- Research
- Open Access
Analysis of three-dimensional boundary-layer nanofluid flow and heat transfer over a stretching surface by means of the homotopy analysis method
- Qingkai Zhao^{1}Email author,
- Hang Xu^{1} and
- Tao Fan^{2}
https://doi.org/10.1186/s13661-015-0327-3
© Zhao et al.; licensee Springer. 2015
- Received: 7 January 2015
- Accepted: 8 April 2015
- Published: 22 April 2015
Abstract
In this paper, an investigation is made of the three-dimensional flow and heat transfer of a nanofluid in the boundary-layer region over a flat sheet stretched continuously in two lateral directions. With the help of a series of similarity transformations, this problem is reduced to a set of three coupled differential equations. The homotopy analysis method (HAM) is then applied to derive the explicit solutions for both the velocity and the temperature distributions. A mathematical analysis shows that these solutions decay exponentially at far field. Besides, the influences of the nanoparticle volume fraction ϕ on the velocity and temperature profiles, as well as the reduced local skin friction coefficients and the reduced local Nusselt numbers are studied. It is found that the heat transfer conductivity of the nanofluid is superior to that of the pure fluids.
Keywords
- boundary layer
- nanofluid flow
- stretching
- heat transfer
- explicit solution
1 Introduction
Flow and heat transfer in the boundary-layer region near a stretching flat surface have been intensively investigated by many researchers during the past several decades owing to their wide applications in a number of industrial processes, for example, polymer extrusion, wire drawing, drawing of plastic films and artificial fibers, hot rolling, glass-fiber, metal extrusion, metal spinning, and so on. Crane [1] investigated the laminar flow over a stretching flat sheet in an otherwise ambient fluid. From then on, many researchers have made contributions to various aspects of this problem such as the effects of the arbitrary velocity of the stretching sheet, magnetic field, blowing or suction, variable thermal conductivity or wall temperature. Typical works can be found in Gupta and Gupta [2], Chakrabarti and Gupta [3], Carragher and Crane [4], Dutta et al. [5], Jeng et al. [6], Dutta [7], Andersson [8], Chiam [9], Vajravelu and Hadjinicolaou [10], etc. It should be noted that Wang [11] initiated the investigation of the three-dimensional boundary-layer flow caused by a stretching flat sheet in two lateral directions in an otherwise ambient fluid. Hayat et al. [12] considered the three-dimensional boundary-layer flow over a stretching surface immersed in a viscoelastic fluid. Takhar et al. [13] investigated the unsteady three-dimensional MHD boundary-layer flow and heat transfer caused by an impulsive motion of a stretching surface in two lateral directions and a suddenly increasing surface temperature from that of the surrounding fluid.
Nanofluids are expected to be a new generation of heat transfer fluids, since they have higher thermal conductivity and single-phase heat transfer coefficients as compared with those of traditional heat transfer fluids. Several types of mathematical models have been proposed for the prediction of the behaviors of nanofluids. One type is the homogeneous models, which were initially proposed by Choi [14] and then developed by Maïga et al. [15]. By means of this approach, it is very convenient to extend the conventional conservation equations for pure fluids to nanofluids. Therefore, all traditional heat transfer correlations involving the computation of the thermophysical properties could be suitable for nanofluids as well. The deficiency of these models is that the obtained results are incompatible with the experimental observation as a large value of the solid volume fraction ϕ is chosen. The valid region for ϕ is usually limited to \(0 \leq\phi\leq0.3\). The dispersion models are another usual approach used for modeling the nanofluids, which were first suggested by Xuan and Roetzel [16]. In this approach, the higher thermal conductivity and the dispersion of nanoparticles are assumed to be the two dominant factors for convective heat transfer enhancement. They therefore treated the effect of the relative velocity between the base fluid and the nanoparticles as a perturbation of the energy equation and then introduced an empirical dispersion coefficient to describe the heat transfer enhancement. Unfortunately, in this approach, the heat transfer enhancement due to nanoparticle dispersion is too small to be considered in nanofluids. The third approach used for modeling nanofluids was proposed by Buongiorno [17] based on the mechanics of the nanoparticle/base-fluid relative velocity. In this model, he, respectively, analyzed the seven slip mechanisms for nanofluids including inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling. He then drew the conclusion that the Brownian diffusion and the thermophoresis are the two most important factors, provided the turbulent effects are not considered. Using the Buongiorno [17] mathematical model, Nield and Kuznetsov [18, 19] extended the well-known Cheng and Minkowycz problem [20] to nanofluids with various physical conditions. Their theoretical results verified the validity of the Buongiorno [17] model for dilute nanofluids. One blemish of this approach is that the physical parameters such as the coefficients of the Brownian diffusion and the thermophoresis are extremely hard to measure experimentally. Another blemish is that, with this model, many well-constructed nonlinear problems in Newtonian fluids cannot be formulated for nanofluids even if the same initial and boundary conditions are given with the boundary-layer approximations and the similarity transformations. This provides the theoretical researchers with difficulties on resolving them numerically or analytically. Recently, some efforts have been made toward understanding of flow and heat transfer characteristics of nanofluids. Among those investigations, some researchers applied the homogeneous models [21–25] to analyze and simplify their physical problems, while other researchers utilized the Buongiorno model [26–29] in their theoretical analysis and numerical simulations.
