Oscillating Flows of Fractionalized Second Grade Fluid

Abstract

New exact solutions for the motion of a fractionalized (this word is suitable when fractional derivative is used in constitutive or governing equations) second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of Laplace and finite Hankel transforms. These solutions are presented in series form in term of generalized 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) functions and satisfy all imposed initial and boundary conditions. In special cases, solutions for ordinary second grade and Newtonian fluids are obtained. Furthermore, other equivalent forms of solutions for ordinary second grade and Newtonian fluids are presented and written as sum of steady-state and transient solutions. The solutions for Newtonian fluid coincide with the well-known classical solutions. Finally, by means of graphical illustrations, the influence of pertinent parameters on fluid motion as well as comparison among different models is discussed.

1. Introduction

In recent years, the non-Newtonian fluids have received considerable attention by scientist and engineers. Such interest is inspired by practical applications of non-Newtonian fluids in industry and engineering applications. The shear stress and shear rate in non-Newtonian fluids are connected by a relation in a nonlinear manner. Because of diverse fluids characteristics in nature, all the non-Newtonian fluids cannot be described by a single constitutive relation [1–6]. Thus, among the several existing non-Newtonian fluid models, there is one which is most famous model called second grade fluid [7]. Although the constitutive equation of second grade fluid is simpler than that for the rate type fluids (those fluids which encounter viscoelastic and memory effects), it has been shown by Walters [8] that, for many types of problems in which the flow is slow enough in the viscoelastic sense, the results given using Oldroyd fluid will be substantially similar to those obtained for second grade fluid. Therefore, if we discuss the result in this manner, it is reasonable to use the second grade fluid to carry out the calculations as compared to other non-Newtonian fluids. This fact seems to be true, not only for exact analytic solutions but even for numerical solutions. The second grade fluid is the simplest subclass of non-Newtonian fluids for which one can reasonably hope to obtain exact analytic solutions. Some recent attempts regarding exact analytic solutions for the flow of a second grade fluid are present in [9–16].

Linear viscoelasticity is certainly the field of most extensive applications of fractional calculus, in view of its ability to model hereditary phenomena with long memory. During the twentieth century, a number of authors have (implicitly or explicitly) used the fractional calculus as an empirical method of describing the properties of viscoelastic materials [17]. A motivation for using fractional order operators in viscoelasticity is that a whole spectrum of viscoelastic mechanisms can be included in a single internal variable [18]. The stress relaxation spectrum for the fractional order model is continuous with the relaxation constant as the most probable relaxation time, while the order of the operator plays the role of a distribution parameter. Note that the spectrum is discrete for the classical model that is based on integer order derivatives. By a suitable choice of material parameters for the classical viscoelastic model, it is observed both numerically and analytically that the classical model with a large number of internal variables (each representing a specific viscoelastic mechanism) converges to the fractional model with a single internal variable [18, 19]. In other cases, it has been shown that the governing equations employing fractional derivatives are also linked to molecular theories [20]. The use of fractional derivatives within the context of viscoelasticity was firstly proposed by Germant [21]. Later, Bagley and Torvik [22] demonstrated that the theory of viscoelasticity of coiling polymers predicts constitutive relations with fractional derivatives, and Makris et al. [23] achieved a very good fit of the experimental data when the fractional derivative Maxwell model has been used instead of the Maxwell model for the silicon gel fluid. Some important recent attempts of fractional derivative approach to non-Newtonian fluids to obtain exact analytic solutions are listed here [24–30].

The oscillating flow of the viscoelastic fluid in cylindrical pipes has been applied in many fields, such as industries of petroleum, chemistry, and bioengineering. In the field of bioengineering, this type of investigation is of particular interest since blood in veins is forced by a periodic pressure gradient. In the petroleum and chemical industries, there are also many problems which involve the dynamic response of the fluid to the frequency of the periodic pressure gradient. An excellent collection of papers on oscillating flow can be found in the paper by Yin and Zhu [31]. We also include some important studies of non-Newtonian fluids, where oscillating boundary value problems are used in cylindrical region [32–40]. Consequently, for completeness and motivated by the above remarks, we solve our problem for fractionalized second grade fluid. The aim of this paper is to find some new and closed-form exact solutions for the oscillating flows of fractionalized second grade fluid. More precisely, our objective is to find the velocity field and the shear stresses corresponding to the motion of a fractionalized second grade fluid through a cylinder due to longitudinal and torsional oscillations of an infinite circular cylinder. The general solutions are obtained using the discrete Laplace and finite Hankel transforms. They are presented in series form in term of the 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) functions in simpler forms as comparison to known results from literature. The solutions for similar motion of ordinary second grade and Newtonian fluids are obtained as spacial cases from general solutions. Equivalent forms of the solutions for ordinary second grade and Newtonian fluids are also constructed and presented as a sum between steady-state and transient solutions. The equivalent forms of general solutions for Newtonian fluid coincide with the well known classical solutions from the literature. Finally, the influence of material and fractional parameters on the motion of fractionalized second grade fluid is underlined by graphical illustrations. The difference among fractionalized, ordinary second grade and Newtonian fluid models is also spotlighted.

