Decay of a potential vortex in a generalized Oldroyd-B fluid

Applied Mathematics and Computation 205 (2008) 497–506

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Decay of a potential vortex in a generalized Oldroyd-B fluid Corina Fetecau a,*, C. Fetecau b, M. Khan c, D. Vieru a a b c

Department of Theoretical Mechanics, Technical University of Iasi, Iasi 700050, Romania Department of Mathematics, Technical University of Iasi, R-6600 Iasi, Romania Department of Mathematics, Quaid-i-Azam University, 45320 Islamabad, Pakistan

a r t i c l e

i n f o

Keywords: Potential vortex Generalized Oldroyd-B fluid Exact solutions

a b s t r a c t The velocity field and the adequate shear stress corresponding to the decay of a potential vortex in a generalized Oldroyd-B fluid are determined by means of Hankel and Laplace transforms. The exact solutions, written in terms of the generalized G and R functions, are presented as a sum of the Newtonian solutions and the adequate non-Newtonian contributions. These solutions can be easy specialized to give the solutions for generalized Maxwell fluids and ordinary Oldroyd-B or Maxwell fluids performing the same motion. The influence of the fractional parameters as well as that of the material parameters on the decay of the vortex is emphasized by graphical illustrations. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction Recently, the interest for motion problems of non-Newtonian fluids has considerably grown because of the wide range of their applications. Among the many models that have been used to describe their behavior, the rate type models have received many attention. The first systematic thermodynamic study of these models seems to be that developed by Rajagopal and Srinivasa [1], within which models for a variety of rate type viscoelastic fluids can be obtained. Amongst them the Oldroyd-B model has had some success in describing the response of some polymeric liquids, it being more amenable to analysis and more importantly experimental. Consequently, a lot of papers regarding these fluids have been published in the last time [2–8]. The Oldroyd-B model contains as a special case the Maxwell model but an inadequacy of this last model has been pointed out by Choi et al. [9]. In a simple shear flow of a real fluid, it predicts a linear relation between shear rate and shear stress. A very good fit of experimental data was achieved when the fractional Maxwell model has been used instead of the Maxwell model [10,11]. Generally, the rheological constitutive equations with fractional derivatives are derived from those for non-Newtonian fluids by replacing the time derivatives of an integer order with the so called Riemann–Liouville fractional operators. These equations have long played an important role in describing the properties of polymeric solutions and melts, proving to be a valuable tool to handle viscoelastic properties. They have been also linked to molecular theories [12]. At least the modified viscoelastic models are appropriate to describe the behavior for Xanthan gum and Sesbania gel [13]. During the last years, many papers dealing with such constitutive equations have been published. We shall here remember, only those corresponding to generalized Oldroyd-B fluids [14–19]. The purpose of this paper is to present a study of the decay of a potential vortex in an incompressible generalized Oldroyd-B fluid (GOF). More exactly we establish the exact solutions for the velocity field and the adequate shear stress corresponding to such a motion of an Oldroyd-B fluid with fractional derivatives. These solutions, presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions, are obtained by means of the Hankel and Laplace * Corresponding author. E-mail address: [email protected] (C. Fetecau). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.08.017

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transforms. They can be specialized to give the similar solutions for generalized Maxwell fluids and ordinary Maxwell or Oldroyd-B fluids. Finally, the influence of the fractional parameters on the decay of the vortex is emphasized by means of the graphical illustrations. 2. Governing equations The constitutive equations for an incompressible GOF, are given by [18]

T ¼ pI þ S;

Sþk

  DS D A; ¼ l 1 þ kr Dt Dt

ð1Þ

where T is the Cauchy stress tensor, pI denotes the indeterminate spherical stress due to the constraint of the incompressibility, S is the extra-stress tensor, A ¼ L þ LT is the first Rivlin–Ericksen tensor, L is the velocity gradient, l is the dynamic viscosity of the fluid, k and kr are the relaxation and retardation times and

DS ¼ Dat S þ ðV  rÞS  LS  SLT ; Dt

DA ¼ Dbt A þ ðV  rÞA  LA  ALT : Dt

ð2Þ

In the above relations V is the velocity, r is the gradient operator, the superscript T indicates the transpose operation and Dat and Dbt are fractional differential operators of order a and b with respect to t, defined as [20]

