Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

3D flow of a generalized Oldroyd-B fluid induced by a constant pressuregradient between two side walls perpendicular to a plateLiancun Zheng a,∗, Zhenlin Guo a, Xinxin Zhang b

a Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, Chinab Mechanical Engineering School, University of Science and Technology Beijing, Beijing 100083, China

a r t i c l e i n f o

Article history:Received 7 November 2010Accepted 6 June 2011

Keywords:Viscoelastic fluidGeneralized Oldroyd-B modelFractional calculusFox H-function

a b s t r a c t

This paper deals with the 3D flow of a generalized Oldroyd-B fluid due to a constantpressure gradient between two side walls perpendicular to a plate. The fractional calculusapproach is used to establish the constitutive relationship of the non-Newtonian fluidmodel. Exact analytic solutions for the velocity and stress fields, in terms of the Fox H-function, are established by means of the finite Fourier sine transform and the Laplacetransform. Solutions similar to those for ordinary Oldroyd-B fluid as well as those forMaxwell and second-grade fluids are also obtained as limiting cases of the resultspresented. Furthermore, 3D figures for velocity and shear stress fields are presented for thefirst time for certain values of the parameters, and the associated transport characteristicsare analyzed and discussed.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, considerable attention has been devoted to the study of non-Newtonian fluids. The main reason for this is thatfluids (such as molten plastics, paints, pulps, slurries, emulsions, petroleum being drilled, blood and other similar entities),which do not obey the Newtonian postulate that the stress tensor is directly proportional to the rate of change of thedeformation tensor, and show flow characteristics quite different to those of the Newtonian fluids, are found in variousengineering applications. Therefore, various constitutive equations have been proposed for these fluids. The models areusually classified as fluids of differential, rate and integral types [1–3], among which the models of rate type have receivedparticular attention. An important class of non-Newtonian fluids is that of the viscoelastic fluids, which exhibit both elasticand viscous properties. It has been found that the viscoelastic generalized Oldroyd-B fluid can be used to approximate theresponse of many dilute polymeric liquids successfully, and this approach has been widely applied to flow problems withsmall relaxation and retardation times, with classical Newtonian and Maxwell fluids being included as special cases.

The fractional derivatives [1–3] were found to be quite flexible for describing the behaviors of viscoelastic fluids. A verygood fit of experimental data was achieved by Song and Jiang in 1998 for the viscoelastic fluid constitutive equation witha fractional derivative [4]. In general, such constitutive equations were modified from the well known fluid models byreplacing the time derivative of integer order with the so-called Riemann–Liouville fractional calculus operators [5,6], fromwhich one can define non-integer order integrals and derivatives.

In recent years, the interest in flows of non-Newtonian fluids has considerably increased and many researchers are en-gaged in obtaining exact solutions by introducing the fractional calculus approach into various rheological problems [7–20].Among these, some problems concerning unsteady flows between two side walls perpendicular to a plate have already

∗ Corresponding author. Tel.: +86 1062332589; fax: +86 10 62332462.E-mail addresses: liancunzheng@sina.com, liancunzheng@163.com (L. Zheng).

1468-1218/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2011.06.010

3500 L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508

been investigated. Fetecau et al. [7,8] and Shah [11] studied some accelerated flows of generalized Oldroyd-B fluid betweentwo side walls perpendicular to a plate. Hayat et al. [14] and Vieru et al. [15] discussed the flow of Maxwell fluid due toa plate, with constant velocity. Fetecau et al. [18] investigated the unsteady flow of a second-grade fluid induced by thetime-dependent motion of a plate between two side walls perpendicular to the plate. Khan and Wang [20] discussed theunsteady flow of a generalized second-grade fluid between two side walls perpendicular to a plate.

