广义Oldroyd-B粘弹性流体Stokes第一问题

Stokes' first problem for a viscoelastic

fluid with the generalized Oldroyd-B

model1

Haitao Qi

Department of Applied Mathematics and Statistics, Shandong University at Weihai

Weihai, P. R. China 264209

htqi@https://www.360docs.net/doc/3017932680.html,

Mingyu Xu

School of Mathematics and Systematical Science, Shandong University

Jinan, P.R. China, 250100

Abstract

The flow near a wall suddenly set in motion for a viscoelastic fluid with the

generalized Oldroyd-B model is studied. The fractional calculus approach has been

taken into account in the constitutive relationship of fluid model. Exact analytical

solutions of velocity and stress are obtained by using the discrete Laplace transform of

the sequential fractional derivative and the Fox-H function. The obtained results

indicate that some well known solutions for the Newtonian fluid, the generalized

second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are

included in our solutions.

Keywords: Generalized Oldroyd-B fluid, Stokes' first problem, Fractional calculus,

Exact solution, Fox- H function.

1Introduction

Navier-Stokes equations are the most fundamental motion equations in fluid dynamics. However, there are few cases in which their exact analytical solutions can be obtained. Exact solutions are very important not only because they are solutions of some fundamental flows, but also because they serve as accuracy checks for experimental, numerical, and asymptotic methods.

The inadequacy of the classical Navier-Stokes theory to describe rheologically complex fluids such as polymer solutions, blood and heavy oils, has led to the development of several theories of non-Newtonian fluids. In order to describe the non-linear relationship between the stress and the rate of strain, numerous models or constitutive equations have been proposed. The model of differential type and those of rate type have received much attention [1]. In recent years, the Oldroyd-B fluid has acquired a special status amongst the many fluids of the rate type, as it includes as special cases the

1 Supported by the National Natural Science Foundation of China (10272067), the Doctoral Program

Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai.

classical Newtonian fluid and the Maxwell fluid. For some special flows the model of second grade fluid is also included. Here, we mention some of the studies such as [2-5].

Recently, fractional calculus has encountered much success in the description of the complex dynamics. In particular it has proved to be a valuable tool to handle viscoelastic properties. The starting point of the fractional derivative model of viscoelastic fluid is usually a classical differential equation which is modified by replacing the classical, time derivatives of an integer order by the fractional calculus operators. This generalization allows one to define precisely non-integer order integrals or derivatives [6]. Fractional calculus has been found to be quite flexible in describing viscoelastic behavior [7-10]. More recently, Huang et al. [11-16] discussed some unsteady flows of the generalized second grade fluid. The unidirectional flow of a viscoelastic fluid with the fractional Maxwell model is considered by Tan et al. [17,18]. The flows with a generalized Jeffreys model in an annular pipe is also studied by Tong et al. [19,20].

On the other hand, almost every one made a study of fluid mechanics is familiar with Stokes’ celebrated paper on pendulums (in 1851) in which he describes a kind of classical problems of the impulsive and oscillatory motion of an infinite plate in its own plane. Nowadays, the flows over a plane wall which is initially at rest and is suddenly moved in its own plane with a constant velocity and with a harmonic vibration are termed Stokes’ first (or Rayleigh-type) and second problems, respectively [21,22]. At present, one paid more and more attention to the Stokes problems due to important theoretical and practice meanings [23-25]. The aim of this paper is to investigate the Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model. The fractional calculus approach has been taken into account in the constitutive relationship of fluid model. Exact analytical solutions of velocity and stress are obtained by using the discrete Laplace transform of the sequential fractional derivative and the Fox-H function. The obtained results indicate that some well known solutions for the Newtonian fluid, the generalized second grade fluid as well as for the ordinary Oldroyd-B fluid, as limiting cases, are included in our solutions.

2 Basic equations

First, we know that the constitutive equation of an incompressible, ordinary Oldroyd-B fluid is of the form [1] ,p =?+T I S (2.1)

D D 1D D t t λ

μθ,?

?+=+????

S S A (2.2) where λ and θ are relaxation and retardation times, μis the dynamic viscosity of fluid, T is the Cauchy stress tensor, p ?I denotes the indeterminate spherical stress, T

=+A L L with the

velocity gradient and (upper convected time derivative) operating on any tensor is defined

by L D /D t B

D ,D T t

t ?=+?????B B

V B LB BL (2.3) where being the velocity vector, is the gradient operator, the superscript T denotes a transpose

operation. V ?

