^{1, 2}

^{3}

^{1}

^{1}

^{2}

^{3}

New exact solutions for the motion of a fractionalized (this word is suitable when fractional derivative is used in constitutive or governing equations) second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of Laplace and finite Hankel transforms. These solutions are presented in series form in term of generalized

In recent years, the non-Newtonian fluids have received considerable attention by scientist and engineers. Such interest is inspired by practical applications of non-Newtonian fluids in industry and engineering applications. The shear stress and shear rate in non-Newtonian fluids are connected by a relation in a nonlinear manner. Because of diverse fluids characteristics in nature, all the non-Newtonian fluids cannot be described by a single constitutive relation [

Linear viscoelasticity is certainly the field of most extensive applications of fractional calculus, in view of its ability to model hereditary phenomena with long memory. During the twentieth century, a number of authors have (implicitly or explicitly) used the fractional calculus as an empirical method of describing the properties of viscoelastic materials [

The oscillating flow of the viscoelastic fluid in cylindrical pipes has been applied in many fields, such as industries of petroleum, chemistry, and bioengineering. In the field of bioengineering, this type of investigation is of particular interest since blood in veins is forced by a periodic pressure gradient. In the petroleum and chemical industries, there are also many problems which involve the dynamic response of the fluid to the frequency of the periodic pressure gradient. An excellent collection of papers on oscillating flow can be found in the paper by Yin and Zhu [

The Cauchy stress

For the problem under consideration, we shall assume a velocity field and an extra-stress of the form

The equation of motion (_{2}, in the absence of a pressure gradient in the axial direction and neglecting body forces, leads to the relevant equations (

Let us consider an incompressible fractionalized second grade fluid at rest, in an infinitely long cylinder of radius

Geometry of the problem for oscillating flows of fractionalized second grade fluid through a cylinder.

Applying the Laplace transform to (

Multiplying now both sides of (

Applying the Laplace transform to (

Making

In practice, the steady-state solutions for unsteady motions of Newtonian or non-Newtonian fluids are important for those who need to eliminate transients from their rheological measurements. Consequently, an important problem regarding the technical relevance of these solutions is to find the approximate time after which the fluid is moving according to the steady-state. More exactly, in practice it is necessary to know the required time to reach the steady-state.

Making the limit

The velocity fields and the adequate shear stresses corresponding to the unsteady motions of an incompressible fractionalized second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder have been determined by means of the Laplace and finite Hankel transforms. The general solutions are written in series form in term of generalized

Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Finally, for comparison, the diagrams of

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

Profiles of the velocity components

The author M. Jamil is extremely grateful and thankful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics & Basic Sciences, NED University of Engineering & Technology, Karachi, Pakistan; also Higher Education Commission of Pakistan for supporting and facilitating this research work. The author N. A. Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi, Pakistan for supporting and facilitating this research work. The author A. Rauf is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan and also Higher Education Commission of Pakistan for generous support and facilitating this research work.