Abstract
In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0+, begins to rotate about its axis with an angular velocity Ωt p. The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions.
The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions.
Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.
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1 Introduction
The motion of a fluid in a rotating or translating cylinder is of interest to both theoretical and practical domains. The flow through rotating cylinders, started from rest, has applications in the food industry and is one of the most important and most interesting problems of motion near rotating bodies. It has been intensively studied since G.I. Taylor (1923) reported the results of his famous investigations [1]. For Newtonian fluids, the velocity distribution for a fluid contained in a circular cylinder can be found in [2], while the case of a fluid contained in an annular region between two cylinders, with a common axis, is given by [3]. Waters and King [4] studied the start-up Poiseuille flow of an Oldroyd-B fluid in a straight circular tube. The exact solution was obtained using the Laplace transform method. Since an integral form of the constitutive equation is used, only one initial condition is required for this unsteady problem. During recent years quite a number of papers on this type of flow have been published. Unsteady, pressure-driven flow of a classical Maxwell fluid in a pipe was studied by Rahaman and Ramkissoon [5]. Exact solutions were obtained as infinite Fourier-Bessel series. Wood [6] has considered the general case of helical flow of an Oldroyd-B fluid, due to the combined action of rotating cylinders (with constant angular velocities) and a constant axial pressure gradient. Hayat et al. [7] obtained the velocity fields for some simple flows of Oldroyd-B fluids using Fourier transform. Recently, Fetecau [8] has established exact solutions for some unidirectional flows of the same fluids in unbounded domains which geometrically are axi-symmetric pipelike.
It is very important to study the mechanism of viscoelastic fluids flow in many industry fields, such as oil exploitation, chemical and food industry and bio-engineering. The first exact solutions for flows of non-Newtonian fluids in such a domain seem to be those of Ting [9] corresponding to Second grade fluids and Srivastava [10] for Maxwell fluids. During recent years quite many papers of this type have been published. The most general solutions corresponding to the helical flow of a Second grade fluid seem to be those of Fetecau and Corina Fetecau [11], in which the cylinder is rotating around its axis and sliding along the same axis with time-dependent velocities. Other interesting solutions for different flows of the same fluids have been also obtained by Hayat et al. [12]. Exact solutions for the helical flows of Oldroyd-B fluid in cylindrical domains have been obtained Fetecau et al. [13]. In the meantime a lot of papers regarding such motions have been published [14, 15].
Nowadays, fractional calculus has encountered much success in the description of viscoelasticity [16–20]. The starting point of the fractional derivative model of a non-Newtonian fluid is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the fractional calculus operators. This generalization allows one to define precisely non-integer order integrals or derivatives. Tan et al. [18] and Xu and Tan [21] examined the velocity field, stress field and vortex sheet of a generalized Second-order fluid with fractional anomalous diffusion. Song and Jiang [22] achieved satisfactory result to apply the constitutive equation with fractional derivative to the experimental data of viscoelasticity. Tan et al. [23] and Tan and Xu [24] applied fractional derivative to the constitutive relationship models of Maxwell viscoelastic fluid and Second grade fluid, and studied some unsteady flows.
The main idea of this work is to establish exact solutions for the velocity field, and the adequate shear stress corresponding to the unsteady rotational flow of an incompressible Oldroyd-B fluid with fractional derivatives through an infinite circular cylinder induced by a time-dependent shear. The motion of the fluid is produced by the cylinder, which after time t=0+, begins to rotate about its axis with a time-dependent angular velocity. The solutions that have been obtained, presented under series form in terms of the generalized G-functions, are established by means of the finite Hankel and Laplace transforms. The similar solutions for the Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions.
2 Governing equations
The flows to be here considered have the velocity v and the extra-stress S of the form [25]
where e θ is the unit vector in the θ-direction of the cylindrical coordinates system r, θ and z. For such flows, the constraint of incompressibility is automatically satisfied.
Furthermore, if initially the fluid is at rest, then
The governing equations, corresponding to such motions of Oldroyd-B fluid, are given by [25]
where μ is the dynamic viscosity, ν=μ/ρ is the kinematic viscosity, ρ being the constant density of the fluid, λ is the relaxation time, λ r is the retardation time, and τ(r,t)=S rθ (r,t) is the non-trivial shear stress.
