Unsteady Free Convection Flow in a Walters’-B Fluid and Heat Transfer Analysis
Ilyas Khan1, Farhad Ali, Sharidan Sha…e1 and Muhammad Qasim2
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia 81310 UTM Skudai.
2Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, 44000, Pakistan
Abstract: In this paper the unsteady free convection ‡ow in a Walters’-B ‡uid with heat transfer analysis is investigated. The governing equations are modelled and exact solutions for velocity and temperature are obtained using the Laplace transform technique.
They satisfy all imposed initial and boundary conditions and for ! 0; can be reduce to the similar solutions for Newtonian ‡uids. The corresponding expressions for skin friction and Nusselt number are also evaluated. Numerical results for velocity and temperature are displayed graphically for various parameters of interest and discussed. This study is of fundamental importance and frequently arises in many practical situations such as chemical engineering and polymer extrusion processes. The exact solutions obtained here can be used as a benchmark for the numerical solvers.
2010 Mathematics Subject Classi…cation: 76A05, 76A10
Keywords: Heat transfer; Walters’-B ‡uid; Free convection; Boundary layer ‡ow; Exact solution.
1 Introduction
Boundary layer ‡ow of Newtonian ‡uids has been studied extensively in the literature (see for instance [1-18] and the related references therein) due to fact that they are relative simple and their solutions are convenient. However, the interest and research activities regarding the boundary layer ‡ow of non-Newtonian ‡uids have increased considerably in the past few decades and it is one of the thrust area of current research [19-26]. Of course, this is because of the industrial and engineering applications of non-Newtonian ‡uids as well as their
1Corresponding author. Tel.: +60167411152 e-mail address: [email protected]
interesting mathematical challenges in the forms of highly non-linear equations governing the ‡ows. One of the sub-class of non-Newtonian ‡uids namely viscoelastic ‡uids have recently gained a considerable attention by researchers. Perhaps, it is due to the fact that such investigations …nd their applications over a broad spectrum of science and engineering disciplines, particularly in the …eld of chemical engineering [27-35].
Among the di¤erent viscoelastic ‡uid models, Walters [36] has presented an elegant model called the Walters’-B ‡uid model for the rheological equation of state of viscoelastic
‡uid and recently has become the central focus of many scientists and engineers. This
‡uid model can accurately simulate the complex ‡ow behavior of various polymer solutions, hydrocarbons, paints and several other industrial liquids. The unsteady ‡ow of Walters’-B
‡uid model generates highly non-linear partial di¤erential equations which are one order higher than the equations governing the ‡ow of Newtonian ‡uids. It also incorporates the elastic properties of the ‡uid which are important in extensional behavior of polymers. The numerical or approximate solutions for both steady and transient ‡ows of Walters’-B ‡uid have been studied at great length in a diverse range of geometries using a wide spectrum of computational or analytical techniques [37-42]. Recently, Chang et al. [43], investigated numerically, the transient free convection mass transfer ‡ow of Walters’-B ‡uid through a vertical porous plate using the …nite di¤erence scheme.
To the best of authors’knowledge so far no study has been reported on the exact solutions of free convection boundary layer ‡ow in a Walters’-B ‡uid. Hence the present problem is based on this motivation. Exact solutions on the other hand are needed not only for the technical relevance of the ‡ows but are also signi…cant for a variety of other reasons such as they can be used as a benchmark for numerical solvers and for checking the stability of their solutions. Consequently, the exact solution for the problem under consideration is desirable.
The rest of paper is arranged as follows. Mathematical formulation of the problem is given in Section 2. Sections 3 comprises the exact solutions for velocity and temperature …elds using the Laplace transform technique. Expressions for skin friction and Nusselt number are also evaluated. Graphical results for various parameters of interest are displayed in Section 4 followed by conclusions in Section 5.
2 Mathematical formulation of the problem
Let us consider the unsteady free convection ‡ow of an incompressible Walters’-B ‡uid over a vertical rigid plate at y= 0;driven by buoyancy force due to the temperature di¤erences, occurs upward along the plate. Thex-axis is taken parallel to the plate in upward direction and y-axis is normal to plane of the plate. Initially for t 0;both the ‡uid and plate are at rest and at uniform temperature T1:At time t > 0; the temperature of the plate is raised or lowered to a constant temperature Tw as shown in Fig. 1.
