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Keywords = equilibrium-based moment closure

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15 pages, 13802 KiB  
Article
Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation
by Stephen Chaffin and Julia Rees
J 2021, 4(3), 341-355; https://doi.org/10.3390/j4030026 - 14 Jul 2021
Cited by 1 | Viewed by 2827
Abstract
Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of [...] Read more.
Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of a mean-field term to account for intermolecular forces in dilute polymer solutions for background profiles of weak shear and elongation. These investigations involved the solution of the Fokker–Planck equation incorporating a constitutive equation for the second moment. In this paper, we extend this analysis to include the effects of large background shear and elongation beyond the Hookean regime. Further, the constitutive equation is solved for the probability density function which permits the computation of any macroscopic variable, allowing direct comparison of the model predictions with molecular dynamics simulations. It was found that if the concentration effects at equilibrium are taken into account, the FENE-P model gives qualitatively the correct predictions, although the over-shoot in extension in comparison to the infinitely dilute case is significantly underpredicted. Full article
(This article belongs to the Section Computer Science & Mathematics)
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22 pages, 10188 KiB  
Article
A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis
by Pawan Kumar and Christina Surulescu
Symmetry 2020, 12(11), 1870; https://doi.org/10.3390/sym12111870 - 13 Nov 2020
Cited by 11 | Viewed by 2896
Abstract
We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the [...] Read more.
We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects. Full article
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