Flow of a fractional Oldroyd-B fluid over a plane wall that applies a time-dependent shear to the fluidConstantin Fetecau, Nazish Shahid, and Masood Khan Citation: AIP Conf. Proc. 1450, 65 (2012); doi: 10.1063/1.4724119 View online: http://dx.doi.org/10.1063/1.4724119 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1450&Issue=1 Published by the American Institute of Physics. Related ArticlesStability of flowing open fluidic channels AIP Advances 3, 022121 (2013) Longitudinal and transverse disturbances in unbounded, gas-fluidized beds Phys. Fluids 25, 023301 (2013) Finite amplitude vibrations of a sharp-edged beam immersed in a viscous fluid near a solid surface J. Appl. Phys. 112, 104907 (2012) Instability in evaporative binary mixtures. II. The effect of Rayleigh convection Phys. Fluids 24, 094102 (2012) Stability of a viscous pinching thread Phys. Fluids 24, 072103 (2012) Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Flow of a fractional Oldroyd-B fluid over a plane wall thatapplies a time-dependent shear to the fluidConstantin Fetecau∗, Nazish Shahid† and Masood Khan∗∗

∗Department of Mathematics, Technical University of Iasi, Iasi 700050, Associatemember of Academy of Romanian Scientists, 050094 Bucuresti, Romania

†Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan∗∗Department of Mathematics, Quaid-i-Azam Universiry, Islamabad 44000,Pakistan

c_fetecau@yahoo.com

Abstract. The unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives induced by a plane wall thatapplies a time-dependent shear stress fta to the fluid is studied using Fourier sine and Laplace transforms. Exact solutionsfor velocity and shear stress distributions are found in integral and series form in terms of generalized G functions. Theyare presented as a sum between the corresponding Newtonian solutions and non-Newtonian contributions and reduce toNewtonian solutions if relaxation and retardation times tend to zero. The solutions for fractional second grade and Maxwellfluids, as well as those for ordinary fluids, are obtained as limiting cases of general solutions. Finally, some special casesare considered and known solutions from the literature are recovered. An important relation with the first problem of Stokesis brought to light. The influence of fractional parameters on the fluid motion, as well as a comparison between models, isgraphically illustrated

Keywords: Fractional Oldroyd-B fluid, Time-dependent shear stress, Exact solutionsPACS: 47.15.-x; 47.50.-d.

INTRODUCTION

Many models have been proposed to describe the re-sponse characteristics of fluids that cannot be describedby classical Navier-Stokes equations. Among them, theOldroyd-B model can describe stress-relaxation, creepand normal stress differences that develop during sim-ple shear flows. This model can be viewed as one of themost successful models for describing the response ofa sub-class of polymeric liquids. It is amenable to anal-ysis and more importantly experimental corroboration.An Oldroyd-B fluid is one which stores energy like a lin-earized elastic solid, its dissipation however being dueto two dissipative mechanisms that implies that it arisesfrom a mixture of two viscous fluids. Recently, there hasbeen considerable interest in describing the behavior ofincompressible Oldroyd-B fluids [1-11]. However, sinceone may expect that the behavior of viscoelastic liquidsto deviate most from that of non-elastic non-Newtonianfluids in transient flows, it seems necessary to investigatenew transient flows in order to have an overall view ofelastic liquid behavior.In view of the above motivation, we are interested tofind exact solutions for the motion of an Oldroyd-Bfluid induced by an infinite flat plate that applies a time-dependent shear stress to the fluid. Such exact solutionsserve a dual purpose, that of providing an explicit so-lution to a problem that has physical relevance and asa means for testing the efficiency of complex numerical

schemes for flows in more complicated flow domains.An interesting aspect of the problem to be studied is thatunlike the usual no slip boundary condition, a boundarycondition on the shear stress is used. This is very impor-tant as in some problems, what is specified is the forceapplied on the boundary. It is also important to bear inmind that the "no slip" boundary condition may not benecessarily applicable to flows of polymeric fluids thatcan slip or slide on the boundary. Thus, the shear stressboundary condition is particularly meaningful. Further-more, in order to include a larger class of fluids, the gen-eral solutions will be established for Oldroyd-B fluidswith fractional derivatives. Particularly, the solutions forOldroyd-B fluids will be obtained as limiting cases.In the last time, the fractional calculus is increasinglyseen as an efficient tool and suitable framework withinwhich useful generalizations of various classical physicalconcepts can be obtained. The list of its applications isquite long and augments almost yearly. It includes frac-tal media [12], fractional wave diffusion [13], fractionalHamiltonian dynamics [14,15] as well as many other top-ics in physics. In other cases, it has been shown that theconstitutive equations employing fractional derivativesare linked to molecular theories [16]. In particular, it hasbeen shown that the predictions of fractional derivativeMaxwell model are in excellent agreement with the lin-ear viscoelastic data in glass transition and α- relaxationzones [17]. The use of fractional derivatives within thecontext of viscoelasticity was firstly proposed by Ger-

The 5th International Conference on Research and Education in MathematicsAIP Conf. Proc. 1450, 65-76 (2012); doi: 10.1063/1.4724119

© 2012 American Institute of Physics 978-0-7354-1049-7/$30.00

65

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mant [18]. Then, Slonimsky [19] introduced fractionalderivatives into Kelvin-Voigt model to describe the re-laxation processes. Subsequently, Bagley and Torvik [20,21] and Koeller [22], among others, extended the the-ory. They demonstrated that the theory of viscoelasticityof coiling polymers and the theory of hereditary solidmechanics predict constitutive relations with fractionalderivatives. As such, these models are consistent with ba-sic theories and are not arbitrary constructions that hap-pen to describe experimental data.It is also important pointing out that the interest for vis-coelastic fluids with fractional derivatives came frompractical problems. In order to predict the dynamic re-sponse of viscous dampers, for instance, Makris et al.[23] firstly used conventional models of viscoelastic-ity. It was not possible to achieve satisfactory fit of theexperimental data over the entire range of frequencies.However, a very good fit of the experimental data wasachieved when the fractional Maxwell model