2. Governing Equations for Fractionalized Second Grade Fluid

The Cauchy stress 𝑇 in an incompressible homogeneous fluid of second grade is related to the fluid motion in the following manner:𝑇=−𝑝𝐼+𝑆,𝑆=𝜇𝐴₁+𝛼₁𝐴₂+𝛼₂𝐴²¹,(2.1) where −𝑝𝐼 is the indeterminate part of the stress due to the constraint of incompressibility, S is the extra-stress tensor, 𝜇 is the dynamic viscosity, 𝛼₁ and 𝛼₂ are the normal stress moduli, and 𝐴₁ and 𝐴₂ are the kinematic tensors defined through𝐴₁=(∇𝑉)+(∇𝑉)^𝑇,𝐴₂=𝑑𝐴¹𝑑𝑡+𝐴¹(∇𝑉)+(∇𝑉)^𝑇𝐴₁.(2.2) In the above equations, 𝑉 is the velocity field, ∇ is the gradient operator, and 𝑑/𝑑𝑡 denotes the material time derivative. Since the fluid is incompressible, it can undergo only isochoric motion, and the equations of motion are∇⋅𝑉=0,∇⋅𝑇=𝜌𝑑𝑉𝑑𝑡+𝜌𝑏,(2.3) where 𝜌 is the constant density of the fluid and 𝑏 is the body force. If the model (2.1) is required to be compatible with thermodynamics in the sense that all motions satisfy the Clausius-Duhem inequality and the assumption that the specific Helmholtz free energy is a minimum in equilibrium, then the material moduli must meet the following restrictions [41]:𝜇≥0,𝛼₁≥0,𝛼₁+𝛼₂=0.(2.4) The sign of the material moduli 𝛼₁ and 𝛼₂ has been the subject of much controversy. A comprehensive discussion on the restrictions given in (2.4) as well as a critical review on the fluids of differential type can be found in the extensive work by Dunn and Rajagopal [42].

For the problem under consideration, we shall assume a velocity field and an extra-stress of the form𝑉=𝑉(𝑟,𝑡)=𝑤(𝑟,𝑡)𝑒_𝜃+𝑣(𝑟,𝑡)𝑒_𝑧,(2.5) where 𝑒_𝜃 and 𝑒_𝑧 are unit vectors in the 𝜃 and 𝑧-directions of the cylindrical coordinate system 𝑟,𝜃 and 𝑧. For such flows the constraint, of incompressibility is automatically satisfied. If the fluid is at rest up to the moment 𝑡=0, then𝑉(𝑟,0)=0,(2.6) and (2.1) implies 𝑆_𝑟𝑟=0 and the meaningful equations𝜏₁(𝑟,𝑡)=𝜇+𝛼₁𝜕𝜕𝑡𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑡),𝜏²(𝑟,𝑡)=𝜇+𝛼₁𝜕𝜕𝑡𝜕𝑣𝜕(𝑟𝑟,𝑡),(2.7) where 𝜏₁=𝑆_𝑟𝜃 and 𝜏₂=𝑆_𝑟𝑧 are the shear stresses that are different of zero.

The equation of motion (2.3) ₂, in the absence of a pressure gradient in the axial direction and neglecting body forces, leads to the relevant equations (𝜕_𝜃𝑝=0 due to the rotational symmetry)𝜌𝜕𝑤(𝑟,𝑡)𝜕𝑡=𝜕𝜕𝑟+2𝑟𝜏¹(𝑟,𝑡),𝜌𝜕𝑣(𝑟,𝑡)𝜕𝑡=𝜕𝜕𝑟+1𝑟𝜏²(𝑟,𝑡).(2.8) Eliminating 𝜏₁ and 𝜏₂ between (2.7) and (2.8), we attain the governing equations𝜕𝑤(𝑟,𝑡)𝜕𝑡=𝜈+𝛼𝜕𝜕𝑡𝜕𝜕𝑟²²+1𝜕𝑣(𝑟,𝑡)𝑟𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑡),𝑟∈(0,𝑅),𝑡>0,𝜕𝑡=𝜈+𝛼𝜕𝜕𝑡𝜕𝜕𝑟²²+1𝑟𝜕𝜕𝑟𝑣(𝑟,𝑡),𝑟∈(0,𝑅),𝑡>0,(2.9) where 𝜈=𝜇/𝜌 is the kinematic viscosity and 𝛼=𝛼₁/𝜌 is the material parameter of the fluid. The governing equations corresponding to an incompressible fractionalized second grade fluid, performing the same motion, are𝜕𝑤(𝑟,𝑡)𝜕𝑡=𝜈+𝛼𝐷^𝛽𝑡𝜕𝜕𝑟²²+1𝑟𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑡),𝑟∈(0,𝑅),𝑡>0,(2.10)𝜕𝑣(𝑟,𝑡)𝜕𝑡=𝜈+𝛼𝐷^𝛽𝑡𝜕𝜕𝑟²²+1𝑟𝜕𝜕𝑟𝑣(𝑟,𝑡),𝑟∈(0,𝑅),𝑡>0,(2.11)𝜏₁(𝑟,𝑡)=𝜇+𝛼₁𝐷^𝛽𝑡𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑡),(2.12)𝜏₂(𝑟,𝑡)=𝜇+𝛼₁𝐷^𝛽𝑡𝜕𝑣(𝑟,𝑡)𝜕𝑟,(2.13) where 0<𝛽<1 is the fractional parameter. Of course, the new material constant 𝛼₁, although for simplicity we keep the same notation, tends to the original 𝛼₁ as 𝛽→1. The fractional differential operator so-called Caputo fractional operator 𝐷^𝛽𝑡 defined by [43, 44]𝐷^𝛽𝑡𝑓(𝑡)=⎧⎪⎪⎨⎪⎪⎩1Γ(1−𝛽)^𝑡0𝑓^(𝜏)(𝑡−𝜏)^𝛽𝑑𝑓(𝑡)𝑑𝜏,0≤𝛽<1,𝑑𝑡,𝛽=1,(2.14) and Γ(⋅) is the Gamma function.

3. Oscillating Flows of Fractionalized Second Grade Fluids

Let us consider an incompressible fractionalized second grade fluid at rest, in an infinitely long cylinder of radius 𝑅  as shown in Figure 1. At time 𝑡=0⁺, the cylinder starts to oscillate according to𝑉(𝑅,𝑡)=𝑊₁𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡𝑒_𝜃+𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡𝑒_𝑧,(3.1) where 𝜔₁ and 𝜔₂ are the frequencies of the velocity of the cylinder and 𝑉₁,𝑉₂,𝑊₁, and 𝑊₂ are constant amplitudes. Owing to the shear, the fluid in cylinder is gradually moved, its velocity being of the form (2.5). The governing equations are given by (2.10)–(2.13) while the associated initial and boundary conditions are𝑤(𝑟,0)=𝑣(𝑟,0)=0,𝑟∈(0,𝑅),(3.2) respectively, and𝑤(𝑅,𝑡)=𝑊₁𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡,𝑣(𝑅,𝑡)=𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡,𝑡≥0,(3.3) where 𝐻(𝑡) is the Heaviside function [45]. In the following, the system of fractional partial differential equations (2.10)–(2.13), with appropriate initial and boundary conditions, will be solved by means of Laplace and finite Hankel transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method will be used [24–30].

