Humans benefit from fluid flow but also need to understand the negative effects in nature, e.g. hurricanes and cyclones. The topographic conditions and the sufficient supply of rain flow is important in the cultivation of crops. Cultivation in deserts can also be achieved with artificial irrigation. Artificial breathing machines, artificial hearts and dialysis systems are designed using fluid dynamics. Hence fluids are vital.
The stress and rate of strain of a fluid is related: fluid is Newtonian if this relationship is linear, if not, the fluid is non-Newtonian. The non-Newtonian fluids, like polymers, plasma, paste and pulps are very important in industry. Therefore, the discussion of non-Newtonian fluids in fluid mechanics is important. All existing fluids cannot be described by the Navier-stocks equations. By categorizing different fluids, we have different mathematical models to represent fluids. Oldroyd-B fluid is one of these rate-type models.
The non-Newtonian fluids possessing shear dependent viscosity are known as viscoelastic fluids. The first rate type fluid model is a Maxwell model that is still in used. Fetecau et. al  derived the results of the velocity field and shear stress of an Oldroyd-B fluid on a plate, accelerating constantly. Hayat et. al  determined the solutions for MHD flow over an infinite oscillatory plate about an axis that is normal to the oscillatory plate. Some authors determined solutions for a viscoelastic fluid model by reproducing a kernel method (RKM) [3,4,5,6]. The unsteady incompressible Oldroyd-B fluid is analyzed by Jamil et. al  and special cases are also obtained for Maxwell, second grade and Newtonian fluids. Similarly, Burdujan  used this idea on Taylor-Couette flow and calculated the velocity field and shear stress of a rotating fluid between annuls of double cylinders. Numerical solutions of fractional fluids are also under attention [9, 10].
A lot of interest is given to a rotating circular domain corresponding to helical flow for second grade, Maxwell fluids and Oldroyd-B [11,12,13,14]. The flow of a polymer in a circular cylinder by considering pulsatile APG was analyzed by Barnes et. al [15, 16]. Davies et. al  and Phan-Thien  presented the same problem for a White-Metzner fluid. To solve the viscoelastic motion of different fluid models, one needs to learn about fractional calculus [19, 20]. Fractional calculus is very useful in fields of mathematics and physics [21,22,23]. Furthermore, the interested reader can consult the references [24,25,26,27,28,29,30].
In this proposal, we consider the fractional Oldroyd-B fluid model and study the flow due to the circulation of two pipes around its axial. Integral transforms are used to obtain the general solutions and the final result is in the form of hypergeometric functions. There are some limiting cases by applying the limits on physical parameters, i.e ς →1, λr →0 and λ → 0, solutions for the ordinary Oldroyd-B, fractional and ordinary Maxwell and Newtonian fluids are discussed respectively.
2 Development of the flow
Consider a fractional Oldroyd-B fluid (OBFFD) between two coaxial pipes. The radii of the pipes are R1 and R2(>R1), after time t = 0+, both pipes and the fluid between them start rotating along their axis. For the considered problem, the z-axis is along the axis of the pipes, the r-axis is perpendicular to the axis of the pipe and θ is along the tangent to the boundary. Let vr, vθ and vz be the velocity components in the direction of the r, θ and z axes respectively.
The continuity and linear momentum equations for incompressible flow is
where V, t, ρ, b, d/dt is velocity, time, density, body force and the material time derivative respectively.
For an incompressible Oldroyd-B fluid 
where p is the hydrostatic pressure, I denotes the identity tensor, S is the extra-stress tensor, λ and λr are relaxation and retardation times, superposed dot indicates the material time derivative, L represents the velocity gradient, A = L + LT is the first Rivlin Ericsen tensor, μ is the dynamic viscosity of the fluid and the superscript T denotes the transpose operation. For the problem in consideration, the velocity field and the extra stress tensor has the following form 
For such flows, (1) is true.
The governing equations for a Oldroyd-B fluid, corresponding to the above defined motions are 
where ν is the kinematic viscosity and τ(r, t) is the non zero shear stress.
3 Initial and boundary conditions
We need three conditions to solve a fractional partial differential equation (9). The relevant initial and boundary conditions are
where ℧1, ℧2 and κ are real numbers.
3.1 Analytic solutions for the velocity field
where the Laplace transform of the function ω(r, t) is ω(r, q). The finite Hankel transform  of the function ω(r, q), can be expressed as
rn are the positive roots of the transcendental equation A(R1, r) = 0, and Jp(·) is Bessel functions of the first kind and Yp(·) is Bessel functions of second kind of order p.
Writing the above equations in a suitable form
The inverse Hankel transform of ωH(rn, q) 
Using the inverse Hankel transform on Eq. (19), and the identity
Using the identity
Eq. (21) can be further simplified to give
4 Limiting cases
and the shear stress
for the ordinary Oldroyd-B fluid.
and the shear stress
corresponding to a fractional Maxwell fluid.
and the shear stress is
for the ordinary Maxwell fluid.
and the shear stress is
corresponding to a Newtonian fluid.
5 Flow through a circular pipe
The finite Hankel transform of the function f(r, t) is
and the inverse Hankel transform of fH is
where J1(R2r) = 0 as rn are the positive roots of the equation.
By making J1(R2rn) = 0, and using the identities J0(z) + J2(z) = (z) in Eq. (15) we get
Writing the Integral
Introducing Eq. (41) and taking R1 = 0, the above integral takes the form
6 Conclusions and results
In this article, exact solutions for a fractional Oldroyd-B fluid between two rotating pipes are determined. Fluid motion is produced due to rotation of the pipes around their axis with time dependent angular velocities. The solutions determined by the use of integral transforms and presented in terms of Bessel functions and hypergeometric functions which are free of integrals. The final result satisfies the initial and boundary conditions. In the limiting cases, the corresponding results for Oldroyd-B fluid, Maxwell and Newtonian fluid, are obtained from general results. Moreover, the solution for the fluid through a circular pipe is obtained as a special case.
The velocity profiles and shear stress are illustrated in Figures 2 — 7 for different values of parameters. Fig. 2 for the effect of t, Fig. 3 for the effect of ν, Fig. 4 for the effect of λ, Fig. 5 for the effect of λr and Fig. 6 shows the effect of the fractional parameter ς on the fluid motion. Clearly, the velocity as well as shear stress are increasing as a function of t and ν, The effect is qualitatively the same for fractional parameter ς and λ, more exactly the velocity ω(r, t) is a decreasing function with regards to ς.
Finally, Fig. 7 is the comparison of ω(r, t) corresponding to the motion of the Newtonian, fractional and ordinary Maxwell, and fractional and ordinary Oldroyd-B for the same values of time and for the same material and fractional parameters. From this comparison, the Oldroyd-B fluid is the slowest and the Newtonian fluid is the fastest on the whole flow domain. SI units are used for all figures, and the roots .
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About the article
Published Online: 2018-04-18
Funding: This research is supported by the Government College University, Faisalabad, Pakistan and the Higher Education Commission Pakistan.
Competing interestCompeting interests: The authors declare that they have no competing interests.
Citation Information: Open Physics, Volume 16, Issue 1, Pages 193–200, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0028.
© 2018 Madeeha Tahir et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0