The governing equations corresponding to an incompressible Oldroyd-B fluid with fractional derivatives (OBFFD), performing the same motion, are obtained by replacing the inner time derivatives with respect to t from Eqs. (3) and (4), by the fractional differential operator [17, 26]
where Γ(⋅) is the Gamma function.
Consequently, the governing equations to be used here are
When β,γ→1, Eqs. (6) and (7) reduce to Eqs. (3) and (4), because \(D^{1}_{t}f=\frac{df}{dt} \). Furthermore, the new material constants λ and λ r (although, for the simplicity, we keep the same notation) reduce to the previous ones for β,γ→1.
3 Starting flow through an infinite circular cylinder
Suppose that an incompressible OBFFD is situated at rest in an infinite circular cylinder of radius R (>0). After time t=0+, the cylinder suddenly begins to rotate about its axis with an angular velocity Ωt p. Owing to the shear the inner fluid is gradually moved, its velocity being of the form (1)1. The governing equations are given by Eqs. (6) and (7), while the appropriate initial and boundary conditions are
where Ω is a constant and \(\emph{N}\) is the set of natural numbers.
Equations (6) and (7) containing fractional derivatives along with initial and boundary conditions can be solved in principle by several methods, i.e. Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), and Adomian Decomposition Method (ADM). Of these methods the integral transform technique is systematic, efficient and powerful tool. In the following, we shall use the Laplace transform to eliminate the time variable, and the finite Hankel transform for the removal of spatial variable. However, in order to avoid the burdensome calculations of residues and contour integrals, we shall apply the discrete inverse Laplace transform method.
3.1 Calculation of the velocity field
Applying the Laplace transform to Eqs. (7) and (9), we get
where \(\overline{w}(r,q)\) and \(\overline{w}(R,q)\) are the Laplace transforms of the functions w(r,t) and w(R,t), respectively.
We shall denote by [27]
the finite Hankel transform of the function \(\overline{w}(r,q)\), and the inverse Hankel transform of \(\overline{w}_{H}(r_{{n}},q)\) is given by [27]
r n being the positive roots of the equation J 1(Rr)=0, and J p (.) is the Bessel function of the first kind of order p. Multiplying now both sides of Eq. (10) by rJ 1(rr n ), then integrating with respect to r from 0 to R, and taking into account Eqs. (11) and (12), and the result which we can easily prove
we find that
It can be also written in the suitable form
where
Applying the inverse Hankel transform to Eqs. (16a) and (16b), and using the known formula
we get
Using the identity
Eq. (17b) can be written as
After taking the inverse Hankel transform of Eq. (15), it leads to
Now taking the inverse Laplace transform of Eq. (20), the velocity field w(r,t) is given by
where the generalized function G a,b,c (⋅,⋅) is defined by [28, Eqs. (97) and (101)]
3.2 Calculation of the shear stress
Applying the Laplace transform to Eq. (6), we find that
where \(\overline{w}(r,q)=\overline{w}_{1}(r,q)+\overline{w}_{2}(r,q)\).
Using Eqs. (17a) and (17b), we can calculate
and by using Eqs. (23) and (24), we have
In view of the identity (18), Eq. (25) can be equivalently written as
Now taking the inverse Laplace transform of both sides of Eq. (26) and using Eq. (22), we find that
4 The special cases
4.1 Ordinary Oldroyd-B fluid
Making β,γ→1 into Eqs. (21) and (27), we obtain the similar solutions for the velocity field
and for the shear stress
for an ordinary Oldroyd-B fluid performing the same motion.
4.2 Maxwell fluid with fractional derivatives
Making λ r →0 into Eqs. (21) and (27), we obtain the velocity field
and the associated shear stress
corresponding to Maxwell fluid with fractional derivatives performing the same motion are recovered [29, Eqs. (21) and (26)].
4.3 Ordinary Maxwell fluid
Making β→1 in Eqs. (30) and (31), we get the expressions for velocity field
and the associated shear stress
corresponding to an ordinary Maxwell fluid.
4.4 Second grade fluid with fractional derivatives
Making λ→0, taking νλ r =α and α 1=αρ (the material constants for Second grade fluid) into Eqs. (21) and (27), and using (A.1) and (A.2), the expressions for the velocity field
and the associated shear stress
corresponding to Second grade fluid with fractional derivatives performing the same motion are recovered [30, Eqs. (21) and (26)].
4.5 Ordinary Second grade fluid
Making γ→1 into Eqs. (34) and (35), we obtain the velocity field
and the associated shear stress
corresponding to an ordinary Second grade fluid performing the same motion.
