Analytical Solutions for the Flow of a Fractional Second Grade Fluid due to a Rotational Constantly Accelerating

Exact analytic solutions are obtained for the flow of a generalized second grade fluid in an annular region between two infinite coaxial cylinders. The fractional calculus approach in the governing equations of a second grade fluid is used. The exact analytic solutions are constructed by means of Laplace and finite Hankel transforms. The motion is produced by the inner cylinder which is rotating about its axis due to a constantly accelerating shear. The solutions that have been obtained satisfy both the governing equations and all imposed initial and boundary conditions. Moreover, they can be easily specialized to give similar solutions for second grade and Newtonian fluids. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between the three models, is underlined by graphical illustrations.


Introduction
The study of the non-Newtonian fluids has recently achieved much importance because of well-established applications in a number of processes that occur in industry.Such applications include the extrusion of polymer fluids, cooling of the metallic plate in a bath, animal bloods, foodstuffs, exotic lubricants and colloidal and suspension solutions.For these fluids, the classical Navier-Stokes theory is inadequate.Because of their complexity, there are several models of non-Newtonian fluids in the literature.One of the most popular models for non-Newtonian fluids is the model that is called second-grade fluid 1 .Although there are some criticisms regarding the applications of this model 2 , it has been shown by Walters 3 that, for many types of problems in which the flow is slow enough in the viscoelastic sense,

Governing Equations
The flows to be here considered have the velocity field of the form 22, 23 v v r, t w r, t e θ , 2.1 where e θ is the unit vector along the θ-direction of the cylindrical coordinate system r, θ, and z.For such flows the constraint of incompressibility is automatically satisfied.The nontrivial shear stress τ r, t S rθ r, t corresponding to such a motion of a second grade fluid is given by 24 where μ is the viscosity and α 1 ia a material modulus.In the absence of a pressure gradient in the flow direction and neglecting the body forces, the balance of the linear momentum leads to the relevant equation 1 r where ν μ/ρ is the kinematic viscosity of the fluid, ρ is its constant density and α α 1 /ρ.Generally, governing equations for generalized fluids with fractional derivatives are derived from those of the ordinary fluids by replacing the inner time derivatives of an integer order with the so-called Riemann-Liouville operator 11, 27 where Γ • is the Gamma function.
Consequently, the governing equations corresponding to the motion 2.1 of a generalized second grade fluid are cf.where the new material constant α 1 for simplicity, we are keeping the same notation goes to the initial α 1 for β → 1.
In this paper, we are interested into the motion of a generalized second grade fluid whose governing equations are given by 2.6 .The fractional partial differential equations 2.6 , with adequate initial and boundary conditions, can be solved in principle by several methods, the integral transforms technique representing a systematic, efficient, and powerful tool.The Laplace transform will be used to eliminate the time variable and the finite Hankel transform to remove the spatial variable.However, in order to avoid the lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform will be used.

Rotational Flow between Two Infinite Cylinders
Consider an incompressible generalized second grade fluid at rest in the annular region between two infinitely long coaxial cylinders.At time t 0 , let the inner cylinder of radius R 1 be set in rotation about its axis by a time-dependent torque per unit length 2πR 1 ft and ISRN Mathematical Physics let the outer cylinder of radius R 2 be held stationary.Owing to the shear, the fluid between cylinders is gradually moved, its velocity being of the form 2.1 .The governing equations are given by 2.6 and the appropriate initial and boundary conditions are see also 7, Equations 5.2 and 5.3 where f is a constant.

