A new version of the nonlinear model for shear flow of tixotropic viscoelastic media with a power dependence of relaxation time on current structuredness

We formulate and analyze a new version of the nonlinear constitutive equation for shear flow of tixotropic viscoelastic media accounting for interaction of deformation and structure evolution (kinetics of its formation and destruction) proposed earlier. Viscosity and shear modulus appearing in the Maxwell-type model are chosen to be power functions of current structuredness instead of exponential functions in the first version of the model. The new model is also governed by an increasing material function and six positive parameters, but the set of two nonlinear differential equations for two unknown functions of time, namely, stress and structuredness (or relative cross-links density), to which the model is reduced in one-dimensional case, differ from the first version. We analyze mathematical properties of the model assuming the material function and six material parameters are arbitrary and show that the new model inherits all valuable basic properties that have been discovered analyzing the first version. We confirm the existence and uniqueness of its equilibrium point and that equilibrium stress and structuredness depend monotonically on every material parameter and on shear rate. We derive general equations for the flow and viscosity curves generated by the model and confirm that the first one increases and the second one decreases while the shear rate grows. Thus, the model describes the basic phenomena observed for the simple shear flow of shear thinning fluids, but a number of qualitative properties of the flow and viscosity curves differ from the first version. We found two applicability indicators for each version of the model which are easy to check in tests. Thus, the new version of the model is a useful additional tool aimed at simulating shear flows of tixotropic viscoelastic media.