Fig. 1: Physical model and coordinates system
Applying the Boussinesq approximation, the free convection boundary layer ‡ow of Walters’- B ‡uid past a rigid vertical plate is governed by the following momentum and energy equa- tions:
@u
@t = @2u
@y2
k0 @3u
@y2@t+g (T T1); (1)
cp@T
@t =k@2T
@y2: (2)
The corresponding initial and boundary conditions are
t 0 :u(y; t) = 0; T (y; t) =T1; y >0;
t >0 :u(0; t) = 0; T(0; t) =Tw;
u(1; t) = 0; T(1; t) = T1; (3) where u is the velocity in the x direction, is the kinematic viscosity, is the ‡uid density, k0 is the Walters’ viscoelasticity parameter, g is the acceleration due to gravity, is the volumetric coe¢ cient of thermal expansion, T is the ‡uid temperature, cp is the speci…c heat of the ‡uid at constant pressure and k is the thermal conductivity of the ‡uid.
The positive sign in the right hand side of equation (1) is corresponding to a second grade
‡uid [27, 28] whereas the negative sign is for a Walters’-B ‡uid [29; 30]. However, here we will construct the solutions for Walters’-B ‡uid only.
Introducing the following dimensionless variables u = u
U0; y = U0y
; t = U02t
; = T T1
Tw T1; (4)
into Eqs. (1) (3);we arrive at the following dimensionless system (* symbols are dropped for simplicity)
@u
@t = @2u
@y2
@3u
@y2@t +Gr ; (5)
Pr@
@t = @2
@y2; (6)
u(y;0) = 0; (y;0) = 0; y >0; (7) u(0; t) = 0; u(1; t) = 0;
(0; t) = 1; (1; t) = 0; (8)
where
= k0U02
2 ; Gr = g (Tw T1)
U03 ; Pr = cp
k ; (9)
are the dimensionless viscoelastic parameter, Grashof number and Prandtl number respec- tively and U0 is the reference velocity.
3 Solution of the problem
Applying Laplace transforms to Eqs. (5) and(6) and using initial conditions(7), we get the following transformed equations
d2u(y; q) dy2
q
1 qu(y; q) = Gr
1 q (y; q); (10)
d2 ( ; q)
d 2 Prq ( ; q) = 0; (11)
where u(y; q) = R1 0
e qtu(y; t)dt and (y; q) = R1
0
e qt (y; t)dt are the Laplace transforms of u(y; t)and (y; t) respectively.
The boundary conditions (8) become
u(0; q) = 0; u(1; q) = 0;
(0; q) = 1
q; (1; q) = 0: (12)
The solutions of Eqs. (10) and (11) subject to the boundary conditions (12) are given by u(y; q) = Gr0
a20 1 q +a0
q2
1 q a0 e
py q q
0 q +Gr
a20 1 q
a0
q2 + 1
q a0 e ypPrq; (13) (y; q) = 1
qe ypPrq; (14)
where
0 = 1
; a0 = Pr 1
Pr ; Gr0 = Gr Pr :
Now taking the inverse Laplace transforms of Eqs. (13) and (14); the exact solutions for velocity and temperature are given by
u(y; t) = Gr0 a20
2
4 u1(y; t) +u2(y; t) +u3(y; t) +u4(y; t) u5(y; t) +u6(y; t) +u7(y; t) u8(y; t)
3
5; (15)
(y; t) = erfc yp Pr 2p
t
!
; (16)
where
u1(y; t) = erfc yp Pr 2p
t
! a0
8<
:
t+y22Pr erfc y
pPr 2p
t yp
Prt
p exp y24tPr 9=
;
u2(y; t) = ea0t 2
8<
:
exp yp
Pra0 erfc y
pPr 2p
t +p
a0t + exp yp
Pra0 erfc y
pPr 2p
t
pa0t 9=
; u3(y; t) = y
2p Z1
0
1 up
uexp( y2
4 u+u)du
u4(y; t) = a0ty 2p
Z1 0
1 up
uexp( y2
4 u+u)du
u5(y; t) = yea0t 2p
Z1 0
1 up
uexp( y2
4 u+u)du
u6(y; t) = yp
0
2p Zt 0
Z1 0
1 up
sexp( y2
4 u +u+ 0s)I1(2p
u 0s)du ds
u7(y; t) = yp
0a0 2p
Zt 0
Z1 0
(t s) up
s exp( y2
4 u+u+ 0s)I1(2pu 0s)du ds
u8(y; t) = yp
0eaot 2p
Zt 0
Z1 0
1 up
sexp( y2
4 u +u+ 0s a0s)I1(2pu 0s)du ds.
