scholarly journals Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

2016 ◽  
Vol 26 (03) ◽  
pp. 469-568 ◽  
Author(s):  
John W. Barrett ◽  
Endre Süli
Keyword(s):  
Stress Tensor ◽  
Weak Solutions ◽  
Center Of Mass ◽  
Momentum Equation ◽  
Polymer Chains ◽  
Navier Stokes ◽  
Diffusion Term ◽  
Quadratic Interaction ◽  
Extra Stress Tensor ◽  
Extra Stress

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain [Formula: see text] in [Formula: see text], [Formula: see text] or [Formula: see text], for the density [Formula: see text], the velocity [Formula: see text] and the pressure [Formula: see text] of the fluid, with an equation of state of the form [Formula: see text], where [Formula: see text] is a positive constant and [Formula: see text]. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term.

scholarly journals EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS

2012 ◽  
Vol 22 (05) ◽  
pp. 1150024 ◽  
Author(s):  
JOHN W. BARRETT ◽  
ENDRE SÜLI
Keyword(s):  
Center Of Mass ◽  
Planck Equation ◽  
Momentum Equation ◽  
Polymer Chains ◽  
Navier Stokes ◽  
Extra Stress Tensor ◽  
The Right ◽  
Extra Stress ◽  
Drag Term ◽  
Fokker Planck

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T>0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

scholarly journals EXISTENCE OF GLOBAL WEAK SOLUTIONS TO DUMBBELL MODELS FOR DILUTE POLYMERS WITH MICROSCOPIC CUT-OFF

2008 ◽  
Vol 18 (06) ◽  
pp. 935-971 ◽  
Author(s):  
JOHN W. BARRETT ◽  
ENDRE SÜLI
Keyword(s):  
Stress Tensor ◽  
Weak Solutions ◽  
Mass Diffusion ◽  
Polymer Chains ◽  
Navier Stokes ◽  
Extra Stress Tensor ◽  
Spring Force ◽  
Extra Stress ◽  
Drag Term ◽  
Fokker Planck

We study the existence of global-in-time weak solutions to a coupled microscopic–macroscopic bead-spring model with microscopic cut-off, which arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function ψ that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term and a cut-off function βL(ψ) = min (ψ,L) in the drag term, where L ≫ 1. We establish the existence of global-in-time weak solutions to the model for a general class of spring-force potentials including, in particular, the widely used finitely extensible nonlinear elastic potential. A key ingredient of the argument is a special testing procedure in the weak formulation of the Fokker–Planck equation, based on the convex entropy function [Formula: see text]. In the case of a corotational drag term, passage to the limit as L → ∞ recovers the Navier–Stokes–Fokker–Planck model with centre-of-mass diffusion, without cut-off.

scholarly journals EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR SOME POLYMERIC FLOW MODELS

2005 ◽  
Vol 15 (06) ◽  
pp. 939-983 ◽  
Author(s):  
JOHN W. BARRETT ◽  
CHRISTOPH SCHWAB ◽  
ENDRE SÜLI
Keyword(s):  
Stress Tensor ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Polymer Chains ◽  
Elastic Model ◽  
Spring Model ◽  
Extra Stress Tensor ◽  
Spring Force ◽  
Extra Stress ◽  
Fokker Planck

We study the existence of global-in-time weak solutions to a coupled microscopic–macroscopic bead-spring model which arises from the kinetic theory of diluted solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2, 3, for the velocity and the pressure of the fluid, with an extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker–Planck type degenerate parabolic equation. Upon appropriate smoothing of the convective velocity field in the Fokker–Planck equation, and in some circumstances, of the extra-stress tensor, we establish the existence of global-in-time weak solutions to this regularised bead-spring model for a general class of spring-force-potentials including in particular the widely used FENE (Finitely Extensible Nonlinear Elastic) model.

scholarly journals EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEAD-SPRING CHAINS

2011 ◽  
Vol 21 (06) ◽  
pp. 1211-1289 ◽  
Author(s):  
JOHN W. BARRETT ◽  
ENDRE SÜLI
Keyword(s):  
Planck Equation ◽  
Momentum Equation ◽  
Polymer Chains ◽  
Navier Stokes ◽  
Centre Of Mass ◽  
Extra Stress Tensor ◽  
The Right ◽  
Extra Stress ◽  
Drag Term ◽  
Fokker Planck

We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term needs not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularisation parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (w.r.t. the measure [Formula: see text] over any time interval [0, T], T > 0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text] norm, at a rate that is independent of (ṵ0, ψ0) and of the centre-of-mass diffusion coefficient.

Normal Stresses on the Surface of a Rigid Body in an Oldroyd-B Fluid

2001 ◽  
Vol 124 (1) ◽  
pp. 279-280 ◽  
Author(s):  
N. A. Patankar ◽  
P. Y. Huang ◽  
D. D. Joseph ◽  
H. H. Hu
Keyword(s):  
Rigid Body ◽  
Stress Tensor ◽  
Normal Component ◽  
Normal Stresses ◽  
Extra Stress Tensor ◽  
Extra Stress

In this note we present a proof showing that the contribution from the extra stress tensor to the normal component of the stress on the surface of a moving rigid body in an incompressible Oldroyd-B fluid is zero.

scholarly journals Extra stress tensor in fiber suspensions: Mechanics and thermodynamics

2011 ◽  
Vol 55 (1) ◽  
pp. 17-42 ◽  
Author(s):  
Miroslav Grmela ◽  
Amine Ammar ◽  
Francisco Chinesta
Keyword(s):  
Stress Tensor ◽  
Fiber Suspensions ◽  
Extra Stress Tensor ◽  
Extra Stress

Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations

2018 ◽  
Vol 52 (5) ◽  
pp. 1947-1980 ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel N. Gatica ◽  
Ricardo Oyarzúa
Keyword(s):  
Stress Tensor ◽  
Variational Formulation ◽  
Strain Tensor ◽  
Heat Equations ◽  
Flux Vector ◽  
Convective Term ◽  
Piecewise Polynomials ◽  
Extra Stress Tensor ◽  
Heat Flux Vector ◽  
Extra Stress

We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are introduced as further unknowns. This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using the constitutive and equilibrium equations, the relation defining the strain and vorticity tensors, and the Dirichlet boundary condition on the temperature. The resulting augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Schauder and Banach theorems, combined with the Lax-Milgram theorem and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point and Lax–Milgram theorems, we employ Raviart–Thomas approximations of order k for the stress tensor and the heat flux vector, continuous piecewise polynomials of order ≤ k + 1 for velocity and temperature, and piecewise polynomials of order ≤ k for the strain tensor and the vorticity. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence.

The Cauchy problem for an Oldroyd-B model in three dimensions

2019 ◽  
Vol 30 (01) ◽  
pp. 139-179
Author(s):  
Wenjun Wang ◽  
Huanyao Wen
Keyword(s):  
Cauchy Problem ◽  
Strong Solutions ◽  
Large Data ◽  
Finite Energy ◽  
Momentum Equation ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Frequency Decomposition ◽  
Extra Stress Tensor ◽  
The Cauchy Problem

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci. 15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to [Formula: see text], in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing [Formula: see text]-norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition.

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