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

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; 𝜆_𝑟→0 and 𝛽→1; 𝜆_𝑟→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 𝜆_𝑟→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)𝐞_𝜽,𝐒=𝐒(𝑟,𝑡),(2.1) 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 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𝜕𝜕𝑡𝜏(𝑟,𝑡)=𝜇1+𝜆^𝑟𝜕1+𝜆𝜕𝜕𝑡𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑡),𝜕𝑡𝜕𝑤𝜕(𝑡𝑟,𝑡)=𝜈1+𝜆^𝑟𝜕𝜕𝑡𝜕𝜕𝑟²²+1𝑟𝜕𝜕𝑟−𝑟1²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, 𝜆_𝑟 is the retardation time, and 𝜏(𝑟,𝑡)=𝑆_𝑟𝜃(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] 𝐷^𝛽𝑡𝑓(𝑡)=⎧⎪⎨⎪⎩Γ(11−𝛽)𝑑𝑑𝑡^𝑡0𝑓(𝜏)(𝑡−𝜏)^𝛽𝑑𝑑𝜏,0≤𝛽<1,𝑑𝑡𝑓(𝑡),𝛽=1,(2.4) where Γ(⋅) is the Gamma function defined by Γ(𝑥)=^∞0𝑡^𝑥−1𝑒^−𝑡𝑑𝑡,𝑥>0.(2.5) Consequently, the governing equations to be used here are 1+𝜆𝐷1_𝛾𝑡+𝜆𝐷^𝛾𝑡𝜏(𝑟,𝑡)=𝜇1+𝜆_𝑟𝐷^𝛽𝑡𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑡),(2.6)𝜕𝑤𝜕(𝑡𝑟,𝑡)=𝜈1+𝜆^𝑟𝐷^𝛽𝑡𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑡).(2.7) We can easily observe that when 𝛾,𝛽→1, (2.6) and (2.7) reduce to (2.3), because 𝐷^1𝑡𝑓=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 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)𝜕𝑤𝑡(𝑅,=𝑡)0=,𝑅Ω𝑡𝜏₂(𝑟,0)=0;𝑟∈[0,𝑅],(3.1),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 𝑞+𝜆𝑞^𝛾+1𝑤(𝑟,𝑞)=𝜈1+𝜆_𝑟𝑞^𝛽𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑞),(3.3)𝑤(𝑅,𝑞)=2𝑅Ω𝑞³,(3.4) where 𝑤(𝑟,𝑞) and 𝑤(𝑅,𝑞) are the Laplace transforms of the functions 𝑤(𝑟,𝑡) and 𝑤(𝑅,𝑡), respectively. We will denote by [25] 𝑤_𝐻𝑟_𝑛,𝑞=^𝑅0𝑟𝑤(𝑟,𝑞)𝐽₁𝑟𝑟_𝑛𝑑𝑟,(3.5) the finite Hankel transform of the function 𝑤(𝑟,𝑞), and the inverse Hankel transform of 𝑤_𝐻(𝑟_𝑛,𝑞) is given by [25] 𝑤(𝑟,𝑞)=2𝑅^2∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽²²𝑅𝑟_𝑛𝑤_𝐻𝑟_𝑛,𝑞,(3.6) where 𝑟_𝑛 is the positive roots of the equation 𝐽₁(𝑅𝑟)=0 and 𝐽_𝑝(⋅) is the Bessel function of the first kind of order 𝑝.

