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Decay of a potential vortex in an Oldroyd-B fluid
International Journal of Engineering Science 43 (2005) 340–351 www.elsevier.com/locate/ijengsci
Decay of a potential vortex in an Oldroyd-B fluid C. Fetecau
a,*
, Corina Fetecau
b
a
b
Department of Mathematics, Technical University of Iasi, R-6600 Iasi, Romania Department of Theoretical Mechanics, Technical University of Iasi, R-6600 Iasi, Romania
Received 12 March 2003; received in revised form 26 July 2004; accepted 12 August 2004
Abstract Analytical expressions for the velocity field and the associated tangential tension corresponding to a potential vortex in an Oldroyd-B fluid are determined by means of the Hankel transform. The well-known solutions for a Navier–Stokes fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams are presented for the circular motion of the glycerine. In each case, the velocity fields as well as the adequate tangential tensions are going to zero for t or r ! 1. Consequently, the potential vortex is damping in time and space. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Potential vortex; Oldroyd fluid; Velocity field; Tangential tension; Limiting cases
1. Introduction The rheological properties of materials are specified in general by their so-called constitutive equations. These equations determine the flow behavior of the respective materials in arbitrary types of motion. The simplest constitutive equation for a fluid is a Newtonian one. The departure from Newtonian behavior of many real fluids, especially those of high molecular weight, manifests
*
Corresponding author. Tel.: +40 232263218. E-mail addresses: [email protected] (C. Fetecau), [email protected] (C. Fetecau).
0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.08.013
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341
itself in these materials in many ways [1]. After much experimenting and theorizing it has been concluded that a new and non-linear constitutive equation for the stress is required. Mechanics of non-linear fluids present a special challenge to engineers, physicists and mathematicians. The non-linearity can manifest itself in a variety of ways. One of the simplest material models, which have been employed to study non-Newtonian fluid behavior, is that of second grade fluids [2,3]. This model has received an especially large (perhaps even an inordinately large [4]) amount of attention. Another interesting class of non-Newtonian fluids is that of Maxwell model. This model, belonging to the rate-type fluids 1, has been applied to a large variety of viscoelastic flows, usually with the small dimensionless relaxation time. The models in which besides time derivatives of the stress tensor also derivatives of the rate of strain tensor are included show relaxation as well as retardation behavior. An important example of this kind is the Jeffreys version of the Oldroyd model [1,6–8] in which the Cauchy stress T has the form d d ð1Þ T ¼ pI þ S; S þ k S ¼ l A þ kr A : dt dt Here A is the first Rivlin–Ericksen tensor, pI denotes the indeterminate spherical stress, S is the extra-stress tensor, k and kr are relaxation and retardation times, l is the dynamic viscosity of the fluid and d/dt represents a convective derivative. The most popular choice is the upper convected derivative dS _ ¼ S LS SLT ; dt
ð2Þ
where L is the velocity gradient and the dot denotes material time differentiation. The associated model is usually referred to as the Oldroyd-B fluid. This model, as well as the Maxwell model, can be derived from a rigorous thermodynamic standpoint [5]. In fact, the Oldroyd-B model is a special case of the model due to Burgers that was introduced in 1939 [9]. For kr = 0 it reduces to the classical model due to Maxwell, for k = 0 it becomes of the type of a second order fluid and for kr = k = 0 it reduces to a Newtonian one. Recently, the Oldroyd-B fluids have received a lot of attention. Existence, uniqueness and stability results for some motions of viscoelastic fluids of this type have been established by Guillope` and Saut [10] and Fontelos and Friedman [11]. The first exact solutions for the flow of an Oldroyd-B fluid seem to be those of Rajagopal and Bhatnagar [12], Fetecau [13,14] and Fetecau and Fetecau [15]. Other analytical results are obtained by Georgiou [16] for small one-dimensional perturbations and for the limiting case of zero Reynolds number. In this note we study the decay of a potential vortex in such a fluid. The velocity field and the associate tangential tension are determined for all values of the material constants. The corresponding solutions for a Navier–Stokes fluid, as well as those for a Maxwell fluid and a second grade one, appear as limiting cases of our solutions.
1
A recent review of these fluid models is given by Rajagopal and Srinivasa [5].
