International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 835398 | 15 pages | https://doi.org/10.5402/2012/835398

Critical Study on Rotational Flow of a Fractional Oldroyd-B Fluid Induced by a Circular Cylinder

Academic Editor: H. Zhou
Received15 Aug 2011
Accepted13 Sep 2011
Published21 Dec 2011

Abstract

We considered the unsteady flow of a fractional Oldroyd-B fluid through an infinite circular cylinder with the help of infinite Hankel and Laplace transforms. The motion of the fluid is produced by the cylinder that, at time 𝑡=0+ is subject to a time-dependent angular velocity. The established solutions have been presented under series form in terms of the generalized 𝐺 functions satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, ordinary and fractional Maxwell, ordinary and fractional second-grade, and Newtonian fluids, performing the same motion, are acquired as limiting cases of general solutions. The keynote points regarding this work to mention are that (1) we extracted the expressions for velocity field and shear stress corresponding to the motion of fractional second-grade fluid as a limiting case of general solutions; (2) the expressions for velocity field and shear stress are in the most simplified form in contrast with the studies of Siddique and Sajid (2011), in which the expression for the velocity field involves the convolution product as well as the integral of the product of generalized 𝐺 functions. Finally, numerical results are presented graphically and discussed in order to reveal some physical aspects of obtained results.

1. Introduction

The motion of a fluid in cylindrical domains has applications in the food industry, oil exploitation, chemistry, and bioengineering [1], it being one of the most important problems of motion near translating or rotating bodies. 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 [2] for second-grade fluids, Srivastava [3] for Maxwell fluids, and Waters and King [4] for Oldroyd-B fluids. In the meantime a lot of papers regarding these motions have been published. However, the most of them deal with motion problems in which the velocity field is given on the boundary. To 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 [5] and Bandelli et al. [6] for second-grade fluids. Other similar solutions have been also obtained in [7–15].

Fractional calculus is a field of applied mathematics that deals with derivatives and integral of arbitrary orders [16, 17]. During the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics. A lot of applications of fractional calculus can be found in turbulence and fluid dynamics, plasma physics and controlled thermonuclear fusion, nonlinear control theory, nonlinear biological systems, and astrophysics [18–21] while its applications to polymer physics, biophysics, and thermodynamics can be found in [22]. The use of the method of fractional derivatives in the theory of viscoelasticity has the advantages that it affords possibilities for obtaining constitutive equations for elastic complex modulus of viscoelastic materials with only few experimentally determined parameters [23].

Our intention here is to establish exact solutions for the velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives through an infinite circular cylinder. The motion of the fluid is produced by the cylinder, which, at 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 an ordinary Oldroyd-B, Maxwell with fractional derivatives, ordinary Maxwell fluids as well as those for the fractional and ordinary second-grade fluids, can be obtained as limiting cases of general solutions for 𝛾,𝛽→1; 𝜆𝑟→0 and 𝛽→1; 𝜆𝑟→0, 𝛾 and 𝛽→1; 𝜆→0 and 𝛾→1; 𝜆→0, 𝛾 and 𝛽→1, respectively. The solutions for a Newtonian fluid performing the same motion are obtained as limiting cases of the solutions for an Oldroyd-B fluid with fractional derivatives when 𝜆𝑟→0, 𝜆→0, and 𝛾,𝛽→1.

At the end with the help of Computer Algebra System (CAS) MATHCAD 2.0, we presented diagrams of velocity as well as of shear stress against different values of radius, time, and of the pertinent parameters. Also in one of the diagrams, we compared the velocity profiles and shear stresses corresponding to the motion of different fluid models, for the same value of time and of the common material parameters.

2. Governing Equations

The flows to be here considered have the velocity 𝐯 and the extrastress 𝐒 of the form [15] 𝐯=𝐯(𝑟,𝑡)=𝑤(𝑟,𝑡)ğžğœ½,𝐒=𝐒(𝑟,𝑡),(2.1) where ğžğœ½ 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 𝐯(𝑟,0)=0,𝐒(𝑟,0)=0.(2.2) The governing equations corresponding to such motions of Oldroyd-B fluid are given by [15] 𝜕1+𝜆𝜕𝑡𝜏(𝑟,𝑡)=𝜇1+𝜆𝑟𝜕𝜕𝜕𝑡−1𝜕𝑟𝑟𝜕𝑤(𝑟,𝑡),1+𝜆𝜕𝑡𝜕𝑤(𝑟,𝑡)𝜕𝑡=𝜈1+𝜆𝑟𝜕𝜕𝜕𝑡2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑡),(2.3) where 𝜇 is the dynamic viscosity, 𝜈=𝜇/𝜌 is the kinematic viscosity, with 𝜌 being the constant density of the fluid, 𝜆 is the relaxation time, 𝜆𝑟 is the retardation time, and 𝜏(𝑟,𝑡)=𝑆𝑟𝜃(𝑟,𝑡) is the nontrivial shear stress.

