Stagnation-point flow over a stretching/shrinking sheet in a nanofluid
- Norfifah Bachok^{1},
- Anuar Ishak^{2}Email author and
- Ioan Pop^{3}
https://doi.org/10.1186/1556-276X-6-623
© Bachok et al; licensee Springer. 2011
Received: 14 August 2011
Accepted: 8 December 2011
Published: 8 December 2011
Abstract
An analysis is carried out to study the steady two-dimensional stagnation-point flow of a nanofluid over a stretching/shrinking sheet in its own plane. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The similarity equations are solved numerically for three types of nanoparticles, namely copper, alumina, and titania in the water-based fluid with Prandtl number Pr = 6.2. The skin friction coefficient, Nusselt number, and the velocity and temperature profiles are presented graphically and discussed. Effects of the solid volume fraction φ on the fluid flow and heat transfer characteristics are thoroughly examined. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique.
Keywords
nanofluids stagnation-point flow heat transfer stretching/shrinking sheet dual solutions.Introduction
Stagnation-point flow, describing the fluid motion near the stagnation region of a solid surface exists in both cases of a fixed or moving body in a fluid. The two-dimensional stagnation-point flow towards a stationary semi-infinite wall was first studied by Hiemenz [1], who used a similarity transformation to reduce the Navier-Stokes equations to nonlinear ordinary differential equations. This problem has been extended by Homann [2] to the case of axisymmetric stagnation-point flow. The combination of both stagnation-point flows past a stretching surface was considered by Mahapatra and Gupta [3, 4]. There are two conditions that the flow towards a shrinking sheet is likely to exist, whether an adequate suction on the boundary is imposed [5] or a stagnation flow is considered [6]. Wang [6] investigated both two-dimensional and axisymmetric stagnation flow towards a shrinking sheet in a viscous fluid. He found that solutions do not exist for larger shrinking rates and non-unique in the two-dimensional case. After this pioneering work, the flow field over a stagnation point towards a stretching/shrinking sheet has drawn considerable attention and a good amount of literature has been generated on this problem [7–10].
All studies mentioned above refer to the stagnation-point flow towards a stretching/shrinking sheet in a viscous and Newtonian fluid. The present paper deals with the problem of a steady boundary-layer flow, heat transfer, and nanoparticle fraction over a stagnation point towards a stretching/shrinking sheet in a nanofluid, with water as the based fluid. Most conventional heat transfer fluids, such as water, ethylene glycol, and engine oil, have limited capabilities in terms of thermal properties, which, in turn, may impose serve restrictions in many thermal applications. On the other hand, most solids, in particular, metals, have thermal conductivities much higher, say, by one to three orders of magnitude, compared with that of liquids. Hence, one can then expect that fluid-containing solid particles may significantly increase its conductivity. The flow over a continuously stretching surface is an important problem in many engineering processes with applications in industries such as the hot rolling, wire drawing, paper production, glass blowing, plastic films drawing, and glass-fiber production. The quality of the final product depends on the rate of heat transfer at the stretching surface. On the other hand, the new type of shrinking sheet flow is essentially a backward flow as discussed by Goldstein [11] and it shows physical phenomena quite distinct from the forward stretching flow [12]. The enhanced thermal behavior of nanofluids could provide a basis for an enormous innovation for heat transfer intensification for the processes and applications mentioned above.
Many of the publications on nanofluids are about understanding of their behaviors so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, nuclear reactors, transportation, electronics as well as biomedicine and food. The broad range of current and future applications involving nanofluids have been given by Wong and Leon [13]. Nanofluid as a smart fluid, where heat transfer can be reduced or enhanced at will, has also been reported. These fluids enhance thermal conductivity of the base fluid enormously, which is beyond the explanation of any existing theory. They are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and non-Newtonian behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. These suspended nanoparticles can change the transport and thermal properties of the base fluid. The comprehensive references on nanofluids can be found in the recent book by Das et al. [14] and in the review papers by Buongiorno [15], Daungthongsuk and Wongwises [16], Trisaksri and Wongwises [17], Ding et al. [18], Wang and Mujumdar [19–21], Murshed et al. [22], and Kakaç and Pramuanjaroenkij [23].