The present paper considers the steady laminar flow and heat transfer of a nanofluid in the boundary-layer region near a flat surface stretched continuously in two lateral directions. A homogeneous nanofluid model is employed here for simplification of the physical problem. With this approach, a set of three coupled differential equations with their appropriate boundary conditions are obtained by means of some similarity transformations. The homotopy analysis method (HAM) [30] is introduced to give explicit solutions with high accuracy for those differential equations. An asymptotic analysis is then provided to examine the mathematical behavior of the reduced velocity and temperature profiles at far field. As far as we know, this problem has not been considered before so that the results are new and original. Note that nanofluids have superior heat transfer properties as compared to conventional heat transfer fluids. This makes nanofluids very attractive for a number of potential engineering applications involved in flow and heat transfer over flat sheets, such as in the design of cooling and heat exchange for nuclear systems, in the configuration of collector of solar energy, and in the thermal processing regarding to the production of paper, polymeric sheets, insulating materials, fine-fiber mattes, etc. Nanofluids provide us with a new opportunity for making new high-efficiency systems and improving the existing systems in industrial applications.
2 Mathematical description
Thermophysical properties of base fluid and nanoparticles [ 33 ]
Physical properties | Base fluid | Cu | \(\mathbf{Al}_{\boldsymbol{2}}\mathbf{O}_{\boldsymbol{3}}\) | \(\mathbf{TiO}_{\boldsymbol{2}}\) |
---|---|---|---|---|
\(C_{p}\) (J/kg K) | 4,179 | 385 | 765 | 686.2 |
ρ (kg/m^{3}) | 997.1 | 8,933 | 3,970 | 4,250 |
k (W/mK) | 0.613 | 400 | 40 | 8.9538 |
3 Explicit solutions
When we set \(A_{0,0}^{0}=1/\lambda\), \(A_{0,1}^{0}=-1/\lambda\), \(B_{0,0}^{0}=c/\lambda\), \(B_{0,1}^{0}=-c/\lambda\), and \(C_{0,1}^{0}=1\), all coefficients in the above formulas can be determined successively in the order \(k=1, 2, 3, \ldots \) .
4 Results
As mentioned before, the explicit solutions for \(f(\eta)\), \(g(\eta)\), and \(\theta(\eta)\) have been obtained. In this part, we shall first provide tools for the choice of the HAM auxiliary parameter ħ. Then we shall give the method for error verification of the obtained results. The behavior at far field of these solutions will be checked subsequently. At the end, the effect of the solid volume fraction of the nanofluid ϕ on the velocity and temperature, as well as the wall friction and the Nusselt number shall be presented and discussed.
