Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

Abstract. The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0 applies a shear stress ft (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalizedG andR functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, . . . can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations


Introduction
The motion of a fluid in cylindrical domains has applications in the food industry, oil exploitation, chemistry and bio-engineering [4].The non-Newtonian fluids are now considered to play a more important and appropriate role in technological applications in comparison with Newtonian fluids.The first exact solutions corresponding to motions of non-Newtonian fluids in cylindrical domains seem to be those of Ting [5] for second grade fluids, Srivastava [6] for Maxwell fluids and Waters and King [7] for Oldroyd-B fluids.To c Vilnius University, 2011 the best of our knowledge, the first exact solutions for motions of non-Newtonian fluids due to a shear stress on the boundary are those of Bandelli and Rajagopal [8] and Bandelli et al. [9] for second grade fluids and Waters and King [10] for Oldroyd-B fluids.Other similar solutions have been recently obtained in [11,12].The solutions from [8], obtained by means of the Laplace transform, give the velocity field corresponding to the motion of second grade fluid between two infinite circular cylinders.
In this paper, our interest is to find the velocity field and the shear stress corresponding to the motion of a Maxwell fluid between two infinite circular cylinders, one of them being subject to a longitudinal time-dependent shear stress f t a , a ≥ 0. In the last time, the fractional calculus has encountered much success in description of complex dynamics such as relaxation, oscillation, wave and viscoelastic behavior.Several authors suggested that the integral-order models for viscoelastic materials seem to be inadequate, especially from the qualitative point of view.In the same time they proposed fractional-order laws of deformation for modeling the viscoelastic behavior of real materials.One of them due to Rouse [13], is used in the molecular theory for dilute polymer solutions.Here σ is the stress, ε the strain, µ s is the steady-flow viscosity of the solvent, µ 0 is the steadyflow viscosity of the solution, n is the number of molecules, K is the Boltzman constant, T is the absolute temperature and D α t is a fractional differential operator to be defined in the next section.Ferry et al. [14], modified the Rouse theory in concentrated polymer solutions and polymer solids with no cross-linking and obtained that where µ is the viscosity, ρ the density, R is the universal gas constant and M the molecular weight.Consequently, the fractional calculus approach to viscoelasticity for the study of viscoelastic material properties is justified, at least for polymer solutions and for polymer solids without cross-linking.In the meantime, a lot of exact solutions corresponding to different motions of non-Newtonian fluids with fractional derivatives have been established, but we mention here only a few in cylindrical domains [15][16][17][18].Furthermore, the one-dimensional fractional derivative Maxwell model has been found very useful in modeling the linear viscoelastic response of some polymers in the glass transition and the glass state [19].It is worth pointing out that Palade et al. [20] developed a fully objective constitutive equation for an incompressible fluid-reducible to the linear fractional derivative Maxwell model under small deformations hypothesis.For a deeper documentation on this subject, one can also see the books [21,22].Consequently, for completeness and motivated by the above remarks, we solve our problem for fractional Maxwell fluids (FMF).The solutions that have been obtained, presented under series form in terms of G a,b,c (. , t) and R a,b (. , t) generalized functions, satisfy both governing equations and all imposed initial and boundary conditions.They can be easy specialized to give the similar solutions for ordinary Maxwell and Newtonian fluids.Some important properties are obtained for the special cases a = 0, 1, 2, and the required time to reach the steady-state or the large-time state is graphically established.

Basic governing equations
The flows to be here studied have the velocity field v and the extra-stress S of the form where e z is the unit vector in the z-direction of the cylindrical coordinate system r, θ and z.For such flows the constraint of incompressibility is automatically satisfied while the governing equations corresponding to fractional Maxwell fluid (FMF) are [18] 1 where τ (r, t) = S rz (r, t) is the non-trivial shear stress, λ is a material constant, µ is the dynamic viscosity, ν = µ/ρ is the kinematic viscosity (ρ being the constant density of the fluid) and the fractional differential operator D α t is defined by [23,24] and Γ(.) is the gamma function.Of course the material constant λ has the dimension of t α and for α → 1 it tends to the relaxation time.In the following the fractional partial differential equations (2), with appropriate initial and boundary conditions, will be solved by means of Hankel and Laplace transforms.In order to avoid lengthy calculations of residues and contours integrals, the discrete inverse Laplace transform method will be used [15][16][17][18].

Axial Couette flow between two infinite cylinders
Let us consider an incompressible FMF at rest in an annular region between two coaxial circular cylinders of radii Re(a − b) > 0, Re(q) > 0, is applied on the boundary of the inner cylinder.Due to the shear, the fluid is gradually moved.Its velocity is of the form (1a) and the governing equations are given by Eqs. ( 2).
The appropriate initial and boundary conditions are and Of course, as we shall later see, τ (R 1 , t) given by Eq. ( 4) is just the solution of the fractional differential equation (7a).For α → 1, Eq. ( 4) takes the simple form corresponding to ordinary Maxwell fluids.