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2. Statement of the problem We proposed to study the circular motion of an Oldroyd-B fluid, whose velocity field in the system of cylindrical coordinates r, h and z has the form vðr; tÞ ¼ xðr; tÞeh ; ð3Þ where eh denotes the unit vector along the h direction. The initial distribution of the velocity is assumed to be that of a potential vortex of circulation C0 [17], i.e. xðr; 0Þ ¼ C0 =ð2prÞ: ð4Þ Substituting (3) in (1) and (2) and having in mind the initial condition (the fluid being at rest up to the moment t = 0) Sðr; 0Þ ¼ 0 ð5Þ and Eqs. (21.1-4) of [12] (see also [18], Eqs. (2.4)) we get Srr = Srz = Shz = Szz = 0 and ð1 þ kot ÞS hh 2kðor x x=rÞS rh ¼ 2lkr ðor x x=rÞ2 ;
ð6Þ
ð1 þ kot ÞS rh ¼ lð1 þ kr ot Þðor x x=rÞ:
ð7Þ
In the same time the equations of motion reduce to (cf. [12], Eqs. (22.1-3)) op 1 þ S hh ¼ qx2 =r or r and or S rh þ 2S rh =r ¼ qot x;
ð8Þ
ð9Þ
where q is the constant density of the fluid. The constraint of incompressibility is automatically satisfied. We also observe that Eqs. (6) and (8) for Shh and p are not coupled with Eqs. (7) and (9), meaning that one can firstly solve the system of the latter two equations and then calculate Shh and p. Eliminating Srh between Eqs. (7) and (9) we attain to the linear partial differential equation (see [12], Eq. (28)) 1 1 2 2 ð10Þ kot xðr; tÞ þ ot xðr; tÞ ¼ mð1 þ kr ot Þ or þ or 2 xðr; tÞ; r r where m = l/q is the kinematic viscosity of the fluid. It is also worth emphasizing that this equation is of a higher order than the similar Navier– Stokes equation or that for a second grade fluid. Thus, in order to obtain an exact solution, in general additional boundary and initial conditions are required. For a detailed discussion about this issue and for some interesting examples we refer the reader to [12] and [19–22]. The flow domain being unbounded the appropriate boundary conditions xðr; tÞ; or xðr; tÞ ! 0
as r ! 1
ð11Þ
assure the fact that the fluid is quiescent at infinity and there is no shear in the free stream [22]. Moreover, we assume that (cf. [18]) ot xðr; tÞ ! 0
when t ! 0:
ð12Þ
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3. Exact solutions 3.1. Calculation of x Multiplying (10) by rJ1(nr), where J1(Æ) is the Bessel function of the first kind of order one, integrating over r from 0 to 1 and taking into account the boundary conditions (11) we find that ko2t xh ðn; tÞ þ ð1 þ an2 Þot xh ðn; tÞ þ mn2 xh ðn; tÞ ¼ 0;
ð13Þ
n; t > 0;
where xh(n, t) is the Hankel transform [23] of the function x(r, t) and a = mkr. From (4) and (12) we also have the initial conditions xh ðn; 0Þ ¼ C0 =ð2pnÞ
and ot xh ðn; 0Þ ¼ 0:
ð14Þ
The solution of the ordinary differential equation (13) with the initial conditions (14) has one of the following three forms: xh ðn; tÞ ¼
C0 r2 expðr1 tÞ r1 expðr2 tÞ r2 r1 2pn
xh ðn; tÞ ¼
C0 mkn2 expðt=kÞ expðmn2 tÞ 2pn mkn2 1
ð15Þ
if k < kr ;
ð16Þ
if k ¼ kr
and 8 C0 r2 expðr1 tÞ r1 expðr2 tÞ > > > ; < 2pn r2 r1 xh ðn; tÞ ¼ > C0 1 þ an2 bt 1 þ an2 bt > > þ ; exp t cos sin : 2k 2k 2pn 2k b
n 2 ð0; aÞ [ ðb; 1Þ; n 2 ða; bÞ if k > kr : ð17Þ
In the above relations we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ an2 Þ ð1 þ an2 Þ2 4mkn2 ; b ¼ 4mkn2 ð1 þ an2 Þ2 ; Pr1 ; r2 ¼ 2k 1 1 and b ¼ pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi : a ¼ pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi mð k þ k kr Þ mð k k kr Þ Inverting (15)–(17) by means of the Hankel inversion formula [23] we find that Z C0 1 r2 expðr1 tÞ r1 expðr2 tÞ xðr; tÞ ¼ J 1 ðrnÞ dn if k < kr ; r2 r1 2p 0 xðr; tÞ ¼
C0 2p
Z
1 0
mkn2 expðt=kÞ expðmn2 tÞ J 1 ðrnÞ dn mkn2 1
if k ¼ kr
ð18Þ
ð19Þ
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and Z
a
r2 expðr1 tÞ r1 expðr2 tÞ J 1 ðrnÞ dn r2 r1 0 t Z b C0 an2 bt 1 þ an2 bt þ exp þ J 1 ðrnÞ dn exp t cos sin 2k a 2k 2k 2p 2k b Z C0 1 r2 expðr1 tÞ r1 expðr2 tÞ þ J 1 ðrnÞ dn if k > kr : r2 r1 2p b
C0 xðr; tÞ ¼ 2p
ð20Þ
The value of the circulation C on a circle of radius r, given by C(r, t) = 2prx(r, t), describes the diffusion of vorticity as function of r and t. 3.2. Calculation of the tangential tension Srh The solution of the ordinary differential equation (7) with the initial condition (5) is t Z t s l xðr; sÞ exp ð1 þ kr os Þ or xðr; sÞ ds: S rh ðr; tÞ ¼ exp k k 0 k r
ð21Þ
Introducing (18)–(20) in (21) and using the identity rJ 2 ðrÞ ¼ J 1 ðrÞ rJ 01 ðrÞ one obtains after a lengthy but straight-computation that Z lC0 1 expðr2 tÞ expðr1 tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn if k < kr ; S rh ðr; tÞ ¼ 2p 0 2 ð1 þ an2 Þ 4mkn2 lC0 S rh ðr; tÞ ¼ 2p and
Z
1
0
Z
expðmn2 tÞ expðt=kÞ nJ 2 ðrnÞ dn mkn2 1
if k ¼ kr
ð23Þ
a
expðr2 tÞ expðr1 tÞ lC0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn p 0 ð1 þ an2 Þ2 4mkn2 t Z b 1 an2 bt
exp exp nJ 2 ðrnÞ dn t sin 2k a b 2k 2k Z lC0 1 expðr2 tÞ expðr1 tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn if k > kr : þ 2p b ð1 þ an2 Þ2 4mkn2
lC0 S rh ðr; tÞ ¼ 2p
ð22Þ
ð24Þ
The normal tension Shh(r, t) as well as the hydrostatic pressure p(r, t) can be easily obtained by means of (5), (6) and (8) as soon as the velocity field x(r, t) and the associated tangential tension Srh(r, t) are known.
