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.

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 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 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.

On convergent schemes for two-phase flow of dilute polymeric solutions

2018 ◽  
Vol 52 (6) ◽  
pp. 2357-2408 ◽  
Author(s):  
Stefan Metzger
Keyword(s):  
Finite Element ◽  
Two Phase Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Phase Flow ◽  
Two Phase ◽  
Extra Stress Tensor ◽  
Spring Force ◽  
Polymeric Solutions ◽  
Fokker Planck

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a recent micro-macro bead-spring model for two-phase flow of dilute polymeric solutions derived by methods from nonequilibrium thermodynamics ([Grün, Metzger, M3AS 26 (2016) 823–866]). The model consists of Cahn-Hilliard type equations describing the evolution of the fluids and the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three spatial dimensions for the velocity and the pressure of the fluids with an elastic extra-stress tensor on the right-hand side in the momentum equation which originates from the presence of dissolved polymer chains. The polymers are modeled by dumbbells subjected to a finitely extensible, nonlinear elastic (FENE) spring-force potential. Their density and orientation are described by a Fokker-Planck type parabolic equation with a center-of-mass diffusion term. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters simultaneously tend to zero, and show that a subsequence of these finite element approximations converges towards a weak solution of the coupled Cahn-Hilliard-Navier-Stokes-Fokker-Planck system. To underline the practicality of the presented scheme, we provide simulations of oscillating dilute polymeric droplets and compare their oscillatory behaviour to the one of Newtonian droplets.

GLOBAL WEAK SOLUTIONS FOR A VLASOV–FOKKER–PLANCK/NAVIER–STOKES SYSTEM OF EQUATIONS

2007 ◽  
Vol 17 (07) ◽  
pp. 1039-1063 ◽  
Author(s):  
A. MELLET ◽  
A. VASSEUR
Keyword(s):  
Weak Solutions ◽  
Stokes System ◽  
Fluid Velocity ◽  
Planck Equation ◽  
Coupled System ◽  
Navier Stokes ◽  
Global Weak Solutions ◽  
Navier Stokes Equation ◽  
Reflection Boundary ◽  
Fokker Planck

We establish the existence of a weak solutions for a coupled system of kinetic and fluid equations. More precisely, we consider a Vlasov–Fokker–Planck equation coupled to compressible Navier–Stokes equation via a drag force. The fluid is assumed to be barotropic with γ-pressure law (γ > 3/2). The existence of weak solutions is proved in a bounded domain of ℝ3 with homogeneous Dirichlet conditions on the fluid velocity field and Dirichlet or reflection boundary conditions on the kinetic distribution function.

scholarly journals Numerical Simulation of Rotor–Wing Transient Interaction for a Tiltrotor in the Transition Mode

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 116 ◽  
Author(s):  
Zhenlong Wu ◽  
Can Li ◽  
Yihua Cao
Keyword(s):  
Numerical Simulation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Transition Mode ◽  
Rans Equations ◽  
Design And Optimization ◽  
Aerodynamic Interaction ◽  
Rotor Wing ◽  
Transition Modes ◽  
The Right

Tiltrotor aerodynamic interaction, especially in the transition mode, is a necessary consideration for tiltrotor aerodynamics, and structural design and optimization. Previous studies have paid much attention to the helicopter mode. However, due to the substantial complexity of the problem, only a small amount of work on the transition mode has been done so far. In this paper, the rotor–wing aerodynamic interaction of a scaled V-22 Osprey tiltrotor, both in the helicopter and transition modes, are studied by computational fluid dynamics (CFD) numerical simulation. The flow field is discretized via the chimera mesh technique and then solved with the Reynolds-averaged Navier–Stokes (RANS) equations. The rotational acceleration of the rotor is considered as a source term added on the right side of the momentum equation of the RANS equations. Both a quasi-steady and a fully transient method are employed to simulate the tilt motion of the rotor in the transition mode. Both qualitative and quantitative results are presented and discussed on the aerodynamic forces, flow physics, and mechanisms. The applicability of the extensively used quasi-steady method for rotor tilt simulation is revealed.

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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