An official website of the United States government
In this article, we introduce a new method for discretizing micro-macro models of dilute polymeric fluids by integrating a finite element method (FEM) discretization for the macroscopic fluid dynamics equation with a deterministic variational particle scheme for the microscopic Fokker-Planck equation. To address challenges arising from micro-macro coupling, we employ a discrete energetic variational approach to derive a coarse-grained micro-macro model with a particle approximation first and then develop a particle-FEM discretization for the coarse-grained model. The accuracy of the proposed method is evaluated for a Hookean dumbbell model in a Couette flow by comparing the computed velocity field with existing analytical solutions. We also use our method to study nonlinear FENE dumbbell models in different scenarios, such as extensional flow, pure shear flow, and lid-driven cavity flow. Numerical examples demonstrate that the proposed deterministic particle approach can accurately capture the various key rheological phenomena in the original FENE model, including hysteresis and δ-function-like spike behavior in extensional flows, velocity overshoot phenomenon in pure shear flows, symmetries breaking, vortex center shifting, and vortices weakening in lid-driven cavity flows, with a small number of particles.
more »
« less
Multiscale Modeling of Viscoelastic Fluids
This article provides an introductory review of the mathematical modeling of viscoelastic fluids (VEFs), which exhibit both viscous and elastic behaviors critical to applications in biological and industrial processes. Focusing on the interplay between microscopic dynamics and macroscopic properties, the paper explores constitutive modeling challenges for VEFs, particularly polymeric fluids. It discusses three levels of description: stochastic differential equations (SDEs) for individual microstructural dynamics, Fokker-Planck equations for ensemble behavior, and macroscopic partial differential equations (PDEs) for continuum flow fields. Using bead-spring models, such as Hookean and FENE dumbbell models, the review illustrates how microstructural configurations influence macroscopic stress responses. Key mathematical challenges, including numerical convergence, equation well-posedness, and closure approximations, are highlighted, with specific attention to the Upper Convected Maxwell and FENE-P models. The article underscores the balance between computational feasibility and physical accuracy in modeling VEFs, offering insights into ongoing research and future directions in rheology and applied mathematics.
more »
« less
- Award ID(s):
- 1751339
- PAR ID:
- 10582523
- Publisher / Repository:
- Notices of the American Mathematical Society by the American Mathematical Society (AMS)
- Date Published:
- Journal Name:
- Notices of the American Mathematical Society
- Volume:
- 71
- Issue:
- 08
- ISSN:
- 0002-9920
- Page Range / eLocation ID:
- 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Modeling the flow of polymer solutions requires knowledge at various length and time scales. The macroscopic behavior is described by the overall velocity, pressure, and stress. The polymeric contribution to the stress requires knowledge of the evolution of polymer chains. In this work, we use a microscopic model, the finitely extensible nonlinear elastic (FENE) model, to capture the polymer’s behavior. The benefit of using microscopic models is that they remain faithful to the polymer dynamics without information loss via averaging. Their downside is the computational cost incurred in solving the thousands to millions of differential equations describing the microstructure. Here, we describe a multiscale flow solver that utilizes GPUs for massively parallel, efficient simulations. We compare and contrast the microscopic model with its macroscopic counterpart under various flow conditions. In particular, significant differences are observed under nonlinear flow conditions, where the polymers become highly stretched and oriented.more » « less
-
Over the last twenty years advances in systems biology have changed our views on microbial communities and promise to revolutionize treatment of human diseases. In almost all scientific breakthroughs since time of Newton, mathematical modeling has played a prominent role. Regulatory networks emerged as preferred descriptors of how abundances of molecular species depend on each other. However, the central question on how cellular phenotypes emerge from dynamics of these network remains elusive. The principal reason is that differential equation models in the field of biology (while so successful in areas of physics and physical chemistry), do not arise from first principles, and these models suffer from lack of proper parameterization. In response to these challenges, discrete time models based on Boolean networks have been developed. In this review, we discuss an emerging modeling paradigm that combines ideas from differential equations and Boolean models, and has been developed independently within dynamical systems and computer science communities. The result is an approach that can associate a range of potential dynamical behaviors to a network, arrange the descriptors of the dynamics in a searchable database, and allows for multi-parameter exploration of the dynamics akin to bifurcation theory. Since this approach is computationally accessible for moderately sized networks, it allows, perhaps for the first time, to rationally compare different network topologies based on their dynamics.more » « less
-
The advent of artificial intelligence and machine learning has led to significant technological and scientific progress, but also to new challenges. Partial differential equations, usually used to model systems in the sciences, have shown to be useful tools in a variety of tasks in the data sciences, be it just as physical models to describe physical data, as more general models to replace or construct artificial neural networks, or as analytical tools to analyse stochastic processes appearing in the training of machine-learning models. This article acts as an introduction of a theme issue covering synergies and intersections of partial differential equations and data science. We briefly review some aspects of these synergies and intersections in this article and end with an editorial foreword to the issue. This article is part of the theme issue ‘Partial differential equations in data science’.more » « less
-
We investigate aspects of the spherical squirmer model employing both large-scale numerical simulations and asymptotic methods when the squirmer is placed in weakly elastic fluids. The fluids are modelled by differential equations, including the upper-convected Maxwell (UCM)/Oldroyd-B, finite-extensibility nonlinear elastic model with Peterlin approximation (FENE-P) and Giesekus models. The squirmer model we examine is characterized by two dimensionless parameters related to the fluid velocity at the surface of the micro-swimmer: the slip parameter $$\xi $$ and the swirl parameter $$\zeta $$ . We show that, for swimming in UCM/Oldroyd-B fluids, the elastic stress becomes singular at a critical Weissenberg number, Wi , that depends only on $$\xi$$ . This singularity for the UCM/Oldroyd-B models is independent of the domain exterior to the swimmer, or any other forces considered in the momentum balance for the fluid – we believe that this is the first time such a singularity has been explicitly demonstrated. Moreover, we show that the behaviour of the solution at the poles is purely extensional in character and is the primary reason for the singularity in the Oldroyd-B model. When the Giesekus or the FENE-P models are utilized, the singularity is removed. We also investigate the mechanism behind the speed and rotation rate enhancement associated with the addition of swirl in the swimmer's gait. We demonstrate that, for all models, the speed is enhanced by swirl, but the mechanism of enhancement depends intrinsically on the rheological model employed. Special attention is paid to the propulsive role of the pressure and elucidated upon. We also study the region of convergence of the perturbation solutions in terms of Wi . When techniques that accelerate the convergence of series are applied, transformed solutions are derived that are in very good agreement with the results obtained by simulations. Finally, both the analytical and numerical results clearly indicate that the low- Wi region is more important than one would expect and demonstrate all the major phenomena observed when swimming with swirl in a viscoelastic fluid.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/noti3001
