Skip to main content
SpringerLink
Log in

A study on the effect of magnetic field and the sinusoidal boundary condition on free convective heat transfer of non-Newtonian power-law fluid in a square enclosure with two constant-temperature obstacles using lattice Boltzmann method

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

In this paper, the flow pattern and thermal characteristics of free convection of a Newtonian magnetohydrodynamic fluid flow inside a square enclosure are compared against two different non-Newtonian power-law (PL) fluids. The enclosure with horizontal insulated walls has two constant-temperature obstacles, while two sidewalls are differentially heated with sinusoidal pattern. Boussinesq approximation is used to consider the effect of hydrodynamic field on the thermal field. The problem is grounded using lattice Boltzmann method, and D2Q9 function is used to describe distribution of the density and energy. The effects of Rayleigh number, PL index, the aspect ratio of constant-temperature obstacles, Hartmann number, and periodicity of sinusoidal boundary condition on the hydrodynamic and thermal characteristics are investigated. The results showed that with increasing Rayleigh number, the Nusselt number increases but with increasing power-law index and Hartmann number, the Nusselt number decreases for shear thinning, Newtonian and shear thickening fluids. In addition, it was shown that increasing the aspect ratio of obstacles increases the Nusselt number. Also, it was reported that the Nusselt number of Newtonian and shear thinning fluids increases with increasing the periodicity of the sinusoidal boundary condition from 2π to 4π, while it reduces by further increase in periodicity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. Finite Difference Lattice Boltzmann Method.

Abbreviations

AR:

Aspect Ratio, \(\left( {{\text{AR}} = \frac{{L_{1} }}{L}} \right)\)

\(B\) :

Magnetic field strength

\(C\) :

Lattice speed

\(C_{\text{p}}\) :

Specific heat (J kg−1 K−1)

\(F\) :

External forces (N)

\(f\) :

Density distribution functions (kg m−3)

\(f_{\text{eq}}\) :

Equilibrium density distribution functions (kg m−3)

\(g\) :

Internal energy distribution functions (K)

\(g_{\text{eq}}\) :

Equilibrium internal energy distribution functions (K)

g :

Gravity (m s−2)

\({\text{Ha}}\) :

Hartmann number, \(\left( {{\text{Ha}} = B_{0} L\sqrt {\frac{{\sigma_{\text{f}} }}{{\rho_{\text{f}} \vartheta_{\text{f}} }}} } \right)\)

\(K\) :

Thermal conductivity (W m−1 K−1)

l 1 :

Obstacles non-dimension length \(\left( {\frac{{L_{1} }}{L}} \right)\)

l 2 :

Non-dimension distance between obstacles \(\left( {\frac{{L_{2} }}{L}} \right)\)

l :

Enclosure non-dimension length \(\left( {L/L} \right)\)

L :

Enclosure length, (m)

\(n\) :

Power-law index

\({\text{Nu}}\) :

Nusselt number (\(hL/k_{\text{f}}\))

\(P\) :

Pressure (Pa)

\(\Pr\) :

Prandtl number, \(\left( {\Pr = \frac{{\mu_{\text{a}} }}{\rho \alpha }} \right)\)

\({\text{Ra}}\) :

Rayleigh number, \(\left( {{\text{Ra}} = \frac{{g\beta_{\text{f}} L^{3} \left( {T_{\text{h}} - T_{\text{c}} } \right)}}{{\alpha_{\text{f}} \vartheta_{\text{f}} }}} \right)\)

\(T\) :

Temperature (K)

\(t\) :

Time (s)

\(u, v\) :

Velocity in x, y direction (ms−1)

\(x,y\) :

Cartesian coordinates (m)

\(\sigma\) :

The electrical conductivity \((\Omega \,{\text{m}})\)

\(\phi\) :

Relaxation time

\(\tau\) :

Shear stress

\(\zeta\) :

Discrete particle speeds

\(\Delta x\) :

Lattice spacing

\(\Delta t\) :

Time increment

\(\rho\) :

Density (kg m−3)

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\psi\) :

Non-dimension stream function value

\({\text{ave}}\) :

Average

\(C\) :

Cold

\(H\) :

Hot

\(x,y\) :

Cartesian coordinates

\(\alpha\) :

The number of the node

References

  1. Qi C, Tang J, Wang G. Natural convection of composite nanofluids based on a two-phase lattice Boltzmann model. J Therm Anal Calorim. 2020;141:277–87.

