Heat transfer enhancement due to nanoparticles, magnetic field, thermal and exponential space-dependent heat source aspects in nanoliquid flow past a stretchable spinning disk


This study explores the heat transfer characteristics of nanoliquid flowing over a rotating disk in the presence of the applied magnetic field and convective boundary condition. The nanoliquid is flowing due to the rotation of the disk with uniform stretching of a disk along the radial direction. Effects of ESHS (exponential space-related heat source) and THS (thermal-related heat source) are the focal concern of this article. The effective thermal conductivity of ethylene glycol (EG)-based graphene oxide (GO) nanoliquid is estimated by using Nan’s model whereas effective dynamic viscosity is calculated through Brinkman model. The partial differential system which governed the problem is transformed by using Von-Karman stretching transformations to the ordinary differential system. The subsequent two-point ODBVP (ordinary differential boundary value problem) is treated numerically. The consequence of effective parameters of the problem on different flow fields is illustrated graphically. The numerical values of shear stress and heat transfer rate (Nusselt number) are also calculated. Further, the slope of the data points is determined to quantify the outcome. Validation of the present results is made by direct comparison with the available results and an excellent agreement is found. It is found that the rate of heat transfer increased with nanoparticle volume fraction at the rate 0.4153 and the friction factor increased by increasing nanoparticle volume fraction at the rate 3.0681. The fluctuation rate of Nusselt number due to the variation of the ESHS parameter is almost three times more than that of THS parameter.

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Angular velocity


Biot number

T f :

Convective fluid temperature (K)

h f :

Convective heat transfer coefficient


Dimensionless normal velocity


Dimensionless radial velocity


Dimensionless tangential velocity

n :

Exponential index

Q*E :

Exponential space-dependent heat source coefficient

Q E :

Exponential space-dependent heat source or sink parameter


Local Reynolds number

B 0 :

Magnetic field strength

M :

Magnetic parameter

Nux :

Nusselt number


Prandtl number

P :


r :

Radial axes

C f :

Skin friction coefficient

S :

Stretching rate

c :

Stretching strength parameter

L ii :

Geometrical factor

T :

Temperature of the fluid (K)

k f :

Thermal conductivity (W m−1 K−1)

Q*T :

Thermal-dependent heat source coefficient

Q T :

Thermal-dependent heat source/sink parameter


Velocity components along r, s, z directions (m s−1)

z :

Vertical axis in the cylindrical coordinates system

T :

Ambient temperature (K)

p :

Constant pressure

ρ :

Density (kg m−3)


Dimensionless temperature

ξ :

Dimensionless variable

µ :

Dynamic viscosity (kg m−1 s−1)

σ :

Electrical conductivity (S m−1)

ν :

Kinematic viscosity (m2 s−1)

κ :

Nanoparticle volume fraction

τ wr :

Radial shear stress at the surface

c p :

Specific heat (J kg−1 K−1)


Stefan-Boltzmann constant (Wm2 K4)

q w :

Surface heat flux

α :

Thermal diffusivity

τ w κ :

Transversal shear stress at the surface

β ii :

Symbol used represents expression

nf :


f :

Base fluid

s :



  1. 1.

    Von Karman, T. Uber laminar and turbulent reibung. ZAMM. 1921:233–35.

  2. 2.

    Turkyilmazoglu M. MHD fluid flow and heat transfer due to a stretching rotating disk. Int J Ther Sci. 2012;51:195–201.

    Article  Google Scholar 

  3. 3.

    Rashidi MM, Ali M, Freidoonimehr N, Nazari F. Parametric analysis and optimization of entropy generation in unsteady MHD flow over a stretching rotating disk using artificial neural network and particle swarm optimization algorithm. Energy. 2013;55:497–510.

    Article  Google Scholar 

  4. 4.

    Asghar S, Jalil M, Hussan M, Turkyilmazoglu M. Lie group analysis of flow and heat transfer over a stretching rotating disk. Int J Heat Mass Transf. 2014;69:140–6.

    Article  Google Scholar 

  5. 5.

    Turkyilmazoglu M. Bödewadt flow and heat transfer over a stretching stationary disk. Int J Mech Sci. 2015;90:246–50.

    Article  Google Scholar 

  6. 6.

    Chamkha AJ, Selimefendigil F, Ismael MA. Mixed convection in a partially layered porous cavity with an inner rotating cylinder. Numer Heat Transf Part A Appl. 2016;69(6):659–75.

    Article  Google Scholar 

  7. 7.

    Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T. Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transf. 2016;99:702–10.

    Article  Google Scholar 

  8. 8.

