Existence, Uniqueness, and Quasilinearization Results for Nonlinear Differential Equations Arising in Viscoelastic Fluid Flow

Because of their various applications during the past several years, generalizations of the Navier-Stokes model to highly nonlinear constitutive laws have been proposed and studied (see [4, 5, 7]). Several different models have been introduced to explain such nonstandard features, as normal stress effect, rod climbing, shear thinning, and shear thickening. Among the differential-typemodels, Oldroydmodels received special attention [2]. These models are rather complex from the point of view of partial differential equations theory. Nevertheless, several authors in fluid mechanics are now engaged with the equations of motion of non-Newtonian fluids of Oldroyd two-, three-, six-, and eight-constant models. Several authors [2, 6] considered an Oldroyd three-constant model which is a special case of the Oldroyd six-constant model. This has been used recently by Baris [1] for dealing with the steady and slow flow in the wedge between intersecting planes, one fixed and the other one moving. The Cauchy stress T in an incompressible Oldroyd six-constant-type fluid is related to the fluid motion by