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1. Master curves for FENE-P fluids in steady shear flow

   https://doi.org/10.1016/j.jnnfm.2022.104944  

   Yamani, Sami; McKinley, Gareth H. (March 2023, Journal of Non-Newtonian Fluid Mechanics)

   Poole, Robert J (Ed.)

   The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number (Wi), i.e., the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter (L), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio. An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available [Sureshkumar et al., Phys Fluids (1997)], but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi/L and it is only in strong shear flows exceeding Wi/L > 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi/L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. 

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2. A deterministic–particle–based scheme for micro-macro viscoelastic flows

   https://doi.org/10.1016/j.jcp.2024.113589  

   Bao, Xuelian; Liu, Chun; Wang, Yiwei (November 2024, Journal of Computational Physics)

   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. 

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3. Multiscale Modeling of Viscoelastic Fluids

   https://doi.org/10.1090/noti3001  

   Vasquez, Paula A (September 2024, Notices of the American Mathematical Society)

   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. 

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4. Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis

   https://doi.org/10.3390/e24010027  

   Zhou, Yuecheng; Latinwo, Folarin; Schroeder, Charles M. (January 2022, Entropy)

   Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions. 

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5. Dimethylformamide-Mediated Polymer Microstructure Dictates the Extensional Rheology of Aqueous PNIPAM Solutions

   https://doi.org/10.1021/acs.macromol.4c01658  

   Zhang, Diana Y; Schwendinger, Alec J; Calabrese, Michelle A (October 2024, Macromolecules)

   Polymer solution processability in extensional-flow dominated operations is strongly influenced by polymer conformation and solution phase behavior. Cosolvent addition can be used to tailor polymer conformation and solution phase behavior to yield formulations that are amenable to processes such as spraying and atomization, coating, and fiber spinning. The addition of N,N-dimethylformamide (DMF) to aqueous poly(N-isopropylacrylamide) (PNIPAM) solutions induces unique phase behavior and microstructure formation, yet the effects on solution processability have remained unexplored. In this work, the effect of DMF cosolvent content on the rheology (both shear and extensional) and microstructure of PNIPAM solutions is investigated. While all examined PNIPAM solutions exhibit nearly Newtonian steady shear behavior regardless of DMF content, the same solutions exhibit varying degrees of extensibility. Surprisingly, the extensional relaxation time increases by more than twenty-fold with increasing DMF content in the water-rich regime. In the DMF-rich regime, however, solution extensibility dramatically decreases. Interestingly, this unique variation in extensional flow behavior does not scale as expected based on changes in the measured intrinsic viscosity and radius of gyration. Instead, a mechanism is proposed that relates the extensional flow behavior to the solution microstructure, which is found to vary with DMF content in light scattering measurements. In the water-rich regime, DMF molecules are proposed to bridge PNIPAM chains via hydrogen bonding and hydrophobic interactions, resulting in physically crosslinked aggregates. In extensional flows, these aggregates behave like a polymer with higher apparent molecular weight, increasing the extensional relaxation time. In the DMF-rich regime, non-bridging DMF molecules increasingly solvate individual PNIPAM chains; consequently, more individual chains are stretched in extensional flows, leading to a reduction in the extensional relaxation time. These findings demonstrate that interactions between components in these ternary systems have unexpected but significant implications in solution extensional flow behavior. Additionally, in the case of PNIPAM/DMF/water, the processability of polymer-containing formulations can be modulated for spraying or for fiber spinning applications just by varying cosolvent (DMF) content. 

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