These solutions can also simplified to give (see also Eqs. (A.3)–(A.4) from Appendix)
Equation (39) doesn’t hold for p=1, for linear case Ωt, we put p=1 in Eqs. (36) and (37), and using Eqs. (A.3) (for p=1) and (A.5) from Appendix, we get the following expression
which are the similar solutions for flows induced by a circular cylinder subject to a constant angular acceleration Ω that have been recently obtained in [13].
4.6 Newtonian fluid
Now, making λ r →0 in Eqs. (38) and (39), we get the velocity field w(r,t) as
and the associated shear stress
corresponding to a Newtonian fluid performing the same motion for p>1.
Now, making λ r →0 into Eqs. (40) and (41), we obtain the velocity field
and the associated shear stress
corresponding to a Newtonian fluid subject to flow induced by a circular cylinder which is subject to a constant angular acceleration Ω.
5 Numerical results and discussion
In the previous sections, exact analytical solutions for the velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible OBFFD through an infinite circular cylinder are obtained. In order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity as well as those of the shear stress are depicted against r for different values of time t and of the pertinent parameters.
Figures 1a and 1b clearly show that both the velocity and the shear stress are increasing functions of t. They are also increasing functions of r, excepting τ(r,t) on a small interval near the boundary. Figure 2 shows the influence of the parameter p on the fluid motion, it shows that the velocity and the shear stress are also increasing functions of p. The influence of the kinematic viscosity ν on the fluid motion is shown in Figs. 3a and 3b, velocity w(r,t) is an increasing function of ν, while shear stress τ(r,t) is a decreasing function of ν.
The influences of relaxation time λ and retardation time λ r on the velocity and shear stress are shown in Figs. 4 and 5. The two parameters, as expected, have opposite effects on the fluid motion. More exactly the velocity and the shear stress are decreasing functions with respect to λ and increasing ones with regards to λ r . The influences of the fractional parameters β and γ on the fluid motion are presented in Figs. 6 and 7. Their effects are also opposite, but they are qualitatively the same as those of λ r and λ, respectively. More exactly the velocity w(r,t) and the shear stress τ(r,t) are increasing functions with regards to β and decreasing ones with regards to γ.
Finally, for comparison, the profiles of w(r,t) and τ(r,t) corresponding to the flow of Oldroyd-B, Maxwell and Second grade fluids with fractional derivatives and ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian) are together depicted in Fig. 8 for the same values of t and of the common material parameters. The Maxwell fluid with fractional derivatives, as it results from these figures, is the slowest and the Second grade fluid with fractional derivatives is the swiftest on the whole flow domain. The units of the material constants are SI units within all figures, and the roots r n have been approximated by (4n+1)π/(4R).
6 Concluding remarks
The purpose of this paper was to establish exact analytical solutions for the velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible OBFFD through an infinite circular cylinder. The motion of the fluid is produced by the cylinder, which after time t=0+, begins to rotate about its axis with a time-dependent angular velocity. The solutions that have been obtained by means of the finite Hankel and Laplace transforms, are presented under series form in terms of the generalized G and Bessel functions satisfy all initial and boundary conditions. The similar solutions for the Maxwell and Second grade fluids with fractional derivatives and ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions. Furthermore, the solutions (30) and (31) corresponding to Maxwell fluid with fractional derivatives, solutions (34) and (35) corresponding to Second grade fluid with fractional derivatives and solutions (40) and (41) corresponding to Second grade fluid are equivalent to those obtained in [11, 29, 30] by a different technique. The results categorically indicate the following findings:
-
The solutions obtained for Maxwell and Second grade fluids with fractional derivatives and ordinary Second grade fluid by two different methods are equivalent.
-
It is noted that the velocity of the fluid is an increasing function with respect to t and r on the whole flow domain.
-
The velocity of the fluid increases for increasing ν while the shear stress has opposite property.
-
It is observed that rheological parameters λ and λ r have a strong influence on the fluid motion, but their effects are opposite.
-
The fractional parameters β and γ have a opposite effects on the fluid motion.
-
The rheological parameters λ and fractional parameters γ qualitatively have the same effects on the fluid motion. It is also observed that rheological parameters λ r and fractional parameters β qualitatively have same effects on the fluid motion, but opposite to λ and γ.