Dpt f ðtÞ ¼

1 d Cð1  pÞ dt

Z 0

t

f ðsÞ ds; ðt  sÞp

0 < p 6 1;

ð3Þ

where CðÞ is the Gamma function. This model contains as special cases the fractional Maxwell model for kr ! 0 and b ¼ 1 as well as the ordinary Oldroyd-B and Maxwell models for a ¼ b ¼ 1, respectively, a ¼ b ¼ 1 and kr ! 0. In the following we are interested in the circular motion of an incompressible GOF, whose velocity field and extra-stress tensor in the system of cylindrical coordinates r, h and z have the forms

V ¼ Vðr; tÞ ¼ xðr; tÞeh ;

S ¼ Sðr; tÞ;

ð4Þ

where eh denotes the unit vector along the h direction. For this motion the constraint of incompressibility is automatically satisfied. The initial distribution of velocity is assumed to be that of a potential vortex of circulation C0 [5], i.e.

xðr; 0Þ ¼ C0 =ð2prÞ:

ð5Þ

Substituting (4) into (1)2 and (2) and having in mind the initial condition

Sðr; 0Þ ¼ 0 ðthe fluid being at rest up to t ¼ 0Þ;

ð6Þ

as well as Eqs. (21.1-4) from [2] we get Srr ¼ Srz ¼ Shz ¼ Szz ¼ 0 and the relevant equation (see Eq. (2) from [14])

ð1 þ kDat Þsðr; tÞ ¼ lð1 þ kr Dbt Þ



 o 1  xðr; tÞ; or r

ð7Þ

where sðr; tÞ ¼ Srh ðr; tÞ is the shear stress. The motion equation of the swirling flow is

q

  oxðr; tÞ 1 o osðr; tÞ ¼ 2 r2 ; ot r or or

ð8Þ

where q is the constant density of the fluid. Eliminating s between Eqs. (7) and (8) we attain to the governing equation (see also [14], Eq. (8))

! oxðr; tÞ o2 1 o 1 b ð1 þ kDt Þ þ xðr; tÞ; ¼ mð1 þ kr Dt Þ  ot or 2 r or r2 a

ð9Þ

where m ¼ l=q is the kinematic viscosity of the fluid. Into Eqs. (7) and (9), for dimensional consistency, k and kr have to be ka and kar . However, for simplicity, we keep the same notation. The fractional differential equations (7) and (9), with appropriate initial and boundary conditions, can be solved in principle by several methods, the integral transforms technique representing a systematic, efficient and powerful tool. The Laplace transform can be used to eliminate the time variable, while the finite Hankel transform can be employed to eliminate the spatial variable. Bandelli and Rajagopal [20] as well as Bandelli et al. [21] showed that the Laplace transform does not work for some problems regarding second grade fluids. More precisely, the obtained solutions by Laplace transform do not satisfy the initial conditions due to the incompatibility between the prescribed data. This is not true for rate type fluids (see [4], for instance). Furthermore, in order to solve a well-posed problem for the fractional differential equation (9), one has to require an additional initial condition. Here, as well as in the previous works [3–8,14–19], we shall assume that

oxðr; tÞ ! 0 when t ! 0: ot

ð10Þ

C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

499

Moreover, the flow domain being unbounded, the natural conditions

oxðr; tÞ ! 0 as r ! 1 or

xðr; tÞ;

ð11Þ

have to be also satisfied. 3. Exact solution for the velocity field Multiplying Eq. (9) by rJ 1 ðrnÞ, integrating with respect to r from 0 to infinity and taking into account the initial and boundary conditions (5), (10) and (11), we find that

ð1 þ kDat Þ

oxH ðn; tÞ þ mn2 ð1 þ kr Dbt ÞxH ðn; tÞ ¼ 0; ot

n; t > 0;

ð12Þ

where the Hankel transform xH ðn; tÞ of xðr; tÞ has to satisfy the conditions

xH ðn; 0Þ ¼

C0 2p n

oxH ðn; 0Þ ¼ 0; ot

;

n > 0:

ð13Þ

Applying the Laplace transform to Eq. (12), using the Laplace transform formula for sequential fractional derivatives [22] and  H ðn; qÞ of having the initial conditions (13) in mind, as well as Eq. (A1) from Appendix, we find for the image function x xH ðn; tÞ, the following expression:

 H ðn; qÞ ¼ x

1 þ kqa þ cn2 qb1

C0

2pn q þ kqaþ1 þ cn2 qb þ mn2

ð14Þ

;

where c ¼ mkr .  H ðn; qÞg and to avoid the burdensome calculations of residues and contour integrals, we In order to obtain xH ðn; tÞ ¼ L1 fx apply the discrete inverse Laplace transform method [14–19]. However, for a more suitable presentation of the final results, we firstly rewrite Eq. (14) in the equivalent form

 H ðn; qÞ ¼ x

C0 1 mnC0 1 kqa þ cn2 qb1 þ : 2pn q þ mn2 2p q þ mn2 q þ kqaþ1 þ cn2 qb þ mn2

ð15Þ

The second factor of the last term from (15) can be written as a double series [19]

kqa þ cn2 qb1 q þ kqaþ1

n2 qb

¼

2

þ mn

þc

1 mþl¼k X 1X mn2  k k¼0 m;lP0 k

!k

ak1þbm þ cn2 qbk2þbm km r k! kq : m!l! ðqa þ 1=kÞkþ1

ð16Þ

Introducing (16) into (15) and inverting the result, applying the discrete inverse Laplace transform and using the property (A2) and (A3), we find that

xH ðn; qÞ ¼

C0 2pn

2

emtn þ

mnC0 2p

where

Fðn; tÞ ¼

1 mþl¼k X X



mn2

!k

k

k¼0 m;lP0

Z

t

emn

2

ðtsÞ

Fðn; sÞds;

ð17Þ

0

"    # km 1 cn2 1 r k! Ga;am ;b  ; t þ Ga;bm ;b  ; t m!l! k k k

ð18Þ

with am ¼ a  k  1 þ bm, bm ¼ b  k  2 þ bm, b ¼ k þ 1 and (see [23] the pages 14 and 15)

Ga;a;b ðc; tÞ ¼

1 X j¼0

Cðj þ bÞt aðjþbÞa1 ðcÞj : Cðj þ 1ÞCðbÞC½aðj þ bÞ  a

ð19Þ

Applying the inverse Hankel transform [24] to Eq. (17), we find xðr; tÞ under the form

xðr; tÞ ¼

C0

Z

2p

0

1

2

emtn J 1 ðrnÞdn þ

mC0 2p

Z

1

0

Z 0

t

n2 emn

2

ðtsÞ

J 1 ðrnÞFðn; sÞds dn:

ð20Þ

Finally, using the integral (A4), the velocity field can be written under the suitable form

xðr; tÞ ¼ xN ðr; tÞ þ

m C0 2p

Z 0

1

Z

t

n2 emn

0

2

ðtsÞ

J 1 ðrnÞFðn; sÞds dn;

ð21Þ

where (cf. [5], Eq. (33) or [25], Eq. (3.8), where a different method has been used)

xN ðr; tÞ ¼

   r2 1  exp  2pr 4mt

C0

is the velocity field corresponding to a Newtonian fluid performing the same motion.

ð22Þ

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Fig. 1. Profiles of the velocity field xðr; tÞ, given by Eq. (21) – curves x1G, x2G and Eq. (23) – curves x1, x2, for m ¼ 0:002, C0 ¼ 1, k ¼ 6, kr ¼ 3, a ¼ 0:9, and b ¼ 0:9.