Motivated by the above mentioned works, in this paper we deal with the 3D flow of a generalized Oldroyd-B fluid dueto a constant pressure gradient between two side walls perpendicular to a plate. The fractional calculus approach is usedto establish the constitutive relationship of the non-Newtonian fluid model. The exact solutions, obtained by means of afinite Fourier sine transform and a discrete Laplace transform [21], are presented using series forms in terms of the FoxH-function [22]. Furthermore, the 3D figures for velocity and shear stress fields are presented for the first time for certainvalues of the parameters and the associated transport characteristics are analyzed and discussed.

2. The governing equation

The constitutive equations for an incompressible generalized Oldroyd-B fluid, as obtained from [7,8], are

T = −pI + S (1)1 + λ

Dα

Dtα

S = µ

1 + θ

Dβ

Dtβ

A1 (2)

where T denotes the Cauchy stress tensor, −pI is the indeterminate spherical stress, S is the extra stress tensor, A1 = L+ LTis the first Rivlin–Ericksen tensor with the velocity gradient L = gradV, µ is the dynamic viscosity of the fluid, λ and θ arerelaxation and retardation times, α and β are the fractional calculus parameters such that 0 ≤ α ≤ β ≤ 1, and

DαSDtα

= Dαt S + (V · ∇) S − LS − SLT (3)

DβA1

Dtβ= Dβ

t A1 + (V · ∇)A1 − LA1 −A1 LT (4)

in which Dαt and Dβ

t are the fractional differentiation operators of order α and β based on the Riemann–Liouville definition,defined as [5,6]

Dpt [f (t)] =

1Γ (1 − p)

ddt

∫ t

0

f (τ )

(t − τ)pdτ , 0 ≤ p ≤ 1. (5)

Γ (·) denotes the Gamma function. This model reduces to the ordinary Oldroyd-B model when α = β = 1.In the following we shall determine a velocity field and an extra stress of the form

V = (u (y, z, t) , 0, 0) , S = S (y, z, t) , (6)

where u (y, z, t) is the velocity in the x-coordinate direction.Substituting Eq. (6) into (2) and taking into account the initial condition S (y, z, 0) = 0 (the fluid being at rest up to the

moment t = 0), we obtain Syy = Syz = Szz = 0 [8,23] and the relevant partial differential equations1 + λDα

t

τ1 = µ

1 + θDβ

t

∂yu (y, z, t) , (7)

1 + λDαt

τ2 = µ

1 + θDβ

t

∂zu (y, z, t) , (8)

in which τ1 = Sxy and τ2 = Sxz are tangential stresses.The balance of linear momentum, in the absence of body forces, reduces to [8]

∂yτ1 + ∂zτ2 − ∂xp = ρ∂tu, (9)

∂yp = ∂zp = 0, (10)

where ρ is the constant density of the fluid and ∂p/∂x is the pressure gradient along x-axis. Eliminating τ1 and τ2 betweenEqs. (7) and (8) and assuming that there is a constant pressure gradient in the flow direction, we find the governing equationin the form

1 + λDαt

∂u (y, z, t)∂t

= ν1 + θDβ

t

∂2

∂y2+

∂2

∂z2

u (y, z, t) −

1ρ

1 + λDα

t

∂p∂x

, (11)

in which ν = µ/ρ is the kinematic viscosity of the fluid.

L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508 3501

3. Statement of the problem

Let us consider an incompressible generalized Oldroyd-B fluid occupying the space above a flat plate perpendicular to they-axis and between two side walls perpendicular to the plate. Initially, the fluid is at rest and at time t = 0+ it is suddenlyset into motion by a constant pressure gradient in the x-direction. Its velocity and stress field are of the form (6) and theassociated initial and boundary conditions are

u (y, z, 0) =∂u (y, z, 0)

∂t= 0, y ≥ 0 and 0 ≤ z ≤ h (12)

u (0, z, t) = 0, t ≥ 0 and 0 ≤ z ≤ h (13)

u (y, z, t) =∂u (y, z, t)

∂y= 0, y, t ≥ 0 and z = 0, h (14)

where h is the distance between the two side walls. Moreover, the natural condition

∂u (y, z, t)∂y

→ 0 as y → ∞ and t ≥ 0, 0 < z < h (15)

also has to be satisfied, which is a consequence of the fact that the velocity of the fluid becomes steady along the y-axis asy → ∞ and there is no shear along the y-axis [24].