-
The Maxwell and Oldroyd-B fluids with fractional derivatives are slower than the ordinary Maxwell and Oldroyd-B fluids while Second grade fluid with fractional derivatives is faster than ordinary Second grade fluid.
References
Taylor GI (1923) Stability of a viscous liquid contained between two rotating cylinders. Philos Trans R Soc Lond A 223:289–298
Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge
Yih CS (1969) Fluid mechanics. McGraw-Hill, New York
Waters ND, King MJ (1971) The unsteady flow of an elasto-viscous liquid in a straight pipe of circular cross section. J Phys D, Appl Phys 4:204–211
Rahaman KD, Ramkissoon H (1995) Unsteady axial viscoelastic pipe flows. J Non-Newton Fluid Mech 57:27–38
Wood WP (2001) Transient viscoelastic helical flows in pipes of circular and annular cross-section. J Non-Newton Fluid Mech 100:115–126
Hayat T, Siddiqui AM, Asghar S (2001) Some simple flows of an Oldroyd-B fluid. Int J Eng Sci 39:135–147
Fetecau C (2004) Analytical solutions for non-Newtonian fluid flows in pipe-like domains. Int J Non-Linear Mech 39:225–231
Ting TW (1963) Certain non-steady flows of second-order fluids. Arch Ration Mech Anal 14:1–26
Srivastava PN (1966) Non-steady helical flow of a visco-elastic liquid. Arch Mech Stosow 18:145–150
Fetecau C, Fetecau C (1985) On the unsteady of some helical flows of a second grade fluid. Acta Mech 57:247–252
Hayat T, Khan M, Ayub M (2006) Some analytical solutions for second grade fluid flows for cylindrical geometries. Math Comput Model 43:16–29
Fetecau C, Fetecau C, Vieru D (2007) On some helical flows of Oldroyd-B fluids. Acta Mech 189:53–63
Vieru D, Akhtar W, Fetecau C, Fetecau C (2007) Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42:573–583
Nadeem S, Asghar S, Hayat T, Hussain M (2008) The Rayleigh Stokes problem for rectangular pipe in Maxwell and second grade fluid. Meccanica 43:495–504
Bagley L, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30:133–155
Friedrich CHR (1991) Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta 30(2):151–158
Tan WC, Feng X, Lan W (2002) Exact solution for the unsteady Couette flow of the generalized second grade fluid. Chin Sci Bull 47(16):1226–1228
Tong D, Shan L (2009) Exact solutions for generalized Burgers fluid in an annular pipe. Meccanica 44:427–431
Shah SHAM (2010) Some helical flows of a Burgers fluid with fractional derivative. Meccanica 45:143–151
Xu MY, Tan WC (2001) Theoretical analysis of the velocity field, stress field and vortex sheet of generalized Second order fluid with fractional anomalous diffusion. Sci China Ser A 44(11):1387–1399
Song DY, Jiang TQ (1998) Study on the constitutive equation with fractional derivative for the vicoelastic fluids—Modified Jeffreys model and its application. Rheol Acta 27:512–517
Tan WC, Wenxiao P, Xu MY (2003) A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int J Non-Linear Mech 38(5):645–650
Tan WC, Xu MY (2004) Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates. Acta Mech Sin 20(5):471–476
Fetecau C, Imran M, Fetecau C, Burdujan I (2010) Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses. Z Angew Math Phys 61:959–969
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Tong D, Liu Y (2005) Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe. Int J Eng Sci 43:281–289
Lorenzo CF, Hartley TT (1999) Generalized functions for the fractional calculus. NASA/TP-1999-209424/REV1
Imran M, Athar M, Kamran M (2011) On the unsteady rotational flow of a generalized Maxwell fluid through a circular cylinder. Arch Appl Mech 81:1659–1666
Athar M, Kamran M, Imran M (2012) On the unsteady rotational flow of a fractional second grade fluid through a circular cylinder. Meccanica 47:603–611
Acknowledgements
The authors would like to express their gratitude to the referees for their careful assessment and constructive comments and corrections.
The author M. Kamran is thankful and grateful to Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan; and especially to Higher Education Commission of Pakistan, for supporting and facilitating the research work.
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Kamran, M., Imran, M. & Athar, M. Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder. Meccanica 48, 1215–1226 (2013). https://doi.org/10.1007/s11012-012-9662-y
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DOI: https://doi.org/10.1007/s11012-012-9662-y