# Oscillating Flows of Fractionalized Second Grade Fluid

Authors: Muhammad Jamil; Najeeb Alam Khan; Abdul Rauf
Journal: ISRN Mathematical Physics
 (2012)

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## 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 generalizedGa,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.


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## Body

## 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 Ga,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 stressT in an incompressible homogeneous fluid of second grade is related to the fluid motion in the following manner:(2.1)T=-pI+S,S=μA1+α1A2+α2A12,
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 through(2.2)A1=(∇V)+(∇V)T,A2=dA1dt+A1(∇V)+(∇V)TA1.
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(2.3)∇⋅V=0,∇⋅T=ρdVdt+ρb,
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]:(2.4)μ≥0,α1≥0,α1+α2=0.
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 form(2.5)V=V(r,t)=w(r,t)eθ+v(r,t)ez,
where eθ and ez 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, then(2.6)V(r,0)=0,
and (2.1) implies Srr=0 and the meaningful equations(2.7)τ1(r,t)=(μ+α1∂∂t)(∂∂r-1r)w(r,t),τ2(r,t)=(μ+α1∂∂t)∂v(r,t)∂r,
where τ1=Srθ and τ2=Srz 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)(2.8)ρ∂w(r,t)∂t=(∂∂r+2r)τ1(r,t),ρ∂v(r,t)∂t=(∂∂r+1r)τ2(r,t).
Eliminating τ1 and τ2 between (2.7) and (2.8), we attain the governing equations(2.9)∂w(r,t)∂t=(ν+α∂∂t)(∂2∂r2+1r∂∂r-1r2)w(r,t),r∈(0,R),t>0,∂v(r,t)∂t=(ν+α∂∂t)(∂2∂r2+1r∂∂r)v(r,t),r∈(0,R),t>0,
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(2.10)∂w(r,t)∂t=(ν+αDtβ)(∂2∂r2+1r∂∂r-1r2)w(r,t),r∈(0,R),t>0,(2.11)∂v(r,t)∂t=(ν+αDtβ)(∂2∂r2+1r∂∂r)v(r,t),r∈(0,R),t>0,(2.12)τ1(r,t)=(μ+α1Dtβ)(∂∂r-1r)w(r,t),(2.13)τ2(r,t)=(μ+α1Dtβ)∂v(r,t)∂r,
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 Dtβ defined by [43, 44](2.14)Dtβf(t)={1Γ(1-β)∫0tf′(τ)(t-τ)βdτ,0≤β<1,df(t)dt,β=1,
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 radiusR  as shown in Figure 1. At time t=0+, the cylinder starts to oscillate according to(3.1)V(R,t)=[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]eθ+[V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)]ez,
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 are(3.2)w(r,0)=v(r,0)=0,r∈(0,R),
respectively, and(3.3)w(R,t)=W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t),v(R,t)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t),t≥0,
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 that(3.4)qw¯(r,q)=(ν+αqβ)(∂2∂r2+1r∂∂r-1r2)w¯(r,q),r∈(0,R),(3.5)qv¯(r,q)=(ν+αqβ)(∂2∂r2+1r∂∂r)v¯(r,q),r∈(0,R),
where the image functions w¯(r,q) and v¯(r,q) of w(r,t) and v(r,t) have to satisfy the conditions(3.6)w¯(R,q)=W1q+W2ω1q2+ω12,v¯(R,q)=V1q+V2ω2q2+ω22.Multiplying now both sides of (3.4) and (3.5) by rJ1(rrm) and rJ0(rrn), 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](3.7)∫0Rr(∂2w¯∂r2+1r∂w¯∂r-w¯r2)J1(rrm)dr=RrmJ2(Rrm)w¯(R,t)-rm2w¯H(rm,t),∫0Rr(∂2v¯∂r2+1r∂v¯∂r)J0(rrn)dr=RrnJ1(Rrn)v¯(R,t)-rn2v¯H(rn,t),
