# Nonlinear fluid dynamics description of non-Newtonian fluids

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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### 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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### "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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1,714 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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1,548 citations