4.1 Techniques for choice of ħ and error verification
Comparison of the 20th-order HAM approximations with Wang’s results [ 35 ] (marked with an asterisk)
c | 0 | 0.25 | 0.50 | 0.75 | 1 |
---|---|---|---|---|---|
\(f''(0)\) | −1 | −1.048811 | −1.093095 | −1.134486 | −1.173721 |
\(f''(0)^{*}\) | −1 | −1.048813 | −1.093097 | −1.134485 | −1.173720 |
\(g''(0)\) | 0 | −0.194564 | −0.465205 | −0.794618 | −1.173721 |
\(g''(0)^{*}\) | 0 | −0.194564 | −0.465205 | −0.794622 | −1.173720 |
f(∞) | 1 | 0.907151 | 0.842386 | 0.792302 | 0.751497 |
\(f(\infty)^{*}\) | 1 | 0.907075 | 0.842360 | 0.792308 | 0.751527 |
g(∞) | 0 | 0.257994 | 0.451678 | 0.612135 | 0.751497 |
\(g(\infty)^{*}\) | 0 | 0.257986 | 0.451671 | 0.612049 | 0.751527 |
Residual errors for various order HAM computation with \(\pmb{\hbar =-1}\) , \(\pmb{c=0.5}\) and \(\pmb{\mathit{Pr}=1}\)
k th-order | X Err | Y Err | T Err |
---|---|---|---|
1 | 1.3778 × 10^{−3} | 2.2158 × 10^{−4} | 4.2787 × 10^{−3} |
5 | 9.6968 × 10^{−8} | 6.7691 × 10^{−8} | 9.1674 × 10^{−6} |
10 | 3.2657 × 10^{−11} | 3.1218 × 10^{−11} | 1.4527 × 10^{−7} |
15 | 4.0317 × 10^{−14} | 4.1084 × 10^{−14} | 4.0237 × 10^{−9} |
20 | 7.0457 × 10^{−17} | 7.3800 × 10^{−17} | 1.5154 × 10^{−10} |
4.2 Asymptotic analysis
4.3 Effects of the nanoparticle volume fraction ϕ on the flows
It is well known that ϕ is the physical quantity for description of the volumetric ratio of solid nanoparticles in nanofluids. \(\phi=0\) denotes the pure fluid (Newtonian fluid) and \(\phi\geq0\) denotes the nanofluids. We now investigate its influence on the velocity and temperature distributions, as well as the reduced local skin friction coefficients and the reduced Nusselt numbers of the considered boundary-layer flows.
5 Conclusions
In this paper, the three-dimensional nanofluid flow and heat transfer in the boundary-layer region over a flat sheet stretched continuously in two lateral directions has been examined in detail. The explicit solutions for this flow problem have been obtained by means of the homotopy analysis method, which exhibits the exponentially decaying behaviors at far field. Besides, the effects of the nanoparticle volume fraction ϕ on the velocity and temperature profiles, as well as the reduced local skin friction coefficients and the reduced local Nusselt numbers have been investigated. It is found that the heat transfer conductivity of the base fluid can be enhanced significantly as the appropriate metallic particles in nano-scale are added into the pure fluids. We believe that this work paves the way for a systematic experimental investigation of the three-dimensional nanofluids flow and heat transfer over a stretching surface. We also believe this work can help researchers to derive explicit solutions of other boundary layer problems.
Declarations
Acknowledgements
We express our sincere appreciations for reviewers’ valuable comments and suggestions. We extend our sincere appreciations to the Program for New Century Excellent Talents in University (Grant No. NCET-12-0347) for its financial support.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645-647 (1970) View ArticleGoogle Scholar
- Gupta, PS, Gupta, AS: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55(6), 744-746 (1977) View ArticleGoogle Scholar
- Chakrabarti, A, Gupta, AS: Hydromagnetic flow and heat transfer over a stretching sheet. Q. Appl. Math. 37(1), 73-78 (1979) MATHGoogle Scholar
- Carragher, P, Crane, LJ: Heat transfer on a continuous stretching sheet. Z. Angew. Math. Mech. 62(10), 564-565 (1982) View ArticleGoogle Scholar
- Dutta, BK, Roy, P, Gupta, AS: Temperature field in flow over a stretching sheet with uniform heat flux. Int. Commun. Heat Mass Transf. 12(1), 89-94 (1985) View ArticleGoogle Scholar
- Jeng, DR, Chang, TCA, De Witt, KJ: Momentum and heat transfer on a continuous moving surface. J. Heat Transf. 108(3), 532-539 (1986) View ArticleGoogle Scholar
- Dutta, BK: Heat transfer from a stretching sheet with uniform suction and blowing. Acta Mech. 78(3-4), 255-262 (1989) View ArticleMATHGoogle Scholar
- Andersson, HI: An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow. Acta Mech. 113(1-4), 241-244 (1995) View ArticleMATHMathSciNetGoogle Scholar