we find that(3.8)w¯H(rm,q)=RrmJ2(Rrm)W1q+W2ω1q2+ω12ν+αqβq+αrm2qβ+νrm2,v¯H(rn,q)=RrnJ1(Rrm)V1q+V2ω2q2+ω22ν+αqβq+αrn2qβ+νrn2,
where [46, 47](3.9)w¯H(rm,q)=∫0Rrw¯(r,q)J1(rrm)dr,v¯H(rn,q)=∫0Rrv¯(r,q)J0(rrn)dr,m,n=1,2,3,…
are the Hankel transforms of w¯(r,q) and v¯(r,q), while rm and rn 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:(3.10)w¯H(rm,q)=RJ2(Rrm)rmW1q+W2ω1q2+ω12-RJ2(Rrm)rnW1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,v¯H(rn,q)=RJ1(Rrn)rnV1q+V2ω1q2+ω12-RJ1(Rrn)rnV1q+V2ω2q2+ω12qq+αrn2qβ+νrn2,
and apply the inverse Hankel transform formulae [46, 47](3.11)w¯(r,q)=2R2∑m=1∞w¯H(rm,q)J1(rrm)J22(Rrm),v¯(r,q)=2R2∑n=1∞v¯H(rn,q)J0(rrn)J12(Rrn).
Taking into account the following results [47]:(3.12)∫0Rr2J1(rrm)dr=R2rmJ2(Rrm),∫0RrJ0(rrn)dr=RrnJ1(Rrn),
we find that(3.13)w¯(r,q)=RrW1q+W2ω1q2+ω12-2R∑m=1∞J1(rrm)rmJ2(Rrm)W1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,v¯(r,q)=V1q+V2ω2q2+ω22-2R∑n=1∞J0(rrn)rnJ2(Rrn)V1q+V2ω2q2+ω22qq+αrn2qβ+νrn2.
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 form(3.14)w¯(r,q)=RrW1q+W2ω1q2+ω12-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+1+W2ω1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+2],v¯(r,q)=V1q+V2ω2q2+ω22-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+1+V2ω2∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+2],
where we used the fact that(3.15)(ki)=(-1)i(-k)ii!,
and (-k)i is the Pochhammer symbol(3.16)(k)i={1,i=0,k(k+1)⋯(k+i-1),i∈N.
In particular (0)0=1, (k)0=1 and (0)i=0, for i∈N. Applying the discrete inverse Laplace transform, we get(3.17)w(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2ji!Γ((i-k)β+k+2j+1)+W2ω1∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2j+1i!Γ((i-k)β+k+2j+2)],v(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)∑i=0∞(-k)i(-ν/α)itk((i-k)β+k+2j)i!Γ((i-k)β+k+2j+1)+V2ω2∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2j+1i!Γ((i-k)β+k+2j+2)].
In terms of the generalized Ga,b,c(·,t) functions [48], we rewrite the above equations in simple forms:(3.18)w(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)Gβ,-k-2j-1,-k(-να,t)+W2ω1Gβ,-k-2j-2,-k(-να,t)],(3.19)v(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)Gβ,-k-2j-1,-k(-να,t)+V2ω2Gβ,-k-2j-2,-k(-να,t)],
where the generalized Ga,b,c(·,t) function is defined by [48](3.20)Ga,b,c(d,t)=∑j=0∞(c)jdjj!t(c+j)a-b-1Γ[(c+j)a-b],Re(ac-b)>0,Re(q)>0,|dqa|<1.


### 3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.12) and (2.13), we find that(3.21)τ1¯(r,q)=(μ+α1qβ)(∂∂r-1r)w¯(r,q),τ2¯(r,q)=(μ+α1qβ)∂v¯(r,q)∂r,
where(3.22)∂w¯(r,q)∂r-1rw¯(r,q)=2R∑m=1∞J2(rrm)J2(Rrm)W1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,∂v¯(r,q)∂r=2R∑m=1∞J1(rrm)J1(Rrm)V1q+V2ω2q2+ω22qq+αrn2qβ+νrn2,
have been obtained form (3.13), using the identities(3.23)rrmJ1′(rrm)-J1(rrm)=-rrmJ2(rrm),J0′(rrn)=-rnJ1(rrn).
