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

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 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.

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.

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 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 A Convergent three Field Quadrilateral Finite Element Method for Simulating Viscoelastic Flow on Irregular Meshes

1992 ◽  
pp. 391-406
Author(s):  
Victoriano Ruas
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fluid Flow ◽  
Stokes System ◽  
Viscoelastic Flow ◽  
Flow Problems ◽  
Extra Stress Tensor ◽  
Irregular Meshes ◽  
Quadrilateral Finite Element ◽  
Extra Stress

In a recent paper a finite elemenJ method for solving the three field Stokes system as a basis for the numerical solution of viscoelastic fluid flow problems was introduced. The method constructed upon a biquadratic velocity, a discontinuous linear pressure and a bilinear extra stress tensor interpolation in quadrilaterals, enriched with fifteen bubble tensors, has been proved to yield second order approximations of these variables, in the case of rectangular meshes. In this work equivalent results are proven to hold in the case of irregular meshes.  

Computational Modelling of Mesophase Pitches’ Shear Rheology

2001 ◽  
Vol 709 ◽  
Author(s):  
Dana Grecov ◽  
Alejandro D. Rey
Keyword(s):  
Normal Stress ◽  
Shear Viscosity ◽  
Computational Modelling ◽  
Simple Shear Flow ◽  
Shear Rates ◽  
Extra Stress Tensor ◽  
Normal Stress Differences ◽  
Macroscopic Orientation ◽  
Arbitrary Flow ◽  
Extra Stress

ABSTRACTFlow modelling of mesophase pitches is performed using a previously formulated mesoscopic viscoelastic rheological theory [1] that takes into account flow-induced texture transformations. A complete extra stress tensor equation is developed from first principles for liquid crystal materials under non-homogeneous arbitrary flow. Predictions for a given simple shear flow, under non-homogeneous conditions, for the apparent shear viscosity and first normal stress differences are presented. The rheological functions are explained using macroscopic orientation effects, which predominate at low shear rates. The predicted normal stress differences and apparent shear viscosity are in agreement with experimental measurements.

scholarly journals NUMERICAL SIMULATION OF A ELASTO-VISCOPLASTIC FLUID FLOW INSIDE A CAVITY

2018 ◽  
Vol 17 (1) ◽  
pp. 73
Author(s):  
G. M. Furtado ◽  
S. Frey ◽  
M. F. Naccache ◽  
P. R. de Souza Mendes
Keyword(s):  
Type Equation ◽  
Surface Topology ◽  
Viscoplastic Fluid ◽  
Material Microstructure ◽  
Finite Element Approximations ◽  
Extra Stress Tensor ◽  
Viscoplastic Flows ◽  
Extra Stress

This article addresses finite element approximations for elasto-viscoplastic flows. Numerical simulations aiming at investigating the role of elasticity for inertialess flows of viscoplastic materials within lid-driven cavity.The mechanical model is made up of the usual governing equations for incompressible fluids coupled with a Oldroyd-B type equation (de Souza Mendes, 2011) modified to incorporate the dependency both of relaxation and retardation time as the viscoplastic viscosity on the strain rate. These parameters depend on the material microstructure, which level is described by an structure parameter . This model is approximated by a multi-field Galerkin least-squares formulation (Behr et al., 1993) in terms of extra-stress tensor, the pressure field and the velocity vector. Results, focused on the determination of yield surface topology, investigate the influence of elastic and viscous governing parameters on the flow pattern.

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