4. Calculation of the velocity field

Multiplying both sides of Eq. (11) by sin( nπzh ), and integrating the result with respect to z from 0 to h, we attain the

differential equations1 + λDα

t

∂un (y, n, t)∂t

= ν1 + θDβ

t

∂2

∂y2un (y, n, t) + A

hnπ

1 + λ

t−α

Γ (1 − α)

1 − (−1)n

− ν

nπh

2 1 + θDβ

t

un (y, n, t) . (16)

Applying the Laplace transform to Eq. (16) and using the Laplace transform formula for sequential fractionalderivatives [21],

us (y, z, s) =

∫∞

0u(y, t)e−stdt, s > 0, (17)

we find that the image functions of usn (y, n, s) are given by

∂2

∂y2usn (y, n, s) −

ξ 2

+s (1 + λsα)

ν1 + θsβ

usn (y, n, s) +Aνξ

1 − (−1)n

(1 + λsα)

s1 + θsβ

= 0, (18)

where ξ =nπh .

The solution of Eq. (18) satisfying the boundary conditions (13) and (15) is of the following form:

usn =−A

1 − (−1)n

ξ s2

νξ 2 (1+θsβ)

s(1+λsα)+ 1

exp−y

ξ 2 +

s (1 + λsα)

ν1 + θsβ

− 1

. (19)

In order to obtain an analytical solution for this problem and to avoid lengthy calculations of residues and contourintegrals, we will apply the discrete inverse Laplace transform method [21] to get the velocity fields. Now we rewriteEq. (19) in series form as

usn = −A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

∞−m=0

(−1)i+j+k+l+myj

j!k!l!m!ν i−kξ 2i+j−2k−1

×Γ−

j2 + k

Γ (i − k + l) Γ (−i + k + m)

Γ−

j2

Γ (i − k) Γ (−i + k)

λ−i+k−lθ i−k−m

si−k+(i−k+l)α−(i−k−m)β+2

+ A1 − (−1)n

∞−i=0

∞−p=0

∞−n=0

(−1)i+p+q

p!q!ν iξ 2i−1 Γ (i + p) Γ (−i + q)

Γ (i) Γ (−i)λ−i−pθ i−q

si+(i+p)α−(i−q)β+2. (20)

3502 L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508

Applying the discrete inverse Laplace transform to Eq. (20), we can get

un = −A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

∞−m=0

(−1)i+j+k+l+myj

j!k!l!m!ν i−kξ 2i+j−2k−1λ−i+k−lθ i−k−m

×Γ−

j2 + k

Γ (i − k + l) Γ (−i + k + m) si−k+(i−k+l)α−(i−k−m)β+1

Γ−

j2

Γ (−i + k) Γ (i − k) Γ (i − k + (i − k + l) α − (i − k − m) β + 2)

+ A1 − (−1)n

∞−i=0

∞−p=0

∞−q=0

(−1)i+p+q

p!q!ν iξ 2i−1λ−i−pθ i−q

×Γ (i + p) Γ (−i + q) si+(i+p)α−(i−q)β+1

Γ (i) Γ (−i) Γ (i + (i + p) α − (i − q) β + 2). (21)

Finally, the inverse finite Fourier sine transform gives the analytical solution for the velocity distribution:

u =2h

∞−n=1

sinnπz

h

un

= −2Ah

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

∞−m=0

(−1)i+j+k+l+myj

j!k!l!m!ν i−kξ 2i+j−2k−1λ−i+k−lθ i−k−m

×Γ−

j2 + k

Γ (i − k + l) Γ (−i + k + m) si−k+(i−k+l)α−(i−k−m)β+1

Γ−

j2

Γ (−i + k) Γ (i − k) Γ (i − k + (i − k + l) α − (i − k − m) β + 2)

+2Ah

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−p=0

∞−q=0

(−1)i+p+q

p!q!ν iξ 2i−1λ−i−pθ i−q

×Γ (i + p) Γ (−i + q) si+(i+p)α−(i−q)β+1

Γ (i) Γ (−i) Γ (i + (i + p) α − (i − q) β + 2). (22)

In terms of the Fox H-function, Eq. (22) takes the simpler form

u = −2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

(−1)i+j+k+lyj

j!k!l!

