{"title":"Quantifying macrostructures in viscoelastic sub-diffusive flows","authors":"T. Chauhan, K. Kalyanaraman, S. Sircar","doi":"10.1063/5.0195666","DOIUrl":null,"url":null,"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.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"253 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1063/5.0195666","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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.
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.