τ +λDα τ = μDβ γ,

has been used. Here τ and γ are the shear stress andstrain, λ and μ are generalized material constants andDα is a fractional derivative operator of order α with re-spect to time. This model is a special case of the moregeneral model of Bagley and Torvik [20]. It collapsesto the conventional Maxwell model with α = β = 1, inwhich case λ and μ become the relaxation time and thedynamic viscosity, respectively. Based on the fact thatat vanishingly small strain rates, the behavior of the vis-coelastic fluid reduces to that of Newtonian fluid, the pa-rameter β was set equal to unity. The other three parame-ters were determined for the silicon gel fluid and the pre-dicted mechanical properties are in excellent agreementswith experimental results. Similar excellent agreementsbetween frequency sweep experimental data obtained onother polymers (e.g. polystyrenes) and theoretical predic-tions of linear fractional derivative models are reported in[24-26].However, despite these successful attempts, it must beemphasized that a constitutive relation should be ex-pressed in a three dimensional setting such that it is alsoframe indifferent. The first objective law which charac-terizes an incompressible fractional derivative Maxwellfluid seems to be that of Palade et al. [27, Eq. (16)].This constitutive relation, under linearization, reduces tothe fractional integral Maxwell model exhibited in [27.Eq. (8)]. Using the definition of a fractional integral,the last equality (8) is equivalent to the present equal-ity proposed by Makris et al. [23]. Consequently, if onewishes to study one-dimensional behavior only, then itwould appear that these models are successful. So in thefollowing we shall establish exact solutions for veloc-ity and shear stress corresponding to the unsteady flow

of an incompressible fractional Oldroyd-B fluid due toan infinite plate that applies a time-dependent shear tothe fluid. These solutions, that satisfy all imposed ini-tial and boundary conditions, are presented as sums ofNewtonian solutions and non-Newtonian contributions.They can easily be specialized to give both the solutionsfor fractional Maxwell and second grade fluids and thosefor ordinary fluids. Finally, the influence of fractional pa-rameters on the fluid motion, as well as a comparison be-tween models, is underlined by graphical illustrations.

GOVERNING EQUATIONS

For the problem under consideration we shall assume thevelocity field v and the extra-stress tensor S of the form

v = v(y, t) = v(y, t)i, S = S(y, t), (1)

where i is the unit vector along the x-direction of theCartesian coordinate system x, y and z. For this flow,the constraint of incompressibility is automatically sat-isfied. Substituting Eq. (1) into the constitutive equationscorresponding to an incompressible Oldroyd-B fluid andassuming that the fluid is at rest till the moment t = 0, weobtain the relevant equation [2,6](

1+λ∂∂ t

)τ(y, t) = μ

(1+λr

∂∂ t

)∂v(y, t)

∂y, (2)

where μ is the viscosity of the fluid, λ and λr are relax-ation and retardation times and τ(y, t) = Sxy(y, t) is thenontrivial shear stress.In the absence of body forces and a pressure gradient inthe flow direction, the balance of linear momentum leadsto the significant equation

∂τ(y, t)∂y

= ρ∂v(y, t)

∂ t, (3)

where ρ is the density of the fluid. Eliminating τ(y, t)between Eqs. (2) and (3), we obtain the following gov-erning equation

(1+λ∂∂ t

)∂v(y, t)

∂ t= ν(1+λr

∂∂ t

)∂ 2v(y, t)

∂y2, (4)

for velocity. Here, ν = μρ is the kinematic viscosity of the

fluid.The governing equations corresponding to incompress-ible fractional Oldroyd-B fluids (FOF), in such motions[28,29]

(1+λ α Dαt )τ(y, t) = μ

(1+λ β

r Dβt

) ∂v(y, t)

∂y, (5)

(1+λ α Dαt )

∂v(y, t)

∂ t= ν(1+λ β

r Dβt )

∂ 2v(y, t)

∂y2, (6)

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are derived from Eqs. (2) and (4) via substituting the in-ner time derivatives by the fractional differential operator(also called Caputo fractional operator with zero initialcondition) [30, 31]

Dpt f (t) =

1Γ(1− p)

∫ t

0

f ′(τ)(t− τ)p

dτ; 0≤ p < 1, (7)

where Γ(.) is the Gamma function. The constraint α ≥ βis explained in [32]. In the following the system of frac-tional partial differential equations (5) and (6) with ap-propriate initial and boundary conditions will be solvedby means of integral transforms.

STATEMENT OF THE PROBLEM ANDITS SOLUTION

Let us consider an incompressible FOF occupying thespace above a flat plate situated in the (x,z) plane. Ini-tially, the fluid as well as the plate is at rest. At timet = 0+ let the plate be pulled with the time-dependentshear

τ(0, t) =f

λ α

∫ t

0(t− s)aGα,0,1

(− 1

λ α ,s

)ds; a≥ 0, (8)

along the x-axis. Here, f and a are constants and (see [33],the pages 14 and 15)

Ga,b,c(d, t) =∞

∑j=0

Γ(c+ j)

Γ( j+1)Γ(c)ta( j+c)−b−1

Γ[a( j+ c)−b]d j.

Owing to the shear the fluid is gradually moved. Itsvelocity is of the form (1)1, the governing equations aregiven by Eqs. (5) and (6) while the appropriate initial andboundry conditions are

v(y,0) =∂v(y,0)

∂ t= 0, τ(y,0) = 0; y > 0, (9)

(1+λ α Dαt )τ(0, t) = μ

(1+λ β

r Dβt

) ∂v(y, t)

∂y

∣∣∣∣y=0

= f ta; t > 0.

(10)Moreover, the natural condition

v(y, t)→ 0 as y→ ∞, (11)

has to be also satisfied. Of course, as we shall see later,τ(0, t) given by Eq. (8) is just the solution of the frac-tional differential equation (10)1.