Figure 1 

Geometry of the problem for oscillating flows of fractionalized second grade fluid through a cylinder.

3.1. Calculation of the Velocity Field

Applying the Laplace transform to (2.10) and (2.11) and having in mind the initial and boundary conditions (3.2) and (3.4), we find that𝑞𝑤(𝑟,𝑞)=𝜈+𝛼𝑞^𝛽𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟−𝑟1²𝑤(𝑟,𝑞),𝑟∈(0,𝑅),(3.4)𝑞𝑣(𝑟,𝑞)=𝜈+𝛼𝑞^𝛽𝜕²𝜕𝑟²+1𝑟𝜕𝜕𝑟𝑣(𝑟,𝑞),𝑟∈(0,𝑅),(3.5) where the image functions 𝑤(𝑟,𝑞) and 𝑣(𝑟,𝑞) of 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) have to satisfy the conditions𝑤(𝑅,𝑞)=𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹,𝑣(𝑅,𝑞)=𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²².(3.6)

Multiplying now both sides of (3.4) and (3.5) by 𝑟𝐽₁(𝑟𝑟_𝑚) and 𝑟𝐽₀(𝑟𝑟_𝑛), respectively, integrating them with respect to 𝑟 from 0 to 𝑅 and taking into account the conditions (3.6) and the known relations [46, 47]^𝑅0𝑟𝜕²𝑤𝜕𝑟²+1𝑟𝜕𝑤𝜕𝑟−𝑤𝑟²𝐽¹𝑟𝑟_𝑚𝑑𝑟=𝑅𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑤(𝑅,𝑡)−𝑟^2𝑚𝑤_𝐻𝑟_𝑚,𝑡_𝑅0,𝑟𝜕²𝑣𝜕𝑟²+1𝑟𝜕𝑣𝜕𝑟𝐽⁰𝑟𝑟_𝑛𝑑𝑟=𝑅𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝑣(𝑅,𝑡)−𝑟^2𝑛𝑣_𝐻𝑟_𝑛,𝑡,(3.7) we find that𝑤_𝐻𝑟_𝑚,𝑞=𝑅𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹𝜈+𝛼𝑞^𝛽𝑞+𝛼𝑟^2𝑚𝑞^𝛽+𝜈𝑟^2𝑚,𝑣_𝐻𝑟_𝑛,𝑞=𝑅𝑟_𝑛𝐽₁𝑅𝑟_𝑚𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²²𝜈+𝛼𝑞^𝛽𝑞+𝛼𝑟^2𝑛𝑞^𝛽+𝜈𝑟^2𝑛,(3.8) where [46, 47]𝑤_𝐻𝑟_𝑚,𝑞=^𝑅0𝑟𝑤(𝑟,𝑞)𝐽₁𝑟𝑟_𝑚𝑑𝑟,𝑣_𝐻𝑟_𝑛,𝑞=^𝑅0𝑟𝑣(𝑟,𝑞)𝐽₀𝑟𝑟_𝑛𝑑𝑟,𝑚,𝑛=1,2,3,…(3.9) are the Hankel transforms of 𝑤(𝑟,𝑞) and 𝑣(𝑟,𝑞), while 𝑟_𝑚 and 𝑟_𝑛 are the positive roots of the transcendental equations 𝐽₁(𝑅𝑟)=0 and 𝐽₀(𝑅𝑟)=0, respectively. In order to determine 𝑤(𝑟,𝑞) and 𝑣(𝑟,𝑞), we must apply the inverse Hankel transforms. However, for a more suitable presentation of final results, we firstly rewrite in (3.8), in the equivalent forms:𝑤_𝐻𝑟_𝑚,𝑞=𝑅𝐽²𝑅𝑟_𝑚𝑟_𝑚𝑊₁𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹−𝑅𝐽²𝑅𝑟_𝑚𝑟_𝑛𝑊₁𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹𝑞𝑞+𝛼𝑟^2𝑚𝑞^𝛽+𝜈𝑟^2𝑚,𝑣_𝐻𝑟_𝑛,𝑞=𝑅𝐽¹𝑅𝑟_𝑛𝑟_𝑛𝑉₁𝑞+𝑉₂𝜔₁𝑞²+𝜔²¹−𝑅𝐽¹𝑅𝑟_𝑛𝑟_𝑛𝑉₁𝑞+𝑉₂𝜔₂𝑞²+𝜔²¹𝑞𝑞+𝛼𝑟^2𝑛𝑞^𝛽+𝜈𝑟^2𝑛,(3.10) and apply the inverse Hankel transform formulae [46, 47]𝑤(𝑟,𝑞)=2𝑅^2∞_𝑚=1𝑤_𝐻𝑟_𝑚,𝑞𝐽¹𝑟𝑟_𝑚𝐽²²𝑅𝑟_𝑚,𝑣(𝑟,𝑞)=2𝑅^2∞_𝑛=1𝑣_𝐻𝑟_𝑛,𝑞𝐽⁰𝑟𝑟_𝑛𝐽²¹𝑅𝑟_𝑛.