× ν i−kξ 2i+j−2k−1λ−i+k−mθ i−ks(i−k)(1+α−β)+mα+1

×H1,33,5

sβ

θ

(1 + i − k, 1) ,

1 +

j2

− k, 0

, (1 − i + k − l, 0)

(1, 0) , ((i − k) (β − α − 1) − 1 − lα, β) ,

1 +

j2, 0

, (1 + i − k, 0) , (1 − i + k, 0)

+

2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−p=0

(−1)i+p

p!ν iξ 2i−1λ−i−pθ i−qsi(1+α−β)+pα+1

×H1,22,4

[sβ

θ

(1 + i, 1) , (1 − i − p, 0)(1, 0) , (i (β − α − 1) − 1 − pα, β) , (1 + i, 0) , (1 − i, 0)

]. (23)

To obtain Eq. (23), the following property of the Fox H-function [22] is used:

H1,pp,q+1

[−χ

(1 − a1, A1) , . . . ,1 − ap, Ap

(1, 0) , (1 − b1, B1) , . . . ,

1 − bq, Bq

] =

∞−k=0

Γ (a1 + A1k) · · · Γap + Apk

k!Γ (b1 + B1k) · · · Γ

bq + Bqk

χ k. (24)

5. Calculation of the shear stress

Applying the Laplace transform to Eqs. (7) and (8), we find that

τ 1 =µ1 + θsβ

1 + λsα

∂u (y, z, s)∂y

, (25)

τ 2 =µ1 + θsβ

1 + λsα

∂u (y, z, s)∂z

. (26)

L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508 3503

The image function u (y, z, s) of u (y, z, t) can be easy obtained from Eq. (23). Consequently, evaluating ∂u(y,z,s)∂y from the

aforementioned equations and introducing it into (25), it results that

τ 1 =2ρν

h

∞−n=1

sinnπz

h

A1 − (−1)n

ν2ξ 3

1 +

s(1+λsα)

νξ2(1+θsβ)

12exp

−y

ξ 2 +

s (1 + λsα)

ν1 + θsβ

. (27)

Finally, the stress field in terms of the Fox H-function is

τ1 =Aρ

h

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

∞−m=0

(−1)i+j+k+lyjξ−2i−2k+j−3

22i−1ν i+k+1i!j!k!l!λi+k−lθ−i−ks(i+k)(−α+β−1)+lα−1

×H1,44,6

sβ

θ

1 +

j2 − k, 0

, (1 − 2i, 0) , (1 + i + k − l, 0) , (1 − i − k,m)

(1, 0) ,1 +

j2 , 0

, (1 + i + k, 0) , (1 − i − k, 0) , (1 − i, 0) , [1 + (i + k) (α − β + 1) − lα, β]

. (28)

As regards τ2 (y, z, t), giving the shear stresses on the side walls, it can be immediately obtained from Eqs. (23) and (26).

6. Special cases

Takingλ → 0 andα = 0 in Eqs. (23) and (28),we attain similar solutions for a generalized second-grade fluid, performingthe same motion. Thus the velocity and the stress fields reduce to

u = −2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

(−1)i+j+kyj

j!k!ν i−kξ 2i+j−2k−1θ i−ks(i−k)(1−β)+1

×H1,22,4

sβ

θ

(1 + i − k, 1) ,

1 +

j2

− k, 0

(1, 0) , [−1 − i + k + (i − k) β, β] ,1 +

j2, 0

, (1 + i − k, 0)

+

2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

(−1)iν iξ 2i−1θ i−qsi−iβ+1

×H1,11,3

[sβ

θ

(1 + i, 1)(1, 0) , (−1 − i + iβ, β) , (1 + i, 0)