From Fig. 1, as it was to be expected, it is clearly seen that for a and b ! 1, the diagrams corresponding to the velocity field (21) are almost identical to those corresponding to the velocity

xðr; tÞ ¼

C0 2p

Z

1

0

r2 expðr 1 tÞ  r 1 expðr2 tÞ J 1 ðrnÞdn; r2  r1

r 1;2 ¼

ð1 þ cn2 Þ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ cn2 Þ2  4mkn2 2k

ð23Þ

obtained in [5] by a different technique. The velocity field (23) can be also obtained from [26], Eq. (30) neglecting the porous effects. 4. Calculation of the shear stress Applying the Laplace transform to Eq. (7) and using the initial condition (6), we find that

sðr; qÞ ¼ l

   ðr; qÞ 1 1 þ kr qb ox  ðr; qÞ : x  or r 1 þ kqa

ð24Þ

 ðr; qÞ, we apply the Laplace transform to Eq. (20), look at the second term as to a convolution product In order to determine x and take into consideration Eqs. (18), (19) and (A5). Using then the known identity

  o 1 1 ½J 1 ðrnÞ ¼ nJ01 ðrnÞ ¼ n J 1 ðrnÞ  J2 ðrnÞ ¼ J1 ðrnÞ  nJ 2 ðrnÞ; or rn r it is not difficult to show that

 ðr; qÞ 1 ox C  ðr; qÞ ¼  0  x or r 2p

Z 0

1

nJ 2 ðrnÞ q þ mn

dn  2

m C0 2p

Z 0

1

n3 J 2 ðrnÞ q þ mn2

Fðn; qÞdn;

ð25Þ

where Fðn; qÞ is the Laplace transform of Fðn; tÞ. Now, we introduce (25) into (24) and apply the inverse Laplace transform. For the first term, we use the identity

l

1 þ kr qb 1 l kr qb  kqa ¼ þl 2 2 a 1 þ kq q þ mn q þ mn q þ kqaþ1 þ mn2 þ mkn2 qa 2 3 1 mþl¼k amþlþb amþlþa X X ð1Þk ðmn2 Þm k! kr l q q 4 5 ¼ þl  k qaþ1 þ mn2 =kkþ1 qaþ1 þ mn2 =kkþ1 q þ mn2 kl m!l! k¼0 m;lP0

and find its contribution to the shear stress as

s1 ðr; tÞ ¼ 

lC0 2p

Z 0

1

2

nemtn J 2 ðrnÞdn 

1 mþl¼k X ð1Þk mm k! Z 1 2mþ1 l C0 X n J 2 ðrnÞ 2p k¼0 m;lP0 kl m!l! 0

" ! !# kr mn2 mn2  Gaþ1;cm ;b  ; t  Gaþ1;dm ;b  ; t dn; k k k

ð26Þ

where cm ¼ am þ b þ l, dm ¼ am þ a þ l and b ¼ k þ 1. Using Eq. (A6), this term can be also written in the suitable form

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C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

s1 ðr; tÞ ¼ sN ðr; tÞ þ s2 ðr; tÞ

"

!

1 mþl¼k X ð1Þk mm k! Z 1 2mþ1 lC0 X kr mn2 mn2 ¼ sN ðr; tÞ  n J 2 ðrnÞ Gaþ1;cm ;b  ; t  Gaþ1;dm ;b  ;t l 2p k¼0 m;lP0 k m!l! k k k 0

!# dn;

ð27Þ

where (cf. [5], Eq. (34))

sN ðr; tÞ ¼

lC0 pr 2

     r2 r2 exp  1 1þ 4mt 4mt

ð28Þ

represents the shear stress for a Newtonian fluid. Lengthy but straightforward computations show that the second part of the shear stress, corresponding to the second term from Eq. (25), can be written in the form (see also Eq. (A7))

s3 ðr; tÞ ¼ 

lmC0 2pk

Z

1

Z

0

t

Z r

0

2

n3 J 2 ðrnÞemn

ðtrÞ

0

  1 Hðn; rÞRa;0  ; 0; r  s ds dr dn; k

ð29Þ

where

!k 1 mþl¼k X X  1j mn2 k!kmr Cðj þ k þ 1Þ Hðn; tÞ ¼   k k m!l! Cðj þ 1ÞCðk þ 1Þ j;k¼0 m;lP0 " # t aðjþkÞbmþk cn2 taðjþbÞbm 1 t aðjþbÞam b1 t aðjþbÞbm b1 þ þ kr þ cn2  C½aðj þ kÞ  bm þ k þ 1 k C½aðj þ bÞ  bm  C½aðj þ bÞ  am  b C½aðj þ bÞ  bm  b