The nanofluid model proposed by Buongiorno [15] was very recently used by Nield and Kuznetsov [24, 25], Kuznetsov and Neild [26, 27], Khan and Pop [28], and Bachok et al. [29] in their papers. The paper by Khan and Pop [28] is the first which considered the problem on stretching sheet in nanofluids. Different from the above model, the present paper considers a problem using the nanofluid model proposed by Tiwari and Das [30], which was also used by several authors (cf. Abu-Nada [31], Muthtamilselvan et al. [32], Abu-Nada and Oztop [33], Talebi et al. [34], Ahmad et al. [35], Bachok et al. [36, 37], Yacob et al. [38]). The model proposed by Buongiorno [15] studies the Brownian motion and the thermophoresis on the heat transfer characteristics, while the model by Tiwari and Das [30] analyzes the behavior of nanofluids taking into account the solid volume fraction. In the present paper, we analyze the effects of the solid volume fraction and the type of the nanoparticles on the fluid flow and heat transfer characteristics of a nanofluid over a stretching/shrinking sheet.
Mathematical formulation
Here, φ is the nanoparticle volume fraction, (ρ C_{ p })_{nf} is the heat capacity of the nanofluid, k _{nf} is the thermal conductivity of the nanofluid, k _{f} and k _{s} are the thermal conductivities of the fluid and of the solid fractions, respectively, and ρ _{f} and ρ _{s} are the densities of the fluid and of the solid fractions, respectively. It should be mentioned that the use of the above expression for k _{nf} is restricted to spherical nanoparticles where it does not account for other shapes of nanoparticles [31]. Also, the viscosity of the nanofluid μ _{nf} has been approximated by Brinkman [40] as viscosity of a base fluid μ _{f} containing dilute suspension of fine spherical particles.
where ε > 0 for stretching and ε < 0 for shrinking.
where Re_{x} = U _{∞} x /ν _{f} is the local Reynolds number.
Results and discussion
Thermophysical properties of fluid and nanoparticles [39]
Physical properties | Fluid phase (water) | Cu | Al_{2}O_{3} | TiO_{2} |
---|---|---|---|---|
C _{ p }(J/kg K) | 4179 | 385 | 765 | 686.2 |
ρ(kg/m^{3}) | 997.1 | 8933 | 3970 | 4250 |
k(W/mK) | 0.613 | 400 | 40 | 8.9538 |
Values of f″(0) for some values of ε and φ for Cu-water working fluid
ε | Wang [6] | Present results | ||
---|---|---|---|---|
φ = 0 | φ = 0 | φ = 0.1 | φ = 0.2 | |
2 | -1.88731 | -1.887307 | -2.217106 | -2.298822 |
1 | 0 | 0 | 0 | 0 |
0.5 | 0.71330 | 0.713295 | 0.837940 | 0.868824 |
0 | 1.232588 | 1.232588 | 1.447977 | 1.501346 |
-0.5 | 1.49567 | 1.495670 | 1.757032 | 1.821791 |
-1 | 1.32882 | 1.328817 | 1.561022 | 1.618557 |
[0] | [0] | [0] | [0] | |
-1.15 | 1.08223 | 1.082231 | 1.271347 | 1.318205 |
[0.116702] | [0.116702] | [0.137095] | [0.142148] | |
-1.2 | 0.932473 | 1.095419 | 1.135794 | |
[0.233650] | [0.274479] | [0.284596] | ||
-1.2465 | 0.55430 | 0.584281 | 0.686379 | 0.711679 |
[0.554297] | [0.651161] | [0.675159] |
Values of ${\mathit{C}}_{\mathbf{f}}{\mathbf{Re}}_{\mathit{x}}^{\mathbf{1}\mathbf{/}\mathbf{2}}$ for some values of ε and φ
ε | φ | Yacob et al.[42] | Present results | ||||
---|---|---|---|---|---|---|---|
Cu-water | Al _{ 2 } O _{ 3 } -water | TiO _{ 2 } -water | Cu-water | Al _{ 2 } O _{ 3 } -water | TiO _{ 2 } -water | ||
-0.5 | 0.1 | 2.2865 | 1.9440 | 1.9649 | |||
0.2 | 3.1826 | 2.4976 | 2.5413 | ||||
0 | 0.1 | 1.8843 | 1.6019 | 1.6192 | 1.8843 | 1.6019 | 1.6192 |