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4. Limiting cases Taking the limit of Eqs. (18) and (22) as k ! 0, we get the velocity field (see also [24], Eq. (13)) Z C0 1 mn2 xðr; tÞ ¼ J 1 ðrnÞ exp t dn ð25Þ 2p 0 1 þ an2 and the associated tangential tension Z lC0 1 nJ 2 ðrnÞ mn2 S rh ðr; tÞ ¼ exp t dn; 2p 0 1 þ an2 1 þ an2
ð26Þ
corresponding to the same problem for a second grade fluid. In the special case when kr (and then a) ! 0, corresponding to a Maxwell fluid, our solutions (20) and (24) reduce to (see [25], Eqs. (3.4) and (4.2)) pffiffiffi Z C0 1=ð2 mkÞ r4 expðr3 tÞ r3 expðr4 tÞ J 1 ðrnÞ dn xðr; tÞ ¼ r4 r3 2p 0 t Z 1 ct 1 ct C0 þ cos ð27Þ þ sin J 1 ðrnÞ dn; exp 2k 1=ð2pffiffiffi 2k c 2k 2p mkÞ respectively, Z
pffiffiffi 1=ð2 mkÞ
t expðr4 tÞ expðr3 tÞ lC0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp nJ ðrnÞ dn 2 2k p 0 1 4mkn2 ct 1 nJ 2 ðrnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin dn; ð28Þ pffiffiffi 2k 1=ð2 mkÞ 4mkn2 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 14mkn2 and c ¼ 4mkn2 1. where r3;4 ¼ 2k Finally, by letting k ! 0 in (19) and (23) or in (27) and (28) as well as a ! 0 in (25) and (26), we attain to Z C0 1 expðmn2 tÞJ 1 ðrnÞ dn ð29Þ xðr; tÞ ¼ 2p 0 lC0 S rh ðr; tÞ ¼ 2p Z
and lC0 S rh ðr; tÞ ¼ 2p
Z
1
expðmn2 tÞnJ 2 ðrnÞ dn
ð30Þ
0
i.e., the similar solutions corresponding to a Navier–Stokes fluid. Having in mind the entry 4 of Table 5 of [23] they can be written in terms of the hypergeometric function 1 X a aða þ 1Þ x2 Cða þ nÞCðbÞ xn F ða; b; xÞ ¼ 1 þ x þ þ ¼ b bðb þ 1Þ 2! CðaÞCðb þ nÞ n! n¼0
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under analytical forms rC0 r2 xðr; tÞ ¼ F 1; 2; ; 8pmt 4mt
ð31Þ
respectively,
lC0 r2 r2 S rh ðr; tÞ ¼ F 2; 3; ; 2p ð4mtÞ2 4mt
ð32Þ
2 from which it clearly results pffiffiffiffi that rx(r, t) and r Srh(r, t) depend of r and t only by means of the similarity variable r=ð2 mtÞ.
3
3
ω1 (r) ω2 (r)
2
ω3 (r) ω4 (r) ω5 (r)
1
ω6 (r)
0
0
(a)
0.2
0.4
0.001
0.6
0.8
1 1
0.6
0.8
1 1
r
0.4 ω1 (r)
0.3
ω2 (r) ω3 (r)
0.2
ω4 (r) ω5 (r) ω6 (r) 0.1
0
(b)
0
0.2 0.001
0.4 r
Fig. 1. Velocity profiles x(r, t) corresponding to an Oldroyd (curves x1, x2, x3) and Navier–Stokes (curves x4, x5, x6) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 8, 14 and 20 s and (b) t = 40, 50 and 60 s.
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347
In view of the identities xF ð1; 2; xÞ ¼ ex 1 and x2 F ð2; 3; xÞ ¼ 2½1 ð1 þ xÞex ; the last two solutions take the simple forms (the first of them being obtained in [17] by means of the similarity by transformation of variables) C0 r2 1 exp ð33Þ xðr; tÞ ¼ 2pr 4mt
0
0
2 S1 (r) S2 (r) S3 (r)
4
S4 (r) S5 (r)
6
S6 (r)
8 9
(a)
0.5
1 r
1.5
2 2
0.5
1 r
1.5
2 2
0.001
0
S1 (r)
0
0.5
S2 (r) S3 (r) S4 (r)
1
S5 (r) S6 (r)
1.5 1.5
(b)
0.001
Fig. 2. Tangential tension profiles Srh(r, t) corresponding to an Oldroyd (curves S1, S2, S3) and Navier–Stokes (curves S4, S5, S6) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 8, 14 and 20 s and (b) t = 40, 50 and 60 s.
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and lC0 S rh ðr; tÞ ¼ 2 pr
r2 r2 1þ exp 1 : 4mt 4mt
ð34Þ
5. Numerical results and conclusions In this paper we have determined the exact solutions corresponding to a circular motion of an Oldroyd-B fluid due to a potential vortex. For k > kr, these solutions contain sine and cosine 3
3
ω1 ( r)
2
ω2 ( r) ω3 ( r) ω4 ( r)
1
0
0
(a)
0.2
0.4
0.001
0.6
0.8
1 1
0.6
0.8
1 1
r
0.4
0.3 ω1 ( r) ω2 ( r) ω3 ( r)
0.2
ω4 ( r)
0.1
0 0
(b)
0.2 0.001
0.4 r
Fig. 3. Velocity profiles x(r, t) corresponding to an Oldroyd (curves x1), second grade (curves x2), Maxwell (curves x3) and Navier–Stokes (curves x4) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 5 s and (b) t = 40 s.
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349
terms. This indicates that, in this case, unlike the Navier–Stokes fluid, oscillations are set up in the fluid. The amplitudes of these oscillations decay exponentially in time, the damping being proportional to exp(t/2k). Direct computations show that x(r, t) and Srh(r, t), given by (18)–(20) and (22)–(24) satisfy both the associate partial differential equations (7), (9), (10) and all imposed initial and boundary conditions. Moreover, they remain finite for t and r > 0 and tend to zero for t or r ! 1. In the special cases when kr or k ! 0 our solutions reduce to those corresponding to a Maxwell fluid or a second grade one, respectively. If both kr and k ! 0, corresponding to a Navier–Stokes fluid, these solutions take the simple forms (33) and (34)pin ffiffiffiffi which rx(r, t) and r2Srh(r, t) depend of r and t only by means of the similarity variable r=ð2 mtÞ.
