Journal of Molecular Liquids 225 (2017) 592–597
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Influence of magnetic field on CNT-Polyethylene nanofluid flow over a permeable cylinder P. Valipour ⁎, F. Shakeri Aski, M. Mirparizi Department of Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Department of Mechanical Engineering, University of Yazd, Yazd, Iran
a r t i c l e
i n f o
Article history: Received 3 September 2016 Received in revised form 24 October 2016 Accepted 25 November 2016 Available online 28 November 2016 Keywords: Nanofluid Stretching cylinder CNT-Polyethylene Boundary layer Injection Magnetic field
a b s t r a c t In this paper, Lorentz force impact on CNT-Polyethylene nanofluid flow characteristics over a stretching permeable cylinder is studied using Runge–Kutta method. Similarity transformation has been applied to reach ODEs. Investigation has been completed by studying the impacts of nanoparticle volume fraction, injection parameter, Reynolds and magnetic numbers on nanofluid flow style. Results show that nanofluid velocity augments with rise of CNT volume fraction and injection parameter but it reduces with rise of magnetic number. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Flow over a stretching cylinder was studied by Wang [1]. Magnetohydrodynamic free convection has several applications such as combustion modeling, geophysics, fire engineering and etc. In recent decade, nanotechnology has been offered as innovative passive method for heat transfer improvement. Sheikholeslami et al. [2] investigated about the effect of radiation of nanofluid free convective heat transfer in presence of magnetic field. They showed that rate of heat transfer decrease with augment of Lorentz forces. Hayat et al. [3] studied the impact of radial magnetic field on nanofluid motion in a non-uniform porous medium. Shehzad et al. [4] studied the radiative heat transfer influence on Oldroyd-B nanofluid. They considered the stretched flow in the existence of an applied magnetic field. Influence of Lorentz forces on 3D Sisko nanofluid was presented by Hayat et al. [5]. They concluded that the impacts of Biot number on the temperature and nanoparticles concentration are quite similar. Ahmad and Mustafa [6] investigated the rotating nanofluid flow induced by an exponentially stretching. Their results revealed that temperature gradient reduces with augment of angular velocity. Hayat et al. [7] presented the influence of radiation on mass transfer of nanofluid. They showed that temperature gradient reduces with augment of ⁎ Corresponding author. E-mail addresses:
[email protected] (P. Valipour),
[email protected] (F. Shakeri Aski).
http://dx.doi.org/10.1016/j.molliq.2016.11.111 0167-7322/© 2016 Elsevier B.V. All rights reserved.
thermal radiation. Hayat et al. [8] investigated Maxwell nanofluid in presence of magnetic field. They considered convective impact on nanofluid behavior. Akbar et al. [9] investigated the CNT nanofluid over plate in presence of convective and slip boundary conditions. Hayat et al. [10] examined 3D viscoelastic nanofluid flow in existence of thermal radiation. Farooq et al. [11] presented stagnation point nanofluid flow in existence of magnetic field. They examined the impact of thermal radiation and found that skin friction augments for a larger magnetic parameter. Nanofluid squeezing flow in existence of magnetic field has been examined by Hayat et al. [12]. They neglected induced magnetic field due to small magnetic Reynolds number. 3D boundary layer nanofluid flow of an Oldroyd-B nanofluid was presented by Hayat et al. [13]. They considered bidirectional stretching surface. Influence of non-uniform Lorentz forces on nanofluid flow style has been studied by Sheikholeslami Kandelousi [14]. He concluded that improvement in heat transfer reduces with rise of Kelvin forces. Water based carbon nanotubes nanofluid mixed convection heat transfer was examined by Hayat et al. [15]. Imtiaz et al. [16] presented the nanofluid flow between two rotating disks. They reported the comparison between the performance of MWCNTs and SWCNTs. Sheikholeslami and Chamkha [17] studied MHD Fe3O4-water flow in a wavy cavity with moving wall. Sheikholeslami et al. [18] examined the influence of Lorentz forces on forced convection. They illustrated that higher lid velocity has more sensible Kelvin forces effect. Sheikholeslami and Ganji [19] examined the Joule heating impact on nanofluid treatment. They indicated that nanofluid motion reduces with rise of Lorentz
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Table 1 Thermo physical properties of base fluid and nanoparticle.