Generally, the constitutive relationship of the generalized Oldroyd-B fluid also has the form (2.1) and (2.2), but D / and are defined as follows [10]:

D t S D /D t A

D ,D T t D t

α=+????S

S V S LS SL (2.4)

D ,D T t D t

β=+????A

A V A LA AL (2.5) where t D α

and t D β

are fractional calculus of order α and β with respect to respectively, and may be defined as [6] t

01d ()

()d ,

01,(1)d ()

t p t p f D f t p p t t τττ=

Γ??∫≤≤ (2.6) where is Gamma function. It should be noted that this model can be simplified as the ordinary Oldroyd-B model when ()Γ?1αβ=

=, the generalized second grade fluid [11-16] when 0α=, 0λ→

and the fractional Maxwell fluid when 0,0βθ=→ [8,17].

The unsteady flow is governed by the equation of motion (in the absence of body force) and continuity equation in the form

d

,

d t

ρ

=

2

2(1)(1),t t u u D D t y

α

λνθ??+=+??β (2.12) where /νμρ=. The initial and boundary conditions are as follows

(,0)(,0)0, 0,t u y u y y =?=> (2.13)

(0,), 0.u t U t => (2.14)

And the natural conditions are

(,),(,)0 as .y u y t u y t y ?→→∞ (2.15)

3 Exact solutions to the Stokes' first problem

Let us introduce the following dimensionless variables

2

22*****

,,,, and ,u yU tU U U u y t U α

β

ρρρρλλθθμμμμ????=====????????

(3.1)

in which U and 2

/U μρ denote the characteristic velocity and time, respectively. The dimensionless

governing equation (2.12) and its initial and boundary conditions (2.13)-(2.15) (the asterisks are dropped) can be written as: 2

(1)(1),t t u u D D t y

α

βλθ??+=+?? (3.2) (,0)(,0)0, 0,t u y u y y =?=> (3.3)

(0,)1, 0.u t t => (3.4)

(,),(,)0 as .y u y t u y t y ?→→∞ (3.5)

Let us suppose that

(3.6) 0

(,){(,),}e (,)d st u

y s L u y t s u y t t ∞?==∫%is the image function of , where s is a transform parameter. Because the fractional order

equation (3.2) has the property of the integer order initial conditions, the fractional derivative must be interpreted as a properly chosen sequential fractional derivative [6]. Using the Laplace transform (,)u y t principle of sequential fractional derivatives, we can obtain the equation of the image function

22

d (1),d 1u

s s u t s

αβλθ+=+%% (3.7) subject to the boundary conditions

1

(0,),u

s s

=% (3.8)

(,),(,)0 as .y u

y s u y s y ?→→%%∞ (3.9) Solving (3.7)-(3.9) yields:

1/21

(1)(,)exp .1s s u y s y s

s αβλθ????+?=?????+????

%? (3.10) In order to obtain an analytical solution for this problem and to avoid the burdensome calculations of

residues and contour integrals, we will apply the discrete inverse Laplace transform method to give the velocity distribution.

Firstly, we rewrite (3.10) as a series form:

10

(1)10

2

1(()(,)!

!()122 .!k

m

k m n k

m n n u

y s s k m k k n m k k n s βααβλθ?∞∞

==?∞

??+++=??=+×?????Γ+Γ?????????????ΓΓ?????????∑∑∑% (3.11) Applying the discrete inverse Laplace transform on (3.11), we obtain

100

(1)2

()(()22(,)1!!! (1)1n k m k m n k

m n k k n m u y t k k k m n t

k m n βααβθλβααβ??∞

∞∞===??++????Γ+Γ?????????=+????ΓΓ?????

????×

??Γ??+++????

∑∑∑???

(1)1

0(1,1),(1,0)1,2122

2,4(0,1),(1,0),(1,0),((1),)

(()1!!

,k

k

m m k m k k m k k k

m t k m H t βααβαβαβλθ?∞

∞

??+==??+??++????=+??

×???

?

∑

∑ (3.12)

in which denotes Fox-H function[10]. In arriving at (3.12), we used the following property of Fox- H function

,,()m n

p q H z

(

)111

1

(1,),,(1,)

11,,1(0,1),(1,),,(1,)

1()().

!()

p p q

q p

n j j a A a A j p

p q q

b B b B

n j j j z a A n H z n b B n ∞

??=+??==?Γ+=Γ+∏∑

∏L L (3.13)

Because the fluid is moved by the action of stress at the plate, the stress field may be calculated. From (2.11) the dimensionless stress can be represented by

(3.14)

(1)(,)(1)(,),t t D F y t D u y t αβλθ+=+?y where 2

(,)xy

S F y t U ρ=.

Using (2.10) and (3.10), the Laplace transform of (3.14) is

1/2

1/2(1)(1)(,)exp .11s s s s F y s y s s α

αβ

λλθθ???

????++?=?????

??++??

????