(3.11) Taking into account the following results [47]:^𝑅0𝑟²𝐽₁𝑟𝑟_𝑚𝑑𝑟=𝑅²𝑟_𝑚𝐽₂𝑅𝑟_𝑚,^𝑅0𝑟𝐽₀𝑟𝑟_𝑛𝑑𝑟=𝑅𝑟_𝑛𝐽₁𝑅𝑟_𝑛,(3.12) we find that𝑤(𝑟,𝑞)=𝑅𝑟𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹𝑞𝑞+𝛼𝑟^2𝑚𝑞^𝛽+𝜈𝑟^2𝑚,𝑣(𝑟,𝑞)=𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²²−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₂𝑅𝑟_𝑛𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²²𝑞𝑞+𝛼𝑟^2𝑛𝑞^𝛽+𝜈𝑟^2𝑛.(3.13) Finally, in order to obtain 𝑤(𝑟,𝑡)=ℒ⁻¹{𝑤(𝑟,𝑞)} and 𝑣(𝑟,𝑡)=ℒ⁻¹{𝑣(𝑟,𝑞)} and to avoid the lengthy calculations of residues and contour integrals, we will apply the discrete inverse Laplace transform method [24–30]. For this, we firstly write (3.13) in series form𝑤(𝑟,𝑞)=𝑅𝑟𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘×𝑊_1∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑖!𝑞1^{(𝑖−𝑘)𝛽+𝑘+2𝑗+1}+𝑊₂𝜔_1∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑖!𝑞1^{(𝑖−𝑘)𝛽+𝑘+2𝑗+2},𝑣(𝑟,𝑞)=𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²²−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘×𝑉_1∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑖!𝑞1^{(𝑖−𝑘)𝛽+𝑘+2𝑗+1}+𝑉₂𝜔_2∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑖!𝑞1^{(𝑖−𝑘)𝛽+𝑘+2𝑗+2},(3.14) where we used the fact that⎛⎜⎜⎝𝑘𝑖⎞⎟⎟⎠=(−1)^𝑖(−𝑘)_𝑖𝑖!,(3.15) and (−𝑘)_𝑖 is the Pochhammer symbol(𝑘)_𝑖=1𝑘,(𝑘+1)⋯(𝑘+𝑖−1),𝑖𝑖=∈N0,.(3.16) In particular (0)₀=1, (𝑘)₀=1 and (0)_𝑖=0, for 𝑖∈N. Applying the discrete inverse Laplace transform, we get𝑤(𝑟,𝑞)=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘×𝑊₁𝐻(𝑡)^∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑡^{(𝑖−𝑘)𝛽+𝑘+2𝑗}𝑖!Γ((𝑖−𝑘)𝛽+𝑘+2𝑗+1)+𝑊²𝜔_1∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑡^{(𝑖−𝑘)𝛽+𝑘+2𝑗+1}𝑣(𝑟,𝑞)=𝑉₁𝑖!Γ((𝑖−𝑘)𝛽+𝑘+2𝑗+2),𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘×𝑉₁𝐻(𝑡)^∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑡^{𝑘((𝑖−𝑘)𝛽+𝑘+2𝑗)}𝑖!Γ((𝑖−𝑘)𝛽+𝑘+2𝑗+1)+𝑉²𝜔_2∞_𝑖=0(−𝑘)_𝑖(−𝜈/𝛼)^𝑖𝑡^{(𝑖−𝑘)𝛽+𝑘+2𝑗+1}𝑖!Γ((𝑖−𝑘)𝛽+𝑘+2𝑗+2).(3.17) In terms of the generalized 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) functions [48], we rewrite the above equations in simple forms:𝑤(𝑟,𝑞)=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘×𝑊₁𝐻(𝑡)𝐺_{𝛽,−𝑘−2𝑗−1,−𝑘}−𝜈𝛼,𝑡+𝑊²𝜔₁𝐺_{𝛽,−𝑘−2𝑗−2,−𝑘}−𝜈𝑣(𝑟,𝑞)=𝑉₁𝛼,𝑡,(3.18)𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘×𝑉₁𝐻(𝑡)𝐺_{𝛽,−𝑘−2𝑗−1,−𝑘}−𝜈𝛼,𝑡+𝑉²𝜔₂𝐺_{𝛽,−𝑘−2𝑗−2,−𝑘}−𝜈𝛼,𝑡,(3.19) where the generalized 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) function is defined by [48]𝐺_{𝑎,𝑏,𝑐}(𝑑,𝑡)=^∞_𝑗=0(𝑐)_𝑗𝑑^𝑗𝑗!Γ[(𝑡𝑐+𝑗)𝑎−^{(𝑐+𝑗)𝑎−𝑏−1}𝑏],Re(𝑎𝑐−𝑏)>0,Re(𝑞)>0,||||𝑞𝑑^𝑎||||<1.(3.20)