Calculation of the velocity field
Applying the Laplace transform to Eq. (2b), using the Laplace transform formula for sequential fractional derivatives [24] and having the initial and boundary conditions ( 6) and ( 7) in mind, we find that where the image function v(r, q) = L{v(r, t)} has to satisfy the conditions In the following we shall denote by [26] v the Hankel transform of v(r, q), where r n are the positive roots of the transcendental equation B(R 2 , r) = 0, while J p (.) and Y p (.) are Bessel functions of the first and second kind of order p.The inverse Hankel transform of v H (r n , q) is given by [26] Multiplying both sides of Eq. ( 9) by rB(r, r n ), integrating with respect to r from R 1 to R 2 and taking into account the conditions (6), and using the result we find that Now, we rewrite Eq. ( 15) in the following equivalent form Applying the inverse Hankel transform to Eq. ( 16), and using the identity we obtain In order to obtain a suitable form for the velocity field v(r, t), we rewrite the last factor from the second term of Eq. ( 18), into the equivalent form Introducing (19) into (18), and applying the discrete inverse Laplace transform, we find the velocity field under the form where the generalized G a,b,c (. , .)functions are defined by [25, Eqs. ( 101) and ( 99)] Re(ac − b) > 0, Re(q) > 0, d q a < 1.

Calculation of the shear stress
Applying the Laplace transform to Eq. (2b) we find that where is obtained from Eq. ( 18) and Introducing ( 23) into (22), we get By means of the identity Eq. ( 25) can be written in the form Now applying again the discrete inverse Laplace transform to Eq. ( 27) and using Eqs.( 5) and ( 21), we find the shear stress τ (r, t) under the form 4 Limiting cases

Classical Maxwell fluid
Making α → 1 into Eqs.( 20) and ( 28), we obtain the velocity field and the associated tangential stress corresponding to an ordinary Maxwell fluid performing the same motion.Direct computations show that v aM (r, t) and τ aM (r, t) satisfy the governing equations (2) with α = 1 and all corresponding initial and boundary conditions.
More exactly, on the boundary we have

Newtonian fluid
By now letting λ → 0 into Eqs.( 29) and (30) and using the limit we obtain the velocity field and the associated shear stress 5 Special cases

Case a = 0
By making a = 0 in Eqs. ( 20), ( 28), ( 29), (30), ( 32) and (33), we recover the solutions ( 28), ( 33), (34), ( 35), ( 37) and (38) obtained in [27].The corresponding velocities are From Fig. 1, it clearly results that v 0M (r, t) given by Eq. ( 35) is equivalent with obtained in [1] by a different technique.In Eq. ( 37) The unsteady motion of Newtonian and Maxwell fluids as it results from Eqs. ( 36) and (37), ultimately becomes steady.Its starting solutions (36) and (37) (or equivalently (35)), describe the motion of the fluid some time after its initiation.that time when the transient disappears, the motion of the fluid is described by the steady solution which is same for both kinds of fluids.The required time to reach the steady state for Maxwell fluids, as it results from Figs. 2 and 3, decreases for increasing λ.Consequently, the steady state is rather obtained for Newtonian fluids in comparison with Maxwell fluids.

Conclusions
In this paper, the unsteady motion of a fractional Maxwell fluid between two infinite coaxial circular cylinders is studied by means of the Laplace and finite Hankel transforms.The motion of the fluid is produced by the inner cylinder that, after the initial moment, is pulled with a time-dependent shear along its axis and the outer cylinder is held fixed.The velocity v(r, t) and the adequate shear stress τ (r, t) are obtained under series form in terms of the generalized G a,b,c (. , t) and R a,b (. , t) functions.They satisfy all imposed initial and boundary conditions, and can immediately be particularized to give the similar solutions for ordinary Maxwell and Newtonian fluids.Furthermore, as it results from Eqs. ( 51) and ( 52), the solutions corresponding to a = 1, 2, 3, . . .can be written as simple or multiple integrals of similar solutions for a = 0.As a check of obtained results, Figs.
Nonlinear Anal.Model.Control, 2011, Vol.16, No. 2, 135-151where f is a real constant and the generalized R a,b (c, t) functions are defined by[25]
1, 4 and 5 clearly show that the diagrams of our solutions (35), (40) and (46) for ordinary Maxwell fluids are almost identical to those obtained correspondingly in [1, 2] and [3] by a different technique.Other new and interesting results are: 1.The decay of the transient component in time, for velocity v 1M (r, t) given inEq.(42), is depicted in Fig.6.It is clearly seen that the required time to reach the large-time state for velocity decreases if λ increases.