The governing equations corresponding to an incompressible fractional Oldroyd-B fluid, performing the same motion, are obtained by replacing the inner time derivatives with respect to 𝑡 from (2.3), by the fractional differential operator given as [22–24] ğ·ğ›½ğ‘¡âŽ§âŽªâŽ¨âŽªâŽ©1𝑓(𝑡)=𝑑Γ(1−𝛽)𝑑𝑡𝑡0𝑓(𝜏)(𝑡−𝜏)𝛽𝑑𝑑𝜏,0≤𝛽<1,𝑑𝑡𝑓(𝑡),𝛽=1,(2.4) where Γ(⋅) is the Gamma function defined by Γ(𝑥)=∞0𝑡𝑥−1𝑒−𝑡𝑑𝑡,𝑥>0.(2.5) Consequently, the governing equations to be used here are 1+𝜆𝐷𝛾𝑡𝜏(𝑟,𝑡)=𝜇1+𝜆𝑟𝐷𝛽𝑡𝜕−1𝜕𝑟𝑟𝑤(𝑟,𝑡),(2.6)1+𝜆𝐷𝛾𝑡𝜕𝑤(𝑟,𝑡)𝜕𝑡=𝜈1+𝜆𝑟𝐷𝛽𝑡𝜕2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,𝑡).(2.7) We can easily observe that when 𝛾,𝛽→1, (2.6) and (2.7) reduce to (2.3), because 𝐷1𝑡𝑓=𝑑𝑓/𝑑𝑡. Furthermore, the new material constants 𝜆 and 𝜆𝑟 (although, for the simplicity, we keep the same notation) reduce to the previous ones for 𝛾,𝛽→1.

3. Starting Flow through an Infinite Circular Cylinder

Suppose that an incompressible fractional Oldroyd-B fluid is lying at rest in an infinite circular cylinder of radius 𝑅(>0). At time 𝑡=0+, the cylinder suddenly begins to rotate about its axis with an angular velocity Ω𝑡2. Owing to the shear the inner fluid is gradually moved, with its velocity being of the form (2.1). The governing equations are given by (2.6) and (2.7), while the appropriate initial and boundary conditions are 𝑤(𝑟,0)=𝜕𝑤(𝑟,0)[],𝜕𝑡=0,𝜏(𝑟,0)=0;𝑟∈0,𝑅(3.1)𝑤(𝑅,𝑡)=𝑅Ω𝑡2,𝑡≥0,(3.2) where Ω is a constant.

The partial differential equations (2.6) and (2.7), also containing fractional derivatives, can be solved in principle by several methods; the integral transforms technique representing a systematic, efficient, and powerful tool. In the following we will use the Laplace transform to eliminate the time variable and the finite Hankel transform for the spatial variable. However, in order to avoid the burdensome calculations of residues and contour integrals, we will apply the discrete inverse Laplace transform method.

3.1. Calculation of the Velocity Field

Applying the Laplace transform to (2.7) and (3.2), we get î€·ğ‘ž+ğœ†ğ‘žğ›¾+1𝑤(𝑟,ğ‘ž)=𝜈1+ğœ†ğ‘Ÿğ‘žğ›½î€¸î‚µğœ•2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,ğ‘ž),(3.3)𝑤(𝑅,ğ‘ž)=2ğ‘…Î©ğ‘ž3,(3.4) where 𝑤(𝑟,ğ‘ž) and 𝑤(𝑅,ğ‘ž) are the Laplace transforms of the functions 𝑤(𝑟,𝑡) and 𝑤(𝑅,𝑡), respectively. We will denote by [25] 𝑤𝐻𝑟𝑛=,ğ‘žğ‘…0𝑟𝑤(𝑟,ğ‘ž)𝐽1𝑟𝑟𝑛𝑑𝑟,(3.5) the finite Hankel transform of the function 𝑤(𝑟,ğ‘ž), and the inverse Hankel transform of 𝑤𝐻(𝑟𝑛,ğ‘ž) is given by [25] 2𝑤(𝑟,ğ‘ž)=𝑅2âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝐽22𝑅𝑟𝑛𝑤𝐻𝑟𝑛,ğ‘ž,(3.6) where 𝑟𝑛 is the positive roots of the equation 𝐽1(𝑅𝑟)=0 and 𝐽𝑝(⋅) is the Bessel function of the first kind of order 𝑝.