The dimensionless form of the skin friction in Walters’-B liquid ‡uid is given by (* symbol is dropped for simplicity)
xy = xy
u0 = @u(0; t)
@y
@2u(0; t)
@y@t : (17)
On substituting from Eq. (16); into Eq. (17); the skin friction becomes
xy = Gr0[ a0p
Prt+p
Prt+p
a0Pr ea0terf p a0t ] a20p
+ Gr0 2a20p
Z1 0
exp (u) up
u du+ Gr0t 2a0p
Z1 0
exp (u) up
u du Gr0ea0t 2a20p
Z1 0
exp (u) up
u du +Gr0p
0
2a20p Zt
0
Z1 0
exp ( 0s+u) up
s I1(2pu 0s)du ds +Gr0p
0
2a0p Zt
0
Z1 0
exp ( 0s+u) (s t) up
s I1(2p
u 0s)du ds Gr0p
0ea0t 2a20p
Zt 0
Z1 0
exp (u a0s+ 0s) up
s I1(2p
u 0s)du ds
+ Gr0[(a0+ 1)p
Prt+ 2a0ea0ttp
a0Pr ea0terf p a0t ] 2a20tp
Gr0 2a0p
Z1 0
exp (u) up
u du+ Gr0ea0t 2a0p
Z1 0
exp (u) up
u du Gr0 0
2a0p Zt
0
Z1 0
exp ( 0s+u)I1 2pu 0s up
s du ds
+Gr0p
0ea0t 2a0p
Zt 0
Z1 0
exp (u a0s+ 0s) up
s I1(2pu 0s)du ds (18)
The rate of heat transfer is given by
N u= rPr
t: (19)
4 Graphical results and discussion
Free convection ‡ow of Walters’-B ‡uid over a vertical ‡at plate with heat transfer is an- alyzed. The non-dimensional boundary layer equations of momentum and heat transfer (equations (5) and (6)) subject to the initial and boundary conditions (equations (8) and (9)) are solved analytically using the Laplace transform technique. The expressions for ve- locity and temperature are obtained. The corresponding skin-friction and Nusselt number are also evaluated. In order to explain the physics of the problem, the obtained analytical solutions are displayed graphically in Figs. 2 7;for several embedded ‡ow parameters. Such parameters include viscoelastic parameter ; Prandtl number Pr; Grashof number Gr and dimensionless time t:
Figures 2 5 are drawn for velocity pro…les whereas Figs. 6 and 7 are prepared for temperature pro…les. The velocity pro…les for di¤erent values of are shown in Fig. 2: It is observed that an increase in viscoelastic parameter produces a signi…cant increase in the momentum boundary layer of the ‡uid and thus the velocity increases. It is true because the non dimensional viscoelastic parameter is inversely proportional to the square of the dynamic viscosity (see Eq. 9). An increase in will therefore correspond to a decrease in
‡uid viscosity, ultimately it will accelerate the ‡ow and hence velocity increases. Further, an increase in causes a rise in velocity near the plate surface, however away from the plate, the trend is reversed and the Newtonian ‡uid ( ! 0) possesses higher velocity. Besides that, it is found that velocity of the ‡uid at the boundary y = 0 is zero. However, velocity increases with increasing distance from the boundary and approaches to its maximum value and from there it starts again the decay motion tending to its minimum value at zero as y ! 1: This behavior of Walters’-B ‡uid for the viscoelastic parameter is quite identical to the already published work in the literature (see for example, [Bhattacharyya et al. [41], Fig. 4b and Chang et al. [43], Fig. 2 and Fig. 3]). More clearly, we can see from this graph that our solutions satisfying the imposed boundary conditions. Hence this provides a useful mathematical check to our calculi.
