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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 207–222 | Cite as

Brownian motion effect on heat transfer of a three-dimensional nanofluid flow over a stretched sheet with velocity slip

  • N. Freidoonimehr
  • Asghar B. RahimiEmail author
Article

Abstract

Present article provides an analytical investigation of the fluid flow and heat and mass transfer for the steady laminar MHD three-dimensional nanofluid flow over a bidirectional stretching sheet with convective surface boundary condition using optimal homotopy analysis method (OHAM) via Mathematica package BVPh2.0. In contrast to the conventional no-slip condition at the surface, Navier’s slip condition has been applied. The governing partial differential equations are transformed into a highly nonlinear coupled ordinary differential equations consisting the momentum, energy and concentration equations via appropriate similarity transformations. The current OHAM solution demonstrates very good correlation with those of the previously published studies in the especial cases. The influence of different physical parameters such as magnetic parameter (M), Prandtl number (\( \Pr \)), Brownian motion parameter (\( {\text{Nb}} \)), thermophoresis parameter (\( {\text{Nt}} \)), Lewis number (Le), velocity slip parameter (γ), stretching rate ratio parameter (λ), and Biot number (Bi) on all fluid velocity components \( \left( {f^{\prime}(\eta ),\,\,g^{\prime}(\eta )} \right) \), temperature distribution \( \left( {\theta \,(\eta )} \right) \) and concentration \( \left( {\phi \,(\eta )} \right) \) as well as the skin friction coefficients in x and y directions \( \left( {C_{\text{fx}} {\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,\,\,C_{\text{fy}} {\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right), \) local Nusselt number \( \left( {{{{\text{Nu}}_{\text{x}} } \mathord{\left/ {\vphantom {{{\text{Nu}}_{\text{x}} } {{\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right. \kern-0pt} {{\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right) \) and local Sherwood number \( \left( {{{{\text{Sh}}_{\text{x}} } \mathord{\left/ {\vphantom {{{\text{Sh}}_{\text{x}} } {{\text{Re}}_{\text{x}}^{{\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right. \kern-0pt} {{\text{Re}}_{\text{x}}^{{\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right) \) are tabulated graphically and discussed in details. This study specifies that nanoparticles in the base fluid offer a potential in increasing the convective heat transfer performance of various liquids.

Keywords

MHD Three-dimensional flow Nanofluid Velocity slip Convective boundary condition Optimal HAM 

List of symbols

a, b

Constants

Bi

Biot number

\( B_{\text{o}} \)

Constant magnetic field

c

Heat capacity

C

Nanoparticle concentration

\( C_{\text{w}} \)

Concentration of nanoparticle

\( C_{\infty } \)

Ambient concentration

\( C_{\text{f}} \)

Friction coefficient

\( C_{\text{i}} \)

Constant in Eq. (24)

D

Brownian diffusion coefficient

\( D_{\text{T}} \)

Thermophoretic diffusion coefficient

\( f(\eta ),\;g(\eta ) \)

Velocity similarity functions

h

Convective heat transfer coefficient

K

Thermal conductivity

k

Constant

\( {\text{Le}} \)

Lewis number

M

Magnetic parameter

Nu

Nusselt number

Nb

Brownian motion parameter

Nt

Thermophoresis parameter

n, m

Constants

Pr

Prandtl number

q

Heat flux and embedding factor in Eqs. (25)–(32)

R

Auxiliary function

Re

Reynolds number

Sh

Sherwood number

T

Temperature

\( T_{\text{f}} \)

Convective surface temperature

\( T_{\infty } \)

Ambient temperature

u, v, w

Velocity components in x, y, z directions

x, y, z

Cartesian coordinates

Greeks

\( \alpha \)

Thermal diffusivity

\( \gamma \)

Velocity slip parameter

\( \gamma_{0} \)

Slip length

\( \varepsilon \)

Total squared residual error

\( \eta \)

Similarity parameter

\( \theta (\eta ) \)

Temperature distribution

\( \lambda \)

Stretching rate ratio parameter

\( \nu \)

Kinematic viscosity

\( \rho \)

Fluid density

\( \sigma \)

Electrical conductivity

\( \tau \)

Skin friction

\( \phi (\eta ) \)

Concentration

\( \chi \)

Auxiliary parameter

\( {\mathcal{L}} \)

Auxiliary linear operator

\( {\mathcal{N}} \)

Nonlinear operator in Eqs. (25)–(32)

Notes

Acknowledgements

Financial support of Ferdowsi University of mashhad under Contract No. 2/40473 is acknowledged.

