The role of relaxation and retardation phenomenon of Oldroyd-B fluid flow through Stehfest's and Tzou's algorithms

Delayed response (delay of the elasticity) and time needed for initial stress can lead to relaxation and retardation phenomenon; this is because of the consistent behavior of viscoelastic fluid on thermodynamic principles. In this context, the aim of this article is to investigate the unsteady, incompressible, and Oldroyd-B viscoelastic fluid under wall slip conditions to know the hidden aspects of relaxation and retardation. The motion of the liquid is assumed over a flat vertical plate which moves through an oscillating velocity. A fractional model is developed by using the modern definition of the non-singular kernel proposed by Caputo and Fabrizio. We have obtained a semi-analytical solution of the non-dimensional model by using the Laplace transformation that satisfies our imposed suitable boundary conditions. We have tackled the Laplace inverse by employing Stehfest's and Tzou's algorithms. The velocity is enhanced by decreasing the estimations of relaxation time lambda as well as slip parameter, and the temperature is also increasing for a considerable measure of the fractional factor. The effects of different fractional and physical parameters are plotted using Mathcad software based on the relaxation and retardation phenomenon of Oldroyd-B viscoelastic fluid.