%β?? (3.15) We apply the discrete inverse Laplace transform method again to give the stress distribution. Firstly,

we rewrite (3.15) as a series form

(1)/20

(1)(1)/20

1()()2(,)1!!1()12 .1!k k

m k m n k m n n k m y F

y s k k m k n k s n βααβ

λλθθ??∞

∞

==?∞

???++=???Γ+??

????

?

?=????????

Γ????

????Γ+????×???Γ??

??

∑∑∑% (3.16)

Applying the discrete inverse Laplace transform to (3.16), we obtain

(1)/2

00

(1)(1)/21

()()()(,)!!!1122 (1)k k

m n k m n k m n y F y t k m n k k n m t k k k m n βααβλλθθβαα???∞

∞

∞===???++??????

=???

????????Γ+Γ+?????

???×????????

ΓΓΓ??++?????

?????

∑

∑∑β???

(1)/21

(1)10

13(,0),(,1)1,2122

3112,4(0,1),(,0),(,0),(1(1),)

()()!! .k k k

m m k m k k m k k k m y t k m H t

βααβαβαβλλθθ???∞

∞

??+?==+????+?++?????

=?????

??

×???

?

∑

∑ (3.17)

In order to obtain the value of the shear stress on the plate, taking 0y = in (3.15), we get the following

expression

1/2

1().p s F s s s β

αθλ??+=??+??

%? (3.18) In the same way as (3.17), we obtain the following formula to calculate the shear stress on the plate

1/213

(,0),(,1)(1)/21,2122

3112,4(0,1),(,0),(,0),(,)

0()().!m m m p m m F t t H t m ααββ

βααβθλθλ?∞

?+????+=??

???

=?????

??

??

∑ (3.19) 4 Limiting cases

Now let us make some analyses for formulas (3.12) and (3.17).

(1) If 0α=, 0λ→, θη=, the medium is the generalized second grade fluid, and (3.12) and

(3.17) tend to the same as obtained by [14,15]

1

2

(1,1)1,112

1,3

(0,1),(1,0),((1))

1

()(,)1,!

k k

k k k k yt

u y t H t k ββ

ββηη?∞

????=??

?=+???

?∑

(4.1)

1(1)(1)

32

1(

,1)1,112

2

3(1)(1)1,3

(0,1),(,0),(1,)

()(,).!k k k

k

k k k y F y t t

H t k ββ

ββη

η????∞

?????+=??

?=?????

∑

(4.2) (2) If 0β=, 0θ→, (3.12) and (3.17) are simplified as

(1)(1,1)1,1122

(1)1,3(0,1),(1,0),(,)

1()(,)1,!k

k

k k k k u y t t H t

k αα

ααλ+∞

?+?++=??

=+???

?∑ (4.3)

1

(1)(1)

1

2

1(

,1)1,112

1

(1)(1)1,3(0,1),(,0),(1,)

()(,).!k k k k

k k k y F y t t

H t k αα

ααλ

λ?+?+∞

??

?++?+=??

?=????

?

∑

(4.4) They are the solutions of velocity and stress for the fractional Maxwell fluid in which the derivatives of the stress and strain are fractional and first order, respectively [8,17].

(3) Setting 0β=, 0θ→ in (4.1) and (4.2) or setting 0α=, 0λ→ in (4.3) and (4.4), we obtain

the classical Rayleigh's similarity solutions of Newtonian fluid [26]

(,)erfc ,u y t ?=?? (4.5)

2(,)/4).F y t y t =? (4.6)

(4) While 1αβ==, (3.12) and (3.17) can be simplified as

(1,1),(1,0)1,21222

2,4(0,1),(1,0),(1,0)(,1)1

0(()(,)1,!!k k k k

m m m k k k m k m u y t t H t k m λθ?∞

∞

???+??+?==????=+????∑

∑ (4.7)

1

113(,0),(,1)1,21222

3112,4(0,1),(,0),(,0),(,1)0

0()()(,).!!k k k k

k

m m m k k k m k m y F y t t H t k m λλθθ?++??∞

∞????++?==??????

=?????

????∑∑ (4.8) which are the solutions for ordinary Oldroyd-B fluid. (5) From (3.10) and (3.15), we also have

1/2

(1)(,)(,).1s s F y s u

y s s β

αθλ??+=???+??

%% (4.9) Using the same method as we used in sec.(3) and the definition of fractional calculus, we have

1200

11()()22(,)(,).11!

!n

m m n t

m m n F y t D u y t n m βα

αβθλ+???∞∞

??+==???

?Γ+Γ???????????=?????ΓΓ?????????

∑ (4.10) If one sets 0α=, 0λ→, θη=, the formula (4.10) can be simplified as the same as obtained

in [14]:

111/2

20

1()2(,)(,).1!n n t

n n F y t D u y t n ββηη?+∞

?+=???Γ?????=????Γ???

??

