Exact Solutions for the Axial Couette Flow of a Fractional Maxwell Fluid in an Annulus

The velocity field and the adequate shear stress corresponding to the rotational flow of a fractional Maxwell fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. The solutions that have been obtained are presented in terms of generalized Ga,b,c ·, t and Ra,b ·, t functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinaryMaxwell andNewtonian fluids are obtained as limiting cases of our general solutions. Finally, the influence of the material parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.


Introduction
Due to the several technological applications, the flow analysis of non-Newtonian fluids is very important in the fields of fluid mechanics.Many investigators have not studied the flow behavior of non-Newtonian fluids in various flow fields due to the complex stressstrain relationship 1 .The study of non-Newtonian fluids has got much attention because of their practical applications.Non-Newtonian characteristics are displayed by a number of industrially important fluids including polymers, molten plastic, pulps, microfluids, and food stuff display.Exact analytic solutions for the flows of non-Newtonian fluids are important provided they correspond to physically realistic problems, and they can be used as checks against complicated numerical codes that have been developed for much more complex flows.Many non-Newtonian models such as differential type, rate type, and integral type fluids have been proposed in recent years.Among them, the rate type fluid models have received special attention.The differential type fluids do not predict stress relaxation, and they are not successful for describing the flows of some polymers.
The flow between rotating cylinders or through a rotating cylinder has applications in the food industry, it being one of the most important and interesting problems of motion near rotating bodies.As early as 1886, Stokes 2 established an exact solution for the rotational oscillations of an infinite rod immersed in a linearly viscous fluid.However, such motions have been intensively studied since G. I. Taylor 1923 reported the results of his famous investigations 3 .For Newtonian fluids, the velocity distribution for a fluid contained in a circular cylinder can be found in 4 .The first exact solutions corresponding to different motions of non-Newtonian fluids, in cylindrical domains, seem to be those of Ting 5 , Srivastava 6 , and Waters and King 7 .A lot of interests and studies were also given to the unidirectional start-up pipe flows, which has a significant practical and mathematical meaning.Zhu  The purpose of this paper is to provide exact solutions of the velocity field and the shear stress corresponding to the motion of a fractional Maxwell fluid between two infinite circular cylinders.The Laplace and finite Hankel transforms are used to solve the problem, and the solutions obtained are presented in terms of generalized G a,b,c •, t and R a,b •, t functions.The solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of our general solutions.Furthermore, the solutions for the motion between the cylinders, when one of them is at rest, are also obtained as special cases from our general results.At the end, obtained solutions are discussed graphically for different values of time and material parameters.

Basic Governing Equations
The constitutive equations of an incompressible Maxwell fluid with fractional calculus are given by 14 where T is the Cauchy stress tensor, −pI denotes the indeterminate spherical stress, S is the extrastress tensor, A L L T with L the velocity gradient, μ is the dynamic viscosity of the fluid, λ is the material constant called relaxation time, and DS/Dt is defined by Here, w is the velocity vector, ∇ is the gradient operator, the superscript T denotes the transpose operation, and the Caputo fractional derivative operator D β t is defined as 26 where Γ • is the Gamma function which is defined as In cylindrical coordinates r, θ, z , the rotational flow velocity is given by w w r, t w r, t e θ , 2.5 where e θ is the unit vector in the θ-direction.For such flows, the constraint of incompressibility is automatically satisfied.Since the velocity field w is independent of θ and z, we also assume that S depends only on r and t.Furthermore, if the fluid is assumed to be at rest at the moment t 0, then w r, 0 0, S r, 0 0.

2.6
Equations 2.1 , 2.5 , and 2.6 imply S rr S zz S θz 0 18 , where τ r, t S rθ r, t is the nontrivial shear stress.In the absence of body forces and a pressure gradient in the axial direction, the equations of motion lead to the relevant equation where ρ is the constant density of the fluid.Eliminating τ between 2.7 and 2.8 , we attain to the governing equation where ν μ/ρ is the kinematic viscosity of the fluid.In the following, the fractional partial differential equations 2.9 and 2.7 , with appropriate initial and boundary conditions, will be solved by means of Laplace and finite Hankel transforms.In order to avoid lengthy calculations of residues and contours integrals, the discrete inverse Laplace method will be used 13, 14 .

Axial Couette Flow between Two Infinite Circular Cylinders
Let us consider an incompressible fractional Maxwell fluid at rest in an annular region between two coaxial circular cylinders of radii R 1 and R 2 > R 1 .At time t 0 , both cylinders with radii R 1 and R 2 begin to rotate along their common axis.Owing to the shear, the fluid is gradually moved, its velocity being of the form 2.5 .The governing equations are given by 2.9 , while the appropriate initial and boundary conditions are where Ω 1 and Ω 2 are constants with dimensions T −2 .