], (29)

τ1 =Aρ

h

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−m=0

(−1)i+j+kyjξ−2i−2k+j−3

22i−1ν i+k+1i!j!k!× θ−i−ks(i+k)(β−1)−1

×H1,33,5

sβ

θ

1 +

j2 − k, 0

, (1 − 2i, 0) , (1 − i − k,m)

(1, 0) ,1 +

j2 , 0

, (1 − i − k, 0) , (1 − i, 0) , [1 + (i + k) (β + 1) , β]

. (30)

If one sends θ → 0 and sets β = 0, Eqs. (23) and (28) can be simplified as

u = −2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

(−1)i+j+k+lyj

j!k!ν i−kξ 2i+j−2k−1si−k+(i−k)α+1λ−i+k

×H1,22,4

sα

λ

1 +

j2

− k, 0

, (1 + i − k, 1)

(1, 0) ,

1 +

j2, 0

, (1 − i + k, 0) , (−1 − i + k − (i − k) α, α)

+

2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−p=0

(−1)iν iξ 2i−1λ−isi+iα+1

×H1,22,3

[sα

λ

(1 − i, 1)(1, 0) , (1 − i, 0) , (−1 − i − iα, α)

], (31)

τ1 =Aρ

h

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

(−1)i+j+kyjξ−2i−2k+j−3

22i−1ν i+k+1i!j!k!λi+ks−(i+k)(α+1)−1

×H1,44,5

sα

λ

1 +

j2 − k, 0

, (1 − 2i, 0) , (1 + i + k, 1)

(1, 0) ,1 +

j2 , 0

, (1 + i + k, 0) , (1 − i, 0) , [1 + (i + k) (α + 1) , α]

, (32)

which represent the velocity field and stress field corresponding to the flow of a generalized Maxwell fluid performing thesame motion.

3504 L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508

Fig. 1. 3D surface of the velocity u (y, z, t) for different values of λ keeping the other parameters fixed (in two views).

Letting now α, β → 1 in Eqs. (23) and (28), we obtain the exact solutions as correspond to an ordinary Oldroyd-B fluidperforming the same motion:

u =2h

∞−n=1

sinnπz

h

u = −2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

(−1)i+j+k+lyj

j!k!l!

× ν i−kξ 2i+j−2k−1λ−i+k−mθ i−ksi−k+m+1

×H1,33,5

sθ

(1 + i − k, 1) ,

1 +

j2

− k, 0

, (1 − i + k − l, 0)

(1, 0) , (k − i − l − 1, 1) ,

1 +

j2, 0

, (1 + i − k, 0) , (1 − i + k, 0)

+

2h

∞−n=1

sinnπz

h

A1 − (−1)n

∞−i=0

∞−p=0

(−1)i+p

p!ν iξ 2i−1λ−i−pθ i−qsi+p+1

×H1,22,4

[sθ

(1 + i, 1) , (1 − i − p, 0)(1, 0) , (−i − 1 − p, 1) , (1 + i, 0) , (1 − i, 0)

], (33)

τ1 =Aρ

h

∞−n=1

sinnπz

h

1 − (−1)n

∞−i=0

∞−j=0

∞−k=0

∞−l=0

∞−m=0

(−1)i+j+k+lyjξ−2i−2k+j−3

22i−1ν i+k+1i!j!k!l!× λi+k−lθ−i−ks−i−k+l−1

×H1,44,6

sθ

1 +

j2 − k, 0

, (1 − 2i, 0) , (1 + i + k − l, 0) , (1 − i − k,m)

(1, 0) ,1 +

j2 , 0

, (1 + i + k, 0) , (1 − i − k, 0) , (1 − i, 0) , [1 + i + k − l, 1]

. (34)

