Energetic Variational Approaches in Complex Fluids and Electrophysiology

复杂的流体在我们的日常生活中无处不在，在物理和生物应用中具有重要意义。一个这样的例子是动物或人类中的离子溶液。生物学和生理学已经表明，了解离子溶液在几乎所有的生物活动中都是必不可少的。在复杂流体中，分子尺度的微观结构与宏观流动之间的机械和电气相互作用可能会产生简单流体或气体中没有遇到的新现象。需要新的数学理论和技术来解决一直存在的基本问题，如微观元素的系综、分子间和扭曲的弹性相互作用、流体动力输运和外加电场或磁场的耦合。本项目研究使用一个非常普遍的能量变分框架来推导和研究复杂流体的动力学，以及它在一般粘弹性流体、电解液和电流变液的流体动力学以及离子溶液在离子通道中的传输中的应用。这项拟议的研究还为学生提供了在数学、生物工程和神经生理学方面进行多学科体验的机会。该项目旨在取得几项不同的数学进展，并整合来自不同领域的这些物理和生物复杂流体的研究。PI建议研究的系统的共同特征是所有这些材料和系统之间潜在的能量变分结构。PI将侧重于了解不同空间或时间尺度上的耦合和竞争效应，其中包括溶液的微观结构和宏观性质，粘弹性材料中的动力学输运和诱导弹性应力，混合物中不同物种之间的相对拖曳和相互作用。能量变分方法在研究这种多尺度的多物理问题方面显示出了希望，例如离子溶液在生物环境中的传输。它们自然地处理具有许多组件的系统，这些组件在固定和移动的域或边界之间流动，并且具有不同的交互。我们将应用这种变分方法来研究生物等离子体、钠、钾和氯离子的离子溶液的输运，以及它们与不同环境和结构的相互作用。

Complex fluids are ubiquitous in our daily life and important in physical and biological applications. One such example is ionic solutions in animals or humans. Biology and physiology has shown that understanding ionic solutions is essential in almost all biological activities. The mechanical and electrical interactions of molecular-scale microstructures with macroscopic flow in complex fluids can give rise to new phenomena not encountered in simple fluids or gases. New mathematical theories and techniques are needed to resolve the ever present fundamental issues such as the ensemble of micro-elements, the intermolecular and distortional elastic interactions, the coupling of hydrodynamic transport and the applied electric or magnetic fields. This project investigates the use of a very general energetic variational framework to derive and study the dynamics of complex fluids, and its applications to general viscoelastic fluids, the hydrodynamics of electrolytes and electro-rheological fluids, and the transport of ionic solutions in ion channels. The proposed research also provides the opportunity for students to have multidisciplinary experiences in mathematics, bioengineering, and neurophysiology.This project aims at making several distinct mathematical advances and to integrate studies of these physical and biological complex fluids from different fields. The common feature of the systems that the PI proposes to study is the underlying energetic variational structure among all these materials and systems. The PI will focus on understanding the coupling and competing effects from different spatial or temporal scales, among them the microstructures and the macroscopic properties of the solutions, the kinetic transport and the induced elastic stresses in viscoelastic materials, the relative drags and interactions between different species in mixtures. Energetic variational methods have shown promise for the study of such multiscale multiphysics problems, for instance the transport of ionic solutions through biological environments. They deal naturally with systems with many components that flow between fixed and moving domains or boundaries, and with different interactions. We will apply such variational methods to study the transport of biological plasma, ionic solutions of sodium, potassium, and chloride ions, and their interaction with different environments and structures.