CALCULATION OF THE VELOCITY

Multiplying Eq. (6) by√

2π cos(yξ ), integrating the re-

sult with respect to y from 0 to infinity and taking into

account the above initial and boundary conditions, wefind that

(1+λ α Dαt )

∂vc(ξ , t)∂ t

+νξ 2(1+λ β

r Dβt

)vc(ξ , t) =

− f

ρta

√2π; ξ , t > 0, (12)

where the Fourier cosine transform vc(ξ , t) of v(y, t) has tosatisfy the initial conditions

vc(ξ ,0) =∂vc(ξ ,0)

∂ t= 0; ξ > 0. (13)

Applying the Laplace transform to Eq. (12), using theLaplace transform formula for sequential fractional derivatives[31] and having in mind the initial conditions (13), we find forthe image function vc(ξ ,q) of vc(ξ , t) the expression

vc(ξ ,q) =− f

ρ

√2π

Γ(a+1)qa+1

1q+λ α qα+1+νξ 2+ γξ 2qβ ,

(14)where q is the transform parameter and γ = νλ β

r . In order toobtain vc(ξ , t) = L−1{vc(ξ ,q)} and to avoid the burdensomecalculations of residues and contour integrals, we apply thediscrete inverse Laplace transform method [34]. However inorder to obtain a more suitable presentation of final results, wefirstly rewrite Eq. (14) in the equivalent form

vc(ξ ,q) =− f

μ

√2π

Γ(a+1)qa+1

1ξ 2 +

f

μ

√2π

1ξ 2 F(ξ ,q)+

+f

ρ

√2π

F(ξ ,q)G(ξ ,q), (15)

where F(ξ ,q) = F1(q)F2(ξ ,q) and

F1(q) =Γ(a+1)

qa, F2(ξ ,q) =

1q+νξ 2

and G(ξ ,q) =λ α qα + γξ 2qβ−1

q+λ α qα+1+νξ 2+ γξ 2qβ . (16)

Denoting by f1(t), f2(ξ , t), f (ξ , t) and g(ξ , t) the inverseLaplace transforms of F1(q), F2(ξ ,q), F(ξ ,q) and G(ξ ,q) andbearing in mind Eq. (A1) from the Appendix A, we can write

vc(ξ , t) =− f

μ

√2π

ta

ξ 2 +f

μ

√2π

1ξ 2 f (ξ , t)+

f

ρ

√2π

h(ξ , t),

(17)where

f (ξ , t) = ( f1 ∗ f2)(t) =

{e−νξ 2t , a = 0a∫ t0 (t− s)a−1e−νξ 2sds, a > 0

(18)and h(ξ , t) = L−1{F(ξ ,q)G(ξ ,q)} = ( f ∗ g)(t) =∫ t0 f (ξ , t− s)g(ξ ,s)ds.

Applying the inverse Fourier transform to Eq. (17) andusing Eqs. (A2) and (A3), we find for the velocity v(y, t), thesimple expression

v(y, t) = vN(y, t)+2 f

ρπ

∫ ∞

0h(ξ , t)cos(yξ )dξ , (19)

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where [34, Eq. (4.1) with α=0]

vN(y, t) =f

μyta− 2 f

μπ

∫ ∞

0{ta− f (ξ , t)cos(yξ )}dξ

ξ 2 , (20)

is the velocity corresponding to a Newtonian fluid performingthe same motion and

h(ξ , t) =∞

∑k=0

k

∑m=0

(−νξ 2

λ α

)kk!λ mβ

r

m!(k−m)!

∫ t

0f (ξ , t− s)

×[

Gα,αm,k+1

(− 1

λ α ,s

)

+νξ 2 λ βr

λ α Gα,βm,k+1

(− 1

λ α ,s

)]ds, (21)

with αm = α +mβ − k−1 and βm = (1+m)β − k−2.The velocity v(y, t), as it results from Eq. (19), is pre-

sented as a sum between the Newtonian solution vN(y, t) andthe non-Newtonian contribution

vnN(y, t) =2 f

ρπ

∫ ∞

0h(ξ , t)cos(yξ )dξ . (22)

Of course, in view of Eq. (A4), it clearly results that forλr and λ → 0, vnN(y, t)→ 0 and therefore v(y, t)→ vN(y, t).

Calculation of the shear stress

Applying the Laplace transform to Eq. (5), we obtain

τ(y,q) = μ1+λ β

r qβ

1+λ α qα∂ v(y,q)

∂y, (23)

where

v(y,q) =− 2 f

ρπΓ(a+1)

qa+1

∫ ∞

0

cos(yξ )q+λ α qα+1+νξ 2+ γξ 2qβ dξ ,

(24)is obtained from Eq. (14). Introducing Eq. (24) in Eq. (23), itresults

τ(y,q) =2 f ν

πΓ(a+1)

qa+11+λ β

r qβ

1+λ α qα

×∫ ∞

0

ξ sin(yξ )q+λ α qα+1+νξ 2+ γξ 2qβ dξ . (25)

In the following, in order to obtain for the shear stressτ(y, t) = L−1{τ(y,q)} a similar form to that of velocity, weshall use the identity

Γ(a+1)qa+1

1+λ βr qβ

1+λ α qα1

q+λ α qα+1+νξ 2+ γξ 2qβ =

=1

νξ 2

[Γ(a+1)

qa+1 −F(ξ ,q)]+F1(q)G1(ξ ,q)G2(ξ ,q),

where F1(.) and F(ξ , .) have been previously defined,

G1(ξ ,q) =1

q+λ α qα+1+νξ 2+νξ 2λ α qα ,

G2(ξ ,q)=λ β

r qβ −2λ α qα −λ 2α q2α − γξ 2λ α qα+β−1−νξ 2λ α qα−1

q+λ α qα+1+νξ 2+ γξ 2qβ

and follow the same way as before. In order to avoid repetition,we give the final result in the simple form

τ(y, t) = τN(y, t)+2 f ν

π

∫ ∞

0ξ sin(yξ )( f1 ∗g1 ∗g2)(t)dξ , (26)

where [34, Eq. (4.2) with α=0]

τN(y, t) = f ta− 2 f

π

∫ ∞

0

sin(yξ )ξ

f (ξ , t)dξ , (27)

represents the shear stress corresponding to Newtonian fluids

f1(t) =

{δ (t), a = 0ata−1, a > 0

,δ (.) being the Dirac delta f unction,

g1(ξ , t) =1

λ α

∞

∑k=0

k

∑m=0

(−νξ 2

λ α

)kk!λ mα

m!(k−m)!