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:(3.24)τ1(r,t)=2αρR∑m=1∞J2(rrm)J2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k[W1H(t)Gβ,-k-2j-1,-k-1(-να,t)+W2ω1Gβ,-k-2j-2,-k-1(-να,t)],τ2(r,t)=2αρR∑n=1∞J1(rrn)J1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k[V1H(t)Gβ,-k-2j-1,-k-1(-να,t)+V2ω2Gβ,-k-2j-2,-k-1(-να,t)].


## 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 that(3.4)qw¯(r,q)=(ν+αqβ)(∂2∂r2+1r∂∂r-1r2)w¯(r,q),r∈(0,R),(3.5)qv¯(r,q)=(ν+αqβ)(∂2∂r2+1r∂∂r)v¯(r,q),r∈(0,R),
where the image functions w¯(r,q) and v¯(r,q) of w(r,t) and v(r,t) have to satisfy the conditions(3.6)w¯(R,q)=W1q+W2ω1q2+ω12,v¯(R,q)=V1q+V2ω2q2+ω22.Multiplying now both sides of (3.4) and (3.5) by rJ1(rrm) and rJ0(rrn), 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](3.7)∫0Rr(∂2w¯∂r2+1r∂w¯∂r-w¯r2)J1(rrm)dr=RrmJ2(Rrm)w¯(R,t)-rm2w¯H(rm,t),∫0Rr(∂2v¯∂r2+1r∂v¯∂r)J0(rrn)dr=RrnJ1(Rrn)v¯(R,t)-rn2v¯H(rn,t),
we find that(3.8)w¯H(rm,q)=RrmJ2(Rrm)W1q+W2ω1q2+ω12ν+αqβq+αrm2qβ+νrm2,v¯H(rn,q)=RrnJ1(Rrm)V1q+V2ω2q2+ω22ν+αqβq+αrn2qβ+νrn2,
where [46, 47](3.9)w¯H(rm,q)=∫0Rrw¯(r,q)J1(rrm)dr,v¯H(rn,q)=∫0Rrv¯(r,q)J0(rrn)dr,m,n=1,2,3,…
are the Hankel transforms of w¯(r,q) and v¯(r,q), while rm and rn 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:(3.10)w¯H(rm,q)=RJ2(Rrm)rmW1q+W2ω1q2+ω12-RJ2(Rrm)rnW1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,v¯H(rn,q)=RJ1(Rrn)rnV1q+V2ω1q2+ω12-RJ1(Rrn)rnV1q+V2ω2q2+ω12qq+αrn2qβ+νrn2,
and apply the inverse Hankel transform formulae [46, 47](3.11)w¯(r,q)=2R2∑m=1∞w¯H(rm,q)J1(rrm)J22(Rrm),v¯(r,q)=2R2∑n=1∞v¯H(rn,q)J0(rrn)J12(Rrn).
Taking into account the following results [47]:(3.12)∫0Rr2J1(rrm)dr=R2rmJ2(Rrm),∫0RrJ0(rrn)dr=RrnJ1(Rrn),
we find that(3.13)w¯(r,q)=RrW1q+W2ω1q2+ω12-2R∑m=1∞J1(rrm)rmJ2(Rrm)W1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,v¯(r,q)=V1q+V2ω2q2+ω22-2R∑n=1∞J0(rrn)rnJ2(Rrn)V1q+V2ω2q2+ω22qq+αrn2qβ+νrn2.
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 form(3.14)w¯(r,q)=RrW1q+W2ω1q2+ω12-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+1+W2ω1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+2],v¯(r,q)=V1q+V2ω2q2+ω22-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+1+V2ω2∑i=0∞(-k)i(-ν/α)ii!1q(i-k)β+k+2j+2],
where we used the fact that(3.15)(ki)=(-1)i(-k)ii!,
and (-k)i is the Pochhammer symbol(3.16)(k)i={1,i=0,k(k+1)⋯(k+i-1),i∈N.