    CAS  Google Scholar 

  2. Shahsavar A, Ali HM, Mahani RB, Talebizadehsardari P. Numerical study of melting and solidification in a wavy double-pipe latent heat thermal energy storage system. J Therm Anal Calorim. 2020.

  3. Khalid SU, Babar H, Ali HM, Janjua MM, Ali MA. Heat pipes: progress in thermal performance enhancement for microelectronics. J Therm Anal Calorim. 2020.

  4. Wang G, Qi C, Tang J. Natural convection heat transfer characteristics of TiO2–H2O nanofluids in a cavity filled with metal foam. J Therm Anal Calorim. 2020;141:15–24.

    CAS  Google Scholar 

  5. Ali HM. Recent advancements in PV cooling and efficiency enhancement integrating phase change materials based systems—a comprehensive review. Solar Energy. 2020;197:163–98.

    Google Scholar 

  6. Tariq HA, Anwar M, Malik A, Ali HM. Hydro-thermal performance of normal-channel facile heat sink using TiO2–H2O mixture (Rutile–Anatase) nanofluids for microprocessor cooling. J Therm Anal Calorim. 2020.

  7. Sriharan G, Harikrishnan S, Ali HM. Experimental investigation on the effectiveness of MHTHS using different metal oxide-based nanofluids. J Therm Anal Calorim. 2020.

  8. Fan F, Qi C, Tang J, Liu Q, Wang X, Yan Y. A novel thermal efficiency analysis on the thermo-hydraulic performance of nanofluids in an improved heat exchange system under adjustable magnetic field. Appl Therm Engi. 2020;179:115688.

    CAS  Google Scholar 

  9. Qi C, Tang J, Fan F, Yan Y. Effects of magnetic field on thermo-hydraulic behaviors of magnetic nanofluids in CPU cooling system. Appl Therm Eng. 2020;179:115717.

    CAS  Google Scholar 

  10. Qi C, Fan F, Pan Y, Liu M, Yan Y. Effects of turbulator with round hole on the thermo-hydraulic performance of nanofluids in a triangle tube. Int J Heat Mass Transf. 2020;146:118897.

    CAS  Google Scholar 

  11. Hybrid nanofluids for convection heat transfer, 1st edn. Amsterdam: Elsevier, pp. 1–300. https://doi.org/10.4018/978-1-7998-1546-4.

  12. Afrand M, Pordanjani AH, Aghakhani S, Oztop HF, Abu-Hamdeh N. Free convection and entropy generation of a nanofluid in a tilted triangular cavity exposed to a magnetic field with sinusoidal wall temperature distribution considering radiation effects. Int Commun Heat Mass Transf. 2020;112:104507.

    CAS  Google Scholar 

  13. Aghakhani S, Ghasemi B, Hajatzadeh Pordanjani A, Wongwises S, Afrand M. Effect of replacing nanofluid instead of water on heat transfer in a channel with extended surfaces under a magnetic field. Int J Numer Methods Heat Fluid Flow. 2019;29:1249–1271.

    Google Scholar 

  14. Yan S-R, Hajatzadeh Pordanjani A, Aghakhani S, Shahsavar Goldanlou A, Afrand M. Managment of natural convection of nanofluids inside a square enclosure by different nano powder shapes in presence of Fins with different shapes and magnetic field effect. Adv Powder Technol. 2020;31:2759–2777.