Calculation of the Velocity Field
Applying the Laplace transform to 2.9 , using the Laplace transform formula for sequential fractional derivatives 26 , and having the initial and boundary conditions 3.1 and 3.2 in mind, we find that where w r, q is the Laplace transform of the function w r, t which is defined as and the image function w r, q has to satisfy the conditions In the following, we denote by 27 and the Hankel transform of w r, q , where and r n are the positive roots of the transcendental equation B R 1 , r 0, while J 1 • and Y 1 • are Bessel functions of the first and second kind of order one.
Multiplying both sides of 3.3 by rB r, r n , integrating with respect to r from R 1 to R 2 , and taking into account the conditions 3.5 and the identity 3.8 we find that 3.9 Now, for a suitable presentation of the final results, we rewrite 3.9 in the following equivalent form:

3.10
Now, applying the inverse Hankel transform formula 27 we obtain the Laplace transform of the velocity field w r, q under the form w r, q writing the last factor of 3.12 in the following equivalent form:

3.13
Introducing 3.13 into 3.12 , applying the discrete inverse Laplace transform, and using the known result 28, equation 97 , where the generalized G a,b,c •, • function is defined by and we find the velocity field under the form w r, t 3.16

Calculation of the Shear Stress
Applying the Laplace transform to 2.7 , we find that τ r, q μ 1 λq β ∂w r, q ∂r − w r, q r , 3.17 where ∂w r, q ∂r − w r, q r is obtained from 3.12 and Thus, 3.17 becomes

3.20
applying again the discrete inverse Laplace transform as well as using the known relation 28, equation 21 , where the generalized R a,b c, t functions are defined by 28 and the expansion

3.23
and we obtain the shear stress τ r, t under the form τ r, t 3.24

Classical Maxwell Fluid
Making β → 1 into 3.16 and 3.24 , we obtain the velocity field and the shear stress the expressions 4.1 and 4.2 can be written in the simplified form 1n e q 2n t − q 2 2n e q 1n t q 2n − q 1n , τ M r, t q 1n e q 2n t − q 2n e q 1n t q 2n − q 1n . 4.4

Newtonian Fluid
By now letting λ → 0 into 4.4 or β → 1 and λ → 0 into 3.16 and 3.24 , using lim we obtain the velocity field and the associated shear stress corresponding to a Newtonian fluid, performing the same motion.

When the Inner Cylinder Is at Rest
Making Ω 1 0 and Ω 2 Ω into 3.16 and 3.24 , for instance, we obtain the velocity field and the shear stress a

When the Outer Cylinder Is at Rest
Making Ω 1 Ω and Ω 2 0 into 3.16 and 3.24 , we obtain the velocity field and the associated shear stress regard to t like Figure 1 a , but it has opposite effect for r, more exact velocity is decreasing with regard to r on the whole flow domain.

Conclusions
In this paper, the velocity w r, t and the shear stress τ r, t corresponding to the flow of an incompressible Maxwell fluid with fractional derivatives, in the annular region between two infinite coaxial circular cylinders, have been determined using the Laplace and finite Hankel transforms.The solutions that have been obtained, written under a series form in terms of generalized G a,b,c •, t -and R a,b •, t -functions, satisfy the governing equations and all imposed initial and boundary conditions.In the limiting cases, when β → 1 or β → 1 and λ → 0, the corresponding solutions for the ordinary Maxwell and Newtonian fluids are obtained.These solutions also satisfy the associated initial and boundary conditions 3.1 and 3.2 , respectively.Moreover, the solutions for the motion between the cylinders, when one of them is at rest, are also obtained from our general results.
In order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field w r, t and the shear stress τ r, t given by 3.16 and 3.24 have been drawn against r for different values of the time t and the material parameters.Figures 2 and 3 show the profile of the fluid motion at different values of time when both inner and outer are rotating with the same angular velocity in the same direction and in the opposite direction, respectively.From these figures, one can clearly see that both velocity and shear stress in absolute values are increasing function of t.From Figure 3 a , one can also observe that fluid has zero velocity nearer to inner cylinder.
In Figure 4, the influence of the relaxation time λ on the fluid motion is shown.As expected, both the velocity and the shear stress in absolute value are decreasing functions  with respect to λ.Effect of fractional parameter β on the fluid motion is represented in Figure 5, and it is clearly seen that both velocity and shear stress in absolute value are increasing with respect to β.Finally, for comparison, the diagrams of w r, t and τ r, t corresponding to the three models fractional Maxwell, ordinary Maxwell, and Newtonian are together depicted in Figure 6 for the same values of the common material constants and time t.The Newtonian fluid is the swiftest, while the fractional Maxwell fluid is the slowest on the whole flow domain.One thing is worth of mentioning that the units of the material constants are SI units in all the figures, and the roots r n have been approximated by nπ/ R 2 − R 1 .
et al. 8 studied the characteristics of the velocity filed and the shear stress field for an ordinary Maxwell fluid, and Yang and Zhu 9 studied it for a fractional Maxwell fluid.In the last decade, the unidirectional flow of viscoelastic fluid with the fractional Maxwell model was studied by Tan et al. 10, 11 and Hayat et al. 12 .Tong et al. 13, 14 discussed the unsteady flow with a generalized Jeffrey's model in an annular pipe.In the meantime, a lot of papers regarding such motions have been published.The interested readers can see for instance the papers 15-25 and their related references.

4 . 2 corresponding
to an ordinary Maxwell fluid, performing the same motion.Of course, in view of the identities

5 . 2 corresponding
to a fractional Maxwell fluid when the inner cylinder is at rest.Figure1 ashows velocity profile corresponding to 5.1 for different values of time, when the inner cylinder is at rest.It shows that velocity is an increasing function with regard to t and r on the whole flow domain.