0.2 | 2.6226 | 2.0584 | 2.0942 | 2.6226 | 2.0584 | 2.0942 | |
0.5 | 0.1 | 1.0904 | 0.9271 | 0.9371 | |||
0.2 | 1.5177 | 1.1912 | 1.2118 |
Values of $\mathbf{N}{\mathbf{u}}_{\mathbf{x}}{\mathbf{Re}}_{\mathbf{x}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}$ for some values of ε and φ
ε | φ | Yacob et al.[42] | Present results | ||||
---|---|---|---|---|---|---|---|
Cu-water | Al _{ 2 } O _{ 3 } -water | TiO _{ 2 } -water | Cu-water | Al _{ 2 } O _{ 3 } -water | TiO _{ 2 } -water | ||
-0.5 | 0.1 | 0.8385 | 0.7272 | 0.7082 | |||
0.2 | 1.0802 | 0.8878 | 0.8423 | ||||
0 | 0.1 | 1.4043 | 1.3305 | 1.3010 | 1.4043 | 1.3305 | 1.3010 |
0.2 | 1.6692 | 1.5352 | 1.4691 | 1.6692 | 1.5352 | 1.4691 | |
0.5 | 0.1 | 1.8724 | 1.8278 | 1.7898 | |||
0.2 | 2.1577 | 2.0700 | 1.9867 |
Conclusions
We have presented an analysis for the flow and heat transfer characteristics of a nanofluid over a stretching/shrinking sheet in its own plane. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point. The resulting system of nonlinear ordinary differential equations is solved numerically for three types of nanoparticles, namely copper (Cu), alumina (Al_{2}O_{3}), and titania (TiO_{2}) in the water-based fluid with Prandtl number Pr = 6.2, to investigate the effect of the solid volume fraction parameter φ on the fluid and heat transfer characteristics. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique. The inclusion of nanoparticles into the base water fluid has produced an increase in the skin friction and heat transfer coefficients, which increases appreciably with an increase of the nanoparticle volume fraction. Nanofluids are capable to change the velocity and temperature profile in the boundary layer. The type of nanofluids is a key factor for heat transfer enhancement. The highest values of the skin friction coefficient and the local Nusselt number were obtained for the Cu nanoparticles compared with the others.
Declarations
Acknowledgements
The authors are indebted to the anonymous reviewers for their constructive comments and suggestions which led to the improvement of this paper. This work was supported by a Research Grant (Project Code: UKM-GGPM-NBT- 080-2010) from the Universiti Kebangsaan Malaysia.
Authors’ Affiliations
References
- Hiemenz K: Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler's Polytech J 1911, 326: 321–324.Google Scholar
- Homann F: Der Einfluss grosser Zahigkeit bei der Stromung um den Zylinder und um die Kugel. Z Angew Math Mech 1936, 16: 153–164. 10.1002/zamm.19360160304View ArticleGoogle Scholar
- Mahapatra TR, Gupta AS: Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Tran 2002, 38: 517–521. 10.1007/s002310100215View ArticleGoogle Scholar
- Mahapatra TR, Gupta AS: Stagnation-point flow towards a stretching surface. Can J Chem Eng 2003, 81: 258–263. 10.1139/v03-027View ArticleGoogle Scholar
- Miklavčič M, Wang CY: Viscous flow due to a shrinking sheet. Quart Appl Math 2006, 64: 283–290.View ArticleGoogle Scholar
- Wang CY: Stagnation flow towards a shrinking sheet. Int J Non Lin Mech 2008, 43: 377–382. 10.1016/j.ijnonlinmec.2007.12.021View ArticleGoogle Scholar
- Lok YY, Ishak A, Pop I: MHD stagnation-point flow towards a shrinking sheet. Int J Numer Meth Heat Fluid Flow 2011, 21: 61–72. 10.1108/09615531111095076View ArticleGoogle Scholar
