Applied Mathematics and Mechanics

, Volume 33, Issue 6, pp 765–780 | Cite as

Thermophoresis and Brownian motion effects on boundary layer flow of nanofluid in presence of thermal stratification due to solar energy

  • N. Anbuchezhian
  • K. Srinivasan
  • K. Chandrasekaran
  • R. KandasamyEmail author


The problem of laminar fluid flow, which results from the stretching of a vertical surface with variable stream conditions in a nanofluid due to solar energy, is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis in the presence of thermal stratification. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations, namely, the scaling group of transformations. An exact solution is obtained for the translation symmetrys, and the numerical solutions are obtained for the scaling symmetry. This solution depends on the Lewis number, the Brownian motion parameter, the thermal stratification parameter, and the thermophoretic parameter. The conclusion is drawn that the flow field, the temperature, and the nanoparticle volume fraction profiles are significantly influenced by these parameters. Nanofluids have been shown to increase the thermal conductivity and convective heat transfer performance of base liquids. Nanoparticles in the base fluids also offer the potential in improving the radiative properties of the liquids, leading to an increase in the efficiency of direct absorption solar collectors.

Key words

solar radiation Brownian motion nanofluid thermophoresis thermal stratification 



nanoparticle volume fraction


skin-fraction coefficient


nanoparticle volume fraction at the wall


ambient nanoparticle volume fraction


specific heat at constant pressure


Brownian diffusion coefficient


thermophoretic diffusion coefficient


dimensionless stream function


acceleration due to gravity


thermal conductive


Lewis number


magnetic parameter


Brownian motion parameter


thermal stratification parameter


thermophoresis parameter


buoyancy ratio


Prandtl number




local Rayleigh number


suction/injection parameter


temperature of the fluid


temperature at the wall


ambient temperature

\(\bar v\)