Nomenclature a c ww f p (r,z)
radius of cylinder positive constant velocity of the stretching cylinder dimensionless stream function pressure cylindrical coordinates in the radial and axial directions, respectively
w ,u Cf Re
velocity components skin friction coefficient Reynolds number
) z, r) directions
Greek symbols η similarity variable surface shear stress τw γ suction/injection parameter ρ fluid density υ kinematic viscosity Subscripts s Nano-solid-particles ∞ Condition at infinity f Base fluid w Condition at the surface nf Nanofluid
Polyethylene CNT
ρ(kg/m3)
Cp(j/kgk)
920 2600
2300 425
Polyethylene (Table 1). Single phase model has been used. The basic PDEs are: ∂ðruÞ ∂ðrwÞ þ ¼ 0; ∂r ∂z
ð1Þ
! 2 ∂w ∂w ∂ w 1 ∂w ¼ μn f −σ nf B0 2 w; ρn f u þ þw r ∂r ∂r ∂z ∂r 2
ð2Þ
! 2 ∂u ∂u ∂P ∂ u 1 ∂u u ¼ − þ μn f ; þ − ρn f w þ u ∂z ∂r ∂r ∂r2 r ∂r r 2
ð3Þ
at r ¼ a : at r ¼ ∞ :
ð4Þ
w ¼ ww ¼ 2cz; w→0;
u ¼ U w ¼ −caγ
where γ and c are constant parameters. μnf and ρnf are: μ nf ¼ ð1−ϕÞ−2:5 μ f ρnf ¼ ρ f ð1−ϕÞ þ ϕ ρs
ð5Þ
Following similarity transformations should be considered [39]: forces. Sheikholeslami et al. [20] investigated the MFD viscosity effect on natural convection of magnetic nanofluid. They indicated that reduction of Nusselt number due to MFD viscosity effect is more sensible for high Rayleigh number and low Hartmann number. Nanotechnology technique has been utilized by several researchers [21–38]. The chief goal of this article is to examine the CNT-Polyethylene nanofluid flow over a permeable cylinder. Runge–Kutta method utilized to solve the ODEs which are obtained by means of similarity transformation. Influences of effective parameter on velocity and temperature are examined.
0
w ¼ 2f ðηÞzc η ¼ ða=r Þ−2 u ¼ η−0:5 f ðηÞð−caÞ
ð6Þ
The final ODEs are: 2 ″ ‴ 0 ″ 0 f þ ηf −A1 A2 Re f −f f −MA2 f ¼ 0 0
f ð1Þ ¼ γ; f ð∞Þ ¼ 0;
ð7Þ
0
f ð1Þ ¼ 1;
ð8Þ
2. Problem statement Dimensionless parameters are defined as Nanofluid laminar flow over a cylinder is depicted in Fig. 1. Radial magnetic field is applied. Working fluid is Carbon nanotubes (CNT)-
M ¼ a2 B20 σ nf = 4 μ f μ nf ρs ϕ; A2 ¼ A1 ¼ ð1−ϕÞ þ ρf μf
Re ¼ a2 c=2 υ f ;
ð9Þ
Pressure formula and skin friction coefficients are: P−P ∞ Re 0 2 ¼ −2f ðηÞ−A3 A1 f ðηÞ υn f ρn f c η
ð10Þ
Table 2 Comparison of skin friction coefficient between present and previous work [39] when ϕ=0.04 for CuO-water.
Fig. 1. Geometry of the problem.