ð30Þ

and

Ra;b ðc; d; tÞ ¼

1 X ðt  dÞðnþ1Þab1 n c : C½ðn þ 1Þa  b n¼0

ð31Þ

Finally, the shear stress corresponding to our problem is given by

sðr; tÞ ¼ sN ðr; tÞ þ s2 ðr; tÞ þ s3 ðr; tÞ;

ð32Þ

where the non-Newtonian contribution s2 ðr; tÞ þ s3 ðr; tÞ, as it results from Eqs. (27) and (29), is given in terms of the generalized G and R functions. 5. Limiting cases 1. Making kr ! 0 and b ¼ 1 into Eqs. (21) and (32), we get the velocity field

xðr; tÞ ¼ xN ðr; tÞ þ

k Z 1  C0 X 1

2p

1



Z

t

ðmn2 Þkþ1 emn

2

  1 J 1 ðrnÞGa;ab;b  ; s ds dn k Z 1 a ðjþkÞþk s 2 ðmn2 Þkþ1 emn ðtsÞ J 1 ðrnÞdn ds C½aðj þ kÞ þ k þ 1 0

ðtsÞ

k 0 0  jþk Z t C0 1 Cðj þ k þ 1Þ  ¼ xN ðr; tÞ þ k 2p j;k¼0 Cðj þ 1ÞCðk þ 1Þ 0 k¼0

1 X

and the associated shear stress

sðr; tÞ ¼ sN ðr; tÞ þ 

ð33Þ

!

jþkþ1 1 mþl¼k 1  X ð1Þk k! Z 1 l C0 X mn2 lC0 X 1 2 m nð m n Þ J ðrnÞG   ; t dn þ a þ1;d ;b 2 m k 2p k¼0 m;lP0 kl m!l! 0 k 2p j;k¼0

Cðj þ k þ 1Þ Cðj þ 1ÞCðk þ 1Þ

Z

1 0

Z

t

0

Z r

nðmn2 Þkþ1 emn

2

ðtrÞ

0

J 2 ðrnÞ

  1 Ra;0  ; 0; r  s ds dr dn k C½aðj þ kÞ þ k þ 1

raðjþkÞþk

ð34Þ

corresponding to a generalized Maxwell fluid. Similar expressions for xðr; tÞ and sðr; tÞ, in terms of the Mittag–Leffler functions, have been recently obtained in [27]. 2. By letting now a ! 1 into (33) and (34), the similar solutions (see also (A8) and (A9)1)

jþk 1  C0 X 1

Z 1 sjþ2k 2 ðmn2 Þkþ1 emn ðtsÞ J 1 ðrnÞdn ds ; ð35Þ k 2p j;k¼0 C ðj þ 2k þ 1Þ 0 0 jþk Z 1 1  lC X 1 Cðj þ k þ 1Þ t 2jþk sðr; tÞ ¼ sN ðr; tÞ þ 0  nðmn2 Þj ð1 þ mkn2 Þk J 2 ðrnÞdn k 2p j;k¼0 Cðj þ 1ÞCðk þ 1Þ Cð2j þ k þ 1Þ 0 jþkþ1 Z t Z 1Z r 1  lC0 X 1 Cðj þ k þ 1Þ rjþ2k 2 þ  nðmn2 Þkþ1 J 2 ðrnÞemn ðtrÞðrsÞ=k ds dn dr k 2p j;k¼0 Cðj þ 1ÞCðk þ 1Þ 0 Cðj þ 2k þ 1Þ 0 0

xðr; tÞ ¼ xN ðr; tÞ þ



Cðj þ k þ 1Þ Cðj þ 1ÞCðk þ 1Þ

Z

t

ð36Þ

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C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

corresponding to the classical Maxwell model are recovered. Of course, these solutions are equivalent to those obtained in [25,27] by a different technique. 3. Making a and b ! 1, Eqs. (21) and (32) can be simplified as (see also Eq. (A9)1 for the second term of the shear stress)