    Hayat T, Qayyum S, Imtiaz M, Alsaedi A. MHD flow and heat transfer between coaxial rotating stretchable disks in a thermally stratified medium. PLoS One. 2016;11(5):e0155899.

    Article  PubMed  PubMed Central  Google Scholar 

  9. 9.

    Selimefendigil F, Öztop HF. Numerical study and pod-based prediction of natural convection in a ferrofluids–filled triangular cavity with generalized neural networks. Numer Heat Transf Part A Appl. 2015;67(10):1136–61.

    Article  Google Scholar 

  10. 10.

    Selimefendigil F, Öztop HF. Forced convection in a branching channel with partly elastic walls and inner L-shaped conductive obstacle under the influence of magnetic field. Int J Heat Mass Transf. 2019;144:118598.

    Article  Google Scholar 

  11. 11.

    Sheikholeslami M, Rashidi MM. Effect of space dependent magnetic field on free convection of Fe3O4–water nanofluid. J Taiw Inst Chem Eng. 2015;56:6–15.

    CAS  Article  Google Scholar 

  12. 12.

    Hayat T, Ahmed B, Abbasi FM, Alsaedi A. Numerical investigation for peristaltic flow of Carreau–Yasuda magneto-nanofluid with modified Darcy and radiation. J Therm Anal Calorim. 2019;137(4):1359–67.

    CAS  Article  Google Scholar 

  13. 13.

    Waqas H, Khan SU, Bhatti MM, Imran M. Significance of bioconvection in chemical reactive flow of magnetized Carreau-Yasuda nanofluid with thermal radiation and second-order slip. J Therm Anal Calorim. 2020;140:1293–306.

    CAS  Article  Google Scholar 

  14. 14.

    Mahanthesh B, Gireesha BJ, Gorla RSR, Makinde OD. Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study. Neural Comput Appl. 2018;30:1557–67.

    Article  Google Scholar 

  15. 15.

    Sheikholeslami M, Hayat T, Muhammad T, Alsaedi A. MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of Lattice Boltzmann method. Int J Mech Sci. 2018;135:532–40.

    Article  Google Scholar 

  16. 16.

    Sheikholeslami M, Shehzad SA. Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition. Int J Heat Mass Transf. 2017;106:1261–9.

    CAS  Article  Google Scholar 

  17. 17.

    Sheikholeslami M, Shehzad SA. Numerical analysis of Fe 3 O 4–H 2 O nanofluid flow in permeable media under the effect of external magnetic source. Int J Heat Mass Transf. 2018;118:182–92.

    CAS  Article  Google Scholar 

  18. 18.

    Mahanthesh B, Mabood F, Gireesha BJ, Gorla RSR. Effects of chemical reaction and partial slip on the three-dimensional flow of a nanofluid impinging on an exponentially stretching surface. Eur Phys J Plus. 2017;132(3):113.

    Article  Google Scholar 

  19. 19.

    Mahanthesh B, Gireesha BJ, Gorla RSR. Heat and mass transfer effects on the mixed convective flow of chemically reacting nanofluid past a moving/stationary vertical plate. Alex Eng J. 2016;55(1):569–81.

    Article  Google Scholar 

  20. 20.

    Chamkha AJ, Rashad AM, Armaghani T, Mansour MA. Effects of partial slip on entropy generation and MHD combined convection in a lid-driven porous enclosure saturated with a Cu–water nanofluid. J Therm Anal Calorim. 2018;132(2):1291–306.

    CAS  Article  Google Scholar 

  21. 21.

    Seyyedi SM, Dogonchi AS, Ganji DD, Hashemi-Tilehnoee M. Entropy generation in a nanofluid-filled semi-annulus cavity by considering the shape of nanoparticles. J Therm Anal Calorim. 2019;138(2):1607–21.

    CAS  Article  Google Scholar 

  22. 22.

    Shamsabadi H, Rashidi S, Esfahani JA. Entropy generation analysis for nanofluid flow inside a duct equipped with porous baffles. J Therm Anal Calorim. 2019;135(2):1009–19.

    CAS  Article  Google Scholar 

  23. 23.

    Sheikholeslami M, Seyednezhad M. Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM. Int J Heat Mass Transf. 2018;120:772–81.

    CAS  Article  Google Scholar 

  24. 24.

    Choi SU, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles (No. ANL/MSD/CP–84938; CONF-951135–29). Argonne National Lab., IL (United States) 1995.

  25. 25.

    Sheikholeslami M. Finite element method for PCM solidification in existence of CuO nanoparticles. J Mol Liq. 2018;265:347–55.

    CAS  Article  Google Scholar 

  26. 26.

    Sheikholeslami M, Ghasemi A. Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM. Int J Heat Mass Transf. 2018;123:418–31.

    CAS  Article  Google Scholar 

  27. 27.

