New exact solutions for an Oldroyd-B fluid with fractional derivatives: Stokes’ first problem.

*(English)*Zbl 1401.76012Summary: The first problem of Stokes for an incompressible Oldroyd-B fluid with fractional derivatives is studied using Fourier sine and Laplace transforms. The solutions that have been obtained, unlike the known solutions from the literature, are presented as a sum between the Newtonian solutions and the non-Newtonian contributions. The non-Newtonian contributions, as expected, tend to zero for \(\alpha = \beta\) and \(\lambda \lambda _{r}\). Furthermore, the solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade fluid, performing the same motion, are also obtained as limiting cases of general solutions. The present solution for ordinary Oldroyd-B and second grade fluids are verified by comparison with the previously well known results. Finally, the influence of material and fractional parameters on the fluid motion, as well as a comparison among fractional Oldroyd-B, fractional Maxwell, fractional second grade and Newtonian fluids, is also analyzed by graphical illustrations.