多游泳的微生物遇到的生物流体与固体一样（弹性）属性以及液体一样（粘性）属性。 例如，从宫颈液中的哺乳动物精子到胃内壁粘液层中的引起溃疡的幽门螺杆菌。尽管这些例子对生育和健康非常重要，但很少有人尝试建立粘弹性流体中游泳的定量理论;几乎所有以前的努力都集中在纯粘性流体如水上。重要的是发展粘弹性游泳者的理论，因为它将加深我们对哺乳动物精子所面临的物理限制的理解，从而更好地了解它们的生物学，并可能为避孕或提高人类或牲畜的生育能力提供新的临床策略。

In this project, we will develop new mathematical models for swimming in a viscoelastic fluid. The project consists of three parts. The first is to combine a model for the sliding filaments of a sperm flagellum with resistive-force theory in the Maxwell model for a linearly viscoelastic fluid, and then to determine the swimming speed. This simple model will capture many of the features which will be present in more complicated but realistic models, and illuminate the fundamental principles of swimming in a viscoelastic fluid. The aim of the second part is to derive the slender-body equations for the forces acting on thin rods in a viscoelastic medium, analogous to the slender-body equations familiar from Stokes flow. Finally, the aim of the third part is to apply these models to the case of sperm swimming in mucus. For example, at a certain point during fertilization, sperm cells change their beat pattern to the asymmetric whip-like "hyperactivated" state. We will study how hyperactivity affects motility in the viscoelastic environments of the uterus and oviduct.This project is motivated by the fact that many swimming microorganisms encounter biological fluids with solid-like (elastic) properties as well as liquid-like (viscous) properties. Examples range from mammalian sperm in cervical fluid to the ulcer-causing bacterium Helicobacter pylori in the mucus layer lining the inside of the stomach. Despite the great importance of these examples for fertility and health, there have been few attempts to develop a quantitative theory for swimming in viscoelastic fluids; almost all previous efforts have focused on purely viscous fluids such as water. It is important to develop a theory for viscoelastic swimmers, since it will deepen our understanding of the physical constraints faced by mammalian sperm, leading to a better understanding of their biology and perhaps new clinical strategies for contraception or enhancing fertility for humans or livestock.