In particular (0)0=1, (k)0=1 and (0)i=0, for i∈N. Applying the discrete inverse Laplace transform, we get(3.17)w(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2ji!Γ((i-k)β+k+2j+1)+W2ω1∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2j+1i!Γ((i-k)β+k+2j+2)],v(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)∑i=0∞(-k)i(-ν/α)itk((i-k)β+k+2j)i!Γ((i-k)β+k+2j+1)+V2ω2∑i=0∞(-k)i(-ν/α)it(i-k)β+k+2j+1i!Γ((i-k)β+k+2j+2)].
In terms of the generalized Ga,b,c(·,t) functions [48], we rewrite the above equations in simple forms:(3.18)w(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)Gβ,-k-2j-1,-k(-να,t)+W2ω1Gβ,-k-2j-2,-k(-να,t)],(3.19)v(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)Gβ,-k-2j-1,-k(-να,t)+V2ω2Gβ,-k-2j-2,-k(-να,t)],
where the generalized Ga,b,c(·,t) function is defined by [48](3.20)Ga,b,c(d,t)=∑j=0∞(c)jdjj!t(c+j)a-b-1Γ[(c+j)a-b],Re(ac-b)>0,Re(q)>0,|dqa|<1.


## 3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.12) and (2.13), we find that(3.21)τ1¯(r,q)=(μ+α1qβ)(∂∂r-1r)w¯(r,q),τ2¯(r,q)=(μ+α1qβ)∂v¯(r,q)∂r,
where(3.22)∂w¯(r,q)∂r-1rw¯(r,q)=2R∑m=1∞J2(rrm)J2(Rrm)W1q+W2ω1q2+ω12qq+αrm2qβ+νrm2,∂v¯(r,q)∂r=2R∑m=1∞J1(rrm)J1(Rrm)V1q+V2ω2q2+ω22qq+αrn2qβ+νrn2,
have been obtained form (3.13), using the identities(3.23)rrmJ1′(rrm)-J1(rrm)=-rrmJ2(rrm),J0′(rrn)=-rnJ1(rrn).
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:(3.24)τ1(r,t)=2αρR∑m=1∞J2(rrm)J2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k[W1H(t)Gβ,-k-2j-1,-k-1(-να,t)+W2ω1Gβ,-k-2j-2,-k-1(-να,t)],τ2(r,t)=2αρR∑n=1∞J1(rrn)J1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k[V1H(t)Gβ,-k-2j-1,-k-1(-να,t)+V2ω2Gβ,-k-2j-2,-k-1(-να,t)].


