活性的、各向异性流体的数学建模分析与计算

In viscoelastic fluids an outstanding open problem is the lack of rigorous foundation for the governing model equations. The obstruction is that the microstructure of a viscoelastic fluid is diverse across biology and engineered materials, whereas the specifics of the microstructure determine not only the moduli, timescales and length scales of elastic behavior, but typically cause the viscous or dissipative properties to vary with forcing frequency, flow type and rate. A high premium is therefore placed on modeling, and especially on identification of a universality class of viscoelastic fluids for which the governing equations can be settled, and the focus can shift to the interplay between experiment, theory, and computation. Here we propose a class of viscoelastic fluids, active, anistropic fluids (AAFs), unifying three apparently diverse fluid systems (catalytic nanorod dispersions, swimming bacterial suspensions, and motor-driven actin filament gels) for which models, analysis, algorithms, and simulations have evolved independently. We propose: a rigorous modeling foundation that spans kinetic to continuum scales; to identify the common, leading-order mathematical structure at scale for all AAFs, and the lower-order structure that distinguishes among activation mechanisms and microstructural details; to illustrate the impact of these results for active nanorod dispersions and actin filament gels in both confined and free surface flows through the development of efficient, structure-preserving and stable numerical algorithms and simulations to discern behavior due to details of the particle scale, aspect ratio, density, and activation mechanism..The intellectual merit of the proposed projects begins with the idea to unify models for a large class of viscoelastic fluids, those with anisotropic microstructure and internal activation, facilitating interplay between analytical and computational advances. This unified strategy links different scale models for different microstructures and activation mechanisms, identifying a common leading order mathematical structure across the class of active, anisotropic fluids, with their discerning features revealed in lower-order mathematical structure. The merit of this strategy lies in the ability to build well-posed boundary conditions, stability analyses, and numerical algorithms that have general applicability across a wide class of viscoelastic fluids. The outcomes of this project has the potential to spread the impact of mathematical advances to the biological and materials science communities, and to inspire the mathematical community to engage in this or similar directions for applied mathematics.

本项目中，我们将对活性的各向异性流体（active anisotropic fluids，简称：AAFs）在数学建模、分析和计算方面进行深入研究。AAFs作为一类具有普遍性的粘弹流体，在生命和材料科学等多个领域均有重要应用，最近被科学界广泛关注。但由于其微观结构的多样性，目前尚缺乏一个严格的、数学结构统一的理论基础。本项目将对AAFs体系建立一整套严格的、从动力学尺度到连续体尺度一致的新的理论框架和计算方法，利用不同尺度下AAFs体系中数学结构的共性和特性，研究多个尺度上的活性机制，开发一系列高效、保结构、稳定的数值算法和相应的高性能计算软件，利用数值计算来阐述限制的和具有自由界面的AAFs体系的性质，探索粒子微观性状和活性机制对集体动力学的影响，并将研究成果用于细胞动力学的模拟，以此推广数学在生命科学和材料科学的应用，同时带动更多数学界的有识之士加入到这一前景无限的研究领域中来。

本项目中，我们对活性物质体系在数学建模、分析和计算方面进行了较深入的研究。活性物质作为一类具有普遍性的自驱软物质，在生命和材料科学等多个领域均有重要应用。但由于其微观结构的多样性和自驱特性，以前缺乏一个严格的、数学结构统一的理论基础。本项目对活性体系建立了一整套严格的、基于非平衡态热力学原理的新的理论模型和相应的计算方法，利用不同尺度下活性物质体系中数学结构的共性和特性，研究了多个尺度上的活性机制，开发了一系列高效、保结构、稳定的数值算法和相应的计算软件，利用数值计算阐述限制在曲面上活性物质体系的性质，探索活性粒子微观性状和活性机制对集体动力学的影响，并将研究成果用于细胞动力学等的模拟。