Multiplying now both sides of (3.3) by 𝑟𝐽₁(𝑟𝑟_𝑛), 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𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑞)𝐽₁𝑟𝑟_𝑛𝑑𝑟=𝑅𝑟_𝑛𝐽₂𝑅𝑟_𝑛𝑤(𝑅,𝑞)−𝑟^2𝑛𝑤_𝐻𝑟_𝑛,𝑞,(3.7) we find that 𝑤_𝐻𝑟_𝑛,𝑞=𝜈+𝜈𝜆_𝑟𝑞^𝛽Ω𝑅²𝑟_𝑛𝐽₂𝑅𝑟_𝑛2𝑞³𝑞+𝜈𝑟^2𝑛+𝜆𝑞^𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽.(3.8) It can be also written in the suitable form 𝑤_𝐻𝑟_𝑛,𝑞=𝑤_1𝐻𝑟_𝑛,𝑞+𝑤_2𝐻𝑟_𝑛,𝑞,(3.9) where 𝑤_1𝐻𝑟_𝑛,𝑞=2Ω𝑅²𝐽₂𝑅𝑟_𝑛𝑟_𝑛1𝑞³,𝑤_2𝐻𝑟_𝑛,𝑞=−2Ω𝑅²𝐽₂𝑅𝑟_𝑛𝑟_𝑛1+𝜆𝑞^𝛾𝑞²𝑞+𝜈𝑟^2𝑛+𝜆𝑞^𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽.(3.10) Applying the inverse Hankel transform to (3.10) and using the known formula ^𝑅0𝑟²𝐽₁𝑟𝑟_𝑛𝑑𝑟=𝑅²𝑟_𝑛𝐽₂𝑅𝑟_𝑛,(3.11) we get𝑤₁(𝑟,𝑞)=2Ω𝑟𝑞³,(3.12a)𝑤₂(𝑟,𝑞)=−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛𝑞²𝑞+𝜈𝑟^2𝑛+1+𝜆𝑞𝜆𝑞^𝛾𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽.(3.12b)Using the identity 1𝑞+𝜈𝑟^2𝑛+𝜆𝑞^𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽=𝜆1^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘𝜆^𝑚𝑟𝑞^{𝛽𝑚−𝑘−1}𝑞^𝛾+𝜆1^𝑘+1,(3.13)(3.12b) can be written as 𝑤₂(𝑟,𝑞)=−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘×𝜆^𝑚𝑟𝑞^{𝛽𝑚−𝑘−3}+𝜆𝑞^{𝛾+𝛽𝑚−𝑘−3}(𝑞^𝛾+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,𝑞)=2Ω𝑟𝑞³−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘×𝜆^𝑚𝑟𝑞^{𝛽𝑚−𝑘−3}+𝜆𝑞^{𝛾+𝛽𝑚−𝑘−3}(𝑞^𝛾+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)=Ω𝑟𝑡²−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘𝜆^𝑚𝑟×𝐺_{𝛾,𝛽𝑚−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆𝐺_{𝛾,𝛾+𝛽𝑚−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡,(3.16) where the generalized function 𝐺 is defined by [26, Equations (97) and (101)] 𝐺_{𝑎,𝑏,𝑐}(𝑑,𝑡)=ℒ⁻¹=_∞(𝑞^𝑎𝑞−^𝑏𝑑)^𝑐_𝑘=0𝑑^𝑘Γ(𝑐+𝑘)Γ(𝑐)Γ(𝑘+1)Γ[(𝑡𝑐+𝑘)𝑎−^{(𝑐+𝑘)𝑎−𝑏−1}𝑏];Re(𝑎𝑐−𝑏)>0,||||𝑞𝑑^𝑎||||<1.(3.17)

