Nonlinear fluid dynamics description of non-Newtonian fluids
TL;DR: In this paper, a generalized hydrodynamic description of viscoelasticity is presented, which replaces the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, and can be used to get a nonlinear dynamic equation for the stress tensor.
Abstract: Nonlinear hydrodynamic equations for visco-elastic 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.
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TL;DR: A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained in this paper, where the authors refer to the theory as GSH.
Abstract: A complete continuum mechanical theory for granular media, including explicit expressions for the energy current and the entropy production, is derived and explained. Its underlying notion is: granular media are elastic when at rest, but turn transiently elastic when the grains are agitated—such as by tapping or shearing. The theory includes the true temperature as a variable, and employs in addition a granular temperature to quantify the extent of agitation. A free energy expression is provided that contains the full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. We refer to the theory as GSH, for granular solid hydrodynamics. In the static limit, it reduces to granular elasticity, shown previously to yield realistic static stress distributions. For steady-state deformations, it is equivalent to hypoplasticity, a state-of-the-art engineering model.
117 citations
TL;DR: Granular elasticity, an elasticity theory useful for calculating static stress distribution in granular media, is generalized to the dynamic case by including the plastic contribution of the strain this paper.
Abstract: Granular elasticity, an elasticity theory useful for calculating static stress distribution in granular media, is generalized to the dynamic case by including the plastic contribution of the strain. A complete hydrodynamic theory is derived based on the hypothesis that granular medium turns transiently elastic when deformed. This theory includes both the true and the granular temperatures, and employs a free energy expression that encapsulates a full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. For the special case of stationary granular temperatures, the derived hydrodynamic theory reduces to {\em hypoplasticity}, a state-of-the-art engineering model.
115 citations
Cites background from "Nonlinear fluid dynamics descriptio..."
...Transiently elastic media such as polymers are under active consideration at present [57, 58, 59, 60]....
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...In this equation, the term (uik∇jvk) + (i ↔ j), important for large strain field and frequently negligible for hard grains, is of geometric origin, see [57, 58, 59, 60] for explanations....
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TL;DR: This work presents a coherent set of robust tools, in three steps, that enable to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics.
Abstract: Discrete rearranging patterns include cellular patterns, for instance liquid foams, biological tissues, grains in polycrystals; assemblies of particles such as beads, granular materials, colloids, molecules, atoms; and interconnected networks. Such a pattern can be described as a list of links between neighbouring sites. Performing statistics on the links between neighbouring sites yields average quantities (hereafter "tools") as the result of direct measurements on images. These descriptive tools are flexible and suitable for various problems where quantitative measurements are required, whether in two or in three dimensions. Here, we present a coherent set of robust tools, in three steps. First, we revisit the definitions of three existing tools based on the texture matrix. Second, thanks to their more general definition, we embed these three tools in a self-consistent formalism, which includes three additional ones. Third, we show that the six tools together provide a direct correspondence between a small scale, where they quantify the discrete pattern's local distortion and rearrangements, and a large scale, where they help describe a material as a continuous medium. This enables to formulate elastic, plastic, fluid behaviours in a common, self-consistent modelling using continuous mechanics. Experiments, simulations and models can be expressed in the same language and directly compared. As an example, a companion paper (P. Marmottant, C. Raufaste, and F. Graner, this issue, 25 (2008) DOI 10.1140/epje/i2007-10300-7) provides an application to foam plasticity.
110 citations
TL;DR: All observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.
Abstract: In this short review we give an overview of selected macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (a) the influence of relative rotations between the director and the strain field on various reorientation instabilities, (b) the nonlinear stress–strain curves for the polydomain–monodomain transition and for the reorientation transition in LCE monodomains and (c) the shear mechanical response of LCEs in the linear regime. We consider only already existing real materials and do not discuss hypothetical “ideal” systems. We conclude that all observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.
78 citations
Journal Article•
TL;DR: In this paper, a short review of macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (i.e., the influence of relative rotations between the director and the strain field on various reorientation instabilities, (ii) the nonlinear stress-strain curves for the polydomain-monodomain transition and (iii) the shear mechanical response of LCEs in the linear regime).
Abstract: In this short review we give an overview of selected macroscopic properties of sidechain liquid crystalline elastomers (LCEs) focusing on three closely related topics (a) the influence of relative rotations between the director and the strain field on various reorientation instabilities, (b) the nonlinear stress-strain curves for the polydomain-monodomain transition and for the reorientation transition in LCE monodomains and (c) the shear mechanical response of LCEs in the linear regime. We consider only already existing real materials and do not discuss hypothetical "ideal" systems. We conclude that all observations reported to date can be accounted for without invoking the concept of soft elasticity, but instead relying on macroscopic dynamics in the linear and the nonlinear domain.
69 citations
References
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Book•
01 Jan 1992TL;DR: A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys as mentioned in this paper, which is not the case in modern physics, since it concerns solely the small particles of matter.
Abstract: Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.
2,644 citations
"Nonlinear fluid dynamics descriptio..." refers background in this paper
...On the other hand, non-Newtonian fluids are believed to show non-universal behavior and a host of different empirical models have been proposed [16–21] to cope with the flow rheology of such substances....
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...33, 239 (1961); [18] C....
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Book•
01 Jan 1980
TL;DR: The Foundations of Statistical Mechanics 7. Equilibrium Statistical Mechanics 8. Order-Disorder Transitions and Renormalization Theory 9. Interacting Fluids 10. Hydrodynamic Processes near Equilbrium 11. Transport Theory 12.
Abstract: 1. Introduction 2. Introduction to Thermodynamics 3. The Thermodynamics of Phase Transitions 4. Elementary Probability Theory and Limit Theorems 5. Stochastic Dynamics and Brownian Motion 6. The Foundations of Statistical Mechanics 7. Equilibrium Statistical Mechanics 8. Order-Disorder Transitions and Renormalization Theory 9. Interacting Fluids 10. Hydrodynamic Processes near Equilbrium 11. Transport Theory 12. Nonequilibrium Phase Transitions Appendices
1,773 citations
"Nonlinear fluid dynamics descriptio..." refers background in this paper
...It is based on (the Gibbsian formulation of) thermodynamics [6, 7], symmetries and well-founded physical principles [8]....
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TL;DR: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed in this article, where the right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material.
Abstract: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed. The right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material, but it is necessary to transform to a fixed frame of reference in order to solve the equations of state simultaneously with the equations of continuity and of motion. An illustration is given of the process of formulating equations of state suitable for universal application, based on non-invariant equations obtained from a simple experiment or structural theory. Anisotropic materials, and materials whose properties depend on previous rheological history, are included within the scope of the paper.
1,714 citations