## 4. Limiting Cases

### 4.1. Ordinary Second Grade Fluid(β→1)

Makingβ→1 into (3.18), (3.19), (3.24), we obtain the solutions(4.1)wOSG(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)G1,-k-2j-1,-k(-να,t)+W2ω1G1,-k-2j-2,-k(-να,t)],vOSG(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)G1,-k-2j-1,-k(-να,t)+V2ω2G1,-k-2j-2,-k(-να,t)],τ1OSG(r,t)=2αρR∑m=1∞J2(rrm)J2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)G1,-k-2j-1,-k-1(-να,t)+W2ω1G1,-k-2j-2,-k-1(-να,t)],τ2OSG(r,t)=2αρR∑n=1∞J1(rrn)J1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)G1,-k-2j-1,-k-1(-να,t)+V2ω2G1,-k-2j-2,-k-1(-να,t)],
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 by(4.2)wOSG(r,t)=wOSS(r,t)+wOST(r,t),vS(r,t)=vOSS(r,t)+vOST(r,t),
where(4.3)wOSS=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2ω1R∑m=1∞J1(rrm)rmJ2(Rrm)×[W1H(t)[ω1(1+αrm2)cos⁡(ω1t)-νrm2sin⁡(ω1t)]ν2rm4+ω12(1+αrm2)2+W2[νrm2cos⁡(ω1t)+ω1(1+αrm2)sin⁡(ω1t)]ν2rm4+ω12(1+αrm2)2],wOST=-2νR∑m=1∞rmJ1(rrm)J2(Rrm)W1H(t)νrm2-W2ω1(1+αrm2)(1+αrm2)[ν2rm4+ω12(1+αrm2)2]exp⁡(-νrm2t1+αrm2),vOSS=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2ω2R∑n=1∞J0(rrn)rnJ1(Rrn)×[V1H(t)[ω2(1+αrn2)cos⁡(ω2t)-νrn2sin⁡(ω2t)]ν2rn4+ω22(1+αrn2)2+V2[νrn2cos⁡(ω2t)+ω2(1+αrn2)sin⁡(ω2t)]ν2rn4+ω22(1+αrn2)2],wOST=-2νR∑n=1∞rnJ0(rrn)J1(Rrn)V1H(t)νrn2-V2ω2(1+αrn2)(1+αrn2)[ν2rn4+ω22(1+αrn2)2]exp⁡(-νrn2t1+αrn2),
are the steady-state and transient solutions. Introducing (4.2) into (2.7), we find that(4.4)τ1OSG(r,t)=τ1OSS(r,t)+τ1OST(r,t),τ2OSG(r,t)=τ2OSS(r,t)+τ2OST(r,t),
where the steady-state and transient components are given by(4.5)τ1OSS=2ρω1R∑m=1∞J2(rrm)J2(Rrm)1ν2rm4+ω12(1+αrm2)2×[W1H(t){νω1cos⁡(ω1t)-[ν2rm2+αω12(1+αrm2)]sin⁡(ω1t)}+W2{[ν2rm2+αω12(1+αrm2)]cos⁡(ω1t)+νω1sin⁡(ω1t)}],(4.6)τ1OST=2ρν2R∑m=1∞rm2J2(rrm)J2(Rrm)W1H(t)νrm2-W2ω1(1+αrm2)(1+αrm2)2[ν2rm4+ω12(1+αrm2)2]exp⁡(-νrm2t1+αrm2),(4.7)τ2OSS=2ρω2R∑n=1∞J1(rrn)J1(Rrm)1ν2rn4+ω22(1+αrn2)2×[V1H(t){νω2cos⁡(ω2t)-[ν2rn2+αω22(1+αrn2)]sin⁡(ω2t)}+V2{[ν2rn2+αω22(1+αrn2)]cos⁡(ω2t)+νω2sin⁡(ω2t)}],(4.8)τ2OST=2ρν2R∑m=1∞rn2J1(rrm)J1(Rrn)V1H(t)νrn2-V2ω2(1+αrn2)(1+αrn2)2[ν2rn4+ω22(1+αrn2)2]exp⁡(-νrn2t1+αrn2).