- Ishak A, Lok YY, Pop I: Stagnation-point flow over a shrinking sheet in a micropolar fluid. Chem Eng Comm 2010, 197: 1417–1427. 10.1080/00986441003626169View ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet. Phys Lett A 2010, 374: 4075–4079. 10.1016/j.physleta.2010.08.032View ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: On the stagnation-point flow towards a stretching sheet with homogeneous-heterogeneous reactions effects. Comm Nonlinear Sci Numer Simulat 2011, 16: 4296–4302. 10.1016/j.cnsns.2011.01.008View ArticleGoogle Scholar
- Goldstein S: On backward boundary layers and flow in converging passages. J Fluid Mech 1965, 21: 33–45. 10.1017/S0022112065000034View ArticleGoogle Scholar
- Fang T-G, Zhang J, Yao S-S: Viscous flow over an unsteady shrinking sheet with mass transfer. Chin Phys Lett 2009, 26: 014703. 10.1088/0256-307X/26/1/014703View ArticleGoogle Scholar
- Wong K-FV, Leon OD: Applications of nanofluids: current and future. Adv Mech Eng 2010., 2010: Article ID 519659:1–11 Article ID 519659:1-11Google Scholar
- Das SK, Choi SUS, Yu W, Pradeep T: Nanofluids: Science and Technology. NJ: Wiley; 2007.View ArticleGoogle Scholar
- Buongiorno J: Convective transport in nanofluids. J Heat Tran 2006, 128: 240–250. 10.1115/1.2150834View ArticleGoogle Scholar
- Daungthongsuk W, Wongwises S: A critical review of convective heat transfer nanofluids. Renew Sustain Energg Rev 2007, 11: 797–817. 10.1016/j.rser.2005.06.005View ArticleGoogle Scholar
- Trisaksri V, Wongwises S: Critical review of heat transfer characteristics of nanofluids. Renew Sustain Energ Rev 2007, 11: 512–523. 10.1016/j.rser.2005.01.010View ArticleGoogle Scholar
- Ding Y, Chen H, Wang L, Yang C-Y, He Y, Yang W, Lee WP, Zhang L, Huo R: Heat transfer intensification using nanofluids. KONA 2007, 25: 23–38.View ArticleGoogle Scholar
- Wang X-Q, Mujumdar AS: Heat transfer characteristics of nanofluids: a review. Int J Thermal Sci 2007, 46: 1–19. 10.1016/j.ijthermalsci.2006.06.010View ArticleGoogle Scholar
- Wang X-Q, Mujumdar AS: A review on nanofluids - part I: theoretical and numerical investigations. Brazilian J Chem Eng 2008, 25: 613–630.Google Scholar
- Wang X-Q, Mujumdar AS: A review on nanofluids - part II: experiments and applications. Brazilian J Chem Eng 2008, 25: 631–648. 10.1590/S0104-66322008000400002View ArticleGoogle Scholar
- Murshed SMS, Leong KC, Yang C: Thermophysical and electrokinetic properties of nanofluids - a critical review. Appl Therm Eng 2008, 28: 2109–2125. 10.1016/j.applthermaleng.2008.01.005View ArticleGoogle Scholar
- Kakaç S, Pramuanjaroenkij A: Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Tran 2009, 52: 3187–3196. 10.1016/j.ijheatmasstransfer.2009.02.006View ArticleGoogle Scholar
- Nield DA, Kuznetsov AV: The Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Tran 2009, 52: 3187–3196. 10.1016/j.ijheatmasstransfer.2009.02.006View ArticleGoogle Scholar
- Nield DA, Kuznetsov AV: The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Tran 2011, 54: 374–378. 10.1016/j.ijheatmasstransfer.2010.09.034View ArticleGoogle Scholar
- Kuznetsov AV, Nield DA: Natural convective boundary layer flow of a nanofluid past a vertical plate. Int J Thermal Sci 2010, 49: 243–247. 10.1016/j.ijthermalsci.2009.07.015View ArticleGoogle Scholar
- Kuznetsov AV, Nield DA: Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Thermal Sci 2011, 50: 712–717. 10.1016/j.ijthermalsci.2011.01.003View ArticleGoogle Scholar
