Some properties of models for generalized Oldroyd-B fluids

A class of generalized Oldroyd-B fluids is studied with reference to properties that are relevant in obtaining energy estimates. The generalization takes the form of a viscosity that depends on shear rate, with shear-thinning behaviour being of particular interest. Conformation stress tensors for the generalized problem are defined, and the conditions for positive-definiteness of the stress, and for its determinant to exceed unity, are obtained. It is shown that these properties, established for standard Oldroyd-B fluids, carry over provided that an expression depending on the Weissenberg number, a relaxation time, and the rate of change of viscosity, is positive. Various alternative energy estimates are obtained. The consequences of the positivity constraint are investigated for a specific viscosity function due to Yeleswarapu et al. [K.K. Yeleswarapu, M.V. Kameneva, K.R. Rajagopal, J.F. Antaki, The flow of blood in tubes: theory and experiment, Mechanics Research Communications 25(3) (1998) 257–262], first in general and then for a set of special flows.