0
0
5 S1 (r) S2 (r)
10
S3 (r) S4 (r)
15
20 23
(a)
0.5
1 r
1.5
2 2
0.5
1 r
1.5
2 2
0.001 0
0
S1 (r) 0.5 S2 (r) S3 (r) S4 (r)
1
1.5 1.5
(b)
0.001
Fig. 4. Tangential tension profiles Srh(r, t) corresponding to an Oldroyd (curves S1), second grade (curves S2), Maxwell (curves S3) and Navier–Stokes (curves S4) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 5 s and (b) t = 40 s.
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In Figs. 1 and 2, for comparison, the diagrams of the velocity fields and of the adequate tangential tensions for the Oldroyd and Navier–Stokes models are plotted for the circular motion of glycerine. It is clearly seen that for a given value of the time t the velocity corresponding to an Oldroyd-B fluid (given by the relation (18)) is larger than that of a Newtonian fluid (given by the relation (33)). As regards the tangential tensions, as it was to be expected, a reverse situation it is generally found and the difference between the two models, Newtonian and non-Newtonian, is strong enough for small values of t. For large values of t the profiles of the velocities as well as those of the tensions are closing by. Figs. 3 and 4 present the velocity and tangential tension profiles corresponding to an OldroydB, second grade, Maxwell and Navier–Stokes fluid as they result from the relations (18), (25), (27), (33) and (22), (26), (28), (34), respectively. For small values of the time t one can observe both the differences between the adequate diagrams and the oscillations corresponding to the non-Newtonian models. In each case, for large values of t and r, x(r, t) and Srh(r, t) are going to zero. Consequently, the vortex decays in time and space. Acknowledgment The authors would like to express their gratitude to the referee for his constructive comments and suggestions regarding the initial shape of this work. References [1] K.R. Rajagopal, Mechanics of non-Newtonian fluids, in: Recent Developments in Theoretical Fluids Mechanics, Pitman Research Notes in Mathematics, vol. 291, Longman, New York, 1993, pp. 129–162. [2] R.S. Rivlin, J.L. Ericksen, Stress deformation relation for isotropic materials, J. Ration. Mech. Anal. 4 (1955) 323– 425. [3] K.R. Rajagopal, A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Non-Linear Mech. 17 (1982) 369–373. [4] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (5) (1995) 689–729. [5] K.R. Rajagopal, A.R. Srinivasa, A thermodynamic frame work for rate type fluid models, J. Non-Newtonian Fluid Mech. 88 (2000) 207–227. [6] J.G. Oldroyd, On the formulation of the rheological equations of state, Proc. Royal Soc. Lond. Ser. A 200 (1950) 523–541. [7] R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworts, Boston–London–Singapore– Sydney–Toronto–Wellington, 1989. [8] D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990. [9] J.M. Burgers, Mechanical considerations––model systems––phenomenological theories of relaxation and of viscosity. First report on viscosity and plasticity. Prepared by the committee for the study of viscosity of the academy of sciences at Amsterdam, second ed., Nordemann Publ, New York, 1939. [10] C. Guillope´, J.C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model. Math. Anal. Numer. 24 (1990) 369–401. [11] M.A. Fontelos, A. Friedman, Stationary non-Newtonian fluid flows in channel-like and pipe-like domains, Arch. Ration. Mech. Anal. 151 (2000) 1–43. [12] K.R. Rajagopal, R.K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113 (1995) 233–239.
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[13] C. Fetecau, The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid, C.R. Acad. Sci. Paris Ser. I 335 (2002) 979–984. [14] C. Fetecau, Analytical solutions for non-Newtonian fluid flows in pipe-like domains, Int. J. Non-Linear Mech. 39 (2004) 225–231. [15] C. Fetecau, Corina Fetecau, The first problem of Stokes for an Oldroyd-B fluid, Int. J. Non-Linear Mech. 38 (2003) 1539–1544. [16] G.C. Georgiou, On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall, Rheol. Acta 35 (1996) 39–47. [17] J. Zierep, Similarity Laws and Modeling, Marcel Dekker, Inc., New York, 1971. [18] P.N. Srivastava, Non-steady helical flow of a visco-elastic liquid, Arch. Mech. Stos. 18 (2) (1966) 145–150. [19] K.R. Rajagopal, A.S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica 19 (1984) 1948–1954. [20] K.R. Rajagopal, On the creeping flow of the second-order fluid, J. Non-Newtonian Fluid Mech. 15 (1984) 239–246. [21] K.R. Rajagopal, P.N. Kaloni, Some remarks on boundary conditions for flows of fluids of the differential type, in: Proceedings of Canadian Applied Mathematics Society, Barnaby, Canada, 1989. [22] K.R. Rajagopal, On boundary conditions for fluids of the differential type, in: Navier–Stokes Equations and Related Nonlinear Problems, Plenum Press, New York, 1995, pp. 273–278. [23] I.N. Sneddon, Fourier Transforms, McGraw Hill Book Company, Inc., New York–Toronto–London, 1951. [24] C. Fetecau, Corina Fetecau, J. Zierep, Decay of a potential vortex and propagation of a heat wave in a second grade fluid, Int. J. Non-Linear Mech. 37 (6) (2002) 1051–1056. [25] C. Fetecau, Corina Fetecau, Decay of a potential vortex in a Maxwell fluid, Int. J. Non-Linear Mech. 38 (7) (2003) 985–990.