    CAS  Google Scholar 

  15. Tian M-W, Rostami S, Aghakhani S, Goldanlou AS, Qi C. Investigation of 2D and 3D configurations of fins and their effects on heat sink efficiency of MHD hybrid nanofluid with slip and non-slip flow. Int J Mech Sci. 2020;105975.

  16. Ghaneifar M, Raisi A, Ali HM, Talebizadehsardari P. Mixed convection heat transfer of Al2O3 nanofluid in a horizontal channel subjected with two heat sources. J Therm Anal Calorim. 2020;1–14.

  17. Aghakhani S, Pordanjani AH, Afrand M, Sharifpur M, Meyer JP. Natural convective heat transfer and entropy generation of alumina/water nanofluid in a tilted enclosure with an elliptic constant temperature: applying magnetic field and radiation effects. Int J Mech Sci. 2020;174:105470.

    Google Scholar 

  18. Pordanjani AH, Aghakhani S, Karimipour A, Afrand M, Goodarzi M. Investigation of free convection heat transfer and entropy generation of nanofluid flow inside a cavity affected by magnetic field and thermal radiation. J Therma Anal Calorim. 2019;137:997–1019.

    Google Scholar 

  19. Muneeshwaran M, Sajjad U, Ahmed T, Amer M, Ali HM, Wang C-C. Performance improvement of photovoltaic modules via temperature homogeneity improvement. Energy. 2020;117:117816.

    Google Scholar 

  20. Tariq SL, Ali HM, Akram MA, Janjua MM, Ahmadlouydarab M. Nanoparticles enhanced phase change materials (NePCMs)–a recent review. Appl Therm Eng. 2020;176:115305.

    Google Scholar 

  21. Ahmadi AA, Arabbeiki M, Ali HM, Goodarzi M, Safaei MR. Configuration and optimization of a minichannel using water-alumina nanofluid by non-dominated sorting genetic algorithm and response surface method. Nanomaterials. 2020;10(5):901.

    CAS  PubMed Central  Google Scholar 

  22. Aghakhani S, Pordanjani AH, Karimipour A, Abdollahi A, Afrand M. Numerical investigation of heat transfer in a power-law non-Newtonian fluid in a C-Shaped cavity with magnetic field effect using finite difference lattice Boltzmann method. Comput Fluids. 2018;176:51–67.

    Google Scholar 

  23. Ali HM. In tube convection heat transfer enhancement: SiO2 aqua based nanofluids. J Mol Liq. 2020;308:113031.

    CAS  Google Scholar 

  24. Zhang R, Aghakhani S, Hajatzadeh Pordanjani A, Vahedi SM, Shahsavar A, Afrand M. Investigation of the entropy generation during natural convection of Newtonian and non-Newtonian fluids inside the L-shaped cavity subjected to magnetic field: application of lattice Boltzmann method. Eur Phys J Plus. 2020;135:184.

    Google Scholar 

  25. Ahmadlouydarab M, Ebadolahzadeh M, Ali HM. Effects of utilizing nanofluid as working fluid in a lab-scale designed FPSC to improve thermal absorption and efficiency. Physica A. 2020;540:123109.

    CAS  Google Scholar 

  26. Ozoe H, Churchill SW. Hydrodynamic stability and natural convection in Ostwald-de Waele and Ellis fluids: the development of a numerical solution. AIChE J. 1972;18(6):1196–207.

    CAS  Google Scholar 

  27. Kim GB, Hyun JM, Kwak HS. Transient buoyant convection of a power-law non-Newtonian fluid in an enclosure. Int J Heat Mass Transf. 2003;46(19):3605–17.

    Google Scholar 

  28. Lamsaadi M, Naimi M, Hasnaoui M. Natural convection heat transfer in shallow horizontal rectangular enclosures uniformly heated from the side and filled with non-Newtonian power law fluids. Energy Convers Manag. 2006;47(15–16):2535–51.

    Google Scholar 

  29. Lamsaadi M, Naimi M, Hasnaoui M, Mamou M. Natural convection in a tilted rectangular slot containing non-Newtonian power-law fluids and subject to a longitudinal thermal gradient. Numer Heat Transf Part A Appl. 2006;50(6):561–83.