The in‡uence of Prandtl number Pr on velocity is exhibited in Fig. 3: Two di¤erent
values of Prandtl number Pr = 0:7 and Pr = 7 are chosen. Physically, they correspond to air and water. We noticed that velocity decreases with increasing values of Pr. A similar behavior was also expected due to the fact that as we increasePr, it results an increase in the viscosity of the ‡uid which makes the ‡uid thick and hence causes a decrease in the velocity of the ‡uid. Figure 4 shows that velocity increases for large values of Grashof number Gr.
Physically, it is due to the fact that an increase in Gr gives rise to buoyancy e¤ects which results in more induced ‡ows. From Fig. 5; we observed that velocity pro…les increase with increasing t.
The deviation in temperature pro…les for di¤erent values of Prandtl numberPris demon- strated in Fig. 6: It is found that the thermal boundary layer thickness decreases rapidly with increasing values of Pr: For small value ofPrheat di¤uses very quickly compare to the velocity. It is because that usually for liquid metals the thickness of the thermal boundary layer is much bigger than momentum (hydrodynamic) boundary layer. The e¤ects of t on the temperature pro…les are quite identical to that on the velocity pro…les. This fact is shown in Fig. 7:
Fig. 2. Pro…les of the dimensionless velocity (15) for di¤erent values of when Pr = 0:71; Gr = 0:5 and t= 0:4:
Fig. 3. Pro…les of the dimensionless velocity (15) for di¤erent values ofPr when = 0:2; Gr= 0:5and t= 0:4:
Fig. 4. Pro…les of the dimensionless velocity (15) for di¤erent values of Gr when = 0:2; Pr = 0:71 and t= 0:4:
Fig. 5. Pro…les of the dimensionless velocity (15) for di¤erent values of t when Pr = 0:71; = 0:2 and Gr= 0:5:
Fig. 6. Pro…les of the dimensionless temperature (16) for di¤erent values of Prwhen = 0:2; t= 0:4 and Gr= 0:5:
Fig. 7. Pro…les of the dimensionless temperature (16) for di¤erent values oft when = 0:2; Pr = 0:71and Gr= 0:5:
5 Conclusions
The main objective of the present work is to study the free convection ‡ow in a Walters’-B
‡uid past a rigid vertical plate together with heat transfer. The governing equations are solved using the Laplace transform technique. Exact solutions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and can be reduce to similar solutions for Newtonian ‡uids. The corresponding expressions for skin friction and Nusselt number are also evaluated. The numerical results for velocity and temperature are computed and then plotted graphically just to see the physical behavior of the emerging ‡ow parameters. We hope that the results obtained here will not only provide useful information for the numerical analysts, but also serve as a complement to the previous studies.
Acknowledgement
The authors would like to acknowledge the Research Management Centre –UTM for the
…nancial support through vote number 4F019 for this research.
References
[1] I. Pop: Unsteady hydrodynamic free convection ‡ow a vertical in…nite ‡at plate. Z.
Angew. Math. Mech. 49 (1969).
[2] M. Katagiri: Unsteady boundary layer ‡ows past an impulsively started circular cylin- der. J. Phys. Soc. Japan. 40 (1976) 1171-1177.
[3] U. N. Das, R. K. Deka and V. M. Soundalgekar: Transient free convection ‡ow past an in…nite vertical plate with periodic temperature. J. Heat Transfer 121 (1999) 1091.
[4] B. K. Jha and R. Prasad: MHD free convection and mass transfer through porous medium with heat source. J. Math. Phys. Sci. 26 (1992) 1-8.
[5] E. Magyari and B. Keller: Heat and mass transfer in the boundary layers on an expo- nentially stretching continuous surface. J. Phys. D, Appl. Phys. 32 (1998) 577-585.
[6] E. Magyari, M. E. Ali and B. Keller: Heat and mass transfer characteristics of the self-similar boundary-layer ‡ows induced by continuous surfaces stretched with rapidly decreasing velocities. Heat Mass Transfer. 38 (2001) 65-74.
[7] I. Pop and D. B. Ingham: Convective heat transfer: mathematical and computational modelling of viscous ‡uids and porous media. Pergamon, Oxford (2001).