3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.6), we find that 𝜏(𝑟,𝑞)=𝜇1+𝜆^𝑟𝑞^𝛽(1+𝜆𝑞^𝛾)𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑞),(3.18) where, in view of (3.12a) and (3.12b), we have 𝑤(𝑟,𝑞)==𝑤2Ω₁(𝑟𝑟,𝑞)+𝑤₂(𝑟,𝑞)𝑞³−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛𝑞²𝑞+𝜈𝑟^2𝑛1++𝜆𝑞𝜆𝑞^𝛾𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽.(3.19) Now differentiating the last expression for 𝑤(𝑟,𝑞) partially with respect to 𝑟, we get 𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑞)=−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛𝑞²𝑞+𝜈𝑟^2𝑛+1+𝜆𝑞𝜆𝑞^𝛾𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽,(3.20) and introducing (3.20) into (3.18), we have 𝜏(𝑟,𝑞)=4𝜇Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛1+𝜆^𝑟𝑞^𝛽𝑞²𝑞+𝜈𝑟^2𝑛+𝜆𝑞^𝛾+1+𝜈𝑟^2𝑛𝜆_𝑟𝑞^𝛽.(3.21) In view of identity (3.13), (3.21) can be equivalently written as 𝜏(𝑟,𝑞)=4𝜇Ω𝜆^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘×𝜆^𝑚𝑟(𝑞^𝛾𝑞+1/𝜆)^{𝛽𝑚−𝑘−3𝑘+1}+𝜆_𝑟𝑞^{𝛽𝑚+𝛽−𝑘−3}(𝑞^𝛾+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)=4𝜇Ω𝜆^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘𝜆^𝑚𝑟×𝐺_{𝛾,𝛽𝑚−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆_𝑟𝐺_{𝛾,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 𝑤_OO(𝑟,𝑡)=Ω𝑟𝑡²−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘×𝜆^𝑚𝑟𝐺_{1,𝑚−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆𝐺_{1,𝑚−𝑘−2,𝑘+1}−𝜆⁻¹,𝑡(4.1) and shear stress 𝜏_OO(𝑟,𝑡)=4𝜇Ω𝜆^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝜆𝑟^2𝑛^𝑘×𝜆^𝑚𝑟𝐺_{1,𝑚−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆_𝑟𝐺_{1,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 𝜆_𝑟→0,𝛽→1 into (3.16) and (3.23), we obtain velocity field 𝑤_FM(𝑟,𝑡)=Ω𝑟𝑡²−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝜆𝑟^2𝑛^𝑘×𝐺_{𝛾,−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆𝐺_{𝛾,𝛾−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡(4.3) and the associated shear stress 𝜏_FM(𝑟,𝑡)=4𝜇Ω𝜆^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝜆𝑟^2𝑛^𝑘𝐺_{𝛾,−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 𝜆_𝑟→0 in (3.16) and (3.23), we get expressions for velocity field 𝑤_OM(𝑟,𝑡)=Ω𝑟𝑡²−4Ω𝜆^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝜆𝑟^2𝑛^𝑘×𝐺_{1,−𝑘−3,𝑘+1}−𝜆⁻¹,𝑡+𝜆𝐺_{1,−𝑘−2,𝑘+1}−𝜆⁻¹,𝑡(4.5) and the associated shear stress 𝜏_OM(𝑟,𝑡)=4𝜇Ω𝜆^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝜆𝑟^2𝑛^𝑘𝐺_{1,−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 𝑤_FS(𝑟,𝑡)=Ω𝑟𝑡²−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛×_∞𝑟_𝑛𝐽₂𝑅𝑟_𝑛_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1}−𝜈𝜆_𝑟𝑟^2𝑛,𝑡(4.7) and associated shear stress 𝜏_FS(𝑟,𝑡)=4𝜇Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘×𝐺_{1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1}−𝜈𝜆_𝑟𝑟^2𝑛,𝑡+𝜆_𝑟𝐺_{1−𝛽,−𝛽𝑘−2,𝑘+1}−𝜈𝜆_𝑟𝑟^2𝑛,𝑡,(4.8) corresponding to a second-grade fluid with fractional derivatives performing the same motion. By taking 𝜈𝜆_𝑟=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 𝑤_FS(𝑟,𝑡)=Ω𝑟𝑡²−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛×^∞_𝑘=0𝜏−𝜈𝑟^2𝑛^𝑘𝐺_{1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1}−𝛼𝑟^2𝑛,𝑡_FS,(𝑟,𝑡)=4Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘×𝜇𝐺_{1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1}−𝛼𝑟^2𝑛,𝑡+𝛼₁𝐺_{1−𝛽,−𝛽𝑘−2,𝑘+1}−𝛼𝑟^2𝑛,𝑡.(4.9)

4.5. Ordinary Second-Grade Fluid

Making 𝛽→1 into (4.9), we obtain velocity field 𝑤_OS(𝑟,𝑡)=Ω𝑟𝑡²−4Ω^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{0,−𝑘−3,𝑘+1}−𝛼𝑟^2𝑛,𝑡(4.10) and the associated shear stress 𝜏_OS(𝑟,𝑡)=4Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝐽₂𝑅𝑟_𝑛^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘×𝜇𝐺_{0,−𝑘−3,𝑘+1}−𝛼𝑟^2𝑛,𝑡+𝛼₁𝐺_{0,−𝑘−2,𝑘+1}−𝛼𝑟^2𝑛,𝑡,(4.11) corresponding to an ordinary second-grade fluid. These solutions can also be simplified to give (see (A3)-(A4)) 𝑤_OS(𝑟,𝑡)=Ω𝑟𝑡²−4Ω𝜈^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟^3𝑛𝐽₂𝑅𝑟_𝑛×𝑡+1+𝜈𝑟𝛼^2𝑛𝑟^2𝑛exp−𝜈𝑟^2𝑛𝑡𝜏1+𝛼𝑟^2𝑛−1_OS,(𝑟,𝑡)=4𝜌Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝑟^2𝑛𝐽₂𝑅𝑟_𝑛𝑡+𝜈1𝑟^2𝑛exp−𝜈𝑟^2𝑛𝑡1+𝛼𝑟^2𝑛−1.(4.12)

4.6. Newtonian Fluid

Making 𝜆_𝑟,𝜆→0 and 𝛾,𝛽→1 into (3.16) and (3.23) and using (A5) we get velocity field as 𝑤_N(𝑟,𝑡)=Ω𝑟𝑡²−4Ω𝜈^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝑟^3𝑛𝐽₂𝑅𝑟_𝑛𝑡+𝜈1𝑟^2𝑛exp−𝜈𝑟^2𝑛𝑡−1(4.13) and the associated shear stress 𝜏_N(𝑟,𝑡)=4𝜌Ω^∞_𝑛=1𝐽₂𝑟𝑟_𝑛𝑟^2𝑛𝐽₂𝑅𝑟_𝑛𝑡+𝜈1𝑟^2𝑛exp−𝜈𝑟^2𝑛𝑡−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; 𝜆_𝑟→0 and 𝛽→1; 𝜆_𝑟→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 𝜆_𝑟→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.