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 limitα1 and then α→0 into (3.13), (3.22), and proceeding as in the last section, the solutions for a Newtonian fluid(4.9)wN(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j×[W1H(t)G1,-2j,1(-νrm2,t)+W2ω2G1,-2j-1,1(-νrm2,t)],vN(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω12)j×[V1H(t)G1,-2j,1(-νrn2,t)+V2ω2G1,-2j-1,1(-νrn2,t)],τ1N(r,t)=2μR∑m=1∞J2(rrm)J2(Rrm)×∑j=0∞(-ω12)j[W1H(t)G1,-2j,1(-νrm2,t)+W2ω1G1,-2j-1,1(-νrm2,t)],τ2N(r,t)=2μR∑n=1∞J1(rrn)J1(Rrn)×∑j=0∞(-ω22)j[V1H(t)G1,-2j,1(-νrn2,t)+V2ω2G1,-2j-1,1(-νrn2,t)],
are obtained. Similarly by making α1 and then α→0 into (4.2)–(4.8), the corresponding solutions(4.10)wN(r,t)=wNS(r,t)+wNT(r,t),vN(r,t)=vNS(r,t)+vNT(r,t),(4.11)τ1N(r,t)=τ1NS(r,t)+τ1NT(r,t),τ2N(r,t)=τ2NS(r,t)+τ2NT(r,t),
where(4.12)wNS=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2ω1R∑m=1∞J1(rrm)rmJ2(Rrm)×[W1H(t)[ω1cos⁡(ω1t)-νrm2sin⁡(ω1t)]+W2[νrm2cos⁡(ω1t)+ω1sin⁡(ω1t)]ν2rm4+ω12],(4.13)wNT=-2νR∑m=1∞rmJ1(rrm)J2(Rrm)W1H(t)νrm2-W2ω1ν2rm4+ω12e-νrm2t,(4.14)vNS=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2ω2R∑n=1∞J0(rrn)rnJ1(Rrn)×[V1H(t)[ω2cos⁡(ω2t)-νrn2sin⁡(ω2t)]+V2[νrn2cos⁡(ω2t)+ω2sin⁡(ω2t)]ν2rn4+ω22],(4.15)vNT=-2νR∑n=1∞rnJ0(rrn)J1(Rrn)V1H(t)νrn2-V2ω2ν2rn4+ω22e-νrm2t,(4.16)τ1NS=2ρω1R∑m=1∞J2(rrm)J2(Rrm)1ν2rm4+ω12×[W1H(t){νω1cos⁡(ω1t)-ν2rm2sin⁡(ω1t)}+W2{ν2rm2cos⁡(ω1t)+νω1sin⁡(ω1t)}],(4.17)τ1NT=2ρν2R∑m=1∞rm2J2(rrm)J2(Rrm)W1H(t)νrm2-W2ω1ν2rm4+ω12e-νrm2t,(4.18)τ2NS=2ρω2R∑n=1∞J1(rrn)J1(Rrm)1ν2rn4+ω22×[V1H(t){νω2cos⁡(ω2t)-ν2rn2sin⁡(ω2t)}+V2{ν2rn2cos⁡(ω2t)+νω2sin⁡(ω2t)}],(4.19)τ2NT=2ρν2R∑m=1∞rn2J1(rrm)J1(Rrn)V1H(t)νrn2-V2ω2ν2rn4+ω22e-νrn2t
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 Ga,b,c(·,t) functions, the solutions(4.20)wN(r,t)=rΩ-2Ω∑m=1∞J1(rrm)rmJ2(Rrm)e-νrm2t,vN(r,t)=U-2UR∑n=1∞J0(rrn)rnJ1(Rrn)e-νrn2t,τ1N(r,t)=2μΩ∑m=1∞J2(rrm)J2(Rrm)e-νrm2t,τ2N(r,t)=2μUR∑n=1∞J1(rrn)J1(Rrn)e-νrn2t,
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).


