Quantifying macrostructures in viscoelastic sub-diffusive flows

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
T. Chauhan, K. Kalyanaraman, S. Sircar
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引用次数: 0

Abstract

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.
量化粘弹性亚扩散流中的宏观结构
我们提出了一种理论,通过对聚合物构象张量进行数学上一致的分解,将其转化为所谓的结构张量,从而量化粘弹性亚扩散流的时空宏观结构(或中等至高流体惯性下的高粘度非均质区域)的形成。我们的方法绕过了标准算术分解中的一个固有问题,即波动构象张量场可能不是正定的,因此无法保留其物理意义。利用矩阵分析的既定结果,通过切线空间的内积定义和构建大地线,将正定矩阵空间转化为黎曼流形。利用这条测地线可以定义结构张量的三个标量不变式,它们不会受到聚合物构象张量常规不变式(如迹和行列式)的影响。首先,我们考虑的问题是利用测地线对结构张量进行扰动展开,这与黎曼流形几何是一致的。在此过程中,扰动解的演化可以用沿欧几里得流形的线性理论很好地近似。通过对线性和非线性动力学的比较,确定了非线性在引发关于中心线的流动变量对称性破坏中的作用。最后,对粘弹性亚扩散通道流进行了全非线性模拟,强调了使用这些不变式有效量化宏观结构的优势。
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来源期刊
Journal of Mathematical Physics
Journal of Mathematical Physics 物理-物理:数学物理
CiteScore
2.20
自引率
15.40%
发文量
396
审稿时长
4.3 months
期刊介绍: Since 1960, the Journal of Mathematical Physics (JMP) has published some of the best papers from outstanding mathematicians and physicists. JMP was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods for such applications and for the formulation of physical theories. The Journal of Mathematical Physics (JMP) features content in all areas of mathematical physics. Specifically, the articles focus on areas of research that illustrate the application of mathematics to problems in physics, the development of mathematical methods for such applications, and for the formulation of physical theories. The mathematics featured in the articles are written so that theoretical physicists can understand them. JMP also publishes review articles on mathematical subjects relevant to physics as well as special issues that combine manuscripts on a topic of current interest to the mathematical physics community. JMP welcomes original research of the highest quality in all active areas of mathematical physics, including the following: Partial Differential Equations Representation Theory and Algebraic Methods Many Body and Condensed Matter Physics Quantum Mechanics - General and Nonrelativistic Quantum Information and Computation Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory General Relativity and Gravitation Dynamical Systems Classical Mechanics and Classical Fields Fluids Statistical Physics Methods of Mathematical Physics.
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