- Chiam, TC: Heat transfer with variable conductivity in a stagnation-point flow towards a stretching sheet. Int. Commun. Heat Mass Transf. 23(2), 239-248 (1996) View ArticleGoogle Scholar
- Vajravelu, K, Hadjinicolaou, A: Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int. J. Eng. Sci. 35(12-13), 1237-1244 (1997) View ArticleMATHGoogle Scholar
- Wang, CY: The three-dimensional flow due to a stretching flat surface. Phys. Fluids 27(8), 1915-1917 (1984) View ArticleMATHMathSciNetGoogle Scholar
- Hayat, T, Sajid, M, Pop, I: Three-dimensional flow over a stretching surface in a viscoelastic fluid. Nonlinear Anal., Real World Appl. 9(4), 1811-1822 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Takhar, HS, Chamkha, AJ, Nath, G: Unsteady three-dimensional MHD-boundary-layer flow due to the impulsive motion of a stretching surface. Acta Mech. 146, 59-71 (2001) View ArticleMATHGoogle Scholar
- Choi, SUS: Enhancing thermal conductivity of fluids with nanoparticle. In: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD 66, pp. 99-105 (1995) Google Scholar
- Maïga, SEB, Palm, SJ, Nguyen, CT, Roy, G, Galanis, N: Heat transfer enhancement by using nanofluids in forced convection flows. Int. J. Heat Fluid Flow 26(4), 530-546 (2005) View ArticleGoogle Scholar
- Xuan, YM, Roetzel, W: Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf. 43(19), 3701-3707 (2000) View ArticleMATHGoogle Scholar
- Buongiorno, J: Convective transport in nanofluids. J. Heat Transf. 128(3), 240-250 (2006) View 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 Transf. 52, 5792-5795 (2009) View ArticleMATHGoogle 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 Transf. 54, 374-378 (2011) View ArticleMATHGoogle Scholar
- Cheng, P, Minkowycz, WJ: Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res. 82, 2040-2044 (1977) View 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 Res. Lett. 6, 314-320 (2011) View ArticleGoogle Scholar
- Bachok, N, Ishak, A, Pop, I: Flow and heat transfer over a rotating porous disk in a nanofluid. Physica B 406, 1767-1772 (2011) View ArticleGoogle Scholar
- Rohni, AM, Ahmad, S, Pop, I: Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids. Int. J. Heat Mass Transf. 55, 1888-1895 (2012) View ArticleGoogle Scholar
- Yacob, NA, Ishak, A, Pop, I: Falkner-Skan problem for a static or moving wedge in nanofluids. Int. J. Therm. Sci. 50, 133-139 (2011) View ArticleGoogle Scholar
- Vajravelu, K, Prasad, KV, Lee, J, Lee, C, Pop, I, Vab Gorder, RA: Convective heat transfer in the flow of viscous Ag-water and Cu-water nanofluids over a stretching surface. Int. J. Therm. Sci. 50, 843-851 (2011) View ArticleGoogle Scholar
- Nield, DA, Kuznetsov, AV: The onset of convection in a horizontal nanofluid layer of finite depth. Eur. J. Mech. B, Fluids 29, 217-223 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Kuznetsov, AV, Nield, DA: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 49, 243-247 (2010) View ArticleGoogle Scholar
- Kuznetsov, AV, Nield, DA: Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 50, 712-717 (2011) View ArticleGoogle Scholar
- Khan, WA, Pop, I: Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53, 2477-2483 (2010) View ArticleMATHGoogle Scholar
- Liao, SJ: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2003) View ArticleGoogle Scholar
- Xu, H, Fan, T, You, XC: Three-dimensional boundary-layer flow and heat transfer of a Cu-water nanofluid over a stretching surface. In: International Conference of Numerical Analysis and Applied Mathematics, Kos, Greece. AIP Conference Proceedings, vol. 1479, pp. 1845-1849 (2012) Google Scholar
- Brinkman, HC: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20, 571-581 (1952) View 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 29, 1326-1336 (2008) View ArticleGoogle Scholar
- Liao, SJ: Homotopy Analysis Method in Nonlinear Differential Equations. Higher Education Press, Beijing (2012) View ArticleMATHGoogle Scholar
- Wang, CY: The three-dimensional flow due to a stretching flat surface. Phys. Fluids 27, 1915-1917 (1984) View ArticleMATHMathSciNetGoogle Scholar