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)eθ,S=S(r,t),
where eθ 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.
The governing equations corresponding to such motions of Oldroyd-B fluid are given by [15] (1+λ∂∂t)τ(r,t)=μ(1+λr∂∂t)(∂∂r-1r)w(r,t),(1+λ∂∂t)∂w(r,t)∂t=ν(1+λr∂∂t)(∂2∂r2+1r∂∂r-1r2)w(r,t),
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] Dtβf(t)={1Γ(1-β)ddt∫0tf(τ)(t-τ)βdτ,0≤β<1,ddtf(t),β=1,
where Γ(·) is the Gamma function defined by Γ(x)=∫0∞tx-1e-tdt,x>0.
Consequently, the governing equations to be used here are (1+λDtγ)τ(r,t)=μ(1+λrDtβ)(∂∂r-1r)w(r,t),(1+λDtγ)∂w(r,t)∂t=ν(1+λrDtβ)(∂2∂r2+1r∂∂r-1r2)w(r,t).
We can easily observe that when γ,β→1, (2.6) and (2.7) reduce to (2.3), because Dt1f=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],w(R,t)=RΩt2,t≥0,
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+λqγ+1)w¯(r,q)=ν(1+λrqβ)(∂2∂r2+1r∂∂r-1r2)w¯(r,q),w¯(R,q)=2RΩq3,
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(rn,q)=∫0Rrw¯(r,q)J1(rrn)dr,
the finite Hankel transform of the function w¯(r,q), and the inverse Hankel transform of w¯H(rn,q) is given by [25] w¯(r,q)=2R2∑n=1∞J1(rrn)J22(Rrn)w¯H(rn,q),
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 ∫0Rr(∂2∂r2+1r∂∂r-1r2)w¯(r,q)J1(rrn)dr=RrnJ2(Rrn)w¯(R,q)-rn2w¯H(rn,q),
we find that w¯H(rn,q)=(ν+νλrqβ)ΩR2rnJ2(Rrn)2q3(q+νrn2+λqγ+1+νrn2λrqβ).
It can be also written in the suitable form w¯H(rn,q)=w¯1H(rn,q)+w¯2H(rn,q),
where w¯1H(rn,q)=2ΩR2J2(Rrn)rn1q3,w¯2H(rn,q)=-2ΩR2J2(Rrn)rn1+λqγq2(q+νrn2+λqγ+1+νrn2λrqβ).
Applying the inverse Hankel transform to (3.10) and using the known formula ∫0Rr2J1(rrn)dr=R2rnJ2(Rrn),
we getw¯1(r,q)=2Ωrq3,w¯2(r,q)=-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)1+λqγq2(q+νrn2+λqγ+1+νrn2λrqβ).Using the identity 1(q+νrn2+λqγ+1+νrn2λrqβ)=1λ∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)kλrmqβm-k-1(qγ+1λ)k+1,(3.12b) can be written as w¯2(r,q)=-4Ωλ∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)k×λrm(qβm-k-3+λqγ+βm-k-3)(qγ+1/λ)k+1.
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=1∞J1(rrn)rnJ2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)k×λrm(qβm-k-3+λqγ+βm-k-3)(qγ+1/λ)k+1.
Now taking the inverse Laplace transform of (3.15), the velocity field w(r,t) is given by w(r,t)=Ωrt2-4Ωλ∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)kλrm×[Gγ,βm-k-3,k+1(-λ-1,t)+λGγ,γ+βm-k-3,k+1(-λ-1,t)],
where the generalized function G is defined by [26, Equations (97) and (101)] Ga,b,c(d,t)=L-1{qb(qa-d)c}=∑k=0∞dkΓ(c+k)Γ(c)Γ(k+1)t(c+k)a-b-1Γ[(c+k)a-b];Re(ac-b)>0,|dqa|<1.

3.2. Calculation of the Shear Stress

Applying the Laplace transform to (2.6), we find that τ¯(r,q)=μ(1+λrqβ)(1+λqγ)(∂∂r-1r)w¯(r,q),
where, in view of (3.12a) and (3.12b), we have w¯(r,q)=w¯1(r,q)+w¯2(r,q)=2Ωrq3-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)1+λqγq2(q+νrn2+λqγ+1+νrn2λrqβ).
Now differentiating the last expression for w¯(r,q) partially with respect to r, we get (∂∂r-1r)w¯(r,q)=-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)1+λqγq2(q+νrn2+λqγ+1+νrn2λrqβ),
and introducing (3.20) into (3.18), we have τ¯(r,q)=4μΩ∑n=1∞J2(rrn)J2(Rrn)(1+λrqβ)q2(q+νrn2+λqγ+1+νrn2λrqβ).
In view of identity (3.13), (3.21) can be equivalently written as τ¯(r,q)=4μΩλ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)k×λrm[qβm-k-3(qγ+1/λ)k+1+λrqβm+β-k-3(qγ+1/λ)k+1].
Now taking the inverse Laplace transform of both sides of (3.22) and using (3.17), we find that τ(r,t)=4μΩλ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)kλrm×[Gγ,βm-k-3,k+1(-λ-1,t)+λrGγ,βm+β-k-3,k+1(-λ-1,t)].