7. Numerical results and discussion

In this paper, we have established the analytical solutions for the 3D flow of a generalized Oldroyd-B fluid due to aconstant pressure gradient between two side walls perpendicular to a plate. By means of the finite Fourier sine transformand the Laplace transform, the analytical solutions are obtained in the form of series. In the limiting cases where α = β = 1,or α = 0 and λ → 0, or β = 0 and θ → 0, our solutions reduce to those corresponding to an ordinary Oldroyd-B fluid,or a generalized second-grade fluid, or a generalized Maxwell one, respectively. Hence, this flow model is more usefulcompared with the ordinary Oldroyd-B model. Some 3D surfaces are also presented to show the characteristics of solutionsfor variation of the emerging parameters. Furthermore, comparisons are performed between generalized Oldroyd-B fluid,ordinary Oldroyd-B fluid, generalized second-grade fluid and generalizedMaxwell fluid. For the sake of simplicity all the 3Dsurfaces are plotted assuming ρ = µ = 1.

Figs. 1 and 2 show the variations of the material parameters λ and θ . The effect of increasing λ (or decreasing θ ) is anincrease of the velocity surfaces for generalized Oldroyd-B fluids.

In Figs. 3 and 4, the variations of the non-integer fractional parameters α and β are shown. The effect of increasing α isto delay the increase of the velocity surfaces for generalized Oldroyd-B fluids; however, an increase in β yields an oppositetrend for the fluids.

Fig. 5 shows the variations of the velocity surfaces for different values of time. It is obvious that the flow velocity increaseswith increasing time.

In Fig. 6, the effect of the constant pressure gradient is shown for the generalized Oldroyd-B fluids. It is shown that, withthe other parameters kept fixed, the velocity of the fluid is increased by increasing the pressure gradient. Furthermore, thevelocity becomes steady with increasing y, which indicates that the boundary condition (11) is satisfied.

L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508 3505

Fig. 2. 3D surface of the velocity u (y, z, t) for different values of θ keeping the other parameters fixed.

Fig. 3. 3D surface of the velocity u (y, z, t) for different values of α keeping the other parameters fixed.

Fig. 4. 3D surface of the velocity u (y, z, t) for different values of β keeping the other parameters fixed.

3506 L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508

Fig. 5. 3D surface of the velocity profiles u (y, z, t) for different values of t keeping the other parameters fixed.

Fig. 6. 3D surface of the velocity u (y, z, t) for different values of A keeping the other parameters fixed.

Fig. 7. 3D surface of the velocity u (y, z, t) for different kinds of fluids: generalized second grade: λ → 0, θ = 4, α = 0, β = 0.8; generalized Maxwell:λ = 12, θ → 0, α = 0.3, β = 0; ordinary Oldroyd-B: λ = 12, θ = 4, α = β = 1; generalized Oldroyd-B: λ = 12, θ = 4, α = 0.3, β = 0.8.

Fig. 7 presents, for comparison, the variations of the velocity fields corresponding to ordinary Oldroyd-B, generalizedOldroyd-B, generalized Maxwell and generalized second-grade fluid in the two side walls perpendicular to the plate. It

L. Zheng et al. / Nonlinear Analysis: Real World Applications 12 (2011) 3499–3508 3507

Fig. 8. 3D surface of the shear stress τ1 (y, z, t) for different values of t keeping the other parameters fixed.

should be noted that the velocity of generalized Maxwell fluid is greater in magnitude than those of all of the other fluids;this is then followed by the velocitymagnitude of generalized Oldroyd-B fluid and ordinary Oldroyd-B fluid, and generalizedsecond-grade fluid has the smallest velocity magnitude.

Moreover, some strong effects on the velocity field resulting from the relaxation and retardation times and the orders offractional parameters can be easily detected from these figures.

Fig. 8 shows the variations of the shear stress for different values of t for the generalized Oldroyd-B fluids. As wasexpected, the strongest shear stress occurs near the plate; however, with increase of the distance from the plate, the shearstress of the generalized Oldroyd-B fluid decays rapidly, which also indicates that the boundary condition (11) is satisfiedaccording to Eq. (7).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 50936003, 51076012) and the OpenProject of the State Key Laboratory for Advanced Metals and Materials (2009Z-02), USTB.

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