    CAS  Google Scholar 

  30. Alloui Z, Vasseur P. Natural convection of Carreau–Yasuda non-Newtonian fluids in a vertical cavity heated from the sides. Int J Heat Mass Transf. 2015;84:912–24.

    Google Scholar 

  31. Ohta M, Ohta M, Akiyoshi M, Obata E. A numerical study on natural convective heat transfer of shear-thinning fluids in a square cavity. Numer Heat Transf Part A Appl. 2002;41(4):357–72.

    CAS  Google Scholar 

  32. Turan O, Chakraborty N, Poole RJ. Laminar natural convection of Bingham fluids in a square enclosure with differentially heated side walls. J Nonnewton Fluid Mech. 2010;165(15–16):901–13.

    CAS  Google Scholar 

  33. Vinogradov I, Khezzar L, Siginer D. Heat transfer of non-Newtonian Dilatant power law fluids in square and rectangular cavities. J Appl Fluid Mech. 2011;4(2, Special Issue):37–42.

    Google Scholar 

  34. Turan O, Poole RJ, Chakraborty N. Aspect ratio effects in laminar natural convection of Bingham fluids in rectangular enclosures with differentially heated side walls. J Nonnewton Fluid Mech. 2011;166(3–4):208–30.

    CAS  Google Scholar 

  35. Turan O, Sachdeva A, Chakraborty N, Poole RJ. Laminar natural convection of power-law fluids in a square enclosure with differentially heated side walls subjected to constant temperatures. J Nonnewton Fluid Mech. 2011;166(17–18):1049–63.

    CAS  Google Scholar 

  36. Guha A, Pradhan K. Natural convection of non-Newtonian power-law fluids on a horizontal plate. Int J Heat Mass Transf. 2014;70:930–8.

    Google Scholar 

  37. Lemus-Mondaca RA, Moraga NO, Riquelme J. Unsteady 2D conjugate natural non-Newtonian convection with non-Newtonian liquid sterilization in square cavity. Int J Heat Mass Transf. 2013;61:73–81.

    CAS  Google Scholar 

  38. Matin MH, Pop I, Khanchezar S. Natural convection of power-law fluid between two-square eccentric duct annuli. J Nonnewton Fluid Mech. 2013;197:11–23.

    Google Scholar 

  39. Kefayati GR. FDLBM simulation of magnetic field effect on natural convection of non-Newtonian power-law fluids in a linearly heated cavity. Powder Technol. 2014;256:87–99.

    CAS  Google Scholar 

  40. Kefayati GR. Mesoscopic simulation of magnetic field effect on natural convection of power-law fluids in a partially heated cavity. Chem Eng Res Des. 2015;94:337–54.

    CAS  Google Scholar 

  41. Barrios G, Rechtman R, Rojas J, Tovar R. The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall. J Fluid Mech. 2005;522:91–100.

    Google Scholar 

  42. Fattahi E, Farhadi M, Sedighi K, Nemati H. Lattice Boltzmann simulation of natural convection heat transfer in nanofluids. Int J Therm Sci. 2012;52:137–44.

    CAS  Google Scholar 

  43. Fattahi E, Farhadi M, Sedighi K. Lattice Boltzmann simulation of natural convection heat transfer in eccentric annulus. Int J Therm Sci. 2010;49(12):2353–62.

    Google Scholar 

  44. Gao D, Chen Z. Lattice Boltzmann simulation of natural convection dominated melting in a rectangular cavity filled with porous media. Int J Therm Sci. 2011;50(4):493–501.

    Google Scholar 

  45. Gabbanelli S, Drazer G, Koplik J. Lattice Boltzmann method for non-Newtonian (power-law) fluids. Phys Rev E. 2005;72(4):046312.

    Google Scholar 

  46. Wang D, Bernsdorf J. Lattice Boltzmann simulation of steady non-Newtonian blood flow in a 3D generic stenosis case. Comput Math Appl. 2009;58(5):1030–4.