xðr; tÞ ¼ xN ðr; tÞ þ

1 mþl¼k X  1jþk km k! C0 X r

2p

"



j;k¼0 m;lP0

Cðj þ k þ 1Þ m!l! Cðj þ 1ÞCðk þ 1Þ

k

Z

1

Z

0

t

2

ðmn2 Þkþ1 emn

ðtsÞ

0

# sjþ2km cn2 sjþ2kmþ1 ds dn; þ Cðj þ 2k  m þ 1Þ k Cðj þ 2k  m þ 2Þ jþk Z 1 1  lC k  k X 1 Cðj þ k þ 1Þ t 2jþk sðr; tÞ ¼ sN ðr; tÞ  0 r  nðmn2 Þj ð1 þ mkn2 Þk J 2 ðrnÞdn k j;k¼0 k 2p Cðj þ 1ÞCðk þ 1Þ Cð2j þ k þ 1Þ 0 Z Z Z m lC0 1 t r 3 2 n J 2 ðrnÞTðn; rÞemn ðtrÞðrsÞ=k ds dr dn;  k 2p 0 0 0  J 1 ðrnÞ

ð37Þ

ð38Þ

where Tðn; rÞ is obtained from Hðn; rÞ making a ¼ b ¼ 1. 4. In the special case when k ¼ 0, Eq. (14) can be written in the simple form

 H ðn; qÞ ¼ x

1 þ cn2 qb1

C0

2pn q þ cn2 qb þ mn2

¼

1 C0 X

2p n

"

qbkb

2 k

ðmn Þ

k¼0

ðq1b þ cn2 Þkþ1

þ cn

qbk1

2

# ð39Þ

ðq1b þ cn2 Þkþ1

and its inverse Laplace transform is (see Eq. (A3))

xH ðn; tÞ ¼

1 C0 X

2pn



ðmn2 Þk G1b;bkb;kþ1 ðcn2 ; tÞ þ cn2 G1b;bk1;kþ1 ðcn2 ; tÞ :

ð40Þ

k¼0

Applying the inverse Hankel transform to this last relation we get the velocity field under the form

xðr; tÞ ¼

1 Z C0 X

2p

k¼0

1



ðmn2 Þk J 1 ðrnÞ G1b;bkb;kþ1 ðcn2 ; tÞ þ cn2 G1b;bk1;kþ1 ðcn2 ; tÞ dn:

ð41Þ

0

Of course, the velocity field (41) is equivalent to that obtained by Sheng et al. [28, Eq. (23)], but unlike their solution, our solution (41) can be specialized to give the Newtonian solution. Indeed, making c ¼ 0 and b ¼ 1 into (41) and taking into account (19), we find that

xðr; tÞ ¼

C0

Z

2p

1

J 1 ðrnÞ

0

1 X ðmn2 Þk k¼0

tk C0 dn ¼ Cðk þ 1Þ 2p

Z

1

2

J 1 ðnrÞemn t dn

ð42Þ

0

represents the velocity field of a Newtonian fluid performing the same motion. As regards the shear stress sðr; tÞ, we have (see Eq. (7))

sðr; tÞ ¼ lð1 þ kr Dbt Þ



 o 1 xðr; tÞ;  or r

ð43Þ

in which xðr; tÞ, as it results from (41) and (A10), has the suitable form

xðr; tÞ ¼

C0

Z

2p

1

J 1 ðrnÞdn þ

0

1 Z C0 X

2p

1



ðmn2 Þk J 1 ðrnÞ G1b;bkb;kþ1 ðcn2 ; tÞ þ cn2 G1b;bk1;kþ1 ðcn2 ; tÞ dn

ð44Þ

0

k¼1

and therefore



 Z 1 Z 1

o 1 C0 1 C0 X nJ 2 ðnrÞdn  nðmn2 Þk J 2 ðrnÞ G1b;bkb;kþ1 ðcn2 ; tÞ þ cn2 G1b;bk1;kþ1 ðcn2 ; tÞ dn:  xðr;tÞ ¼  or r 2p 0 2p k¼1 0 ð45Þ