[8] R. Nazar, N. Amin, D. Filip and I. Pop: Unsteady boundary layer ‡ow in the region of the stagnation point on a stretching sheet. Int. J. Eng. Sci. 42 (2004) 1241-1253.
[9] M. Narahari: E¤ects of thermal radiation and free convection currents on the unsteady Couette ‡ow between two parallel plates with constant heat ‡ux on the boundary,Weseas Trans. Heat Mass Transfer. 1 (2010) 21-30.
[10] M. Narahari and A. Ishak: Radiation e¤ects on free convection ‡ow near a moving vertical plate with Newtonian heating. J. Appl. Sci. 11 (2011) 1096-1104.
[11] O. A. Bég, S. K. Ghosh, M. Narahari and T. A. Bég: Mathematical modelling of thermal radiation e¤ects on transient gravity driven optically thick gray convection ‡ow along an inclined plane with pressure gradient. Che. Eng. Commun. 198 (2011) 1630-1644.
[12] M. Narahari, O. A. Bég and S. K. Ghosh: Mathematical modelling of mass transfer and free convection current e¤ects on unsteady viscous ‡ow with ramped wall temperature.
World J. Mech. 1 (2011) 176-184.
[13] A. Ishak: Similarity solutions for ‡ow and heat transfer over a permeable surface with convective boundary conditions. Appl. Math. Comput. 217 (2010) 837-842.
[14] M. Z. Salleh and R. Nazar: Free convection boundary layer ‡ow on a vertical surface with prescribed wall temperature and heat ‡ux. J. Quality Measur. Anal. 4 (2008) 57-69.
[15] M. Z. Salleh, R. Nazar and I. Pop: Modeling of free convection boundary layer ‡ow on a sphere with Newtonian heating. Acta Appl. Mathe. 112 (2010) 263-274.
[16] I. Khan, F. Ali, S. Sharidan and M. Norzieha: E¤ects of Hall current and mass transfer on the unsteady MHD ‡ow in a porous channel. J. Phys. Soc. Jpn. 80 (2011) 06440.
[17] I. Khan, K. Fakhar and S. Sharidan: Magnetohydrodynamic free convection ‡ow past an oscillating plate embedded in a porous medium. J. Phys. Soc. Jpn. 80 (2011) 104401.
[18] Samiulhaq, I. Khan, F. Ali and S. Sharidan: MHD free convection ‡ow in a porous medium with thermal di¤usion and ramped wall temperature. J. Phys. Soc. Jpn. 81 (2012) 044401.
[19] N. Bachok, A. Ishak and I. Pop: Boundary-layer ‡ow of nano‡uids over a moving surface in a ‡owing ‡uid. Int. J. Therm. Sci. 49, (2010a). 1663-1668.
[20] N. Bachok, A. Ishak, R. Nazar and I. Pop: Flow and heat transfer at a general three- dimensional stagnation point in a nano‡uid. Physica B, 405 (2010b) 4914-4918.
[21] N. Bachok, A. Ishak, and R. Nazar, Flow and heat transfer over an unsteady stretching sheet in a micropolar ‡uid with prescribed surface heat ‡ux. Int. J. Math. Model.
Method Appl. Sci., 4 (2010) 167-176.
[22] N. Bachok, A. Ishak and I. Pop: Flow and heat transfer over a rotating disk in a nano‡uid. Physica B, 406 (2011) 1767-1772.
[23] S. Yao, T. Fang and Y. Zhong: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun. Nonlinear Sci. Numer.
Simulat. 16 (2011) 752-760.
[24] P. G. Siddheshwar and U. S. Mahabaleshwar: E¤ects of radiation and heat source on MHD ‡ow of a viscoelastic liquid and heat transfer over a stretching sheet. Int. J.
Non-linear Mech. 40 (2005) 807-821.
[25] I. Khan, K. Fakhar and S. Sha…e: Magnetohydrodynamic rotating ‡ow of a general- ized Burgers’‡uid in a porous medium with Hall current. Trans. Porous Media, DOI:
10.1007/s11242-011-9832-8.
[26] P. Anuradha and S. Krishnambal: Lie group analysis of magnetohydrodynamic ‡ow and mass transfer of a visco-elastic ‡uid over a porous stretching sheet. Bull. Malays.
Math. Sci. Soc. 33 (2010) 295-309.