## 4.1. Ordinary Second Grade Fluid(β→1)

Makingβ→1 into (3.18), (3.19), (3.24), we obtain the solutions(4.1)wOSG(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)G1,-k-2j-1,-k(-να,t)+W2ω1G1,-k-2j-2,-k(-να,t)],vOSG(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)G1,-k-2j-1,-k(-να,t)+V2ω2G1,-k-2j-2,-k(-να,t)],τ1OSG(r,t)=2αρR∑m=1∞J2(rrm)J2(Rrm)∑j=0∞(-ω12)j∑k=0∞(-αrm2)k×[W1H(t)G1,-k-2j-1,-k-1(-να,t)+W2ω1G1,-k-2j-2,-k-1(-να,t)],τ2OSG(r,t)=2αρR∑n=1∞J1(rrn)J1(Rrn)∑j=0∞(-ω22)j∑k=0∞(-αrn2)k×[V1H(t)G1,-k-2j-1,-k-1(-να,t)+V2ω2G1,-k-2j-2,-k-1(-να,t)],
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 by(4.2)wOSG(r,t)=wOSS(r,t)+wOST(r,t),vS(r,t)=vOSS(r,t)+vOST(r,t),
where(4.3)wOSS=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2ω1R∑m=1∞J1(rrm)rmJ2(Rrm)×[W1H(t)[ω1(1+αrm2)cos⁡(ω1t)-νrm2sin⁡(ω1t)]ν2rm4+ω12(1+αrm2)2+W2[νrm2cos⁡(ω1t)+ω1(1+αrm2)sin⁡(ω1t)]ν2rm4+ω12(1+αrm2)2],wOST=-2νR∑m=1∞rmJ1(rrm)J2(Rrm)W1H(t)νrm2-W2ω1(1+αrm2)(1+αrm2)[ν2rm4+ω12(1+αrm2)2]exp⁡(-νrm2t1+αrm2),vOSS=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2ω2R∑n=1∞J0(rrn)rnJ1(Rrn)×[V1H(t)[ω2(1+αrn2)cos⁡(ω2t)-νrn2sin⁡(ω2t)]ν2rn4+ω22(1+αrn2)2+V2[νrn2cos⁡(ω2t)+ω2(1+αrn2)sin⁡(ω2t)]ν2rn4+ω22(1+αrn2)2],wOST=-2νR∑n=1∞rnJ0(rrn)J1(Rrn)V1H(t)νrn2-V2ω2(1+αrn2)(1+αrn2)[ν2rn4+ω22(1+αrn2)2]exp⁡(-νrn2t1+αrn2),
are the steady-state and transient solutions. Introducing (4.2) into (2.7), we find that(4.4)τ1OSG(r,t)=τ1OSS(r,t)+τ1OST(r,t),τ2OSG(r,t)=τ2OSS(r,t)+τ2OST(r,t),
where the steady-state and transient components are given by(4.5)τ1OSS=2ρω1R∑m=1∞J2(rrm)J2(Rrm)1ν2rm4+ω12(1+αrm2)2×[W1H(t){νω1cos⁡(ω1t)-[ν2rm2+αω12(1+αrm2)]sin⁡(ω1t)}+W2{[ν2rm2+αω12(1+αrm2)]cos⁡(ω1t)+νω1sin⁡(ω1t)}],(4.6)τ1OST=2ρν2R∑m=1∞rm2J2(rrm)J2(Rrm)W1H(t)νrm2-W2ω1(1+αrm2)(1+αrm2)2[ν2rm4+ω12(1+αrm2)2]exp⁡(-νrm2t1+αrm2),(4.7)τ2OSS=2ρω2R∑n=1∞J1(rrn)J1(Rrm)1ν2rn4+ω22(1+αrn2)2×[V1H(t){νω2cos⁡(ω2t)-[ν2rn2+αω22(1+αrn2)]sin⁡(ω2t)}+V2{[ν2rn2+αω22(1+αrn2)]cos⁡(ω2t)+νω2sin⁡(ω2t)}],(4.8)τ2OST=2ρν2R∑m=1∞rn2J1(rrm)J1(Rrn)V1H(t)νrn2-V2ω2(1+αrn2)(1+αrn2)2[ν2rn4+ω22(1+αrn2)2]exp⁡(-νrn2t1+αrn2).