Multiplying now both sides of (3.3) by 𝑟𝐽1(𝑟𝑟𝑛), integrating with respect to 𝑟 from 0 to 𝑅, and taking into account (3.4), (3.5), and the result which we can easily prove 𝑅0𝑟𝜕2𝜕𝑟2+1𝑟𝜕−1𝜕𝑟𝑟2𝑤(𝑟,ğ‘ž)𝐽1𝑟𝑟𝑛𝑑𝑟=𝑅𝑟𝑛𝐽2𝑅𝑟𝑛𝑤(𝑅,ğ‘ž)−𝑟2𝑛𝑤𝐻𝑟𝑛,ğ‘ž,(3.7) we find that 𝑤𝐻𝑟𝑛=,ğ‘žğœˆ+ğœˆğœ†ğ‘Ÿğ‘žğ›½î€¸Î©ğ‘…2𝑟𝑛𝐽2𝑅𝑟𝑛2ğ‘ž3î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸.(3.8) It can be also written in the suitable form 𝑤𝐻𝑟𝑛=,ğ‘žğ‘¤1𝐻𝑟𝑛+,ğ‘žğ‘¤2𝐻𝑟𝑛,ğ‘ž,(3.9) where 𝑤1𝐻𝑟𝑛=,ğ‘ž2Ω𝑅2𝐽2𝑅𝑟𝑛𝑟𝑛1ğ‘ž3,𝑤2𝐻𝑟𝑛,ğ‘ž=−2Ω𝑅2𝐽2𝑅𝑟𝑛𝑟𝑛1+ğœ†ğ‘žğ›¾ğ‘ž2î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸.(3.10) Applying the inverse Hankel transform to (3.10) and using the known formula 𝑅0𝑟2𝐽1𝑟𝑟𝑛𝑅𝑑𝑟=2𝑟𝑛𝐽2𝑅𝑟𝑛,(3.11) we get𝑤1(𝑟,ğ‘ž)=2Î©ğ‘Ÿğ‘ž3,(3.12a)𝑤2(𝑟,ğ‘ž)=−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2𝑅𝑟𝑛1+ğœ†ğ‘žğ›¾ğ‘ž2î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸.(3.12b)Using the identity 1î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸=1ğœ†âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2ğ‘›ğœ†î‚¶ğ‘˜ğœ†ğ‘šğ‘Ÿğ‘žğ›½ğ‘šâˆ’ğ‘˜âˆ’1î‚€ğ‘žğ›¾+1𝜆𝑘+1,(3.13)(3.12b) can be written as 𝑤2(𝑟,ğ‘ž)=−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2ğ‘›ğœ†î‚¶ğ‘˜Ã—ğœ†ğ‘šğ‘Ÿî€·ğ‘žğ›½ğ‘šâˆ’ğ‘˜âˆ’3+ğœ†ğ‘žğ›¾+𝛽𝑚−𝑘−3(ğ‘žğ›¾+1/𝜆)𝑘+1.(3.14) After taking the inverse Hankel transform of (3.9) and with the help of (3.12a) and (3.14), it leads to 𝑤(𝑟,ğ‘ž)=2Î©ğ‘Ÿğ‘ž3−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2ğ‘›ğœ†î‚¶ğ‘˜Ã—ğœ†ğ‘šğ‘Ÿî€·ğ‘žğ›½ğ‘šâˆ’ğ‘˜âˆ’3+ğœ†ğ‘žğ›¾+𝛽𝑚−𝑘−3(ğ‘žğ›¾+1/𝜆)𝑘+1.(3.15) Now taking the inverse Laplace transform of (3.15), the velocity field 𝑤(𝑟,𝑡) is given by 𝑤(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2𝑛𝜆𝑘𝜆𝑚𝑟×𝐺𝛾,𝛽𝑚−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝐺𝛾,𝛾+𝛽𝑚−𝑘−3,𝑘+1−𝜆−1,,𝑡(3.16) where the generalized function 𝐺 is defined by [26, Equations (97) and (101)] ğºğ‘Ž,𝑏,𝑐(𝑑,𝑡)=ℒ−1î‚»ğ‘žğ‘(ğ‘žğ‘Žâˆ’ğ‘‘)𝑐=âˆžî“ğ‘˜=0𝑑𝑘Γ(𝑐+𝑘)𝑡Γ(𝑐)Γ(𝑘+1)(𝑐+𝑘)ğ‘Žâˆ’ğ‘âˆ’1Γ[]||||𝑑(𝑐+𝑘)ğ‘Žâˆ’ğ‘;Re(ğ‘Žğ‘âˆ’ğ‘)>0,ğ‘žğ‘Ž||||<1.(3.17)