[27] R. Cortell, Similarity solutions for ‡ow and heat transfer of a viscoelastic ‡uid over a stretching sheet. Int. J. Non-Linear Mech., 29 (1994) 155-161.
[28] R. Cortell, A note on ‡ow and heat transfer of a viscoelastic ‡uid over a stretching sheet. Int. J. Non-Linear Mech., 41 (2006) 78-85.
[29] S. K. Khan, M. S. Abel and R. M. Sonth: Viscoelastic MHD ‡ow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work. Heat Mass Transfer, 40 (2003) 47-57.
[30] K. V. Prasad, M. S. Abel and S. K. Khan, Momentum and heat transfer in viscoelastic
‡uid ‡ow in a porous medium over a non-isothermal stretching sheet. Int. J. Num. Meth.
Heat Fluid Flow, 10 (2000) 786-801.
[31] I. Khan, K. Fakhar and M. I. Anwar, Hydromagnetic rotating ‡ows of an Oldroyd-B
‡uid in a porous medium. Special Topics and Review in Porous Media. 3 (2012) 89-95.
[32] F. Ali, N. Mustapha, S. Sharidan, I. Khan and T. Hayat, New exact solutions of Stokes’
second problem for MHD second grade ‡uid in a porous space. Int. J. Non-Linear Mech.
doi:10.1016/j.ijnonlinmec.2011.09.027.
[33] D. Vieru and A. Rauf: Stokes ‡ows of a Maxwell ‡uid with wall slip condition. Can. J.
Phys. 89 (2011) 1-12.
[34] T. R. Mahapatra, S. K. Nandy and A. S. Gupta: Magnetohydrodynamic stagnation- point ‡ow of a power-law ‡uid towards a stretching surface. Int. J. Non-linear Mech. 44 (2009) 123-128.
[35] T. R. Mahapatra, S. K. Nandy and A. S. Gupta: Heat transfer in the magnetohydro- dynamic ‡ow of a power-law ‡uid past a porous ‡at plate with suction or blowing. Int.
Commun. Heat Mass Transfer 39 (2012) 17-23.
[36] D. W. Beard and K. Walters: Elasto-viscous boundary layer ‡ow. Proc. Camb. Philos.
Soc. 60 (1964) 667-74.
[37] D. Kumar and R. K. Srivastava: E¤ects of chemical reaction on MHD ‡ow of dusty vis- coelastic (Walter’s liquid model-B) liquid with heat source/sink. Proceeding of National Seminar on Mathematics and Computer Science, Meerut. (2005) 105-112.
[38] M. M. Nandeppanavar, M. S. Abel and J. Tawade: Heat transfer in a Walter’s Liquid B ‡uid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation. Commun. Non-linear Sci. Numer. Simulat. 15 (2010) 1791-1802.
[39] O. Prakash, D. Kumar and Y. K. Dwivedi: E¤ects of thermal di¤usion and chemical reaction on MHD ‡ow of dusty viscoelastic (Walter’s Liquid Model-B) ‡uid. J. Electro.
Anal. Appli. 2 (2010) 581-587.
[40] E. Ghasemi, M. Bayat and M. Bayat: Viscoelastic MHD ‡ow of Walters liquid B ‡uid and heat transfer over a non-isothermal stretching sheet. Int. J. Phys. Sci. 6 (2011) 5022-5039.
[41] K. Bhattacharyya, M. S. Uddin, G. C. Layek and W. Ali Pk.: Analysis of boundary layer ‡ow and heat transfer for two classes of viscoelastic ‡uid over a stretching sheet with heat generation or absorption. Bangladesh J. Sci. Ind. Res. 64 (2011) 451-456.
[42] I. J. Uwanta, B. Y. Isah and M. O. Ibrahim, Viscoelastic ‡uid ‡ow past an in…nite vertical plate with heat dissipation. Int. J. Comput. Appl. 36 (2011) 0975-8887.
[43] T. B. Chang, A. Mehmood, O. A. Beg, M. Narahari, M. N. Islam and F. Ameen:
Numerical study of transient free convective mass transfer in a Walters-B viscoelastic
‡ow with wall suction. Commun. Non-linear Sci. Numer. Simulat. 16 (2011) 216-225.