    Google Scholar 

  47. Kefayati GR. Simulation of magnetic field effect on natural convection of non-Newtonian power-law fluids in a sinusoidal heated cavity using FDLBM. Int Commun Heat Mass Transf. 2014;53:139–53.

    Google Scholar 

  48. Kefayati GR, Tang H. Double-diffusive natural convection and entropy generation of Carreau fluid in a heated enclosure with an inner circular cold cylinder (part I: heat and mass transfer). Int J Heat Mass Transf. 2018;120:731–50.

    Google Scholar 

  49. Kefayati GR. Simulation of non-Newtonian molten polymer on natural convection in a sinusoidal heated cavity using FDLBM. J Mol Liq. 2014;195:165–74.

    CAS  Google Scholar 

  50. Pordanjani AH, Jahanbakhshi A, Nadooshan AA, Afrand M. Effect of two isothermal obstacles on the natural convection of nanofluid in the presence of magnetic field inside an enclosure with sinusoidal wall temperature distribution. Int J Heat Mass Transf. 2018;121:565–78.

    CAS  Google Scholar 

  51. Habibi Matin M, Pop I, Khanchezar S. Natural convection of power-law fluid between two-square eccentric duct annuli. J Non-Newt Fluid Mech. 2013;197:11–23.

    CAS  Google Scholar 

  52. Matin MH, Khan WA. Laminar natural convection of non-Newtonian power-law fluids between concentric circular cylinders. Int Commun Heat Mass Transf. 2013;43:112–21.

    Google Scholar 

  53. Fu SC, So RMC, Leung WWF. Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows. J Comput Phys. 2010;229(17):6084–103.

    CAS  Google Scholar 

  54. Succi S. The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford: Oxford University Press; 2001.

    Google Scholar 

  55. Fu SC, So RMC, Leung WWF. Linearized-Boltzmann-type-equation-based finite difference method for thermal incompressible flow. Comput Fluids. 2012;69:67–80.

    Google Scholar 

  56. Benzi R, Succi S, Vergassola MJPR. The lattice Boltzmann equation: theory and applications. 1992;222(3):145–97.

    Google Scholar 

  57. Kefayati GR, Tang H. Double-diffusive laminar natural convection and entropy generation of Carreau fluid in a heated enclosure with an inner circular cold cylinder (part II: entropy generation). Int J Heat Mass Transf. 2018;120:683–713.

    Google Scholar 

  58. Chhabra RP, Richardson JF. Non-Newtonian flow and applied rheology: engineering applications. Oxford: Butterworth-Heinemann; 2011.

    Google Scholar 

  59. Khezzar L, Siginer D, Vinogradov I. Natural convection of power law fluids in inclined cavities. Int J Therm Sci. 2012;53:8–171.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Magnetism and Magnetic Materials, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Sara Rostami

  2. Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Sara Rostami

  3. Department of Mathematics and Statistics, IIUI, Islamabad, Pakistan

    R. Ellahi

  4. Department of Mechanical Engineering, University of California Riverside, Riverside, USA

    R. Ellahi

  5. Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey

    Hakan F. Oztop

  6. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Aysan Shahsavar Goldanlou

  7. Faculty of Electrical – Electronic Engineering, Duy Tan University, Da Nang, 550000, Vietnam

    Aysan Shahsavar Goldanlou

Authors
  1. Sara Rostami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ellahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hakan F. Oztop
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Aysan Shahsavar Goldanlou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aysan Shahsavar Goldanlou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rostami, S., Ellahi, R., Oztop, H.F. et al. A study on the effect of magnetic field and the sinusoidal boundary condition on free convective heat transfer of non-Newtonian power-law fluid in a square enclosure with two constant-temperature obstacles using lattice Boltzmann method. J Therm Anal Calorim 144, 2557–2573 (2021). https://doi.org/10.1007/s10973-020-10202-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-020-10202-2

Keywords

Search

Navigation