×Gα,mα−k−1,k+1

(− 1

λ α , t

),

g2(ξ , t) =1

λ α

∞

∑k=0

k

∑m=0

(−νξ 2

λ α

)kk!λ mβ

r

m!(k−m)!×

[λ β

r Gα,βm+1,k+1

(− 1

λ α , t

)−2λ α Gα,αm,k+1

(− 1

λ α , t

)

−λ 2α Gα,αm+α,k+1

(− 1

λ α , t

)− γξ 2λ α Gα,βm+α,k+1

(− 1

λ α , t

)

−νξ 2λ α Gα,αm−1,k+1

(− 1

λ α , t

)].

A simple analysis clearly shows that τ(y, t)→ τN(y, t) for λr

and λ → 0.

Special cases a = 0,1,2,3, ...

By making a = 0 into Eqs. (19) and (26) and having inmind Eq. (18), (A5) and the entry 6 of Table 5 from [35], wefind that

v0(y, t) = v0N(y, t)+2 f

ρπ

∞

∑k=0

k

∑m=0

k!λ mβr

m!(k−m)!

×∫ ∞

0

(−νξ 2

λ α

)k

cos(yξ )∫ t

0e−νξ 2(t−s)

×{

Gα,αm,k+1(− 1

λ α ,s)

+νξ 2 λ βr

λ α Gα,βm,k+1(− 1

λ α ,s)

dsdξ ,

}(28)

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τ0(y, t) = τ0N(y, t)+2 f ν

π

∫ ∞

0ξ sin(yξ )

×∫ t

0g1(ξ , t− s)g2(ξ ,s)dsdξ , (29)

where the expressions of

v0N(y, t) =f

μy− 2 f

μπ

∫ ∞

0{1− e−νξ 2t cos(yξ )}dξ

ξ 2

=f y

μer f c

(y

2√

νt

)− 2 f

μ

√νt

πexp

(− y2

4νt

)(30)

and

τ0N(y, t) = f − 2 f

π

∫ t

0

sin(yξ )ξ

e−νξ 2t dξ

= f er f c

(y

2√

νt

), (31)

are identical to those obtained in [3, Eqs. (4.3) and (4.4)].The solutions corresponding to a = 1, namely,

v1(y, t) = v1N(y, t)+2 f

ρπ

∞

∑k=0

k

∑m=0

k!λ mβr

m!(m− k)!

×∫ ∞

0

(−νξ 2

λ α

)k

cos(yξ )∫ t

0e−νξ 2(t−s)

×{

Gα,αm−1,k+1(− 1

λ α ,s)+

νξ 2 λ βr

λ α Gα,βm−1,k+1(− 1

λ α ,s)

dsdξ ,

}(32)

τ1(y, t) = τ1N(y, t)+2 f ν

π

∫ ∞

0ξ sin(yξ )

∫ t

0(g1∗g2)(s)dsdξ ,

(33)where

v1N(y, t) =f

μyt− 2 f

μπ

∫ ∞

0

{t− 1− e−νξ 2t

νξ 2 cos(yξ )

}dξξ 2 , (34)

τ1N(y, t) = f t− 2 f

π

∫ ∞

0

1− e−νξ 2t

ξ 3 sin(yξ )dξ

= f

∫ t

0er f c

(y

2√

νs

)ds, (35)

are also identical to those obtained in [36, Eqs. (21) and (22)]by a different technique.In order to get Eq. (32), for instance, we made an integrationby parts into Eq. (21) and used Eqs. (18) and (A6). A simpleanalysis shows that

v1(y, t) =∫ t

0v0(y,s)ds and τ1(y, t) =

∫ t

0τ0(y,s)ds. (36)

Lengthy but straightforward computations allow us to provethat

vn(y, t) = n!∫ t

0

∫ s1

0

∫ s2

0...∫ sn−1

0v0(y,sn)dsndsn−1...ds1, (37)

τn(y, t) = n!∫ t

0

∫ s1

0

∫ s2

0...∫ sn−1

0τ0(y,sn)dsndsn−1...ds1. (38)

LIMITING CASES

The case λr → 0 (fractional Maxwell fluids)

Making λr → 0 into Eqs. (19) and (26) we get the solutions

vFM(y, t) = vN(y, t)+2 f

ρπ

∫ ∞

0cos(yξ )hFM(ξ , t)dξ , (39)

τFM(y, t) = τN(y, t)+2 f ν

π

∫ ∞

0ξ sin(yξ )( f1 ∗g1 ∗g2FM)(t)dξ ,

(40)corresponding to a Maxwell fluid with fractional derivativesperforming the same motion. Here f1(.) and g1(ξ , .) are thesame as before and

hFM(ξ , t)=∞

∑k=0

(−νξ 2

λ α

)k ∫ t

0f (ξ , t− s)Gα,α−k−1,k+1

(− 1

λ α ,s

)ds,

g2FM(ξ , t)=−∞

∑k=0

(−νξ 2

λ α

)k⎧⎨⎩

2Gα,α−k−1(− 1

λ α , t)

+λ α Gα,2α−k−1(− 1λ α , t)

+νξ 2Gα,α−k−2,k+1(− 1

λ α , t)⎫⎬⎭ .

Of course, in view of Eq. (A4), vFM(y, t) → vN(y, t) andτFM(y, t)→ τN(y, t) if λ → 0.