Decay of a potential vortex in an Oldroyd-B fluid C. Fetecau
a,*
, Corina Fetecau
b
a
b
Department of Mathematics, Technical University of Iasi, R-6600 Iasi, Romania Department of Theoretical Mechanics, Technical University of Iasi, R-6600 Iasi, Romania
Received 12 March 2003; received in revised form 26 July 2004; accepted 12 August 2004
Abstract Analytical expressions for the velocity field and the associated tangential tension corresponding to a potential vortex in an Oldroyd-B fluid are determined by means of the Hankel transform. The well-known solutions for a Navier–Stokes fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams are presented for the circular motion of the glycerine. In each case, the velocity fields as well as the adequate tangential tensions are going to zero for t or r ! 1. Consequently, the potential vortex is damping in time and space. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Potential vortex; Oldroyd fluid; Velocity field; Tangential tension; Limiting cases
1. Introduction The rheological properties of materials are specified in general by their so-called constitutive equations. These equations determine the flow behavior of the respective materials in arbitrary types of motion. The simplest constitutive equation for a fluid is a Newtonian one. The departure from Newtonian behavior of many real fluids, especially those of high molecular weight, manifests
*
Corresponding author. Tel.: +40 232263218. E-mail addresses: [email protected] (C. Fetecau), [email protected] (C. Fetecau).
0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2004.08.013
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itself in these materials in many ways [1]. After much experimenting and theorizing it has been concluded that a new and non-linear constitutive equation for the stress is required. Mechanics of non-linear fluids present a special challenge to engineers, physicists and mathematicians. The non-linearity can manifest itself in a variety of ways. One of the simplest material models, which have been employed to study non-Newtonian fluid behavior, is that of second grade fluids [2,3]. This model has received an especially large (perhaps even an inordinately large [4]) amount of attention. Another interesting class of non-Newtonian fluids is that of Maxwell model. This model, belonging to the rate-type fluids 1, has been applied to a large variety of viscoelastic flows, usually with the small dimensionless relaxation time. The models in which besides time derivatives of the stress tensor also derivatives of the rate of strain tensor are included show relaxation as well as retardation behavior. An important example of this kind is the Jeffreys version of the Oldroyd model [1,6–8] in which the Cauchy stress T has the form d d ð1Þ T ¼ pI þ S; S þ k S ¼ l A þ kr A : dt dt Here A is the first Rivlin–Ericksen tensor, pI denotes the indeterminate spherical stress, S is the extra-stress tensor, k and kr are relaxation and retardation times, l is the dynamic viscosity of the fluid and d/dt represents a convective derivative. The most popular choice is the upper convected derivative dS _ ¼ S LS SLT ; dt
ð2Þ
where L is the velocity gradient and the dot denotes material time differentiation. The associated model is usually referred to as the Oldroyd-B fluid. This model, as well as the Maxwell model, can be derived from a rigorous thermodynamic standpoint [5]. In fact, the Oldroyd-B model is a special case of the model due to Burgers that was introduced in 1939 [9]. For kr = 0 it reduces to the classical model due to Maxwell, for k = 0 it becomes of the type of a second order fluid and for kr = k = 0 it reduces to a Newtonian one. Recently, the Oldroyd-B fluids have received a lot of attention. Existence, uniqueness and stability results for some motions of viscoelastic fluids of this type have been established by Guillope` and Saut [10] and Fontelos and Friedman [11]. The first exact solutions for the flow of an Oldroyd-B fluid seem to be those of Rajagopal and Bhatnagar [12], Fetecau [13,14] and Fetecau and Fetecau [15]. Other analytical results are obtained by Georgiou [16] for small one-dimensional perturbations and for the limiting case of zero Reynolds number. In this note we study the decay of a potential vortex in such a fluid. The velocity field and the associate tangential tension are determined for all values of the material constants. The corresponding solutions for a Navier–Stokes fluid, as well as those for a Maxwell fluid and a second grade one, appear as limiting cases of our solutions.
1
A recent review of these fluid models is given by Rajagopal and Srinivasa [5].