(a)

(b)

Figure 1 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑅=0.3, Ω=2, 𝜆=3, 𝜆𝑟=1, 𝜈=0.0031, 𝜇=2.96, 𝛾=0.6, 𝛽=0.6, and different values of 𝑡.

(a)

(b)

Figure 2 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑡=5 s, 𝑅=0.3, Ω=2, 𝜆=3, 𝜆𝑟=1, 𝜇=2.96, 𝛾=0.7, 𝛽=0.3, and different values of 𝜈.

The influences of the relaxation time 𝜆 and of the retardation time 𝜆_𝑟 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 𝜆_𝑟. 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 𝜆_𝑟 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.

(a)

(b)

Figure 3 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑡=5 s, 𝑅=0.3, Ω=2, 𝜆𝑟=1, 𝜈=0.003, 𝜇=2.96, 𝛾=0.8, 𝛽=0.2, and different values of 𝜆.

(a)

(b)

Figure 4 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑡=5 s, 𝑅=0.3, Ω=2, 𝜆=4, 𝜈=0.003, 𝜇=2.96, 𝛾=0.8, 𝛽=0.2, and different values of 𝜆𝑟.

(a)

(b)

Figure 5 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑡=7 s, 𝑅=0.3, Ω=1, 𝜆=5, 𝜆𝑟=2, 𝜈=0.003, 𝜇=2.96, 𝛽=0.4, and different values of 𝛾.

(a)

(b)

Figure 6 

Profiles of the velocity 𝑤(𝑟,𝑡) and shear stress 𝜏(𝑟,𝑡) given by (3.16) and (3.23) for 𝑡=7 s, 𝑅=0.3, Ω=2, 𝜆=3, 𝜆𝑟=1, 𝜈=0.0031, 𝜇=2.96, 𝛾=0.7, and different values of 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 𝑟_𝑛 have been approximated by (4𝑛+1)𝜋/(4𝑅).

Figure 7 

Profiles of the velocity 𝑤(𝑟,𝑡) corresponding to the Newtonian, ordinary second grade, fractional second-grade, Maxwell, fractional Maxwell, Oldroyd-B and fractional Oldroyd-B fluids, for 𝑡=5 s, 𝑅=0.3, Ω=1, 𝜆=4, 𝜆𝑟=2, 𝜈=0.004, 𝜇=2, 𝛾=0.7, and 𝛽=0.5.

Appendix

One has^∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝑟^2𝑛^𝑘𝜆^𝑚𝑟𝑡^{−𝛽𝑚+𝑘+2}Γ(−𝛽𝑚+𝑘+3)=^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{1−𝛽,−𝛽𝑘−𝛽−2,𝑘+1}−𝜈𝜆_𝑟𝑟^2𝑛,𝑡,(A1)_∞_𝑘𝑘=0_𝑚=0𝑘!𝑚!(𝑘−𝑚)!−𝜈𝑟^2𝑛^𝑘𝜆^𝑚𝑟𝑡^{−𝛽𝑚−𝛽+𝑘+2}Γ(−𝛽𝑚−𝛽+𝑘+3)=^∞_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{1−𝛽,−𝛽𝑘−2,𝑘+1}−𝜈𝜆_𝑟𝑟^2𝑛,𝑡,(A2)_∞_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{0,−𝑘−2,𝑘+1}−𝛼𝑟^2𝑛,𝑡=1𝜈𝑟^2𝑛1−exp−𝜈𝑟^2𝑛𝑡1+𝛼𝑟^2𝑛,(A3)_∞_𝑘=0−𝜈𝑟^2𝑛^𝑘𝐺_{0,−𝑘−3,𝑘+1}−𝛼𝑟^2𝑛,𝑡=1𝜈𝑟^2𝑛𝑡+1+𝜈𝑟𝛼^2𝑛𝑟^2𝑛exp−𝜈𝑟^2𝑛𝑡lim_𝜆→01+𝛼𝑟^2𝑛−1,(A4)1𝜆^𝑘𝐺_{𝑎,𝑏,𝑘}−𝜆⁻¹,𝑡=𝑡^−𝑏−1Γ(−𝑏),𝑏<0.(A5)