The case λ → 0 (fractional second gradefluids)

The solutions corresponding to second grade fluids withfractional derivatives can also be obtained as limiting cases ofgeneral solutions using Eq. (A4). However, simpler but equiva-lent forms of these solutions, namely

vFSG(y, t) = vN(y, t)+2 f

ρπγ∫ ∞

0ξ 2 cos(yξ )hFSG(ξ , t)dξ ,

(41)

τFSG(y, t) = τN(y, t)+2 f

πγ∫ ∞

0ξ sin(yξ )gFSG(ξ , t)dξ , (42)

are obtained making λ → 0 into Eqs. (14) and (25) and usingthe identity

1q+νξ 2+ γξ 2qβ =

∞

∑k=0

(−γξ 2)k qβk

(q+νξ 2)k+1 .

The two functions from Eqs. (41) and (42) are given by

hFSG(ξ , t) =∑∞

k=0(−γξ 2)k∫ t0 f (ξ , t− s)

×G1,βk+β−1,k+1(−νξ 2,s)ds,

gFSG(ξ , t) =Γ(a+1)∑∞

k=0(−γξ 2)k∫ t0 e−νξ 2(t−s)

×G1,β (k+1)−a,k+1(−νξ 2,s)ds.

The case α = β = 1 (Oldroyd-B fluids)

By making α = β = 1 into Eqs. (19) and (26) we obtainthe similar solutions for Oldroyd-B fluids. The velocity field

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v0(y, t), for instance, has the form

v0(y, t) = vN(y, t)+2 f

ρπ

∞

∑k=0

k

∑m=0

k!λ mr

m!(k−m)!

×∫ ∞

0

(−νξ 2

λ

)k

cos(yξ )∫ t

0f (ξ , t− s)

×{

G1,m−k,k+1(− 1

λ ,s)

νξ 2 λr

λ G1,m−k−1,k+1(− 1

λ ,s)}

dsdξ , (43)

where f (ξ , t) is given by Eq. (18). For a= 0, the correspondingsolution

v00(y, t) = vN(y, t)+2 f

ρπ

∞

∑k=0

k

∑m=0

k!λ mr

m!(k−m)!

×∫ ∞

0

(−νξ 2

λ

)k

cos(yξ )∫ t

0e−νξ 2(t−s)

×{

G1,m−k,k+1(− 1

λ ,s)

+νξ 2 λr

λ G1,m−k−1,k+1(− 1

λ ,s)}

dsdξ , (44)

as it results from Fig. 1, is equivalent to the known solution

v00(y, t) =f y

μ− 2 f

μπ

∫ ∞

0

[1− q2exp(q1t)−q1exp(q2t)

q2−q1

×cos(yξ )

]1

ξ 2 dξ , (45)

obtained in [3] by a different technique. In the last relation q1and q2 are the roots of the algebraic equation of second gradeλq2+(1+ γξ 2)q+νξ 2 = 0.

Equivalent but simpler expressions for v0(y, t) and τ0(y, t)can be obtained using new suitable decompositions for thecorresponding functions G(ξ , .), G1(ξ , .) and G2(ξ , .). Usingthe identity

λq+ γξ 2

λq2+(1+ γξ 2)q+νξ 2 =q+ α

2λ

(q+ α2λ )

2− ( β2λ )

2

−1− γξ 2

β

β2λ

(q+ α2λ )

2− ( β2λ )

2,

for instance, we find that

v0(y, t) = vN(y, t)+2 f

ρπ

∫ ∞

0cos(yξ )

∫ t

0f (ξ , t− s)

×[

ch

(β s

2λ

)− 1− γξ 2

βsh

(β s

2λ

)]exp

(−αs

2λ

)dsdξ , (46)

where α = 1+ γξ 2 and β =√(1+ γξ 2)2−4νλξ 2. The

equivalence of the solutions given by Eqs. (45) and (46) (witha = 0) is shown by Fig.2. It can also be proved by directcomputations.

The case λr → 0 and α = 1 (Maxwell fluids)

The solutions corresponding to Maxwell fluids performingthe same motion are immediately obtained from Eqs. (19) and(26) by making λr → 0 and α = 1. However, they can alsobe obtained from Eqs. (39) and (40) for α = 1 or from thesolutions corresponding to Oldroyd-B fluids for λr → 0.

The case λ → 0 and β = 1 (Second gradefluids)

By letting now β = 1 into Eqs. (41) and (42) we get thesimilar solutions for second grade fluids. They can also beobtained from general solutions (for λ → 0 and β = 1) or fromthe solutions of the Oldroyd-B fluids (for λ → 0). The solutionscorresponding to a = 0 and 1, for instance,

v0SG(y, t) = v0N(y, t)+2 f γρπ

∞

∑k=0

∫ ∞

0ξ 2(−γξ 2)k cos(yξ )

×∫ t

0e−νξ 2(t−s)G1,k,k+1(−νξ 2,s)dsdξ , (47)

τ0SG(y, t) = τ0N(y, t)+2 f γπ

∞

∑k=0

∫ ∞

0ξ (−γξ 2)k sin(yξ )

×∫ t

0e−νξ 2(t−s)G1,k+1,k+1(−νξ 2,s)dsdξ , (48)

v1SG(y, t) = v1N(y, t)+2 f

ρπγ

∞

∑k=0

∫ ∞

0ξ 2(−γξ 2)k cos(yξ )

×∫ t

0e−νξ 2(t−s)G1,k−1,k+1(−νξ 2,s)dsdξ , (49)

τ1SG(y, t) = τ1N(y, t)+2 f

πγ

∞

∑k=0

∫ ∞

0ξ (−γξ 2)k sin(yξ )

×∫ t

0e−νξ 2(t−s)G1,k,k+1(−νξ 2,s)dsdξ , (50)

are immediately obtained from Eqs. (41) and (42) for a = 0or 1 and β = 1. Finally, it is worth pointing out that in viewof Eqs. (B1),(B2),(B3) and (B4) from the Appendix B, thesesolutions take the simplified forms

v0SG(y, t) =f

μyt− 2 f

μπ

∫ ∞

0

[1− cos(yξ )exp

(− νξ 2t

1+ γξ 2

)]