References

  1. 1.
    Choi SUS, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles. Mater Sci. 1995;231:99–105.Google Scholar
  2. 2.
    Eastman JA, Choi SUS, Li S, Soyez G, Thompson LJ, DiMelfi RJ. Novel thermal properties of nanostructured materials. Mater Sci Forum. 1999;312–314:629–34.Google Scholar
  3. 3.
    Xuana Y, Roetzel W. Conceptions for heat transfer correlation of nanofluids. Int J Heat Mass Transf. 2000;43:3701–7.Google Scholar
  4. 4.
    Masuda H, Ebata A, Teramae K, Hishinuma H. Alteration of Thermal conductivity and viscosity of liquid by dispersing ultra-fine particles (dispersion of 7-A1203, SiO2 and TiO2 ultra-fine particles). Netsu Bussei. 1993;4:227–33.Google Scholar
  5. 5.
    Xuan Y, Li Q. Investigation on convective heat transfer and flow features of nanofluids. J Heat Transf. 2003;125:151–5.Google Scholar
  6. 6.
    Hatami M, Sheikholeslami M, Ganji DDJ. Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method. Powder Technol. 2014;253:769–79.Google Scholar
  7. 7.
    Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf. 2010;53:2477–83.Google Scholar
  8. 8.
    Rashidi MM, Abelman S, Freidoonimehr N. Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Int J Heat Mass Transf. 2013;62:515–25.Google Scholar
  9. 9.
    Rana P, Bhargava R, Bég OA. Numerical solution for mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium. Comput Math Appl. 2012;64:2816–32.Google Scholar
  10. 10.
    Rana P, Bhargava R. Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study. Commun Nonlinear Sci Numer Simul. 2012;17:212–26.Google Scholar
  11. 11.
    Sheikholeslami M, Gorji-Bandpy M, Ganji DDJ. Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid. Powder Technol. 2014;254:82–93.Google Scholar
  12. 12.
    Toghraie D, Alempour SM, Afrand M. Experimental determination of viscosity of water based magnetite nanofluid for application in heating and cooling systems. J Magn Magn Mater. 2016;417:243–8.Google Scholar
  13. 13.
    Amani M, Amani P, Kasaeian A, Mahian O, Wongwises S. Thermal conductivity measurement of spinal-type ferrite MnFe2O4 nanofluids in the presence of a uniform magnetic field. J Mol Liq. 2017;230:121–8.Google Scholar
  14. 14.
    Amani M, Amani P, Kasaeian A, Mahian O, Kasaeian F, Wongwises S. Experimental study on viscosity of spinal-type manganese ferrite nanofluid in attendance of magnetic field. J Magn Magn Mater. 2017;428:457–63.Google Scholar
  15. 15.
    Esfe MH, Ahangar MRH, Toghraie D, Hajmohammad MH, Rostamian H, Tourang H. Designing artificial neural network on thermal conductivity of Al2O3-water—EG (60–40%) nanofluid using experimental data. J Therm Anal Calorim. 2016;126(2):837–43.Google Scholar
  16. 16.
    Ameri M, Amani M, Amani P. Thermal performance of nanofluids in metal foam tube: thermal dispersion model incorporating heterogeneous distribution of nanoparticles. Adv Powder Technol. 2017;28(10):2747–55.Google Scholar
  17. 17.
    Amani M, Amani P, Kasaeian A, Mahian O, Pop I, Wongwises S. Modelling and optimization of thermal conductivity and viscosity of MnFe2O4 nanofluid under magnetic field using an ANN. Sci Rep. 2017;7:Article number:17369.Google Scholar
  18. 18.
    Esfe MH, Afrand M, Rostamian SH, Toghraie D. Examination of rheological behavior of MWCNTs/ZnO-SAE40 hybrid nano-lubricants under various temperatures and solid volume fractions. Exp Therm Fluid Sci. 2017;80:384–90.Google Scholar
  19. 19.
    Esfe MH, Rostamian H, Toghraie D, Yan WM. Using artificial neural network to predict thermal conductivity of ethylene glycol with alumina nanoparticle. J Therm Anal Calorim. 2016;126(2):643–8.Google Scholar
  20. 20.
    Amani M, Ameri M, Kasaeian A. The experimental study of convection Heat transfer characteristics and pressure drop of magnetic nanofluid in a porous metal foam tube. Transp Porous Media. 2017;16(2):959–74.Google Scholar
  21. 21.
    Sajadifar SA, Karimipour A, Toghraie D. Fluid flow and heat transfer of non- Newtonian nanofluid in a microtube considering slip velocity and temperature jump boundary conditions. Eur J Mech B Fluids. 2017;61:25–32.Google Scholar
  22. 22.
    Esfe MH, Razi P, Hajmohammad MH, Rostamian SH, Sarsam WS, Arani AA, Dahari M. Optimization, modeling and accurate prediction of thermal conductivity and dynamic viscosity of stabilized ethylene glycol and water mixture Al2O3 nanofluids by NSGA-II using ANN. Int Commun Heat Mass Transf. 2017;82:154–60.Google Scholar
  23. 23.
    Amani M, Amani P, Mahian O, Estelle P. Multi objection optimization of thermophysical properties of eco-friendly organic nanofluid. J Clean Prod. 2017;166:350–9.Google Scholar
  24. 24.