3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.6), we find that 𝜏(𝑟,ğ‘ž)=𝜇1+ğœ†ğ‘Ÿğ‘žğ›½î€¸(1+ğœ†ğ‘žğ›¾)𝜕−1𝜕𝑟𝑟𝑤(𝑟,ğ‘ž),(3.18) where, in view of (3.12a) and (3.12b), we have 𝑤(𝑟,ğ‘ž)=𝑤1(𝑟,ğ‘ž)+𝑤2=(𝑟,ğ‘ž)2Î©ğ‘Ÿğ‘ž3−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2𝑅𝑟𝑛1+ğœ†ğ‘žğ›¾ğ‘ž2î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸.(3.19) Now differentiating the last expression for 𝑤(𝑟,ğ‘ž) partially with respect to 𝑟, we get 𝜕−1𝜕𝑟𝑟𝑤(𝑟,ğ‘ž)=−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2𝑅𝑟𝑛1+ğœ†ğ‘žğ›¾ğ‘ž2î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸,(3.20) and introducing (3.20) into (3.18), we have 𝜏(𝑟,ğ‘ž)=4ğœ‡Î©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2𝑅𝑟𝑛1+ğœ†ğ‘Ÿğ‘žğ›½î€¸ğ‘ž2î€·ğ‘ž+𝜈𝑟2𝑛+ğœ†ğ‘žğ›¾+1+𝜈𝑟2ğ‘›ğœ†ğ‘Ÿğ‘žğ›½î€¸.(3.21) In view of identity (3.13), (3.21) can be equivalently written as 𝜏(𝑟,ğ‘ž)=4ğœ‡Î©ğœ†âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2ğ‘›ğœ†î‚¶ğ‘˜Ã—ğœ†ğ‘šğ‘Ÿî‚¸ğ‘žğ›½ğ‘šâˆ’ğ‘˜âˆ’3(ğ‘žğ›¾+1/𝜆)𝑘+1+ğœ†ğ‘Ÿğ‘žğ›½ğ‘š+𝛽−𝑘−3(ğ‘žğ›¾+1/𝜆)𝑘+1.(3.22) Now taking the inverse Laplace transform of both sides of (3.22) and using (3.17), we find that 𝜏(𝑟,𝑡)=4ğœ‡Î©ğœ†âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2𝑛𝜆𝑘𝜆𝑚𝑟×𝐺𝛾,𝛽𝑚−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝑟𝐺𝛾,𝛽𝑚+𝛽−𝑘−3,𝑘+1−𝜆−1.,𝑡(3.23)

4. The Special Cases

4.1. Ordinary Oldroyd-B Fluid

Making 𝛾,𝛽→1 into (3.16) and (3.23), we obtain the similar solutions for velocity field 𝑤OO(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2𝑛𝜆𝑘×𝜆𝑚𝑟𝐺1,𝑚−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝐺1,𝑚−𝑘−2,𝑘+1−𝜆−1,𝑡(4.1) and shear stress 𝜏OO(𝑟,𝑡)=4ğœ‡Î©ğœ†âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!−𝑚!(𝑘−𝑚)!𝜈𝑟2𝑛𝜆𝑘×𝜆𝑚𝑟𝐺1,𝑚−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝑟𝐺1,𝑚−𝑘−2,𝑘+1−𝜆−1,,𝑡(4.2) for an ordinary Oldroyd-B fluid performing the same motion.

4.2. Maxwell Fluid with Fractional Derivatives

Making 𝜆𝑟→0,𝛽→1 into (3.16) and (3.23), we obtain velocity field 𝑤FM(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝜆𝑘×𝐺𝛾,−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝐺𝛾,𝛾−𝑘−3,𝑘+1−𝜆−1,𝑡(4.3) and the associated shear stress 𝜏FM(𝑟,𝑡)=4ğœ‡Î©ğœ†âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝜆𝑘𝐺𝛾,−𝑘−3,𝑘+1−𝜆−1,𝑡,(4.4) corresponding to a Maxwell fluid with fractional derivatives performing the same motion.

