A note on an unsteady flow of an Oldroyd-B fluid

The laminar flow of a number of fluids such as polymeric liquids, food products, paints, slurries, foams, and so forth cannot be adequately described by the help of the classical linearly viscous Newtonian model. Thus, there is a need to have at hand an arsenal of non-Newtonian fluid models and over the past several decades a variety of models have been developed. While the fluids of the differential type can explain, for instance, normal stress differences in a simple shear flow, shear thinning/shear thickening, and nonlinear creep, characteristics exhibited by some non-Newtonian fluids, they cannot describe the stress relaxation exhibited by certain liquids. Rate-type fluid models can be constructed that can describe stress relaxation, nonlinear creep, shear thinning/shear thickening, and normal stress differences in simple shear flows. Maxwell [8] developed the first rate-type one-dimensional model that could describe stress relaxation. Another important study of rate-type fluids is due to Burgers [2] who developed a one-dimensional model that has proved useful in describing the response of a class of geomaterials. The model due to Burgers includes the one-dimensional version of a popular rate-type model due to Oldroyd, namely, the Oldroyd-B model. Oldroyd [9] was the first to develop systematically three-dimensional rate-type models that met the requirements of frame indifference. While the Oldroyd-B model can describe stress relaxation and normal stress differences in a simple shear flow, it is incapable of describing shear thinning/shear thickening. It can however be easily generalized to describe shear thinning/thickening. Here, we study the flow of an Oldroyd-B fluid due to an impulsively started plate and we obtain an exact solution for the velocity field and stress in the form of a definite integral. Such solutions in addition to providing a solution to an interesting problem can also be used to check