Theory of non-Newtonian viscosity and normal stress coefficients of fluids

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Abstract
The evolution equation for the traceless symmetric part P̂ of the stress tensor derived previously for dense simple fluids is further investigated of its applicability to non-Newtonian fluid behaviors and normal stress effects. Nonlinear differential equations for three independent components of the stress tensor are obtained in the case of a unidirectional flow in an incompressible fluid. The asymptotic stability of the steady state (equilibrium) is studied with linearized equations and the stability conditions are derived. Approximate steady state solutions are calculated from the evolution equations for the three components and viscometric functions are obtained therefrom. By taking the evolution equations as a set of phenomenological evolution equations, we show that it is possible to apply them even to nonsimple fluids such as polymer solutions, by analyzing experimental data on the shear rate dependencies of the viscometric functions of some polymer solutions. We also show that the approximate viscometric functions used for analysis can be reduced to a set of corresponding-state viscometric functions to a good accuracy. Relations of the evolution equations for P̂ to some of the existing rheological equations of state are discussed. It is shown that suitable linearizations of the present nonlinear evolution equation for P̂ yield the same forms as those for the existing rheological equations of state.