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2. Statement of the problem We proposed to study the circular motion of an Oldroyd-B fluid, whose velocity field in the system of cylindrical coordinates r, h and z has the form vðr; tÞ ¼ xðr; tÞeh ; ð3Þ where eh denotes the unit vector along the h direction. The initial distribution of the velocity is assumed to be that of a potential vortex of circulation C0 [17], i.e. xðr; 0Þ ¼ C0 =ð2prÞ: ð4Þ Substituting (3) in (1) and (2) and having in mind the initial condition (the fluid being at rest up to the moment t = 0) Sðr; 0Þ ¼ 0 ð5Þ and Eqs. (21.1-4) of [12] (see also [18], Eqs. (2.4)) we get Srr = Srz = Shz = Szz = 0 and ð1 þ kot ÞS hh 2kðor x x=rÞS rh ¼ 2lkr ðor x x=rÞ2 ;
ð6Þ
ð1 þ kot ÞS rh ¼ lð1 þ kr ot Þðor x x=rÞ:
ð7Þ
In the same time the equations of motion reduce to (cf. [12], Eqs. (22.1-3)) op 1 þ S hh ¼ qx2 =r or r and or S rh þ 2S rh =r ¼ qot x;
ð8Þ
ð9Þ
where q is the constant density of the fluid. The constraint of incompressibility is automatically satisfied. We also observe that Eqs. (6) and (8) for Shh and p are not coupled with Eqs. (7) and (9), meaning that one can firstly solve the system of the latter two equations and then calculate Shh and p. Eliminating Srh between Eqs. (7) and (9) we attain to the linear partial differential equation (see [12], Eq. (28)) 1 1 2 2 ð10Þ kot xðr; tÞ þ ot xðr; tÞ ¼ mð1 þ kr ot Þ or þ or 2 xðr; tÞ; r r where m = l/q is the kinematic viscosity of the fluid. It is also worth emphasizing that this equation is of a higher order than the similar Navier– Stokes equation or that for a second grade fluid. Thus, in order to obtain an exact solution, in general additional boundary and initial conditions are required. For a detailed discussion about this issue and for some interesting examples we refer the reader to [12] and [19–22]. The flow domain being unbounded the appropriate boundary conditions xðr; tÞ; or xðr; tÞ ! 0
as r ! 1
ð11Þ
assure the fact that the fluid is quiescent at infinity and there is no shear in the free stream [22]. Moreover, we assume that (cf. [18]) ot xðr; tÞ ! 0
when t ! 0:
ð12Þ
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3. Exact solutions 3.1. Calculation of x Multiplying (10) by rJ1(nr), where J1(Æ) is the Bessel function of the first kind of order one, integrating over r from 0 to 1 and taking into account the boundary conditions (11) we find that ko2t xh ðn; tÞ þ ð1 þ an2 Þot xh ðn; tÞ þ mn2 xh ðn; tÞ ¼ 0;
ð13Þ
n; t > 0;
where xh(n, t) is the Hankel transform [23] of the function x(r, t) and a = mkr. From (4) and (12) we also have the initial conditions xh ðn; 0Þ ¼ C0 =ð2pnÞ
and ot xh ðn; 0Þ ¼ 0:
ð14Þ
The solution of the ordinary differential equation (13) with the initial conditions (14) has one of the following three forms: xh ðn; tÞ ¼
C0 r2 expðr1 tÞ r1 expðr2 tÞ r2 r1 2pn
xh ðn; tÞ ¼
C0 mkn2 expðt=kÞ expðmn2 tÞ 2pn mkn2 1
ð15Þ
if k < kr ;
ð16Þ
if k ¼ kr
and 8 C0 r2 expðr1 tÞ r1 expðr2 tÞ > > > ; < 2pn r2 r1 xh ðn; tÞ ¼ > C0 1 þ an2 bt 1 þ an2 bt > > þ ; exp t cos sin : 2k 2k 2pn 2k b
n 2 ð0; aÞ [ ðb; 1Þ; n 2 ða; bÞ if k > kr : ð17Þ
In the above relations we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ an2 Þ ð1 þ an2 Þ2 4mkn2 ; b ¼ 4mkn2 ð1 þ an2 Þ2 ; Pr1 ; r2 ¼ 2k 1 1 and b ¼ pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi : a ¼ pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi mð k þ k kr Þ mð k k kr Þ Inverting (15)–(17) by means of the Hankel inversion formula [23] we find that Z C0 1 r2 expðr1 tÞ r1 expðr2 tÞ xðr; tÞ ¼ J 1 ðrnÞ dn if k < kr ; r2 r1 2p 0 xðr; tÞ ¼
C0 2p
Z
1 0
mkn2 expðt=kÞ expðmn2 tÞ J 1 ðrnÞ dn mkn2 1
if k ¼ kr
ð18Þ
ð19Þ
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and Z
a
r2 expðr1 tÞ r1 expðr2 tÞ J 1 ðrnÞ dn r2 r1 0 t Z b C0 an2 bt 1 þ an2 bt þ exp þ J 1 ðrnÞ dn exp t cos sin 2k a 2k 2k 2p 2k b Z C0 1 r2 expðr1 tÞ r1 expðr2 tÞ þ J 1 ðrnÞ dn if k > kr : r2 r1 2p b
C0 xðr; tÞ ¼ 2p
ð20Þ
The value of the circulation C on a circle of radius r, given by C(r, t) = 2prx(r, t), describes the diffusion of vorticity as function of r and t. 3.2. Calculation of the tangential tension Srh The solution of the ordinary differential equation (7) with the initial condition (5) is t Z t s l xðr; sÞ exp ð1 þ kr os Þ or xðr; sÞ ds: S rh ðr; tÞ ¼ exp k k 0 k r
ð21Þ
Introducing (18)–(20) in (21) and using the identity rJ 2 ðrÞ ¼ J 1 ðrÞ rJ 01 ðrÞ one obtains after a lengthy but straight-computation that Z lC0 1 expðr2 tÞ expðr1 tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn if k < kr ; S rh ðr; tÞ ¼ 2p 0 2 ð1 þ an2 Þ 4mkn2 lC0 S rh ðr; tÞ ¼ 2p and
Z
1
0
Z
expðmn2 tÞ expðt=kÞ nJ 2 ðrnÞ dn mkn2 1
if k ¼ kr
ð23Þ
a
expðr2 tÞ expðr1 tÞ lC0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn p 0 ð1 þ an2 Þ2 4mkn2 t Z b 1 an2 bt
exp exp nJ 2 ðrnÞ dn t sin 2k a b 2k 2k Z lC0 1 expðr2 tÞ expðr1 tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nJ 2 ðrnÞ dn if k > kr : þ 2p b ð1 þ an2 Þ2 4mkn2
lC0 S rh ðr; tÞ ¼ 2p
ð22Þ
ð24Þ
The normal tension Shh(r, t) as well as the hydrostatic pressure p(r, t) can be easily obtained by means of (5), (6) and (8) as soon as the velocity field x(r, t) and the associated tangential tension Srh(r, t) are known.
