An Oldroyd-B fluid suddenly disturbed by relatively moving half-planes is theoretically studied in this paper. This new problem, extended from the traditional Stokes' problem, recently attracts a great deal of attention due to its potential applications in engineering. Using integral transformations and dividing the original system into two subsystems, the exact solution is derived and shown in a series form. In addition to the general understanding of velocity developments, the effects of the relaxation time and the retardation time on the velocity profiles are examined in a series of figures. It is found that above two rheological parameters influence the induced flow in an opposite way. Other interesting characteristics are also elucidated from present results.

The most classical problem in fluid mechanics might be a viscous flow driven by an impulsive or oscillatory plate below. This problem usually bears Stokes’ name [

For the rapidly growing applications either in academic studies or engineering fields, the studies on the Stokes’ problems traditionally for a Newtonian fluid were extended to cases of non-Newtonian fluids. Since the rheological properties vary within a quite wide range, how to choose a suitable model to accurately describe fluid properties always becomes an important issue in related studies. To this end, the Maxwell model which is one of the simplest models for viscoelastic fluids was studied to understand basic fluid characteristics. Fetecau and Fetecau [

The Oldroyd-B fluid model [

In addition to studies contributing to various fluids based on various models, more boundary conditions were involved to meet the practical requirements. The most common modified boundary condition might be a porous plate which allows the mass injection and suction through the plate. The corresponding solutions have been obtained for different fluids [

In this paper, the flow of an Oldroyd-B fluid driven by relatively moving half-planes with constant speed is analyzed. This problem is named as the extended Stokes’ first problem. The organization of present paper is as follows. The constitutive equations for the present problem are firstly derived in Section

The flow field considered is depicted as Figure

Diagram of an Oldroyd-B fluid driven by relatively moving half-planes.

For the flow system shown in Figure

In this section, the total solution consisting of

Velocity profiles at various

Velocity profiles for

In addition to a general understanding given by Figures

Velocity developments at

Velocity developments at

The extended Stokes’ first problem of an Oldroyd-B fluid driven by relatively moving half-planes is theoretically studied in this paper. Using integral transformations and dividing the original problem into two subsystems, the total solution is derived and shown in a series form. Velocity profiles and their developments influenced by the relaxation time and the retardation time are examined. From the present results, the following conclusions are summarized: (1) the velocity profiles above both half-planes will move with that at the central plane (

In this appendix, the derivation of displaying (

The author is indebted to the financial support from National Science Council of Taiwan with the Grant no. NSC 97-2221-E-270-013-MY3.