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

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.

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

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2018027 ◽

2018 ◽

Vol 52 (5) ◽

pp. 1947-1980 ◽

Cited By ~ 1

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.

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EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR SOME POLYMERIC FLOW MODELS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202505000625 ◽

2005 ◽

Vol 15 (06) ◽

pp. 939-983 ◽

Cited By ~ 51

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.

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Non integrable extra stress tensor solution for a flow in a bounded domain of an Oldroyd fluid

Acta Mechanica ◽

10.1007/bf01179049 ◽

1999 ◽

Vol 135 (1-2) ◽

pp. 95-99 ◽

Cited By ~ 6

Author(s):

D. Sandri

Keyword(s):

Stress Tensor ◽

Bounded Domain ◽

Oldroyd Fluid ◽

Extra Stress Tensor ◽

Extra Stress

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

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202508002917 ◽

2008 ◽

Vol 18 (06) ◽

pp. 935-971 ◽

Cited By ~ 21

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.

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On the Normal Component of Centralized Frictionless Collision Sequences

Journal of Applied Mechanics ◽

10.1115/1.2712237 ◽

2001 ◽

Vol 74 (5) ◽

pp. 908-915 ◽

Cited By ~ 6

Author(s):

Pieter J. Mosterman

Keyword(s):

Rigid Body ◽

Time Scale ◽

Normal Component ◽

Expansion Behavior ◽

Compression And Expansion ◽

Multiple Impact

A typical assumption for rigid body collisions with multiple impact points is that all collisions occur simultaneously and are synchronized in their compression/expansion behavior, a useful assumption given the microscopic time scale at which collisions occur. In the case in which collisions are dependent upon one another, however, there is interaction between and within compression and expansion phases. Instead of treating the collisions as separate consecutive impacts or by activating all constraints at the same time, a rule is presented that orders the collisions as a sequence of interacting events at a point in time to handle the normal component of the collisions.

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Stress

Physics of Elasticity and Crystal Defects ◽

10.1093/oso/9780198860785.003.0002 ◽

2020 ◽

pp. 9-28

Author(s):

Adrian P. Sutton

Keyword(s):

Shear Stress ◽

Stress Tensor ◽

Elastic Energy ◽

Functional Derivative ◽

Shear Stresses ◽

Mechanical Equilibrium ◽

Normal Stresses ◽

Atomic Interactions ◽

Insightful Analysis ◽

Von Mises

The concept of stress is introduced in terms of interatomic forces acting through a plane, and in the Cauchy sense of a force per unit area on a plane in a continuum. Normal stresses and shear stresses are defined. Invariants of the stress tensor are derived and the von Mises shear stress is expressed in terms of them. The conditions for mechanical equilibrium in a continuum are derived, one of which leads to the stress tensor being symmetric. Stress is also shown to be the functional derivative of the elastic energy with respect to strain,which enables the stress tensor to be derived in models of interatomic forces. Adiabatic and isothermal stresses are distinguished thermodynamically and anharmonicity of atomic interactions is identified as the reason for their differences. Problems set 2 containsfour problems, one of which is based on Noll’s insightful analysis of stress and mechanical equilibrium.

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

European Journal of Computational Mechanics ◽

10.13052/ejcm2642-2085.141 ◽

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.

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