Eur. Phys. J. Plus (2018) 133: 397
https://doi.org/10.1140/epjp/i2018-12186-7

Regular Article

A mathematical analysis of a circular pipe in rate type fluid via Hankel transform

¹ Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan
² Basic Engineering Sciences Department, College of Engineering Majmaah University, Al Majma’ah, Saudi Arabia
³ CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

^* e-mail: jgomez@cenidet.edu.mx

Received: 18 July 2018
Accepted: 3 August 2018
Published online: 3 October 2018

Abstract

In this paper, helices of a generalized Oldroyd-B fluid have been analyzed through a horizontal circular pipe. The circular pipe is taken in the form of a circular cylinder. The analytical solutions are determined for velocities and shear stresses due to the unsteady helical flow of a generalized Oldroyd-B fluid. The general solutions are derived by using finite Hankel and discrete Laplace transforms to satisfy the imposed conditions and the governing equations. The special cases of our general solutions are also perused performing the same motion for fractional and ordinary Maxwell fluid, fractional and ordinary second-grade fluid and fractional and ordinary viscous fluid as well. The graphical illustration is depicted in order to explore how the two velocities and shear stresses profiles are impacted by different rheological parameters, for instance, fractional parameter, relaxation time, retardation time, material non-zero constant, dynamic viscosity and few others. Finally, ordinary and fractional operators have various similarities and differences on a circular pipe for helicoidal behavior of fluid flow.