γ
Re
Sheikholeslami [39]
Present work
0 0 0 1 1 1
0.1 1 2 0.1 1 2
0.679741 1.194617 1.579317 0.730548 1.765922 2.82033
0.680923 1.198356 1.592929 0.732728 1.777492 2.847328
594
P. Valipour et al. / Journal of Molecular Liquids 225 (2017) 592–597
1.8
with the boundary conditions (Eq. (8)), which are solved numerically using the fourth order Runge-Kutta integration scheme featuring a shooting technique with the MAPLE package. Runge-Kutta Method is a method of numerically integrating ordinary differential equations using a trial step at the midpoint of an interval to cancel out lowerorder error terms. The fourth-order formula is [40]:
f
1.6
1.4
φ=0 φ = 0.01 φ = 0.02 φ = 0.04
1.2
1
1
2
3
η
4
5
6
k1 ¼ h f ðxn ; yn Þ
ð12Þ
1 1 k2 ¼ h f xn þ h; yn þ k1 2 2
ð13Þ
1 1 k3 ¼ h f xn þ h; yn þ k2 2 2
ð14Þ
k4 ¼ h f ðxn þ h; yn þ k3 Þ
ð15Þ
1 1 1 1 5 ynþ1 ¼ yn þ k1 þ k2 þ k3 þ k4 þ O h 6 3 3 6
ð16Þ
Fig. 2. Effect of nanoparticle volume fraction for velocity profile when γ=1,Re=1,M=1.
4. Results and discussion C f ¼ ðRez=ð0:5aÞÞ C f ¼
1
″
ð1−ϕÞ2:5
f ð1Þ
CNT-Polyethylene MHD nanofluid flow over a cylinder is investigated using Runge–Kutta method. Table 2 indicates that the Maple code has good agreement with previous work [39]. Influence of active parameters on flow style is examined. Fig. 2 is presented to investigate the impact of CNT-Polyethylene volume fraction on flow style. Adding nanoparticles in the base fluid enhances the fluid motion. So, nanofluid velocity augments with augment of ϕ. Fig. 3 depicts the influence of injection parameter on
ð11Þ
3. Numerical method The governing partial differential Eqs.(1) to (3) and their boundary conditions (Eq. (4)) are transformed to ordinary differential Eq.(7)
1
4
γ=0
γ=0 γ = 0.5 γ=1 3
γ = 0.5 γ=1
0.8
γ=2
γ=2
f
f'
0.6 2
0.4
1
0
0.2
0 1
2
3
4
η
5
1
6
2
3
η
(a)
4
5
(b) 0
(P-P∞ )/( ρnf cvnf )
-1.5
-3
-4.5
γ=0 γ = 0.5 γ=1 γ=2
-6
-7.5
1
2
3
η
4
5
6
(c) Fig. 3. Effect of injection parameter on (a, b) velocity profiles, (c) Pressure distribution when Re=1,M=1,ϕ=0.04.
6
P. Valipour et al. / Journal of Molecular Liquids 225 (2017) 592–597 2
1
1.8
0.8
595
Re = 0.5 Re = 1 Re = 1.5 Re = 2
0.6
f
f
'
1.6
0.4
1.4
Re = 0.5 Re = 1 Re = 1.5
1.2
0.2
Re = 2 0 1
1
2
3
4
η
5
1
6
2
3
η
(a)
4
5
6
(b)
(P-P∞ )/( ρnf cvnf )
0
-1.5
Re = Re = Re = Re =
-3
-4.5
1
2
3
4
η
0.5 1 1.5 2
5
6
(c) Fig. 4. Effect of Reynolds number on (a, b) velocity profiles, (c) Pressure distribution when γ=1,M=1,ϕ=0.04. 1
2
M= 0 M = 0.5 0.8
1.6
0.6
M= 1 M= 2
f
f'
1.8
0.4
1.4
M= 0 M = 0.5 1.2
0.2
M= 1 M= 2
1
0 1
2
3
4
η
5
1
6
2
3
η
(a)
4
5
(b)
(P-P∞ )/( ρnf cvnf )
0
-1
M= 0 M = 0.5
-2
M= 1 M= 2
-3
1
2
3
η
4
5
6
(c) Fig. 5. Effect of magnetic parameter on (a, b) velocity profiles, (c) Pressure distribution when γ=1,Re=1,ϕ=0.04.