    Turkyilmazoglu M. Nanofluid flow and heat transfer due to a rotating disk. Comput Fluids. 2014;94:139–46.

    CAS  Article  Google Scholar 

  28. 28.

    Yin C, Zheng L, Zhang C, Zhang X. Flow and heat transfer of nanofluids over a rotating disk with uniform stretching rate in the radial direction. Propuls Pow Res. 2017;6(1):25–30.

    Article  Google Scholar 

  29. 29.

    Mahanthesh B, Gireesha BJ, Shashikumar NS, Shehzad SA. Marangoni convective MHD flow of SWCNT and MWCNT nanoliquids due to a disk with solar radiation and irregular heat source. Phys E. 2017;94:25–30.

    CAS  Article  Google Scholar 

  30. 30.

    Mahanthesh B, Gireesha BJ, Prasannakumara BC, Kumar PS. Magneto–thermo–marangoni convective flow of Cu-H2O nanoliquid past an infinite disk with particle shape and exponential space based heat source effects. Res Phys. 2017;7:2990–6.

    Google Scholar 

  31. 31.

    Mahanthesh B, Gireesha BJ, Shashikumar NS, Hayat T, Alsaedi A. Marangoni convection in Casson liquid flow due to an infinite disk with exponential space dependent heat source and cross-diffusion effects. Res Phys. 2018;9:78–85.

    Google Scholar 

  32. 32.

    Mahanthesh B, Gireesha BJ, Shehzad SA, Rauf A, Kumar PS. Nonlinear radiated MHD flow of nanoliquids due to a rotating disk with irregular heat source and heat flux condition. Phys B. 2018;537:98–104.

    CAS  Article  Google Scholar 

  33. 33.

    Sheikholeslami M. New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media. Comput Methos Appl Mech Eng. 2019;344:319–33.

    Article  Google Scholar 

  34. 34.

    Sheikholeslami M, Rezaeianjouybari B, Darzi M, Shafee A, Li Z, Nguyen TK. Application of nano-refrigerant for boiling heat transfer enhancement employing an experimental study. Int J Heat Mass Transf. 2019;141:974–80.

    CAS  Article  Google Scholar 

  35. 35.

    Sheikholeslami M, Haq RU, Shafee A, Li Z, Elaraki YG, Tlili I. Heat transfer simulation of heat storage unit with nanoparticles and fins through a heat exchanger. Int J Heat Mass Transf. 2019;135:470–8.

    CAS  Article  Google Scholar 

  36. 36.

    Sheikholeslami M, Haq RU, Shafee A, Li Z. Heat transfer behavior of nanoparticle enhanced PCM solidification through an enclosure with V shaped fins. Int J Heat Mass Transf. 2019;130:1322–42.

    CAS  Article  Google Scholar 

  37. 37.

    Sheikholeslami M, Jafaryar M, Li Z. Nanofluid turbulent convective flow in a circular duct with helical turbulators considering CuO nanoparticles. Int J Heat Mass Transf. 2018;124:980–9.

    CAS  Article  Google Scholar 

  38. 38.

    Dogonchi AS, Alizadeh M, Ganji DD. Investigation of MHD Go-water nanofluid flow and heat transfer in a porous channel in the presence of thermal radiation effect. Adv Pow Tech. 2017;28(7):1815–25.

    CAS  Article  Google Scholar 

  39. 39.

    Yu W, Xie H, Wang X, Wang X. Significant thermal conductivity enhancement for nanofluids containing Graphene nanosheets. Phys Lett A. 2011;375:1323–8.

    CAS  Article  Google Scholar 

  40. 40.

    Nan CW, Birringer R, Clarke DR, Gleiter H. Effective thermal conductivity of particulate composites with inter-facial thermal resistance. J Appl Phys. 1997;81:6692. https://doi.org/10.1063/1.365209.

    CAS  Article  Google Scholar 

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The author (B.Mahanthesh) gratefully acknowledges the support of the Management, CHRIST (Deemed to be University), Bangalore, INDIA for pursuing this work. Also, we are very grateful for the Editor and reviewer for their constructive suggestions.

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Correspondence to Giulio Lorenzini.

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Mahanthesh, B., Shashikumar, N.S. & Lorenzini, G. Heat transfer enhancement due to nanoparticles, magnetic field, thermal and exponential space-dependent heat source aspects in nanoliquid flow past a stretchable spinning disk. J Therm Anal Calorim (2020). https://doi.org/10.1007/s10973-020-09927-x

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  • Exponential space-based heat source (ESHS)
  • Nanoliquid
  • Magnetic field
  • Thermal-based heat source (THS)
  • Rotating disk
  • Nanoparticles