A Modified Functional Expansion for Viscoelastic Fluids

A rheological equation for viscoelastic fluids is derived as an expansion of Noll's functional for simple fluids. The present expansion is developed in terms of the deformation rate referred to a co-rotational material frame. The resulting series of approximations for the stress differs significantly from those proposed previously by Green and Rivlin and by Coleman and Noll. Applications to both simple-extensional and viscometric flows are considered. For viscometric flows the present model exhibits non-Newtonian effects similar to those observed in real fluids, with relatively few terms in the functional expansion. The first term in the expansion yields a “first-order” fluid similar to the model proposed by Pao. Expressions are also given for the viscometric material functions of the present “third-order” fluid. As with higher-order fluids, these functions are Fourier integrals of certain hereditary or memory functions for the fluid.