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4. Limiting cases Taking the limit of Eqs. (18) and (22) as k ! 0, we get the velocity field (see also [24], Eq. (13)) Z C0 1 mn2 xðr; tÞ ¼ J 1 ðrnÞ exp t dn ð25Þ 2p 0 1 þ an2 and the associated tangential tension Z lC0 1 nJ 2 ðrnÞ mn2 S rh ðr; tÞ ¼ exp t dn; 2p 0 1 þ an2 1 þ an2
ð26Þ
corresponding to the same problem for a second grade fluid. In the special case when kr (and then a) ! 0, corresponding to a Maxwell fluid, our solutions (20) and (24) reduce to (see [25], Eqs. (3.4) and (4.2)) pffiffiffi Z C0 1=ð2 mkÞ r4 expðr3 tÞ r3 expðr4 tÞ J 1 ðrnÞ dn xðr; tÞ ¼ r4 r3 2p 0 t Z 1 ct 1 ct C0 þ cos ð27Þ þ sin J 1 ðrnÞ dn; exp 2k 1=ð2pffiffiffi 2k c 2k 2p mkÞ respectively, Z
pffiffiffi 1=ð2 mkÞ
t expðr4 tÞ expðr3 tÞ lC0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp nJ ðrnÞ dn 2 2k p 0 1 4mkn2 ct 1 nJ 2 ðrnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin dn; ð28Þ pffiffiffi 2k 1=ð2 mkÞ 4mkn2 1 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 14mkn2 and c ¼ 4mkn2 1. where r3;4 ¼ 2k Finally, by letting k ! 0 in (19) and (23) or in (27) and (28) as well as a ! 0 in (25) and (26), we attain to Z C0 1 expðmn2 tÞJ 1 ðrnÞ dn ð29Þ xðr; tÞ ¼ 2p 0 lC0 S rh ðr; tÞ ¼ 2p Z
and lC0 S rh ðr; tÞ ¼ 2p
Z
1
expðmn2 tÞnJ 2 ðrnÞ dn
ð30Þ
0
i.e., the similar solutions corresponding to a Navier–Stokes fluid. Having in mind the entry 4 of Table 5 of [23] they can be written in terms of the hypergeometric function 1 X a aða þ 1Þ x2 Cða þ nÞCðbÞ xn F ða; b; xÞ ¼ 1 þ x þ þ ¼ b bðb þ 1Þ 2! CðaÞCðb þ nÞ n! n¼0
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under analytical forms rC0 r2 xðr; tÞ ¼ F 1; 2; ; 8pmt 4mt
ð31Þ
respectively,
lC0 r2 r2 S rh ðr; tÞ ¼ F 2; 3; ; 2p ð4mtÞ2 4mt
ð32Þ
2 from which it clearly results pffiffiffiffi that rx(r, t) and r Srh(r, t) depend of r and t only by means of the similarity variable r=ð2 mtÞ.
3
3
ω1 (r) ω2 (r)
2
ω3 (r) ω4 (r) ω5 (r)
1
ω6 (r)
0
0
(a)
0.2
0.4
0.001
0.6
0.8
1 1
0.6
0.8
1 1
r
0.4 ω1 (r)
0.3
ω2 (r) ω3 (r)
0.2
ω4 (r) ω5 (r) ω6 (r) 0.1
0
(b)
0
0.2 0.001
0.4 r
Fig. 1. Velocity profiles x(r, t) corresponding to an Oldroyd (curves x1, x2, x3) and Navier–Stokes (curves x4, x5, x6) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 8, 14 and 20 s and (b) t = 40, 50 and 60 s.
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347
In view of the identities xF ð1; 2; xÞ ¼ ex 1 and x2 F ð2; 3; xÞ ¼ 2½1 ð1 þ xÞex ; the last two solutions take the simple forms (the first of them being obtained in [17] by means of the similarity by transformation of variables) C0 r2 1 exp ð33Þ xðr; tÞ ¼ 2pr 4mt
0
0
2 S1 (r) S2 (r) S3 (r)
4
S4 (r) S5 (r)
6
S6 (r)
8 9
(a)
0.5
1 r
1.5
2 2
0.5
1 r
1.5
2 2
0.001
0
S1 (r)
0
0.5
S2 (r) S3 (r) S4 (r)
1
S5 (r) S6 (r)
1.5 1.5
(b)
0.001
Fig. 2. Tangential tension profiles Srh(r, t) corresponding to an Oldroyd (curves S1, S2, S3) and Navier–Stokes (curves S4, S5, S6) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 8, 14 and 20 s and (b) t = 40, 50 and 60 s.
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and lC0 S rh ðr; tÞ ¼ 2 pr
r2 r2 1þ exp 1 : 4mt 4mt
ð34Þ
5. Numerical results and conclusions In this paper we have determined the exact solutions corresponding to a circular motion of an Oldroyd-B fluid due to a potential vortex. For k > kr, these solutions contain sine and cosine 3
3
ω1 ( r)
2
ω2 ( r) ω3 ( r) ω4 ( r)
1
0
0
(a)
0.2
0.4
0.001
0.6
0.8
1 1
0.6
0.8
1 1
r
0.4
0.3 ω1 ( r) ω2 ( r) ω3 ( r)
0.2
ω4 ( r)
0.1
0 0
(b)
0.2 0.001
0.4 r
Fig. 3. Velocity profiles x(r, t) corresponding to an Oldroyd (curves x1), second grade (curves x2), Maxwell (curves x3) and Navier–Stokes (curves x4) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 5 s and (b) t = 40 s.
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349
terms. This indicates that, in this case, unlike the Navier–Stokes fluid, oscillations are set up in the fluid. The amplitudes of these oscillations decay exponentially in time, the damping being proportional to exp(t/2k). Direct computations show that x(r, t) and Srh(r, t), given by (18)–(20) and (22)–(24) satisfy both the associate partial differential equations (7), (9), (10) and all imposed initial and boundary conditions. Moreover, they remain finite for t and r > 0 and tend to zero for t or r ! 1. In the special cases when kr or k ! 0 our solutions reduce to those corresponding to a Maxwell fluid or a second grade one, respectively. If both kr and k ! 0, corresponding to a Navier–Stokes fluid, these solutions take the simple forms (33) and (34)pin ffiffiffiffi which rx(r, t) and r2Srh(r, t) depend of r and t only by means of the similarity variable r=ð2 mtÞ.
