About the structural stability of Maxwell fluids: convergence toward elastodynamics

TL;DR

This paper rigorously proves the convergence of Maxwell's viscoelastic models to elastodynamics as relaxation time and viscosity grow large, unifying elastic and viscous behaviors in continuum mechanics.

Contribution

It provides a rigorous mathematical framework showing the stability and convergence of Maxwell fluids to elastic solids in the limit of large relaxation time and viscosity.

Findings

Convergence of 1D shear waves to elastic wave solutions as parameters grow large.

Embedding Maxwell's model into symmetric-hyperbolic PDE systems for multi-dimensional analysis.

Continuous dependence and convergence of multi-dimensional flows to elastodynamics as relaxation parameters tend to infinity.

Abstract

Maxwell's models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But rigorous proofs are lacking. The present note is a contribution toward well-defined viscoelastic flows proved to encompass both solid and (liquid) fluid regimes. In a first part, we consider the structural stability of particular viscoelastic flows: 1D shear waves solutions to damped wave equations. We show the convergence toward purely elastic 1D shear waves solutions to standard wave equations, as the relaxation time $\lambda$ and the viscosity $\mu$ grow unboundedly $\lambda$ $\not\equiv$ $\mu$/G $\rightarrow$ $\infty$ in Maxwell's constitutive equation $\lambda$ $\tau$ +$\tau$ = 2 $\mu$D(u) for the stress $\tau$ of viscoelastic fluids with velocity u. In a second part, we consider the structural stability of general multi-dimensional viscoelastic flows. To that aim, we embed Maxwell's constitutive equation in a symmetric-hyperbolic system of PDEs which we proposed in our previous publication [ESAIM:M2AN 55 (2021) 807-831]so as to define multi-dimensional viscoelastic flows unequivocally. Next, we show the continuous dependence of multi-dimensional viscoelastic flows on $\lambda$ $\not\equiv$ $\mu$/ G using the relative-entropy tool developped for symmetric-hyperbolic systems after C. M. Dafermos. It implies convergence of the viscoelastic flows defined in [ESAIM:M2AN 55 (2021) 807-831] toward compressible neo-Hookean elastodynamics when $\lambda$ $\rightarrow$ $\infty$.