Role of non-integer and integer order differentiations on the relaxation phenomena of viscoelastic fluid

The dynamics of elastic, anelastic responses and thermoelastic relaxation mechanism can be exhibited by the viscoelastic materials. This manuscript aims to present the analytical study of viscoelastic fluid based on non-integer and integer order differentiations. The mathematical modeling of viscoelastic fluid has been established in terms of partial differential equations for the velocity field corresponding to the shear stress subject to the fractional approaches. The partial differential equations govern the viscoelastic fluid have been fractionalized and then solved via Laplace transform method. The solutions are obtained via two types of approaches namely non-fractional (classical) and fractional (Caputo–Fabrizio and Atangana–Baleanu) in Caputo sense. The solutions for velocity field and shear stress are expressed in the property of special function so called Fox- H function satisfying the imposed conditions. Finally, the graphical illustration has disclosed the amorphous and non-amorphous behavior of fluid flow which represents the effective role of singular, non-singular and local, non-local kernels on viscoelastic fluid.