    Esfahani MA, Toghraie D. Experimental investigation for developing a new model for the thermal conductivity of Silica/Water-Ethylene glycol (40%–60%) nanofluid at different temperatures and solid volume fractions. J Mol Liq. 2017;232:105–12.Google Scholar
  25. 25.
    Akbari OA, Afrouzi HH, Marzban A, Toghraie D, Malekzade H, Arabpour A. Investigation of volume fraction of nanoparticles effect and aspect ratio of the twisted tape in the tube. J Therm Anal Calorim. 2017;129:1–12.Google Scholar
  26. 26.
    Zadkhast M, Karimipour A, Toghraie D. Developing a new correlation to estimate the thermal conductivity of MWCNT-CuO/water hybrid nanofluid via an experimental investigation. J Therm Anal Calorim. 2017;129:859–67.Google Scholar
  27. 27.
    Sharma P, Baek IH, Cho T, Park S, Lee KB. Measurement of thermal conductivity of ZnO-TiO2/EG hybrid nanofluid. J Therm Anal Calorim. 2016;125(1):527–35.Google Scholar
  28. 28.
    Toghraie D, Chaharsoghi VA, Afrand M. Measurement of thermal conductivity of ZnO–TiO2/EG hybrid nanofluid. J Therm Anal Calorim. 2016;125(1):527–35.Google Scholar
  29. 29.
    Esfe MH, Saedodin S, Wongwises S, Toghraie D. An experimental study on the effect of diameter on thermal conductivity and dynamic viscosity of Fe/water nanofluids. J Therm Anal Calorim. 2015;119(3):1817–24.Google Scholar
  30. 30.
    Esfe MH, Saedodin S, Bahiraei M, Toghraie D, Mahian O, Wongwises S. Thermal conductivity modeling of MgO/EG nanofluids using experimental data and artificial neural network. J Therm Anal Calorim. 2014;118(1):287–94.Google Scholar
  31. 31.
    Wang CY. Flow due to a stretching boundary with partial slip—an exact solution of the Navier-Stokes equations. Chem Eng Sci. 2002;57:3745–7.Google Scholar
  32. 32.
    Sparrow E, Beavers G, Hung L. Flow about a porous-surfaced rotating disk. Int J Heat Mass Transf. 1971;14:993–6.Google Scholar
  33. 33.
    Turkyilmazoglu M, Senel P. Heat and mass transfer of the flow due to a rotating rough and porous disk. Int J Therm Sci. 2013;63:146–58.Google Scholar
  34. 34.
    Sahoo B. Effects of partial slip, viscous dissipation and joule heating on von kármán flow and heat transfer of an electrically conducting non-newtonian fluid. Commun Nonlinear Sci Numer Simul. 2009;14:2982–98.Google Scholar
  35. 35.
    Mohammadein AA, Gorla RSR. Heat transfer in a micropolar fluid over a stretching sheet with viscous dissipation and internal heat generation. Int J Numer Methods Heat Fluid Flow. 2001;11:50–8.Google Scholar
  36. 36.
    Rashidi MM, Mohimanianpour SA. Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method. Nonlinear Anal Model Control. 2010;15:83–95.Google Scholar
  37. 37.
    Bhargava R, Kumar L, Takhar HS. Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet. Int J Eng Sci. 2003;41:2161–78.Google Scholar
  38. 38.
    Bachok N, Ishak A, Pop I. Boundary-layer flow of nanofluids over a moving surface in a flowing fluid. Int J Therm Sci. 2010;49:1663–8.Google Scholar
  39. 39.
    Rashidi MM, Freidoonimehr N, Hosseini A, Bég OA, Hung TK. Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica. 2013;49:1–14.Google Scholar
  40. 40.
    Freidoonimehr N, Rahimi AB. Investigation of MHD nano-fluid flow over a stretching surface with velocity slip and convective surface boundary conditions. Modares Mech Eng. 2015;15(3):208–18.Google Scholar
  41. 41.
    Liao SJ. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul. 2010;15:2003–16.Google Scholar
  42. 42.
  43. 43.
    Hayat T, Shehzad SA, Qasim M, Asghar S. Three-dimensional stretched flow via convective boundary condition and heat generation/absorption. Int J Numer Methods Heat Fluid Flow. 2012;24:342–58.Google Scholar
  44. 44.
    Wang CY. The three-dimensional flow due to a stretching sheet. Phys Fluids. 1984;27:1915–7.Google Scholar
  45. 45.
    Hayat T, Shehzad TS, Alsaedi A. Three-dimensional flow of jeffrey fluid over a bidirectional stretching surface with heat source/sink. J Aerosp Eng. 2014;27:04014007.Google Scholar
  46. 46.
    Hayat T, Muhammad T, Alsaedi A, Ahmad AB. Three-dimensional flow of nano-fluid with Cattaneo-Christov double diffusion. Results Phys. 2016;6:897–903.Google Scholar
  47. 47.
    Rashidi MM, Freidoonimehr N, Hosseini A, Bég OA, Hung TK. Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica. 2014;49:469–82.Google Scholar
  48. 48.
    Khan JA, Mustafa M, Hayat T, Farooq MA, Alsaedi A, Liao SJ. On model for three-dimensional flow of nanofluid: an application to solar energy. J Mol Liq. 2014;194:41–7.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of EngineeringFerdowsi University of MashhadMashhadIran

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