4.3. Ordinary Maxwell Fluid

Making 𝛾→1, 𝛽→1, and 𝜆𝑟→0 in (3.16) and (3.23), we get expressions for velocity field 𝑤OM(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœ†âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝜆𝑘×𝐺1,−𝑘−3,𝑘+1−𝜆−1,𝑡+𝜆𝐺1,−𝑘−2,𝑘+1−𝜆−1,𝑡(4.5) and the associated shear stress 𝜏OM(𝑟,𝑡)=4ğœ‡Î©ğœ†âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝜆𝑘𝐺1,−𝑘−3,𝑘+1−𝜆−1,𝑡,(4.6) corresponding to an ordinary Maxwell fluid.

4.4. Second-Grade Fluid with Fractional Derivatives

Making 𝜆→0, 𝛾→1 into (3.16) and (3.23), and using (A1) and (A2), we obtain velocity field 𝑤FS(𝑟,𝑡)=Ω𝑟𝑡2−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸Ã—âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1−𝜈𝜆𝑟𝑟2𝑛,𝑡(4.7) and associated shear stress 𝜏FS(𝑟,𝑡)=4ğœ‡Î©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘×𝐺1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1−𝜈𝜆𝑟𝑟2𝑛,𝑡+𝜆𝑟𝐺1−𝛽,−𝛽𝑘−2,𝑘+1−𝜈𝜆𝑟𝑟2𝑛,,𝑡(4.8) corresponding to a second-grade fluid with fractional derivatives performing the same motion. By taking 𝜈𝜆𝑟=𝛼 and 𝛼1=𝛼𝜌 (the material constants for second grade fluid) into (4.7) and (4.8), we will get the similar solutions for fractional second-grade fluid as we get directly from the governing equations of fractional second-garde fluid as under 𝑤FS(𝑟,𝑡)=Ω𝑟𝑡2−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸Ã—âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1−𝛼𝑟2𝑛,𝜏,𝑡FS(𝑟,𝑡)=4Î©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘×𝜇𝐺1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1−𝛼𝑟2𝑛,𝑡+𝛼1𝐺1−𝛽,−𝛽𝑘−2,𝑘+1−𝛼𝑟2𝑛.,𝑡(4.9)

4.5. Ordinary Second-Grade Fluid

Making 𝛽→1 into (4.9), we obtain velocity field 𝑤OS(𝑟,𝑡)=Ω𝑟𝑡2−4Î©âˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺0,−𝑘−3,𝑘+1−𝛼𝑟2𝑛,𝑡(4.10) and the associated shear stress 𝜏OS(𝑟,𝑡)=4Î©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝐽2î€·ğ‘…ğ‘Ÿğ‘›î€¸âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘×𝜇𝐺0,−𝑘−3,𝑘+1−𝛼𝑟2𝑛,𝑡+𝛼1𝐺0,−𝑘−2,𝑘+1−𝛼𝑟2𝑛,,𝑡(4.11) corresponding to an ordinary second-grade fluid. These solutions can also be simplified to give (see (A3)-(A4)) 𝑤OS(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœˆâˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟3𝑛𝐽2𝑅𝑟𝑛×𝑡+1+𝛼𝑟2𝑛𝜈𝑟2𝑛−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛,𝜏−1OS(𝑟,𝑡)=4ğœŒÎ©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝑟2𝑛𝐽2𝑅𝑟𝑛1𝑡+𝜈𝑟2𝑛−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛.−1(4.12)

4.6. Newtonian Fluid

Making 𝜆𝑟,𝜆→0 and 𝛾,𝛽→1 into (3.16) and (3.23) and using (A5) we get velocity field as 𝑤N(𝑟,𝑡)=Ω𝑟𝑡2−4Î©ğœˆâˆžî“ğ‘›=1𝐽1𝑟𝑟𝑛𝑟3𝑛𝐽2𝑅𝑟𝑛1𝑡+𝜈𝑟2𝑛exp−𝜈𝑟2𝑛𝑡−1(4.13) and the associated shear stress 𝜏N(𝑟,𝑡)=4ğœŒÎ©âˆžî“ğ‘›=1𝐽2𝑟𝑟𝑛𝑟2𝑛𝐽2𝑅𝑟𝑛1𝑡+𝜈𝑟2𝑛exp−𝜈𝑟2𝑛𝑡−1,(4.14) corresponding to a Newtonian fluid performing the same motion. Of course, the Newtonian solutions (4.13) and (4.14) can be also obtained by making 𝜆→0 into (4.5) and (4.6) and using (A5) or making 𝛼→0 into (4.12).