Introducing (45) into (43) and using (19) and (A11)1, we find that

sðr; tÞ ¼  



l C0 kr ð1  bÞ b 1þ t 2p Cð2  bÞ 1 X ðcn2 Þj j¼0

Z 0

1

nJ 2 ðrnÞdn  

1 X lC0 ð1 þ kr Dbt Þ 2p k¼1

Z 0

1

nðmn2 Þk J 2 ðrnÞ

Cðj þ k þ 1Þ tkþð1bÞj t kþð1bÞð1þjÞ dn: þ cn2 Cðj þ 1ÞCðk þ 1Þ C½k þ ð1  bÞj þ 1 C½k þ ð1  bÞð1 þ jÞ þ 1

ð46Þ

C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

503

Finally, making b ¼ 1 and using (A9)2, (A11)2 and (A12), we obtain, after lengthy but straightforward computations, the wellknown expression ([5], Eq. (26))

sðr; tÞ ¼ 

l C0 2p

Z

1

0

nJ 2 ðrnÞ 1 þ cn2

!

exp 

mn2 t dn 1 þ cn2

ð47Þ

corresponding to a second grade fluid. The adequate velocity field ([5], Eq. (25))

xðr; tÞ ¼

C0 2p

Z 0

1

!

mn2 t dn J 1 ðrnÞ exp  1 þ cn2

ð48Þ

can be also obtained from (41) by making b ¼ 1: Certainly, this last result is in accordance with that obtained by Rajagopal [29] by a different technique. More exactly, the vortex decays faster as the viscosity m increases, while its decay is slower if the relaxation time kr is larger. The first study regarding the decay of vortices in viscous fluids seems to be that of Taylor [30].

6. Conclusions and numerical results The aim of this paper is to provide exact solutions for the velocity field xðr; tÞ and the shear stress sðr; tÞ corresponding to the circular motion of a generalized Oldroyd-B fluid due to a potential vortex. These solutions, presented under integral and series forms in terms of generalized G and R functions, have been determined by means of the Laplace and Hankel transforms. Furthermore, in order to avoid the lengthy calculations of residues and contour integrals, the discrete Laplace transform method has been used. Making kr ! 0 and b ¼ 1 or kr ! 0, b ¼ 1 and a ¼ 1 into Eqs. (21) and (32), the similar solutions for generalized and ordinary Maxwell fluids are obtained. In the special case when a and b ¼ 1, the corresponding solutions for an ordinary Oldroyd-B fluid, performing the same motion, are recovered. These solutions, as it results from Fig. 1, are identical to those obtained in [5] by a different technique. It is worthy pointing out that all solutions that have been here obtained are presented as a sum between the similar Newtonian solutions and the adequate non-Newtonian contributions. Finally, the solutions for generalized second grade fluids are also established. These solutions, unlike the previous solutions obtained by Sheng et al. [28], can be easily specialized to give the Newtonian solutions. Finally, in order to reveal some relevant physical effects of the obtained results, the graphs of the velocity field xðr; tÞ corresponding to Eqs. (21) and (22) have been drawn against r for different values of t and of the material parameters. From these graphs, we can see not only the difference between the behavior of the two types of fluids (Newtonian and non-Newtonian) but also the influence of these parameters on the decay of the vortex. From Figs. 2 and 3, as well as from Figs. 4 and 5, it clearly results that the vortex is stronger in a non-Newtonian fluid and it increases if the fractional parameters a or b or the material parameters k or kr decrease. For large values of t the profiles of the non-Newtonian velocities draw near to those for Newtonian fluids. The influence of the viscosity m, on the decay of the vortex, it is shown in Fig. 6. The diagrams clearly show that, in accordance with our expectations, the vortex increases if the viscosity m decreases. Of course, in all cases, the vortex is damping in time and space, for large times the difference between the two models being almost imperceptible.

