Mathematical modeling of viscoelastic fluids

Abstract: A class of continuum mechanical models aimed at describing the behaviorof viscoelastic materials will be presented. These models are obtained by incorporating concepts originated in the theory of solid plasticity in a fluid mechanics context [1]. Within this class, even a simple model with constant material parameters is able to qualitatively reproduce a number of experimental observations in both simple shear and extensional flows, including linear viscoelastic properties, the rate dependence of steady-state material functions, the stress overshoot in incipient shear flows, and the difference in shear and extensional rheological curves.

These constitutive models are based on a logarithmic relation between the elastic strain measure and the stress tensor and on evolution equations for a local representative of the elastically-relaxed strain state. Importantly, it can be shown that classical models are recovered by expanding the evolution equation for the elastic stress around the null solution. The mathematical analysis of such tensorial transport equations leads to the definition of the notion of charted weak solutions [2]. These are based on non-standard a priori estimates that involve both viscous and plastic energy dissipation. The main aspects and open problems of the theoretical analysis of the evolution equations will be presented.

[1] M. A. H Alrashdi, G. G. Giusteri, Evolution of local relaxed states and the modeling of viscoelastic fluids, Phys. Fluids, 36, 093129, 2024. [2] G. Ciampa, G. G. Giusteri, A. G. Soggiu, Viscoelasticity, logarithmic stresses, and tensorial transport equations, Math. Meth. Appl. Sci. 48, 2934--2953, 2025.

Computer scienceMathematics

Audience: researchers in the topic

Modelling of materials - theory, model reduction and efficient numerical methods (UNCE MathMAC)