3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.12) and (2.13), we find that𝜏₁(𝑟,𝑞)=𝜇+𝛼₁𝑞^𝛽𝜕𝜕𝑟−1𝑟𝑤(𝑟,𝑞),𝜏₂(𝑟,𝑞)=𝜇+𝛼₁𝑞^𝛽𝜕𝑣(𝑟,𝑞)𝜕𝑟,(3.21) where𝜕𝑤(𝑟,𝑞)𝜕𝑟−1𝑟𝑤(𝑟,𝑞)=2𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝑞+𝑊₂𝜔₁𝑞²+𝜔²¹𝑞𝑞+𝛼𝑟^2𝑚𝑞^𝛽+𝜈𝑟^2𝑚𝜕,𝑣(𝑟,𝑞)𝜕𝑟=𝑅2^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝐽₁𝑅𝑟_𝑚𝑉¹𝑞+𝑉₂𝜔₂𝑞²+𝜔²²𝑞𝑞+𝛼𝑟^2𝑛𝑞^𝛽+𝜈𝑟^2𝑛,(3.22) have been obtained form (3.13), using the identities𝑟𝑟_𝑚𝐽^′₁𝑟𝑟_𝑚−𝐽₁𝑟𝑟_𝑚=−𝑟𝑟_𝑚𝐽₂𝑟𝑟_𝑚,𝐽^0𝑟𝑟_𝑛=−𝑟_𝑛𝐽₁𝑟𝑟_𝑛.(3.23) Substituting (3.22) into (3.21), respectively, and applying again the discrete inverse Laplace transform method, we find that the shear stresses 𝜏₁(𝑟,𝑡) and 𝜏₂(𝑟,𝑡) have the following forms:𝜏₁(𝑟,𝑡)=2𝛼𝜌𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘+𝑊𝑊¹₂𝐻(𝑡)𝐺_{𝛽,−𝑘−2𝑗−1,−𝑘−1}−𝛼𝜈,𝑡𝜔₁𝐺_{𝛽,−𝑘−2𝑗−2,−𝑘−1}−𝜈𝜏𝛼,𝑡₂,(𝑟,𝑡)=2𝛼𝜌𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘𝑉+¹𝑉𝐻₂(𝑡)𝐺_{𝛽,−𝑘−2𝑗−1,−𝑘−1}−𝛼𝜈,𝑡𝜔₂𝐺_{𝛽,−𝑘−2𝑗−2,−𝑘−1}−𝜈𝛼,𝑡.(3.24)