Fig. 2. Decay of the potential vortex in a Newtonian (curves xN1, xN2) and non-Newtonian fluid (curves x1, x2, x3, x4), for kr ¼ 3, b ¼ 0:6 and different values of a and t.

m ¼ 0:002, C0 ¼ 1, k ¼ 6,

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Fig. 3. Decay of the potential vortex in a Newtonian (curves xN1, xN2) and non-Newtonian fluid (curves x1, x2, x3, x4), for kr ¼ 3, a ¼ 0:6 and different values of b and t.

m ¼ 0:002, C0 ¼ 1, k ¼ 6,

Fig. 4. Decay of the potential vortex in a Newtonian (curves xN1, xN2) and non-Newtonian fluid (curves x1, x2, x3, x4), for a ¼ 0:6, b ¼ 0:8 and different values of k and t.

m ¼ 0:002, C0 ¼ 1, kr ¼ 3,

Fig. 5. Decay of the potential vortex in a Newtonian (curves xN1, xN2) and non-Newtonian fluid (curves x1, x2, x3, x4), for a ¼ 0:6, b ¼ 0:8 and different values of kr , and t.

m ¼ 0:002, C0 ¼ 1, k ¼ 6,

C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

505

Fig. 6. Profiles of the velocity field xðr; tÞ, given by Eq. (21) – curves x1, x2, x3 and Eq. (22) – curves xN1, xN2, xN3, for C0 ¼ 1, k ¼ 10, kr ¼ 3, a ¼ 0:6, b ¼ 0:8, t ¼ 5 and different values of m.

Acknowledgements The authors Corina Fetecau, C. Fetecau and D. Vieru acknowledge the support from the Ministry of Education and Research, through PN II-Ideas, Grant No. 26/28-09-2007, CNCSIS Code ID_593. The authors would like to express their gratitude to reviewers both for their careful assessment and fruitful suggestions regarding the previous form of this work. Appendix

LfDpt f ðtÞg ¼ qp Lff ðtÞg  qp1 f ð0Þ; 0 < p 6 1; Z t Z t  1 ðqÞu  2 ðqÞg ¼ ðu1  u2 ÞðtÞ ¼ L1 fu u1 ðt  sÞu2 ðsÞds ¼ u1 ðsÞu2 ðt  sÞds 0

0

 1 ðqÞg and u2 ðtÞ ¼ L1 fu  2 ðqÞg; if u1 ðtÞ ¼ L1 fu b

ðA2Þ



d

< 1;

qa

q g; Reðac  bÞ > 0; ReðqÞ > 0; Ga;b:c ðd; tÞ ¼ L1 f a ðq  dÞc " ! # Z 1 2 1 b 2 2 1  exp  2 ; a; b > 0; ea x J 1 ðbxÞdx ¼ b 4a 0  a t 1 L ¼ aþ1 ; ReðaÞ > 1; ReðqÞ > 0; q Cða þ 1Þ " ! !# Z 1 2 2 2 b b 2 2 xea x J 2 ðbxÞdx ¼ 2 1  1 þ 2 exp  2 ; a; b > 0; 4a 4a 0 b  b q L1 a ¼ Ra;b ðc; 0; tÞ; Reða  bÞ > 0; ReðqÞ > 0; q c k   X 1  1 1 tk  R1;0  ; 0; t ¼ ¼ et=k ; k k C ðk þ 1Þ k¼0 k X m¼0

k!am ¼ ð1 þ aÞk ; m!ðk  mÞ!

1 X j¼0

ðA1Þ

Cðj þ k þ 1Þ 1 ; ðan2 Þj ¼ Cðj þ 1ÞCðk þ 1Þ ð1 þ an2 Þkþ1

G1b;b;1 ðcn2 ; tÞ þ cn2 G1b;1;1 ðcn2 ; tÞ ¼ 1; ( 0; b¼1 Cða þ 1Þ ab ; Dbt ðt a Þ ¼ t ; Dbt ðcÞ ¼ b c C ða  b þ 1Þ t ; b 2 ð0; 1Þ Cð1bÞ !k ! 1 X 1 mtn2 mtn2 : ¼ exp k! 1 þ an2 1 þ an2 k¼0

ðA3Þ ðA4Þ ðA5Þ ðA6Þ ðA7Þ ðA8Þ

ðA9Þ ðA10Þ ðA11Þ

ðA12Þ

506

C. Fetecau et al. / Applied Mathematics and Computation 205 (2008) 497–506

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