0
0
5 S1 (r) S2 (r)
10
S3 (r) S4 (r)
15
20 23
(a)
0.5
1 r
1.5
2 2
0.5
1 r
1.5
2 2
0.001 0
0
S1 (r) 0.5 S2 (r) S3 (r) S4 (r)
1
1.5 1.5
(b)
0.001
Fig. 4. Tangential tension profiles Srh(r, t) corresponding to an Oldroyd (curves S1), second grade (curves S2), Maxwell (curves S3) and Navier–Stokes (curves S4) fluid for C0 = 1.5, k = 4, kr = 6 and a = 0.008. (a) t = 5 s and (b) t = 40 s.
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In Figs. 1 and 2, for comparison, the diagrams of the velocity fields and of the adequate tangential tensions for the Oldroyd and Navier–Stokes models are plotted for the circular motion of glycerine. It is clearly seen that for a given value of the time t the velocity corresponding to an Oldroyd-B fluid (given by the relation (18)) is larger than that of a Newtonian fluid (given by the relation (33)). As regards the tangential tensions, as it was to be expected, a reverse situation it is generally found and the difference between the two models, Newtonian and non-Newtonian, is strong enough for small values of t. For large values of t the profiles of the velocities as well as those of the tensions are closing by. Figs. 3 and 4 present the velocity and tangential tension profiles corresponding to an OldroydB, second grade, Maxwell and Navier–Stokes fluid as they result from the relations (18), (25), (27), (33) and (22), (26), (28), (34), respectively. For small values of the time t one can observe both the differences between the adequate diagrams and the oscillations corresponding to the non-Newtonian models. In each case, for large values of t and r, x(r, t) and Srh(r, t) are going to zero. Consequently, the vortex decays in time and space. Acknowledgment The authors would like to express their gratitude to the referee for his constructive comments and suggestions regarding the initial shape of this work. References [1] K.R. Rajagopal, Mechanics of non-Newtonian fluids, in: Recent Developments in Theoretical Fluids Mechanics, Pitman Research Notes in Mathematics, vol. 291, Longman, New York, 1993, pp. 129–162. [2] R.S. Rivlin, J.L. Ericksen, Stress deformation relation for isotropic materials, J. Ration. Mech. Anal. 4 (1955) 323– 425. [3] K.R. Rajagopal, A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Non-Linear Mech. 17 (1982) 369–373. [4] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci. 33 (5) (1995) 689–729. [5] K.R. Rajagopal, A.R. Srinivasa, A thermodynamic frame work for rate type fluid models, J. Non-Newtonian Fluid Mech. 88 (2000) 207–227. [6] J.G. Oldroyd, On the formulation of the rheological equations of state, Proc. Royal Soc. Lond. Ser. A 200 (1950) 523–541. [7] R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworts, Boston–London–Singapore– Sydney–Toronto–Wellington, 1989. [8] D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990. [9] J.M. Burgers, Mechanical considerations––model systems––phenomenological theories of relaxation and of viscosity. First report on viscosity and plasticity. Prepared by the committee for the study of viscosity of the academy of sciences at Amsterdam, second ed., Nordemann Publ, New York, 1939. [10] C. Guillope´, J.C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model. Math. Anal. Numer. 24 (1990) 369–401. [11] M.A. Fontelos, A. Friedman, Stationary non-Newtonian fluid flows in channel-like and pipe-like domains, Arch. Ration. Mech. Anal. 151 (2000) 1–43. [12] K.R. Rajagopal, R.K. Bhatnagar, Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113 (1995) 233–239.
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[13] C. Fetecau, The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid, C.R. Acad. Sci. Paris Ser. I 335 (2002) 979–984. [14] C. Fetecau, Analytical solutions for non-Newtonian fluid flows in pipe-like domains, Int. J. Non-Linear Mech. 39 (2004) 225–231. [15] C. Fetecau, Corina Fetecau, The first problem of Stokes for an Oldroyd-B fluid, Int. J. Non-Linear Mech. 38 (2003) 1539–1544. [16] G.C. Georgiou, On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall, Rheol. Acta 35 (1996) 39–47. [17] J. Zierep, Similarity Laws and Modeling, Marcel Dekker, Inc., New York, 1971. [18] P.N. Srivastava, Non-steady helical flow of a visco-elastic liquid, Arch. Mech. Stos. 18 (2) (1966) 145–150. [19] K.R. Rajagopal, A.S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica 19 (1984) 1948–1954. [20] K.R. Rajagopal, On the creeping flow of the second-order fluid, J. Non-Newtonian Fluid Mech. 15 (1984) 239–246. [21] K.R. Rajagopal, P.N. Kaloni, Some remarks on boundary conditions for flows of fluids of the differential type, in: Proceedings of Canadian Applied Mathematics Society, Barnaby, Canada, 1989. [22] K.R. Rajagopal, On boundary conditions for fluids of the differential type, in: Navier–Stokes Equations and Related Nonlinear Problems, Plenum Press, New York, 1995, pp. 273–278. [23] I.N. Sneddon, Fourier Transforms, McGraw Hill Book Company, Inc., New York–Toronto–London, 1951. [24] C. Fetecau, Corina Fetecau, J. Zierep, Decay of a potential vortex and propagation of a heat wave in a second grade fluid, Int. J. Non-Linear Mech. 37 (6) (2002) 1051–1056. [25] C. Fetecau, Corina Fetecau, Decay of a potential vortex in a Maxwell fluid, Int. J. Non-Linear Mech. 38 (7) (2003) 985–990.