Existence Results for the Flow of Viscoelastic Fluids with an Integral Constitutive Law (original) (raw)

On asymptotic stability for linear viscoelastic fluids

Differential and Integral Equations

The constitutive equation of the classical theory of infinitesimal viscometric fluids is considered, and some restrictions for this equation are found as a direct consequence of thermodynamics principles. These restrictions are proved to be sufficient to prove existence, uniqueness and stability theorems for the boundary-initial history value problem. Subsequently it is shown that the asymptotic stability of the rest state fails when these conditions are not satisfied. Introduction. This paper presents an evolution problem in a bounded domain n for viscoelastic fluids of the kind discussed in [2], [15], [21] and [22] 1. We confine our attention to the classical theory of infinitesimal visco-elasticity and consider isotropic, homogeneous, incompressible fluids. For these materials the linearized constitutive theory states that the symmetric stress tensor T is determined by the infinitesimal stmin history Et(x, s) = E(x, ts) through the hereditary law 2 T(x, t) =-p(x, t)I + 21 00 tt'(s)[Et(x, s)-E(x, t)] ds, (0.1)

A Physicists' View on Constitutive Equations

Nonlinear hydrodynamic equations for viscoelastic media are discussed. We start from the recently derived fully hydrodynamic nonlinear description of permanent elasticity that utilizes the (Eulerian) strain tensor. The reversible quadratic nonlinearities in the strain tensor dynamics are of the 'lower convected' type, unambiguously. Replacing the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, a generalized hydrodynamic description of viscoelasticity is obtained. This can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. The form of this equation and in particular the form of the nonlinear convective term is not universal but depends on various material parameters. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from.

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids

Continuum Mechanics and Thermodynamics, 2005

Conservation laws and constitutive relations for a density-gradient-dependent viscous fluid as a multipolar continuum obeying an entropy inequality with generalized entropy flux and supply density are considered in this paper. A decomposition of the rate of work of dipolar stress, which reveals the contribution of various parts of this stress to the energy equation, is used to discuss the inconsistencies between the results obtained here and those obtained by Bluestein and Green [1] on the basis of the pioneering work of Green and Rivlin [8]. Furthermore, we discuss the connection between the model presented here and the materials of Korteweg type considered by Dunn and Serrin [6]. In particular, we relate the rate of work of dipolar stress and the interstitial working introduced by Dunn and Serrin [6].

Existence results for integral viscoelastic fluids

2011

We consider the flows of viscoelastic fluid which obey a constitutive law of integral type. The existence and uniqueness results for solutions of the initial boundary value problem are proved, and the stationary case is studied.

VISCOELASTIC FLUIDS: FREE ENERGIES, DIFFERENTIAL PROBLEMS AND ASYMPTOTIC BEHAVIOUR

Some expressions for the free energy in the case of incompressible viscoelastic uids are given. These are derived from free energies already in- troduced for other viscoelastic materials, adapted to incompressible uids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these di erent functionals are also derived. Two equiv- alent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible uids. Fi- nally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.

Stability of Non-Linear Constitutive Formulations for Viscoelastic Fluids

SpringerBriefs in Applied Sciences and Technology, 2014

Controversy about the frame indifference principle, the concept of non-local continuum field theories, local constitutive formulations, differential constitutive equations of linear viscoelasticity, Oldroyd, K-BKZ, FENE (Finitely Extensible Non-linear Elastic) class of constitutive equations, Smoluchowski and Fokker-Planck diffusion equations, constant stretch history flows, fading memory and nested integral representations of the stress, order fluids of the integral and differential type, constitutive formulations consistent with thermodynamics, maximization of the rate of dissipation in formulating thermodynamics compatible constitutive structures, Burgers equation which is finding a gradually widening niche in applications, minimum free energy and maximum recoverable work in the case of linearized viscoelastic constitutive structures, implicit constitutive theories, which define the stress field when the viscosity depends for instance on the constitutively undetermined pressure field, and which have found new focus in applications such as elastohydrodynamic lubrication are discussed and progress made is summarized. Canonical forms of Maxwell-like constitutive differential equations and single integral constitutive equations are presented and commented on together with the Hadamard and dissipative type of instabilities they may be subject to.

Global Existence Results for Some Viscoelastic Models with an Integral Constitutive Law

SIAM Journal on Mathematical Analysis, 2014

We provide a proof of global regularity of solutions of some models of viscoelastic flow with an integral constitutive law, in the two spatial dimensions and in a periodic domain. Models that are included in these results are classical models for flow memory: for instance some K-BKZ models, the PSM model or the Wagner model. The proof is based on the fact that these models naturally give a L ∞-bound on the stress and that they allow to control the spatial gradient of the stress. The main result does not cover the case of the Oldroyd-B model.