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 limitα1 and then α→0 into (3.13), (3.22), and proceeding as in the last section, the solutions for a Newtonian fluid(4.9)wN(r,q)=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2R∑m=1∞J1(rrm)rmJ2(Rrm)∑j=0∞(-ω12)j×[W1H(t)G1,-2j,1(-νrm2,t)+W2ω2G1,-2j-1,1(-νrm2,t)],vN(r,q)=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2R∑n=1∞J0(rrn)rnJ1(Rrn)∑j=0∞(-ω12)j×[V1H(t)G1,-2j,1(-νrn2,t)+V2ω2G1,-2j-1,1(-νrn2,t)],τ1N(r,t)=2μR∑m=1∞J2(rrm)J2(Rrm)×∑j=0∞(-ω12)j[W1H(t)G1,-2j,1(-νrm2,t)+W2ω1G1,-2j-1,1(-νrm2,t)],τ2N(r,t)=2μR∑n=1∞J1(rrn)J1(Rrn)×∑j=0∞(-ω22)j[V1H(t)G1,-2j,1(-νrn2,t)+V2ω2G1,-2j-1,1(-νrn2,t)],
are obtained. Similarly by making α1 and then α→0 into (4.2)–(4.8), the corresponding solutions(4.10)wN(r,t)=wNS(r,t)+wNT(r,t),vN(r,t)=vNS(r,t)+vNT(r,t),(4.11)τ1N(r,t)=τ1NS(r,t)+τ1NT(r,t),τ2N(r,t)=τ2NS(r,t)+τ2NT(r,t),
where(4.12)wNS=rR[W1H(t)cos⁡(ω1t)+W2sin⁡(ω1t)]-2ω1R∑m=1∞J1(rrm)rmJ2(Rrm)×[W1H(t)[ω1cos⁡(ω1t)-νrm2sin⁡(ω1t)]+W2[νrm2cos⁡(ω1t)+ω1sin⁡(ω1t)]ν2rm4+ω12],(4.13)wNT=-2νR∑m=1∞rmJ1(rrm)J2(Rrm)W1H(t)νrm2-W2ω1ν2rm4+ω12e-νrm2t,(4.14)vNS=V1H(t)cos⁡(ω2t)+V2sin⁡(ω2t)-2ω2R∑n=1∞J0(rrn)rnJ1(Rrn)×[V1H(t)[ω2cos⁡(ω2t)-νrn2sin⁡(ω2t)]+V2[νrn2cos⁡(ω2t)+ω2sin⁡(ω2t)]ν2rn4+ω22],(4.15)vNT=-2νR∑n=1∞rnJ0(rrn)J1(Rrn)V1H(t)νrn2-V2ω2ν2rn4+ω22e-νrm2t,(4.16)τ1NS=2ρω1R∑m=1∞J2(rrm)J2(Rrm)1ν2rm4+ω12×[W1H(t){νω1cos⁡(ω1t)-ν2rm2sin⁡(ω1t)}+W2{ν2rm2cos⁡(ω1t)+νω1sin⁡(ω1t)}],(4.17)τ1NT=2ρν2R∑m=1∞rm2J2(rrm)J2(Rrm)W1H(t)νrm2-W2ω1ν2rm4+ω12e-νrm2t,(4.18)τ2NS=2ρω2R∑n=1∞J1(rrn)J1(Rrm)1ν2rn4+ω22×[V1H(t){νω2cos⁡(ω2t)-ν2rn2sin⁡(ω2t)}+V2{ν2rn2cos⁡(ω2t)+νω2sin⁡(ω2t)}],(4.19)τ2NT=2ρν2R∑m=1∞rn2J1(rrm)J1(Rrn)V1H(t)νrn2-V2ω2ν2rn4+ω22e-νrn2t
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 Ga,b,c(·,t) functions, the solutions(4.20)wN(r,t)=rΩ-2Ω∑m=1∞J1(rrm)rmJ2(Rrm)e-νrm2t,vN(r,t)=U-2UR∑n=1∞J0(rrn)rnJ1(Rrn)e-νrn2t,τ1N(r,t)=2μΩ∑m=1∞J2(rrm)J2(Rrm)e-νrm2t,τ2N(r,t)=2μUR∑n=1∞J1(rrn)J1(Rrn)e-νrn2t,
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 generalizedGa,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 componentsw(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.Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsv(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.
(a)(b)Finally, for comparison, the diagrams ofw(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 rm and rn have been approximated by (4m+1)π/4R and (4n-1)π/4R, respectively.Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)Profiles of the velocity componentsw(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.
(a)(b)



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*Source: 908386-2012-04-04.xml*