6
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P. Valipour et al. / Journal of Molecular Liquids 225 (2017) 592–597 -0.5
-1.6
γ=0
φ=0 φ = 0.02
γ=1
-1
γ=2
φ = 0.04
-1.8
-1.5 -2
Cf
Cf
-2
-2.2
-2.5 -2.4
-2.6
-3
0
0.5
1
1.5
2
-3.5
0
0.5
1
M
1.5
2
M
(a) γ = 1 , Re = 1
(b) Re = 1 , φ = 0.04
-0.5
Re = 0.5 Re = 1 -1
Re = 1.5
-1.5
Cf -2
-2.5
-3
0
0.5
1
1.5
2
M
(c) γ = 1 , φ = 0.04 Fig. 6. Effects of the Reynolds number, injection parameter, magnetic parameter and nanoparticle volume fraction on skin friction coefficient.
f , P distributions. The radial velocity curves increases as injection parameter augments but opposite trend are observed for pressure. Impact of Re on f , P distributions is shown in Fig. 4. As Re rises, nanofluid velocity reduces. Increasing Reynolds number leads hydraulic boundary layer thickness to augment. Pressure reduces with rise of Reynolds number and it reach to its constant value at higher η. Influence of magnetic number on f , P distributions is depicted in Fig. 5. Adding magnetic field generates Lorentz forces. Lorentz forces make the nanofluid flow to retard. Nanofluid velocity reduces with augment of magnetic number due to impact of Lorentz force on flow style. Fig. 6 illustrates the impacts of Re , M , γ and ϕ on Cf. Stretching cylinder uses a dragging force on nanofluid and in turn Cf has negative values. Cf enhances with rise of ϕ , γ, Re but it reduces with rise of M. 5. Conclusions CNT-Polyethylene nanofluid over a stretching porous pipe is investigated considering Lorentz force. 4th-order Runge–Kutta method is selected to solve obtained ODEs. Influences of injection parameter, Reynolds and magnetic numbers and CNT volume fraction on the flow style are studied. Velocity profile augments with enhance of CNT volume fraction and injection parameter but it reduces with augment of magnetic and Reynolds numbers. References [1] C.Y. Wang, Phys. Fluids 31 (1988) 466–468. [2] M. Sheikholeslami, T. Hayat, A. Alsaedi, Int. J. Heat Mass Transf. 96 (2016) 513–524. [3] T. Hayat, S. Farooq, A. Alsaedi, B. Ahmad, Int. J. Heat Mass Transf. 103 (2016) 1133–1143. [4] S.A. Shehzad, Z. Abdullah, F.M. Abbasi, T. Hayat, A. Alsaedi, J. Magn. Magn. Mater. 399 (2016) 97–108. [5] T. Hayat, T. Muhammad, B. Ahmad, S.A. Shehzad, J. Magn. Magn. Mater. 413 (2016) 1–8.
[6] R. Ahmad, M. Mustafa, J. Mol. Liq. 220 (2016) 635–641. [7] T. Hayat, Z. Nisar, H. Yasmin, A. Alsaedi, J. Mol. Liq. 220 (2016) 448–453. [8] T. Hayat, T. Muhammad, S.A. Shehzad, G.Q. Chen, I.A. Abbas, J. Magn. Magn. Mater. 389 (2015) 48–55. [9] N.S. Akbar, Z.H. Khan, S. Nadeem, Journal of molecular liquids, 196 (2014) 21–25. [10] T. Hayat, T. Muhammad, A. Alsaedi, M.S. Alhuthali, J. Magn. Magn. Mater. 385 (2015) 222–229. [11] M. Farooq, M. Ijaz Khan, M. Waqas, T. Hayat, A. Alsaedi, M. Imran Khan, J. Mol. Liq. 221 (2016) 1097–1103. [12] T. Hayat, T. Muhammad, A. Qayyum, A. Alsaedi, M. Mustafa, J. Mol. Liq. 213 (2016) 179–185. [13] T. Hayat, T. Muhammad, S.A. Shehzad, M.S. Alhuthali, J. Lu, J. Mol. Liq. 212 (2015) 272–282. [14] M.S. Kandelousi, Eur. Phys. J. Plus (2014) 129–248. [15] T. Hayat, B. Ahmed, F.M. Abbasi, B. Ahmad, Comput. Methods Programs Biomed. 