× 1ξ 2 dξ , (51)

τ0SG(y, t) = f H(t)− 2 f H(t)

π

∫ ∞

0

sin(yξ )ξ (1+ γξ 2)

×exp

(− νξ 2t

1+ γξ 2

)dξ , (52)

70

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and

v1SG(y, t) =f

μyt− 2 f

μπ

∫ ∞

0

[t− 1+ γξ 2

νξ 2

×[1− exp

(− νξ 2t

1+ γξ 2

)]cos(yξ )

]1

ξ 2 dξ , (53)

τ1SG(y, t) = f t− 2 f

νπ

∫ ∞

0

[1− exp

(− νξ 2t

1+ γξ 2

)]

× sin(yξ )ξ 3 dξ , (54)

obtained in [3, Eq. (4.1)], respectively [36, Eqs. (19) and (20)]by a different technique.

CONCLUSIONS AND NUMERICALRESULTS

The main purpose of this paper is to provide exact solu-tions for velocity and shear stress corresponding to the unsteadymotion of an Oldroyd-B fluid due to an infinite plate that ap-plies a shear stress fta to the fluid. However, for generality,these solutions have been established for a larger class of fluids,namely Oldroyd-B fluids with fractional derivatives. They arepresented as a sum of Newtonian solutions and non-Newtoniancontributions and satisfy all imposed initial and boundary con-ditions. The non-Newtonian contributions, as expected, tend tozero for λ and λr → 0. Furthermore, the similar solutions forfractional Maxwell and second grade fluids as well as those forordinary Oldroyd-B , Maxwell and second grade fluids are alsoobtained as limiting cases of general solutions for λr → 0 orλ → 0 , respectively α = β = 1, λr → 0 and α = 1 or λ → 0and β = 1.

Finally, in order to establish a relation with the motionover a moving plate, let us remember the velocity fields (see[37, Eq. (3)] and [6, Eq. (23)])

v0(y, t) =V H(t)

[1− 2

π

∫ ∞

0

sin(yξ )ξ (1+ γξ 2)

exp

(− νξ 2t

1+ γξ 2

)dξ], (55)

v1(y, t) = At− 2A

νπ

∫ ∞

0

{1− exp

(− νξ 2t

1+ γξ 2

)}sin(yξ )

ξ 3 dξ ,

(56)corresponding to the unsteady motion of a second grade fluiddue to a suddenly moved plate or a constantly acceleratingplate (a plate that slides in its plane with a velocity V or At).As form, these expressions are identical to those of the shearstresses τ0SG(y, t) and τ1SG(y, t) given by Eqs. (52) and (54)(corresponding to the motion induced by a plate that applies ashear stress f or ft to the fluid). This is not a surprise because asimple analysis of the equations (2) and (3) with λ = 0 showsthat the shear stress τ(y, t) in such motions of second gradefluids satisfies the same governing equation as velocity, i.e.

∂τ(y, t)∂ t

=(ν+γ∂∂ t

)∂ 2τ(y, t)

∂y2like

∂v(y, t)

∂ t=(ν+γ

∂∂ t

)∂ 2v(y, t)

∂y2.

Consequently, the present results regarding second grade fluidsbring about exact solutions for the velocity v(y, t) correspond-ing to the unsteady motion due to an infinite plate that slides inits plane with a velocity Ata.

Furthermore, eliminating v(y, t) between Eqs. (2) and (3),we obtain for the shear stress τ(y, t) a governing equation

(1+λ∂∂ t

)∂τ(y, t)

∂ t= ν(1+λr

∂∂ t

)∂ 2τ(y, t)

∂y2; t > 0,

of the same form as Eq. (4) for velocity. Consequently, thepresent results also allow us to present close form solutions forthe velocity of Maxwell and Oldroyd-B fluids over an infiniteplate that is moving in its plane according to the boundarycondition.

v(0, t) =A

λ

∫ t

0(t− s)aG1,0,1(− 1

λ,s)ds; a≥ 0.

Now, in order to bring to light some relevant physicalaspects of the obtained results, the influence of fractionalparameters on the fluid velocity is underlined by graphicalillustrations. A series of calculations were performed fordifferent solutions with typical values. The velocity of thefluid, as it results from Fig. 3 is an increasing function withrespect to α . Consequently, a fractional Maxwell fluid flowsslower in comparison with an ordinary Maxwell fluid. Theinfluence of fractional parameter β on velocity is shown in Fig.4. The velocity of the fluid is an increasing function of β in arelative small neighborhood of the plate only. Therefore, in thevicinity of the plate the fractional second grade fluid also flowsslower in comparison with an ordinary second grade fluid.A comparison between Oldroyd-B and fractional Oldroyd-Bmodels is realized in Fig. 5. As it was to be expected, for αand β → 1 the diagrams of velocity tend to that correspondingto the Oldroyd-B fluid. The units of all material constants inFigs. 1-5 are ISI units.

