Equilibrium Point and Phase Portrait of a Model for Flow of Tixotropic Media Accounting for Structure Evolution

Abstract

We continue the systematic analytical study of a nonlinear Maxwell-type constitutive equation for shear flow for thixotropic viscoelastic media accounting for interaction of deformation process and structure evolution, namely, the influence of the kinetics formation and breakage of chain cross-links, agglomerations of molecules and crystallites on viscosity and shear modulus and deformation influence on the kinetics. We formulated it in the previous article and reduced it to the set of two nonlinear autonomous differential equations for two unknown functions (namely, the stress and relative cross-links density). We examine the phase portrait of the system for arbitrary (increasing) material function and six (positive) material parameters governing the model and prove that the (unique) equilibrium point is stable and the only three cases are realized: the equilibrium point is either a stable sink, or a degenerated stable sink, or a stable spiral sink. We found criteria for every case in the form of explicit restrictions on the material function and parameters and shear rate.