∑ (4.11) The physical meaning of (4.10) is that the stress at a given point at any time is dependent on the time history of the velocity profile at that point, and the history can be depicted by the fractional calculus [9].

5 Conclusion

Fractional calculus approach is introduced to the constitutive relationship model of generalized Oldroyd-B fluid in this paper. The flow model is more useful compared with the traditional model. Exact analytical solutions of velocity and stress for the Stokes' first problem are obtained by using the discrete Laplace transform of the sequential fractional derivatives and the Fox-H function. We hope that our results can provide new models and analytical solutions for studying the complicated fluid in rheology. References

[1]Han Shifang, Constitutive equation and computational analytical theory of non-Newtonian fluids, Science

Press, Beijing, 2000.

[2] K.R. Rajagopal, R.K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113

(1995) 233-239.

[3] T. Hayat, A.M. Siddiqui, S. Asghar, Some simple flows of an Oldroyd-B fluid, Int. J. Eng. Sci. 39 (2001)

135-147.

[4] C. Fetecau, C. Fetecau, The first problem of Stokes for an Oldroyd-B fluid, Int. J. Non-Linear Mech. 38 (2003)

1539-1544.

[5] T. Hayat, M. Khan, M. Ayub, Exact solutions of flow problems of an Oldroyd-B fluid, Appl. Math. Comp. 151

(2004) 105-119.

[6] I. Podlubny, Fractional differential equations, Academic Press, San Diego 1999.

[7] R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol. 30 (1986)

133--155.

[8] C. Friedrich, Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol.

Acta 30 (1991) 151-158.

[9] D.Y. Song, T.Q. Jiang, Study on the constitutive equation with fractional derivative for the viscoelastic fluids

– modified Jeffreys model and its application, Rheol. Acta 37 (1998) 512-517.

[10] R. Hilfer, Applications of fractional calculus in physics, World Scientific Press, Singapore, 2000.

[11] Huang Junqi, He Guangyu, Liu Ciqun, Analysis of general second-order fluid flow in double cylinder

rheometer, Sci. China (Ser. A) 40 (1997) 183--190.

[12] Xu Mingyu, Tan Wenchang, Theoretical analysis of the velocity field, stress field and vortex sheet of

generalized second order fluid with fractional anomalous diffusion, Sci. China (Ser. A) 31 (2001) 626-638.

[13] Tan Wenchang, Xian Feng, Wei Lan, An exact solution of unsteady Couette flow of generalized second grade

fluid, Chinese Science Bulletin 47 (2002) 1783-1785.

[14] Tan Wenchang, Xu Mingyu, The impulsive motion of flat plate in a generalized second grade fluid, Mech.

Res. Comm. 29 (2002) 3-9.

[15] M. El-Shahed, On the impulsive motion of flat plate in a generalized second grade fluid, Z. Naturforsch 59a

(2004) 829-837.

[16] M. Khan, S. Nadeem, T. Hayat, A.M. Siddiqui, Unsteady motions of a generalized second-grade fluid, Math.

Comput. Modelling 41 (2005) 629-637.

[17] Tan Wenchang, Pan Wenxiao, Xu Mingyu, A note on unsteady flows of a viscoelastic fluid with the fractional

Maxwell model between two parallel plates, Int. J. Non-Linear Mech. 38 (2003) 645-650.

[18] T. Hayat, S. Nadeem, S. Asghar, Periodic unidirectional flows of a viscoelastic fluid with the fractional

Maxwell model, Appl. Math. Comp. 151 (2004) 153-161.

[19] Tong Dengke, Liu Yusong, Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an

annular pipe, Int. J. Eng. Sci. 43 (2005) 281-289.

[20] Tong Dengke, Wang Ruihe, Yang Heshan, Exact solutions for the flow of non-Newtonian fluid with

fractional derivative in an annular pipe, Sci. China (Ser. G) 48 (2005) 485-495.

[21] M.E. Erdogan, A note on an unsteady flow of a viscous fluid due to an oscillating plane wall, Int. J.

Non-Linear Mech. 35 (2000) 1-6.

[22] Y. Zeng, S. Weinbaum, Stokes problems for moving half-planes, J. Fluid Mech. 287 (1995) 59-74.

[23] D. Volfson, J. Vinals, Flow induced by a randomly vibrating boundary, J. Fluid Mech. 432 (2001) 387-408.

[24] P.M. Jordan, A. Puri, G. Boros, On a new exact solution to Stokes' first problem for Maxwell fluids, Int. J.

Non-Linear Mech. 39 (2004) 1371-1377.

[25] P.M. Jordan, A. Puri, Revisiting Stokes' first problem for Maxwell fluids, Q. J. Mechanics Appl. Math. 58

(2005) 213-227.

[26] C.S. Yih, Fluid mechanics: a concise introduction to the theory, McGraw-Hill, New York, 1969.