135 (2016) 141–150. [16] M. Imtiaz, T. Hayat, A. Alsaedi, B. Ahmad, Int. J. Heat Mass Transf. 101 (2016) 948–957. [17] M. Sheikholeslami, A.J. Chamkha, Numer. Heat Transfer, Part A 69 (10) (2016) 1186–1200. [18] M. Sheikholeslami, K. Vajravelu, M.M. Rashidi, Int. J. Heat Mass Transf. 92 (2016) 339–348. [19] M. Sheikholeslami, D.D. Ganji, J. Mol. Liq. 224 (2016) 526–537. [20] M. Sheikholeslami, M.M. Rashidi, T. Hayat, D.D. Ganji, J. Mol. Liq. 218 (2016) 393–399. [21] Mohsen Sheikholeslami, Ali J. Chamkha, Numer. Heat Transfer, Part A, 2016, vol. 69, NO. 7, 781–793, http://dx.doi.org/10.1080/10407782.2015.1090819 [22] M. Sheikholeslami, H.R. Ashorynejad, P. Rana, J. Mol. Liq. 214 (2016) 86–95. [23] M. Sheikholeslami, S. Soleimani, D.D. Ganji, J. Mol. Liq. 213 (2016) 153–161. [24] M. Sheikholeslami, M.M. Rashidi, D.D. Ganji, J. Mol. Liq. 212 (2015) 117–126. [25] M. Sheikholeslami, R. Ellahi, Int. J. Heat Mass Transf. 89 (2015) 799–808. [26] M. Sheikholeslami, S. Abelman, IEEE Trans. Nanotechnol. 14 (3) (2015) 561–569. [27] M.S. Kandelousi, Phys. Lett. A 378 (45) (2014) 3331–3339. [28] M. Sheikholeslami, D.D. Ganji, J. Mol. Liq. 194 (2014) 13–19. [29] Junaid Ahmad Khan, M. Mustafa, T. Hayat, M. Sheikholeslami, A. Alsaedi, PLoS One 10(3): e0116603. doi:10.1371/journal.pone.0116603 [30] M. Sheikholeslami, Influence of Lorentz forces on nanofluid flow in a porous cylinder considering Darcy model, J. Mol. Liq. (2016), http://dx.doi.org/10.1016/j. molliq.2016.11.022 (In Press). [31] M. Sheikholeslami, H.B. Rokni, Nanofluid two phase model analysis in existence of induced magnetic field, Int. J. Heat Mass Transf. 107 (2017) 288–299.
P. Valipour et al. / Journal of Molecular Liquids 225 (2017) 592–597 [32] M. Sheikholeslami, S.A. Shehzad, Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition, Int. J. Heat Mass Transf. (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.10.107 (In Press). [33] M. Sheikholeslami, Int. J. Hydrogen Energy, (2016), DOI information: 10.1016/j. ijhydene.2016.09.185 [34] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical study for external magnetic source influence on water based nanofluid convective heat transfer, Int. J. Heat Mass Transf. (2016), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.077 (In Press).
597
[35] M. Sheikholeslami, D.D. Ganji, Int. J. Hydrogen Energy (2016), http://dx.doi.org/10. 1016/j.ijhydene.2016.09.121. [36] M. Sheikholeslami, D.D. Ganji, J. Taiwan Inst. Chem. Eng. 65 (2016) 43–77. [37] M.M. Bhatti, A. Zeeshan, N. Ijaz, J. Mol. Liq. 218 (2016) 240–245. [38] R. Ellahi, A. Zeeshan, M. Hassan, Int. J. Numer. Methods Heat Fluid Flow 26 (7) (2016) 2160–2174. [39] M. Sheikholeslami, J. Braz. Soc. Mech. Sci. Eng. 37 (2015) 1623–1633. [40] M. Sheikholeslami, D.D. Ganji, M. Younus Javed, R. Ellahi, J. Magn. Magn. Mater. 374 (2015) 36–43.