4. Limiting Cases

4.1. Ordinary Second Grade Fluid (𝛽→1)

Making 𝛽→1 into (3.18), (3.19), (3.24), we obtain the solutions𝑤_OSG(𝑟,𝑞)=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘×𝑊₁𝐻(𝑡)𝐺_{1,−𝑘−2𝑗−1,−𝑘}−𝜈𝛼,𝑡+𝑊²𝜔₁𝐺_{1,−𝑘−2𝑗−2,−𝑘}−𝜈𝑣𝛼,𝑡_OSG,(𝑟,𝑞)=𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘×𝑉₁𝐻(𝑡)𝐺_{1,−𝑘−2𝑗−1,−𝑘}−𝜈𝛼,𝑡+𝑉²𝜔₂𝐺_{1,−𝑘−2𝑗−2,−𝑘}−𝜈𝜏𝛼,𝑡_1OSG,(𝑟,𝑡)=2𝛼𝜌𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗∞_𝑘=0−𝛼𝑟^2𝑚^𝑘×𝑊₁𝐻(𝑡)𝐺_{1,−𝑘−2𝑗−1,−𝑘−1}−𝜈𝛼,𝑡+𝑊²𝜔₁𝐺_{1,−𝑘−2𝑗−2,−𝑘−1}−𝜈𝜏𝛼,𝑡_2OSG,(𝑟,𝑡)=2𝛼𝜌𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²²^𝑗∞_𝑘=0−𝛼𝑟^2𝑛^𝑘×𝑉₁𝐻(𝑡)𝐺_{1,−𝑘−2𝑗−1,−𝑘−1}−𝜈𝛼,𝑡+𝑉²𝜔₂𝐺_{1,−𝑘−2𝑗−2,−𝑘−1}−𝜈𝛼,𝑡,(4.1) corresponding to an ordinary second grade fluid, performing the same motion. Other equivalent forms of solutions for ordinary second grade fluids can be directly obtained from (3.10) by substituting 𝛽=1, and performing the inverse Laplace transform. The expressions for velocity field are given by𝑤_OSG(𝑟,𝑡)=𝑤_OSS(𝑟,𝑡)+𝑤_OST(𝑟,𝑡),𝑣_𝑆(𝑟,𝑡)=𝑣_OSS(𝑟,𝑡)+𝑣_OST(𝑟,𝑡),(4.2) where𝑤_OSS=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝜔¹𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚×𝑊₁𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝐻(𝑡)𝜔₁1+𝛼𝑟^2𝑚cos𝜔₁𝑡−𝜈𝑟^2𝑚sin𝜔₁𝑡𝜈²𝑟^4𝑚+𝜔²¹1+𝛼𝑟^2𝑚²+𝑊²𝜈𝑟^2𝑚cos𝜔₁𝑡+𝜔₁1+𝛼𝑟^2𝑚sin𝜔₁𝑡𝑤_OST𝜈²𝑟^4𝑚+𝜔²¹1+𝛼𝑟^2𝑚²,=−2𝜈𝑅^∞_𝑚=1𝑟_𝑚𝐽₁𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝐻(𝑡)𝜈𝑟^2𝑚−𝑊₂𝜔₁1+𝛼𝑟^2𝑚1+𝛼𝑟^2𝑚𝜈²𝑟^4𝑚+𝜔²¹1+𝛼𝑟^2𝑚²exp−𝜈𝑟^2𝑚𝑡𝑣_OSS1+𝛼𝑟^2𝑚,=𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝜔²𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛×𝑉₁𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝐻(𝑡)𝜔₂1+𝛼𝑟^2𝑛cos𝜔₂𝑡−𝜈𝑟^2𝑛sin𝜔₂𝑡𝜈²𝑟^4𝑛+𝜔²²1+𝛼𝑟^2𝑛²+𝑉²𝜈𝑟^2𝑛cos𝜔₂𝑡+𝜔₂1+𝛼𝑟^2𝑛sin𝜔₂𝑡𝑤_OST𝜈²𝑟^4𝑛+𝜔²²1+𝛼𝑟^2𝑛²,=−2𝜈𝑅^∞_𝑛=1𝑟_𝑛𝐽₀𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝑉¹𝐻(𝑡)𝜈𝑟^2𝑛−𝑉₂𝜔₂1+𝛼𝑟^2𝑛1+𝛼𝑟^2𝑛𝜈²𝑟^4𝑛+𝜔²²1+𝛼𝑟^2𝑛²exp−𝜈𝑟^2𝑛𝑡1+𝛼𝑟^2𝑛,(4.3) are the steady-state and transient solutions. Introducing (4.2) into (2.7), we find that𝜏_1OSG(𝑟,𝑡)=𝜏_1OSS(𝑟,𝑡)+𝜏_1OST(𝑟,𝑡),𝜏_2OSG(𝑟,𝑡)=𝜏_2OSS(𝑟,𝑡)+𝜏_2OST(𝑟,𝑡),(4.4) where the steady-state and transient components are given by𝜏_1OSS=2𝜌𝜔¹𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝜈²𝑟^4𝑚+𝜔²¹11+𝛼𝑟^2𝑚²×+𝑊𝑊₁𝐻₂(𝑡)𝜈𝜔₁cos𝜔₁𝑡−𝜈²𝑟^2𝑚+𝛼𝜔²¹1+𝛼𝑟^2𝑚sin𝜔₁𝑡𝜈²𝑟^2𝑚+𝛼𝜔²¹1+𝛼𝑟^2𝑚cos𝜔₁𝑡+𝜈𝜔₁sin𝜔₁𝑡,(4.5)𝜏_1OST=2𝜌𝜈²𝑅^∞_𝑚=1𝑟^2𝑚𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝐻(𝑡)𝜈𝑟^2𝑚−𝑊₂𝜔₁1+𝛼𝑟^2𝑚1+𝛼𝑟^2𝑚²𝜈²𝑟^4𝑚+𝜔²¹1+𝛼𝑟^2𝑚²exp−𝜈𝑟^2𝑚𝑡1+𝛼𝑟^2𝑚,(4.6)𝜏_2OSS=2𝜌𝜔²𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑚𝜈²𝑟^4𝑛+𝜔²²11+𝛼𝑟^2𝑛²×+𝑉₁𝑉𝐻₂(𝑡)𝜈𝜔₂cos𝜔₂𝑡−𝜈²𝑟^2𝑛+𝛼𝜔²²1+𝛼𝑟^2𝑛sin𝜔₂𝑡𝜈²𝑟^2𝑛+𝛼𝜔²²1+𝛼𝑟^2𝑛cos𝜔₂𝑡+𝜈𝜔₂sin𝜔₂𝑡,(4.7)𝜏_2OST=2𝜌𝜈²𝑅^∞_𝑚=1𝑟^2𝑛𝐽₁𝑟𝑟_𝑚𝐽₁𝑅𝑟_𝑛𝑉¹𝐻(𝑡)𝜈𝑟^2𝑛−𝑉₂𝜔₂1+𝛼𝑟^2𝑛1+𝛼𝑟^2𝑛²𝜈²𝑟^4𝑛+𝜔²²1+𝛼𝑟^2𝑛²exp−𝜈𝑟^2𝑛𝑡1+𝛼𝑟^2𝑛.(4.8)

