On the flow of Oldroyd-B fluids with fractional derivatives over a plate that applies shear to the fluid

Abstract

The motion of incompressible fractional Oldroyd-B fluids between two parallel walls perpendicular to a plate that applies time-dependent shear stresses to the fluid is studied by means of integral transforms. In the special cases of Newtonian and second grade fluids, these shear stresses reduces to fH(t)sin(wt) or fH(t)cos(wt). General solutions for velocity are presented as a sum of Newtonian solutions and the corresponding non-Newtonian contributions. They reduce to the similar solutions corresponding to the motion over an infinite plate if the distance between walls tend to infinity and can be easy particularized to give the similar solutions for ordinary and fractional Maxwell or second grade fluids performing the same motions. As a check of general results some known solutions from the literature are recovered as limiting cases. Finally, the influence of fractional parameters on the fluid motion and the distance between walls for which the measured value of the velocity in the middle of the channel is unaffected by their presence (more exactly, it is equal to the velocity corresponding to the motion over an infinite plate) are graphically determined.