5. Conclusions

The underlying idea of this research paper is to calculate exact solutions for velocity field and the adequate shear stress corresponding to the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives through an infinite circular cylinder. The motion of the fluid is produced by the cylinder, which, at time 𝑡=0+, begins to rotate about its axis subject to 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 an ordinary Oldroyd-B, Maxwell with fractional derivatives, ordinary Maxwell fluids as well as those for fractional and ordinary second-grade fluids, can be obtained as limiting cases of general solutions for 𝛾,𝛽→1; 𝜆𝑟→0 and 𝛽→1; 𝜆𝑟→0, 𝛾 and 𝛽→1; 𝜆→0 and 𝛾→1; 𝜆→0, 𝛾 and 𝛽→1, respectively. The solutions for a Newtonian fluid performing the same motion are obtained as limiting cases of the solutions for an Oldroyd-B fluid with fractional derivatives when 𝜆𝑟→0, 𝜆→0, and 𝛾,𝛽→1.

6. Numerical Results

Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity 𝑤(𝑟,𝑡) as well as those of the shear stress 𝜏(𝑟,𝑡), are depicted against 𝑟 for different values of time 𝑡 and of the pertinent parameters. Figures 1(a) and 1(b) clearly show that both 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. The influence of the kinematic viscosity 𝜈 on the fluid motion is shown in Figures 2(a) and 2(b), 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 the relaxation time 𝜆 and of the retardation time 𝜆𝑟 on the velocity and shear stress are shown in Figures 3 and 4. The two parameters, as expected, have opposite effects on the fluid motion. More exactly, velocity and the shear stress are decreasing functions with respect to 𝜆 and increasing ones with regards to 𝜆𝑟. The influences of the fractional parameters 𝛾 and 𝛽 on the fluid motion are presented in Figures 5 and 6. Their effects are also opposite, but they are qualitatively the same as those of 𝜆𝑟 and 𝜆, respectively. More exactly velocity 𝑤(𝑟,𝑡) and the shear stress 𝜏(𝑟,𝑡) are increasing functions with regards to 𝛾 and decreasing with regards to 𝛽.

At the end, for comparison, the profiles of 𝑤(𝑟,𝑡) corresponding to the motion of the seven models (Newtonian, ordinary second grade, fractional second grade, ordinary Maxwell, fractional Maxwell, ordinary Oldroyd-B, fractional Oldroyd-B) are together depicted in Figure 7, for the same values of 𝑡 and of the common material parameters. The fractional Maxwell fluid, as it results from figure, is the slowest while the fractional second-grade fluid is the swiftest on the whole flow domain. The units of the material constants are SI units within all figures, and the roots 𝑟𝑛 have been approximated by (4𝑛+1)𝜋/(4𝑅).

Appendix

One hasâˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝑟2𝑛𝑘𝜆𝑚𝑟𝑡−𝛽𝑚+𝑘+2Γ=(−𝛽𝑚+𝑘+3)âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1−𝜈𝜆𝑟𝑟2𝑛,,𝑡(A1)âˆžî“ğ‘˜ğ‘˜=0𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝑟2𝑛𝑘𝜆𝑚𝑟𝑡−𝛽𝑚−𝛽+𝑘+2=Γ(−𝛽𝑚−𝛽+𝑘+3)âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺1−𝛽,−𝛽𝑘−2,𝑘+1−𝜈𝜆𝑟𝑟2𝑛,𝑡,(A2)âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺0,−𝑘−2,𝑘+1−𝛼𝑟2𝑛=1,𝑡𝜈𝑟2𝑛−1−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛,(A3)âˆžî“ğ‘˜=0−𝜈𝑟2𝑛𝑘𝐺0,−𝑘−3,𝑘+1−𝛼𝑟2𝑛=1,𝑡𝜈𝑟2𝑛𝑡+1+𝛼𝑟2𝑛𝜈𝑟2𝑛−exp𝜈𝑟2𝑛𝑡1+𝛼𝑟2𝑛−1,(A4)lim𝜆→01ğœ†ğ‘˜ğºğ‘Ž,𝑏,𝑘−𝜆−1=𝑡,𝑡−𝑏−1Γ(−𝑏),𝑏<0.(A5)

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