- Khan AV, Pop I: Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Tran 2010, 53: 2477–2483. 10.1016/j.ijheatmasstransfer.2010.01.032View ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: Boundary layer flow of nanofluids over a moving surface in a flowing fluid. Int J Thermal Sci 2010, 49: 1663–1668. 10.1016/j.ijthermalsci.2010.01.026View ArticleGoogle Scholar
- Tiwari RK, Das MK: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Tran 2007, 50: 2002–2018. 10.1016/j.ijheatmasstransfer.2006.09.034View ArticleGoogle Scholar
- Abu-Nada E: Application of nanofluids for heat transfer enhancement of separated flow encountered in a backward facing step. Int J Heat Fluid Flow 2008, 29: 242–249. 10.1016/j.ijheatfluidflow.2007.07.001View ArticleGoogle Scholar
- Muthtamilselvan M, Kandaswamy P, Lee J: Heat transfer enhancement of Copper-water nanofluids in a lid-driven enclosure. Comm Nonlinear Sci Numer Simulat 2010, 15: 1501–1510. 10.1016/j.cnsns.2009.06.015View ArticleGoogle Scholar
- Abu-Nada E, Oztop HF: Effect of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. Int J Heat Fluid Flow 2009, 30: 669–678. 10.1016/j.ijheatfluidflow.2009.02.001View ArticleGoogle Scholar
- Talebi F, Houshang A, Shahi M: Numerical study of mixed convection flows in a squre lid-driven cavity utilizing nanofluid. Int Comm Heat Mass Tran 2010, 37: 79–90. 10.1016/j.icheatmasstransfer.2009.08.013View ArticleGoogle Scholar
- Ahmad S, Rohni AM, Pop I: Blasius and Sakiadis problems in nanofluids. Acta Mech 2011, 218: 195–204. 10.1007/s00707-010-0414-6View ArticleGoogle Scholar
- Bachok N, Ishak A, Nazar R, Pop I: Flow and heat transfer at a general three-dimensional stagnation point flow in a nanofluid. Physica B 2010, 405: 4914–4918. 10.1016/j.physb.2010.09.031View ArticleGoogle Scholar
- Bachok N, Ishak A, Pop I: Flow and heat transfer over a rotating porous disk in a nanofluid. Physica B 2011, 406: 1767–1772. 10.1016/j.physb.2011.02.024View ArticleGoogle Scholar
- Yacob NA, Ishak A, Pop I, Vajravelu K: Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid. Nanoscale Research Letters 2011, 6: 314. 10.1186/1556-276X-6-314View ArticleGoogle Scholar
- Oztop HF, Abu-Nada E: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int J Heat Fluid Flow 2008, 29: 1326–1336. 10.1016/j.ijheatfluidflow.2008.04.009View ArticleGoogle Scholar
- Brinkman HC: The viscosity of concentrated suspensions and solutions. J Chem Phys 1952, 20: 571–581. 10.1063/1.1700493View ArticleGoogle Scholar
- Khanafer K, Vafai K, Lightstone M: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Tran 2003, 46: 3639–3653. 10.1016/S0017-9310(03)00156-XView ArticleGoogle Scholar
- Yacob NA, Ishak A, Pop I: Falkner-Skan problem for a static or moving wedge in nanofluids. Int J Thermal Sci 2011, 50: 133–139. 10.1016/j.ijthermalsci.2010.10.008View ArticleGoogle Scholar
- Weidman PD, Kubitschek DG, Davis AMJ: The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int J Eng Sci 2006, 44: 730–737. 10.1016/j.ijengsci.2006.04.005View ArticleGoogle Scholar
- Merkin JH: A note on the similarity equations arising in free convection boundary layers with blowing and suction. J Appl Math Phys (ZAMP) 1994, 45: 258–274. 10.1007/BF00943504View ArticleGoogle Scholar
- Postelnicu A, Pop I: Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge. Appl Math Comp 2011, 217: 4359–4368. 10.1016/j.amc.2010.09.037View ArticleGoogle Scholar
- Chiam TC: Stagnation point flow towards a stretching plate. J Phys Soc Japan 1994, 63: 2443–2444. 10.1143/JPSJ.63.2443View ArticleGoogle Scholar
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