Appendix A

L−1{

Γ(a+1)qa

}=

{δ (t), a = 0ata−1, a > 0

;

L

{1

q+νξ 2

}= e−νξ 2t , (A1)

where δ (.) is the Dirac delta function and (δ ∗ f )(t) = f (t).∫ ∞

0

1− cos(yξ )ξ 2 dξ =

π2

y;

1z+a

=∞

∑k=0

(−1)k zk

ak+1 ;

(b+1)k =k

∑m=0

k!bm

m!(k−m)!. (A2)

Ga,b,c(d, t) = L−1{

qb

(qa−d)c

};

Re(ac−b)> 0, Re(q)> 0 and | p

qa|< 1. (A3)

71

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limλ→0

1λ αk

Ga,b,k

(− 1

λ α , t

)=

t−b−1

Γ(−b)i f b < 0. (A4)

2π

∫ ∞

0

[1− e−νξ 2t cos(yξ )

] dξξ 2 = 2

√νt

πexp

(− y2

4νt

)

+yer f

(y

2√

νt

). (A5)

∫ t

0Ga,b,c(d,s)ds = Ga,b−1,c(d, t). (A6)

Appendix B

γξ 2∞

∑k=0

(−γξ 2)k∫ t

0e−νξ 2(t−s)G1,k,k+1(−νξ 2,s)ds

=1

νξ 2

[exp

(− νξ 2t

1+ γξ 2

)− e−νξ 2t

]. (B1)

λr

∞

∑k=0

(−γξ 2)k∫ t

0e−νξ 2(t−s)G1,k+1,k+1(−νξ 2,s)ds

=1

νξ 2

[e−νξ 2t − 1

1+ γξ 2 exp

(− νξ 2t

1+ γξ 2

)]. (B2)

γξ 2∞

∑k=0

(−γξ 2)k∫ t

0e−νξ 2(t−s)G1,k−1,k+1(−νξ 2,s)ds

=1

ν2ξ 4

[γξ 2+ e−νξ 2t − (1+ γξ 2)exp

(− νξ 2t

1+ γξ 2

)]. (B3)

λr

∞

∑k=0

(−γξ 2)k∫ t

0e−νξ 2(t−s)G1,k,k+1(−νξ 2,s)ds

=1

ν2ξ 4

[exp(− νξ 2t

1+ γξ 2 )− e−νξ 2t

]. (B4)

In order to prove the last two relations, we use the nextidentities

γξ 2

q+νξ 2

∞

∑k=0

(−γξ 2)k qk−1

(q+νξ 2)k+1

=1

q+νξ 2γξ 2

q[(q+νξ 2)+ γξ 2q]

=1

q+νξ 2γξ 2

1+ γξ 21

q(

q+ νξ 2

1+γξ 2

) ,

λr

q+νξ 2

∞

∑k=0

(−γξ 2)k qk

(q+νξ 2)k+1

=1

q+νξ 2λr

q+νξ 2+ γξ 2q

=1

ν2ξ 4

⎡⎣ 1

q+ νξ 2

1+γξ 2

− 1q+νξ 2

⎤⎦ .

Applying the inverse Laplace transform to the last iden-tity, for instance, and using (A1)2 and the property( f1 ∗ f2)(t) =

∫ t0 f1(t− s) f2(s)ds = L−1{F1(q)F2(q)}, we

immediately obtain (B4).

ACKNOWLEDGMENTS

The first author thanks the School of Graduate Studies(GSO), Universiti Putra Malaysia for the financial Aid toattend international conference.

REFERENCES

1. T. Hayat, M. Khan, M. Ayub, Exact solutions of flowproblems of an Oldroyd-B fluid, Appl. Math. Comput. 151(2004) 105-119..

2. W. C. Tan, T. Masuoka, Stokes’ first problem for anOldroyd-B fluid in a porous half space, Phys. Fluids 17(2005) 023101-7..

3. C. Fetecau, K. Kannan, A note on an unsteady flow ofan Oldroyd-B fluid, Int. J. Math. Math. Sci. 19 (2005)3185-3194..

4. T. Hayat, M. Hussain, M. Khan, Hall effect on flows ofan Oldroyd-B fluid through porous medium for cylindricalgeometries, Comput. Math. Appl. 52 (2006) 269-282.

5. N. Aksel, C. Fetecau, M. Scholle, Starting solutions forsome unsteady unidirectional flows of Oldroyd-B fluids, Z.Angew. Math. Phys. 57 (2006) 1-17.

6. C. Fetecau, S. C. Prasad, K. R. Rajagopal, A note onthe flow induced by a constantly accelerating plate in anOldroyd-B fluid, Appl. Math. Model. 31 (2007) 647-654.

7. T. Hayat, S. B. Khan, M. Khan, The influence of Hallcurrent on the rotating flow of an Oldroyd-B fluid in aporous medium, Nonlinear Dynam. 47 (2007) 353-362.

8. Corina Fetecau, T. Hayat, C. Fetecau, Starting solutions forOscillating motions of Oldroyd-B fluids, J. Non-Newton.Fluid Mech. 153 (2008) 191-201.

9. M. Hussain, T. Hayat, C. Fetecau, S. Asghar, On acceleratedflows of an Oldroyd-B fluid in a porous medium, NonlinearAnal. Real World Appl. 9 (2008) 1394-1408.

10. S. McGinty, S. McKee, R. McDermott, Analytic solutionsof Newtonian and non-Newtonian pipe flows subject to ageneral time-dependent pressure gradient, J. Non-Newton.Fluid. Mech. 162 (2009) 54-77.

11. Asia Anjum, M. Ayub, M. Khan, Starting solutions foroscillating motions of an Oldroyd-B fluid over a plane wall,Commun Nonlinear Sci Number Simulat, doi: 10.1016/j.cnsns.2011.05.004.

12. A. Carpinteri, F. Mainardi, Fractals and FractionalCalculus in Continuum Mechanics, Springer, New York,1997.

13. O. P. Agrawal, Solution for a fractional diffusion-waveequation defined in a bounded domain, Nonlinear Dynam.29 (2002) 145-155.

14. M. Seredynska, A. Hanyga, Nonlinear differentialequations with fractional damping with applications to the1dof and 2dof pendulum, Acta Mech. 176 (2005) 169-183.

72

Downloaded 15 Mar 2013 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

15. D. Baleanu, S. I. Muslih, K. Tas, Fractional Hamiltoniananalysis of higher order derivatives systems, J. Math. Phys.47, 10 (2006) art. no. 103503, 8 pp.

16. C. Friedrich, Relaxation and retardation functions of theMaxwell model with fractional derivatives, Rheol. Acta 30(1991) 151-158.

17. L. I. Palade, V. Verney, P. Attané, A modified fractionalmodel to describe the entire viscoelastic behavior ofpolybutadienes from flow to glassy regime, Rheol. Acta 35(1996) 265-273.

