Energetic Variational Approaches in Complex Fluids and Electrophysiology

复杂流体在我们的日常生活中无处不在，在物理和生物应用中非常重要。其中一个例子是动物或人类的离子溶液。生物学和生理学表明，了解离子溶液对于几乎所有生物活动都是至关重要的。复杂流体中分子尺度微观结构与宏观流动的机械和电相互作用可以产生简单流体或气体中未遇到的新现象。需要新的数学理论和技术来解决一直存在的基本问题，例如微元素的集合、分子间和畸变弹性相互作用、流体动力传输和施加的电场或磁场的耦合。该项目研究使用非常通用的能量变分框架来推导和研究复杂流体的动力学，及其在一般粘弹性流体、电解质和电流变流体的流体动力学以及离子通道中离子溶液的传输中的应用。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获得多学科经验的机会。该项目旨在取得几项独特的数学进展，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将重点了解不同空间或时间尺度的耦合和竞争效应，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传输和诱导弹性应力、混合物中不同物质之间的相对阻力和相互作用。能量变分方法已显示出研究此类多尺度多物理问题的前景，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血浆、钠、钾和氯离子的离子溶液的传输，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.