4. The Special Cases4.1. 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)=Ωrt2-4Ωλ∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)k×λrm[G1,m-k-3,k+1(-λ-1,t)+λG1,m-k-2,k+1(-λ-1,t)]
and shear stress τOO(r,t)=4μΩλ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞∑m=0kk!m!(k-m)!(-νrn2λ)k×λrm[G1,m-k-3,k+1(-λ-1,t)+λrG1,m-k-2,k+1(-λ-1,t)],
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)=Ωrt2-4Ωλ∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞(-νrn2λ)k×[Gγ,-k-3,k+1(-λ-1,t)+λGγ,γ-k-3,k+1(-λ-1,t)]
and the associated shear stress τFM(r,t)=4μΩλ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞(-νrn2λ)kGγ,-k-3,k+1(-λ-1,t),
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)=Ωrt2-4Ωλ∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞(-νrn2λ)k×[G1,-k-3,k+1(-λ-1,t)+λG1,-k-2,k+1(-λ-1,t)]
and the associated shear stress τOM(r,t)=4μΩλ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞(-νrn2λ)kG1,-k-3,k+1(-λ-1,t),
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)=Ωrt2-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)×∑k=0∞(-νrn2)kG1-β,-βk-β-2,k+1(-νλrrn2,t)
and associated shear stress τFS(r,t)=4μΩ∑n=1∞J2(rrn)J2(Rrn)∑k=0∞(-νrn2)k×[G1-β,-βk-β-2,k+1(-νλrrn2,t)+λrG1-β,-βk-2,k+1(-νλrrn2,t)],
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)=Ωrt2-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)×∑k=0∞(-νrn2)kG1-β,-βk-β-2,k+1(-αrn2,t),τFS(r,t)=4Ω∑n=1∞J2(rrn)J2(Rrn)∑k=0∞(-νrn2)k×[μG1-β,-βk-β-2,k+1(-αrn2,t)+α1G1-β,-βk-2,k+1(-αrn2,t)].

4.5. Ordinary Second-Grade Fluid

Making β→1 into (4.9), we obtain velocity field wOS(r,t)=Ωrt2-4Ω∑n=1∞J1(rrn)rnJ2(Rrn)∑k=0∞(-νrn2)kG0,-k-3,k+1(-αrn2,t)
and the associated shear stress τOS(r,t)=4Ω∑n=1∞J2(rrn)J2(Rrn)∑k=0∞(-νrn2)k×[μG0,-k-3,k+1(-αrn2,t)+α1G0,-k-2,k+1(-αrn2,t)],
corresponding to an ordinary second-grade fluid. These solutions can also be simplified to give (see (A3)-(A4)) wOS(r,t)=Ωrt2-4Ων∑n=1∞J1(rrn)rn3J2(Rrn)×[t+1+αrn2νrn2{exp(-νrn2t1+αrn2)-1}],τOS(r,t)=4ρΩ∑n=1∞J2(rrn)rn2J2(Rrn)[t+1νrn2{exp(-νrn2t1+αrn2)-1}].

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)=Ωrt2-4Ων∑n=1∞J1(rrn)rn3J2(Rrn)[t+1νrn2{exp(-νrn2t)-1}]
and the associated shear stress τN(r,t)=4ρΩ∑n=1∞J2(rrn)rn2J2(Rrn)[t+1νrn2{exp(-νrn2t)-1}],
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.

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.

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 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.

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.

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.

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.

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 rn have been approximated by (4n+1)π/(4R).

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∑k=0∞∑m=0kk!m!(k-m)!(-νrn2)kλrmt-βm+k+2Γ(-βm+k+3)=∑k=0∞(-νrn2)kG1-β,-βk-β-2,k+1(-νλrrn2,t),∑k=0∞∑m=0kk!m!(k-m)!(-νrn2)kλrmt-βm-β+k+2Γ(-βm-β+k+3)=∑k=0∞(-νrn2)kG1-β,-βk-2,k+1(-νλrrn2,t),∑k=0∞(-νrn2)kG0,-k-2,k+1(-αrn2,t)=1νrn2{1-exp(-νrn2t1+αrn2)},∑k=0∞(-νrn2)kG0,-k-3,k+1(-αrn2,t)=1νrn2[t+1+αrn2νrn2{exp(-νrn2t1+αrn2)-1}],limλ→01λkGa,b,k(-λ-1,t)=t-b-1Γ(-b),b<0.

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