In practice, the steady-state solutions for unsteady motions of Newtonian or non-Newtonian fluids are important for those who need to eliminate transients from their rheological measurements. Consequently, an important problem regarding the technical relevance of these solutions is to find the approximate time after which the fluid is moving according to the steady-state. More exactly, in practice it is necessary to know the required time to reach the steady-state.

4.2. Newtonian Fluids (𝛼₁→0)

Making the limit 𝛼₁ and then 𝛼→0 into (3.13), (3.22), and proceeding as in the last section, the solutions for a Newtonian fluid𝑤_𝑁(𝑟,𝑞)=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚^∞_𝑗=0−𝜔²¹^𝑗𝑣×𝑊₁𝐻(𝑡)𝐺_1,−2𝑗,1−𝜈𝑟^2𝑚,𝑡+𝑊₂𝜔₂𝐺_{1,−2𝑗−1,1}−𝜈𝑟^2𝑚,𝑡_𝑁,(𝑟,𝑞)=𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛^∞_𝑗=0−𝜔²¹^𝑗𝜏×𝑉₁𝐻(𝑡)𝐺_1,−2𝑗,1−𝜈𝑟^2𝑛,𝑡+𝑉₂𝜔₂𝐺_{1,−2𝑗−1,1}−𝜈𝑟^2𝑛,𝑡_1𝑁,(𝑟,𝑡)=2𝜇𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚×_∞𝐽₂𝑅𝑟_𝑚_𝑗=0−𝜔²¹^𝑗𝑊₁𝜏𝐻(𝑡)𝐺_1,−2𝑗,1−𝜈𝑟^2𝑚,𝑡+𝑊₂𝜔₁𝐺_{1,−2𝑗−1,1}−𝜈𝑟^2𝑚,𝑡_2𝑁,(𝑟,𝑡)=2𝜇𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛×_∞𝐽₁𝑅𝑟_𝑛_𝑗=0−𝜔²²^𝑗𝑉₁𝐻(𝑡)𝐺_1,−2𝑗,1−𝜈𝑟^2𝑛,𝑡+𝑉₂𝜔₂𝐺_{1,−2𝑗−1,1}−𝜈𝑟^2𝑛,𝑡,(4.9) are obtained. Similarly by making 𝛼₁ and then 𝛼→0 into (4.2)–(4.8), the corresponding solutions𝑤_𝑁(𝑟,𝑡)=𝑤_𝑁𝑆(𝑟,𝑡)+𝑤_𝑁𝑇(𝑟,𝑡),𝑣_𝑁(𝑟,𝑡)=𝑣_𝑁𝑆(𝑟,𝑡)+𝑣_𝑁𝑇(𝑟,𝑡),(4.10)𝜏_1𝑁(𝑟,𝑡)=𝜏_1𝑁𝑆(𝑟,𝑡)+𝜏_1𝑁𝑇(𝑟,𝑡),𝜏_2𝑁(𝑟,𝑡)=𝜏_2𝑁𝑆(𝑟,𝑡)+𝜏_2𝑁𝑇(𝑟,𝑡),(4.11) where𝑤_𝑁𝑆=𝑟𝑅𝑊¹𝐻(𝑡)cos𝜔₁𝑡+𝑊₂sin𝜔₁𝑡−2𝜔¹𝑅^∞_𝑚=1𝐽₁𝑟𝑟_𝑚×𝑊₁𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝐻(𝑡)𝜔₁cos𝜔₁𝑡−𝜈𝑟^2𝑚sin𝜔₁𝑡+𝑊₂𝜈𝑟^2𝑚cos𝜔₁𝑡+𝜔₁sin𝜔₁𝑡𝑤𝜈²𝑟^4𝑚+𝜔²¹,(4.12)_𝑁𝑇=−2𝜈𝑅^∞_𝑚=1𝑟_𝑚𝐽₁𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝐻(𝑡)𝜈𝑟^2𝑚−𝑊₂𝜔₁𝜈²𝑟^4𝑚+𝜔²¹𝑣𝑒^{−𝜈𝑟2𝑚𝑡},(4.13)_𝑁𝑆=𝑉₁𝐻(𝑡)cos𝜔₂𝑡+𝑉₂sin𝜔₂𝑡−2𝜔²𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛×𝑉₁𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝐻(𝑡)𝜔₂cos𝜔₂𝑡−𝜈𝑟^2𝑛sin𝜔₂𝑡+𝑉₂𝜈𝑟^2𝑛cos𝜔₂𝑡+𝜔₂sin𝜔₂𝑡𝑣𝜈²𝑟^4𝑛+𝜔²²,(4.14)_𝑁𝑇=−2𝜈𝑅^∞_𝑛=1𝑟_𝑛𝐽₀𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝑉¹𝐻(𝑡)𝜈𝑟^2𝑛−𝑉₂𝜔₂𝜈²𝑟^4𝑛+𝜔²²𝜏𝑒^{−𝜈𝑟2𝑚𝑡},(4.15)_1𝑁𝑆=2𝜌𝜔¹𝑅^∞_𝑚=1𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝜈²𝑟^4𝑚1+𝜔²¹𝜏×𝑊₁𝐻(𝑡)𝜈𝜔₁cos𝜔₁𝑡−𝜈²𝑟^2𝑚sin𝜔₁𝑡+𝑊₂𝜈²𝑟^2𝑚cos𝜔₁𝑡+𝜈𝜔₁sin𝜔₁𝑡,(4.16)_1𝑁𝑇=2𝜌𝜈²𝑅^∞_𝑚=1𝑟^2𝑚𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑊¹𝐻(𝑡)𝜈𝑟^2𝑚−𝑊₂𝜔₁𝜈²𝑟^4𝑚+𝜔²¹𝜏𝑒^{−𝜈𝑟2𝑚𝑡},(4.17)_2𝑁𝑆=2𝜌𝜔²𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑚𝜈²𝑟^4𝑛1+𝜔²²𝜏×𝑉₁𝐻(𝑡)𝜈𝜔₂cos𝜔₂𝑡−𝜈²𝑟^2𝑛sin𝜔₂𝑡+𝑉₂𝜈²𝑟^2𝑛cos𝜔₂𝑡+𝜈𝜔₂sin𝜔₂𝑡,(4.18)_2𝑁𝑇=2𝜌𝜈²𝑅^∞_𝑚=1𝑟^2𝑛𝐽₁𝑟𝑟_𝑚𝐽₁𝑅𝑟_𝑛𝑉¹𝐻(𝑡)𝜈𝑟^2𝑛−𝑉₂𝜔₂𝜈²𝑟^4𝑛+𝜔²²𝑒^{−𝜈𝑟2𝑛𝑡}(4.19) for Newtonian fluids are obtained. Substituting 𝑊₁=𝑅Ω,𝑊₂=0,𝑉₁=𝑈,𝑉₂=0, 𝜔₁=0, and 𝜔₂=0 in (4.9) and using the definition of generalized 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) functions, the solutions𝑤_𝑁(𝑟,𝑡)=𝑟Ω−2Ω^∞_𝑚₌₁𝐽₁𝑟𝑟_𝑚𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑒^{−𝜈𝑟2𝑚𝑡}𝑣_𝑁,(𝑟,𝑡)=𝑈−2𝑈𝑅^∞_𝑛=1𝐽₀𝑟𝑟_𝑛𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝑒^{−𝜈𝑟2𝑛𝑡}𝜏_1𝑁,(𝑟,𝑡)=2𝜇Ω^∞_𝑚₌₁𝐽₂𝑟𝑟_𝑚𝐽₂𝑅𝑟_𝑚𝑒^{−𝜈𝑟2𝑚𝑡}𝜏_2𝑁,(𝑟,𝑡)=2𝜇𝑈𝑅^∞_𝑛=1𝐽₁𝑟𝑟_𝑛𝐽₁𝑅𝑟_𝑛𝑒^{−𝜈𝑟2𝑛𝑡},(4.20) obtained in [49, equations (36)–(39)] by a different technique are recovered. Of course the above expressions can also be obtained form (4.10)–(4.19).

5. Numerical Results and Conclusions

The velocity fields and the adequate shear stresses corresponding to the unsteady motions of an incompressible fractionalized second grade fluid due to longitudinal and torsional oscillations of an infinite circular cylinder have been determined by means of the Laplace and finite Hankel transforms. The general solutions are written in series form in term of generalized 𝐺_{𝑎,𝑏,𝑐}(⋅,𝑡) functions and satisfy all imposed initial and boundary conditions. The solutions for ordinary second grade and Newtonian fluids performing the same motion are obtain as special cases of general solutions. Furthermore, another equivalent solutions for ordinary second grade and Newtonian fluids are presented, in terms of steady-state and transient solutions. They describe the motion of the fluid sometime after its initiation. After that time, when the transients disappear, they tend to the steady-state solutions, which are periodic in time and independent of the initial conditions. It is also shown that for 𝑊₁=𝑅Ω, 𝑊₂=0, 𝑉₁=𝑈, 𝑉₂=0, 𝜔₁=0 and 𝜔₂=0, (4.9) reduce to the well-known classical solutions [49, equations (36)–(39)]. The similar solutions corresponding to the sine and cosine oscillations of the boundary are immediately obtained by making 𝑉₂=𝑊₂=0, respectively, 𝑉₁=𝑊₁=0 into general solutions.

Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) are depicted against 𝑟 for different values of 𝑡 and the pertinent parameters. Figure 2 contains the diagrams of the velocity components for four different times 𝑡; it is obvious to see the impact of rigid boundary of the cylinder on the motion of the fluid. Further, the amplitude of oscillations for the two components of the velocity decreases with increasing values of 𝑡. However, this conclusion cannot be generalized. The influence of the frequency of oscillations 𝜔₁ and 𝜔₂, on fluid motion is shown in Figure 3. The amplitudes of both components of velocity are decreasing functions of frequency of oscillations 𝜔₁ and 𝜔₂ respectively. The effect of material parameter 𝛼 on fluid motion is discussed in Figure 4. The influence of material parameter 𝛼 is quite similar to that of Figures 2 and 3. Nowadays fractional derivative approach in viscoelastic fluid plays an important role to describe the behavior of complex fluid. Therefore, it is important to see the effect of fractional parameter on oscillating fluid. Figure 5 depict the influence of fractional parameter 𝛽 on fluid motion. It is again clear that the amplitude of fluid oscillations decreases with respect to fractional parameter 𝛽. The viscosity is an important property of the fluid. It is observed that the amplitude of oscillations is an increasing function of kinematic viscosity 𝜈 in this geometry as shown in Figure 6. The influence of radius 𝑟 against time 𝑡 is shown in Figure 7. The oscillating behavior of fluid motion clearly results from these figures. As expected, the amplitude of oscillations increases with the increasing values of 𝑟. The influence of 𝑊₁ and 𝑊₂ on rotational component 𝑤(𝑟,𝑡) and the effect of 𝑉₁ and 𝑉₂ on longitudinal component 𝑣(𝑟,𝑡) are presented in Figures 8 and 9. The influence of 𝑊₁ and 𝑊₂ on 𝑤(𝑟,𝑡) is quite opposite. For instance, 𝑤(𝑟,𝑡) is a decreasing function with respect to 𝑊₁ and an increasing one of 𝑊₂. The longitudinal component 𝑣(𝑟,𝑡) is an increasing function of 𝑉₁ near the center of cylinder and of 𝑉₂ on the whole flow domain.

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Figure 2 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1, 𝜈=0.5566,𝜇=33, 𝛼=0.5,𝛽=0.5, and different values of 𝑡.

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Figure 3 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝜈=0.5566,𝜇=33, 𝛼=0.5,𝛽=0.5, and different values of 𝑤1 and 𝑤2, respectively.

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Figure 4 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1, 𝜈=0.5566,𝜇=33, 𝛽=0.5,𝑡=2.5s, and different values of 𝛼.

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Figure 5 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1, 𝜈=0.5566,𝜇=33,𝛼=0.5,𝑡=2.5s, and different values of 𝛽.

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Figure 6 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1, 𝜌=59.289,𝛼=0.1,𝛽=0.8,𝑡=2.5s, and different values of 𝜈.

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Figure 7 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) given by (3.18) and (3.19), for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1,𝜈=0.5566, 𝜇=33,𝛼=0.5,𝛽=0.5, and different values of 𝑟.

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Figure 8 

Profiles of the velocity components 𝑤(𝑟,𝑡) given by (3.18), for 𝑅=3,𝑤1=1, 𝜈=0.5566,𝜇=33,𝛼=0.5, 𝛽=0.5,𝑡=2.5s, and different values of 𝑊1 and 𝑊2.

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Figure 9 

Profiles of the velocity components 𝑣(𝑟,𝑡) given by (3.19), for 𝑅=3,𝑤2=1, 𝜈=0.5566,𝜇=33,𝛼=0.5, 𝛽=0.5,𝑡=2.5s, and different values of 𝑉1 and 𝑉2.

Finally, for comparison, the diagrams of 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) corresponding to the three models, fractionalized second grade (𝛽=0.3 and 𝛽=0.6), ordinary second grade (𝛽=1), and Newtonian fluids (𝛼=0 and 𝛽=1) are presented in Figures 10–13. It is clearly seen from Figure 10 that the fractionalized second grade fluid is the swiftest and the Newtonian one is the slowest. However, the behavior of these models is quite opposite at time 𝑡=4s as shown by Figure 11. The large time effect on oscillating fluid is shown in Figures 12 and 13. It is observed that for large time the non-Newtonian effects can be neglected for rotational component of velocity 𝑤(𝑟,𝑡). This seems to be not true for the longitudinal component 𝑣(𝑟,𝑡) of the velocity. The units of the material constants in Figures 2–13 are SI units and the roots 𝑟_𝑚 and 𝑟_𝑛 have been approximated by (4𝑚+1)𝜋/4𝑅 and (4𝑛−1)𝜋/4𝑅, respectively.

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Figure 10 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) for fractionalized second grade, ordinary second grade, and Newtonian fluids, for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1,𝜈=0.5566,𝜇=33,𝛼=0.9,𝛽=0.3 and 0.6 and different values of 𝑡=1s.

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Figure 11 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) for fractionalized second grade, ordinary second grade, and Newtonian fluids, for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1,𝜈=0.5566, 𝜇=33,𝛼=0.9,𝛽=0.3 and 0.6 and different values of 𝑡=4s.

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Figure 12 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) for fractionalized second grade, ordinary second grade, and Newtonian fluids, for 𝑅=3, 𝑊1=𝑊2=𝑉1=𝑉2=1,𝑤1=𝑤2=1,𝜈=0.5566, 𝜇=33,𝛼=0.9,𝛽=0.3 and 0.6 and for large time 𝑡.

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Figure 13 

Profiles of the velocity components 𝑤(𝑟,𝑡) and 𝑣(𝑟,𝑡) for fractionalized second grade, ordinary second grade, and Newtonian fluids, for 𝑅=3,𝑊1=𝑊2=𝑉1=𝑉2=1, 𝑤1=𝑤2=1,𝜈=0.5566,𝜇=33,𝛼=0.9,𝛽=0.3 and 0.6 and for large time 𝑡.

Acknowledgments

The author M. Jamil is extremely grateful and thankful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics & Basic Sciences, NED University of Engineering & Technology, Karachi, Pakistan; also Higher Education Commission of Pakistan for supporting and facilitating this research work. The author N. A. Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi, Pakistan for supporting and facilitating this research work. The author A. Rauf is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan and also Higher Education Commission of Pakistan for generous support and facilitating this research work.