18. A. Germant, On fractional differentials, philosophicalMagazine 25 (1938) 540-549.

19. G. L. Slonimsky, Laws of mechanical relaxation processesin polymer, J. Polym. Pol. Sym. 16 (1967) 1667-1672.

20. R. L. Bagley, P. J. Torvik, Fractional Calculus- adifferential approach to the analysis of viscoelasticitydamped structures, AIAA J. 21 (1983) 742-748.

21. R. L. Bagley, P. J. Torvik, A theoretical basis for theapplications of fractional calculus of viscoelasticity, J.Rheol. 27 (1983) 201-210.

22. R. C. Koeller, Applications of fractional calculus to thetheory of viscoelasticity, J. Appl. Mech. 51 (1984) 299-307.

23. N. Makris, G. F. Dargush, M. C. Constantinou, Dynamicanalysis of generlized viscoelastic fluids, J. Eng. Mech. 119(1993) 1663-1679.

24. C. Friedrich, Relaxation functions of rheologicalconstitutive equations with fractional derivatives:Thermodynamical constraints, in: J. Casas-Vasquez, D. Jou(Eds.), Lecture Notes in Phys., vol. 381, Springer-Verlag,New York, 1991.

25. H. H. Kausch, N. Heymans, C. J. Plummer, P. Decroly,Matériaux Polymers: Propriétés Mécaniques et Physiques,Presses Polytechniques et Universitaires Romandes,Lausanne, 2001, p. 195.

26. F. Mainardi, Fractional Calculus and Waves in linearviscoelasticity. An introduction to Mathematical Models,Imperial College Press, London, 2010.

27. L. I. Palade, P. Attané, R. R. Huilgol, B. Mena, Anomalousstability behavior of a properly invariant constitutiveequation which generalizes fractional derivative models,Int. J. Eng. Sci. 37 (1999) 315-329.

28. D. Vieru, Corina Fetecau, C. Fetecau, Flow of ageneralized Oldroyd-B fluid due to a costantly acceleratingplate, Appl. Math. Comput. 201 (2008) 834-842.

29. H. T. Qi, M. Y. Xu, Some unsteady unidirectional flows ofa generalized Oldroyd-B fluid with fractional derivatives,Appl. Math. Model. 33 (2009) 4184-4191.

30. S. G. Samko, A. A. Kilbas, O. I. Marichev, FractionalIntegrals and derivatives: Theory and Applications, Gordonand Breach, Amsterdam, 1993.

31. I. Podlubny, Fractional Differential Equations, AcademicPress, San Diego, 1999.

32. D. Y. Song, T. Q. Jiang, Study on the constitutive equationwith fractional derivative for the viscoelastic fluids.Modified Jeffreys model and its application, Rheol. Acta 37(1998) 512-517.

33. C. F. Lorenzo, T. T. Hartley, Generalized Functions forFractional Calculus, NASA/TP-1999-209424, 1999.

34. D. Vieru, Corina Fetecau, A. Sohail, Flow due to a platethat applies an accelerated shear to a seond grade fluidbetween two parallel walls perpendicular to the plate, Z.Angew. Math. Phys. 62 (2011) 161-172.

35. I. N. Sneddon, Functional Analysis, Encyclopedia ofPysics, Vol. 2, Springer, Berlin, Gottingen, Heidelberg,1955.

36. Nazish Shahid, Mehwish Rana, M. A. Imran, Flow ofan Oldroyd-B fluid over an infinite plate subject to atime-dependent shear stress, sent for publication.

37. I. C. Christov, C. I. Christov, Comment on ”On a classof exact solutions of the equations of motion of a secondgrade fluid” by C. Fetecau and J. Zierep (Acta Mech. 150,135-138, 2001), Acta Mech.

73

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Fig. 1. Profiles of the velocity v(y,t) given by Eq. (44) - curves , v12 , v13

and Eq. (45) - curves

v11(y)

= 0.003974, = 3,902,

(y) (y)

v21(y), v22(y), v23(y), for

f = -2, = 2, = 1 and different values of t. r

Fig. 2. Profiles of the velocity v(y,t) given by Eq. (45) - curves , v12 , v13

and Eq. (46) - curves

v11(y)

= 0.003974, = 3,902,

(y) (y)

v21(y), v22(y), v23(y), for

f = -2, = 2, = 1 and different values of t. r

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

v11 y( )

v12 y( )

v13 y( )

v21 y( )

v22 y( )

v23 y( )

y

t = 5s

t = 10s

t = 15s

0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

v11 y( )

v12 y( )

v13 y( )

v21 y( )

v22 y( )

v23 y( )

y

t = 5s

t = 10s

t = 15s

74

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Fig. 3. Profiles of the velocity v(y,t) given by Eq. (19) - curves , v2 , v3v1(y)

= 0.003974, = 3,902, (y) (y)

for a = 1, f = -2, = 2, = 1, = 0.2, t = 10s

and different values of .

r

Fig. 4. Profiles of the velocity v(y,t) given by Eq. (19) - curves , v2 , v3v1(y)

= 0.003974, = 3,902, (y) (y)

for f = -2, = 2, = 1, = 0.95, t = 10s

and different values of .

ra = 1,

= 0.65

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

v1 y( )

v2 y( )

v3 y( )

y

= 0.75

= 0.95

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

v1 y( )

v2 y( )

v3 y( )

y

= 0.01

= 0.50

= 0.90

75

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Fig. 5. Profiles of the velocity v(y,t) for fractional Oldroyd-B fluid - curves v1FOF(y),

v2FOF(y) and Oldroyd-B fluid - curve , for a = 1, withvOF(y) = 0.003974,

= 3,902,

f = -2, = 2, = 1, t = 10s and different values of and . r

= 0.70, = 0.60

= 0.85, = 0.85

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

v1FOF y( )

v2FOF y( )

vOF y( )

y

76

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