Calculation of the Velocity Field
Applying the Laplace transform to 2.6 1 and 3.2 , we get where w r, q and τ R 1 , q are the Laplace transforms of the functions w r, t and τ R 1 , t , respectively.We denote by 22, Equation 34 the finite Hankel transform of the function w r, q , where where r n are the positive roots of the equation B R 2 r 0, and J p • , Y p • are the Bessel functions of the first and second kind of order p.
The inverse Hankel transform of w H r n , q is given by 22, Equation 35w r, q π 2 2 By means of 3.4 2 and of the identity we can easily prove that 3.9 Combining 3.3 , 3.4 , and 3.9 , we find that

3.10
Writing w H r n , q under the equivalent form and applying the inverse Hankel transform and using the identities 3.14

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Now applying the inverse Laplace transform to 3.14 , we find for the velocity field the expression w r, t f 2μ

3.15
where the generalized function G a,b,c d, t is defined by 28, Equations 97 and 101 3.16

Calculation of the Shear Stress
Applying the Laplace transform to 2.6 2 , we find that τ r, q μ α 1 q β ∂ ∂r − 1 r w r, q . 3.17 In order to get a suitable form for τ r, t , we rewrite 3.10 under the equivalent form

3.18
Applying the inverse Hankel transform to 3.18 and using 3.7 and the identity 3.13 , we find that Introducing 3.19 into 3.17 , it results that τ r, q R 1 r 3.20 or equivalently see also 3.12 τ r, q R 1 r where B 1 rr n J 2 rr n Y 2 R 1 r n − J 2 R 1 r n Y 2 rr n .Now taking the inverse Laplace transform of both sides of 3.21 , we get

The Special Case β → 1
Making β → 1 into 3.15 and 3.22 , we obtain the similar solutions 4.4 The velocity field can be also processed to give the equivalent form

4.5
Making α 1 and then α → 0 into 4.3 and 4.4 , the velocity field and the associated shear stress corresponding to a Newtonian fluid, are obtained.

Conclusions
The purpose of this note is to provide exact analytic solutions for the velocity field w r, t and the shear stress τ r, t corresponding to the unsteady rotational flow of a generalized second grade fluid between two infinite coaxial cylinders, the inner cylinder being set in rotation about its axis by a constantly accelerating shear.The solutions that have been obtained, presented under series form in terms of usual Bessel J 1  Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity w r, t and of the shear stress τ r, t are depicted against r for different values of the time t and of the pertinent parameters.From Figures 1 a and 1 b , containing the diagrams of the velocity and the shear stress at several times, it clearly results in the influence of the rigid boundary on the fluid motion.The velocity is an increasing function of t.For the same values of the parameters, the shear stress, in absolute value, is also an increasing function of t.The influence of the kinematic viscosity ν on the fluid motion is shown in Figures 2 a and 2 b .The velocity is a decreasing function of ν.The shear stress, in absolute value, on the first part of the flow domain, near the moving cylinder, is a decreasing function of ν.It is an increasing function of ν in the neighborhood of the stationary cylinder.Figure 3 shows the influence of the parameter α on the flow motion.Both the velocity and the shear stress, in absolute value, on the first part of the flow domain, near the moving cylinder, are increasing functions of α.They are decreasing functions of α in the neighborhood of the stationary cylinder.The influence of the fractional parameter β on the fluid motion is shown in Figure 4. Its effect on the fluid motion is qualitatively opposite to that of parameter α.
Finally, for comparison, the profiles of w r, t and τ r, t corresponding to the motion of the three models Newtonian, ordinary second grade, and generalized second grade are together depicted in Figure 5, for the same values of t and of the common material parameters.In all the cases the velocity of the fluid is a decreasing function with respect to r, and the Newtonian fluid is the swiftest, while the generalized second grade fluid is the slowest in the region near the moving cylinder.The units of the material constants are SI units within all figures, and the roots r n have been approximated by 2 n − 1 π/ 2 R 2 − R 1 .

4 . 1 corresponding
to a second grade fluid performing the same motion.Now, in view of the identity ∞ k 0 −νr 2 n k G 0,− k 2 ,k 1 −αr
• , J 2 • , Y 1 • , and Y 2 • and generalized G a,b,c•, t functions, satisfy all imposed initial and boundary conditions.They can be easily specialized to give the similar solutions for ordinary second grade and Newtonian fluids.Furthermore, in view of some recent results 29, Equation 3.15 , our velocity field 4.3 is in accordance with that obtained in 7, Equation5.17by a different technique.