velocity vector

u, υ

velocity components along the x- and y-axes


uniform velocity of the free stream flow


velocity of suction/injection

Greek symbols


thermal conductivity


coefficient of thermal expansion


dimensionless temperature


dimensionless nanoparticle volume fraction


similarity variable


dynamic viscosity


density of the base fluid


nanoparticle mass density


heat capacity of the base fluid


heat capacity ratio


kinematic viscosity


stream function

Chinese Library Classification


2010 Mathematics Subject Classification

75F15 73D50 


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  1. [1]
    Todd, P., Otanicar, P. P. E., and Jay Golden, S. Optical properties of liquids for direct absorption solar thermal energy systems. Solar Energy, 83(4), 969–977 (2009)Google Scholar
  2. [2]
    Richard, K. S. and Lee, S. M. 800 hours of operational experience from a 2 kW solar dynamic system. Space Technology and Application International Forum, Mohamed S EI-Genk, New Mexico, 1426–1431 (1999)Google Scholar
  3. [3]
    Odeh, S. D., Behnia, M., and Morrison, G. L. Performance evaluation of solar thermal electric generation systems. Energy Conversion and Management, 44(4), 2425–2443 (2003)CrossRefGoogle Scholar
  4. [4]
    Clausing, A. Analysis of convective losses from cavity solar central receivers. Solar Energy, 27(1), 295–300 (1981)CrossRefGoogle Scholar
  5. [5]
    Dehghan, A. A. and Behnia, M. Combined natural convection conduction and radiation heat transfer in a discretely heated open cavity. Journal of Heat Transfer, 118(1), 56–65 (1996)CrossRefGoogle Scholar
  6. [6]
    Muftuoglu, A. and Bilgen, E. Heat transfer in inclined rectangular receivers for concentrated solar radiation. International Communications in Heat and Mass Transfer, 35(5), 551–556 (2008)CrossRefGoogle Scholar
  7. [7]
    Kennedy, C. E. Reciew in Mid-to High-Ternperature Solar Selective Absorber Materials, National Renewable Energy Laboratory, Coloraclo (2002)CrossRefGoogle Scholar
  8. [8]
    Trieb, F. and Nitsch, J. Recommendations for the market introduction of solar thermal power stations. Renewable Energy, 14(1), 17–22 (1998)CrossRefGoogle Scholar
  9. [9]
    Lin, P. F. and Lin, J. Z. Prediction of nanoparticle transport and deposition in bends. Applied Mathematics and Mechanics (English Edition), 30(8), 957–968 (2009) DOI 10.1007/s10483-009-0802-zzbMATHCrossRefGoogle Scholar
  10. [10]
    Lin, J. Z., Lin, P. F., and Chen, H. J. Research on the transport and deposition of nanoparticles in a rotating curved pipe. Physics of Fluids, 21, 122001 (2009)CrossRefGoogle Scholar
  11. [11]
    Masuda, H., Ebata, A., Teramae, K., and Hishinuma, N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei, 7(2), 227–233 (1993)CrossRefGoogle Scholar
  12. [12]
    Buongiorno, J. and Hu, W. Nanofluid coolants for advanced nuclear power plants. Proceedings of International Congress on Advances in Nuclear Power Plants, Carran Associate, lnc., Seoul, 5705 (2005)Google Scholar
  13. [13]
    Buongiorno, J. Convective transport in nanofluids. ASME Journal of Heat Transfer, 128(1), 240–250 (2006)CrossRefGoogle Scholar
  14. [14]
    Kuznetsov, A. V. and Nield, D. A. Natural convective boundary layer flow of a nanofluid past a vertical plate. International Journal of Thermal Sciences, 49(2), 243–247 (2010)MathSciNetCrossRefGoogle Scholar
  15. [15]
    Nield, D. A. and Kuznetsov, A. V. The Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid. International Journal of Heat and Mass Transfer, 52(25–26), 5792–5795 (2009)zbMATHCrossRefGoogle Scholar
  16. [16]
    Cheng, P. and Minkowycz, W. J. Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. Journal of Geophysical Research, 82, 2040–2044 (1977)CrossRefGoogle Scholar
  17. [17]
    Oberlack, M. Similarity in non-rotating and rotating turbulent pipe flows. Journal of Fluid Mechanics, 379, 1–22 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Bluman, G. W. and Kumei, M. Symmetries and Differential Equations, Springer-Verlag, New York (1989)zbMATHCrossRefGoogle Scholar
  19. [19]
    Pakdemirli, M. and Yurusoy, M. Similarity transformations for partial differential equations. SIAM Review, 40, 96–101 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Aminossadati, S. M. and Ghasemi, B. Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure. European Journal of Mechanics-B/Fluids, 28, 630–640 (2009)zbMATHCrossRefGoogle Scholar
  21. [21]
    Akira, N. and Hitoshi, K. Similarity solutions for buoyancy induced flows over a non-isothermal curved surface in a thermally stratified porous medium. Applied Scientific Research, 46(1), 309–314 (1989)zbMATHGoogle Scholar
  22. [22]
    Brewster, M. Q. Thermal Radiative Transfer Properties, John Wiley and Sons, New York (1972)Google Scholar
  23. [23]
    Gill, S. A process for the step-by-step integration of differential equations in an automatic digital computing machine. Mathematical Proceedings of the Cambridge Philosophical Society, 47(1), 96–108 (1951)zbMATHCrossRefGoogle Scholar
  24. [24]
    Khan, W. A. and Pop, I. Boundary layer flow of a nanofluid past a stretching sheet. International Journal of Heat and Mass Transfer, 53(5), 2477–2483 (2010)zbMATHCrossRefGoogle Scholar
  25. [25]
    Wang, C. Y. Free convection on a vertical stretching surface. ZAMM Journal of Applied Mathematics and Mechanics, 69(3), 418–420 (1989)zbMATHCrossRefGoogle Scholar
  26. [26]
    Gorla, R. S. R. and Sidawi, I. Free convection on a vertical stretching surface with suction and blowing. Journal of Applied Science Research, 52(1), 247–257 (1994)zbMATHCrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • N. Anbuchezhian
    • 1
  • K. Srinivasan
    • 2
  • K. Chandrasekaran
    • 3
  • R. Kandasamy
    • 4
    Email author
  1. 1.Department of Mechanical EngineeringSri Guru Institute of TechnologyCoimbatoreIndia
  2. 2.Department of Mechanical EngineeringAnna UniversityChennaiIndia
  3. 3.Department of Mechanical EngineeringR. M. K. Engineering CollegeChennaiIndia
  4. 4.Research Centre for Computational Mathematics, Faculty of Science, Technology and Human DevelopmentUniversiti Tun Hussein Onn MalaysiaJohorMalaysia

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