Journal of Thermal Analysis and Calorimetry

, Volume 136, Issue 4, pp 1769–1779 | Cite as

Effects of binary chemical reaction and Arrhenius activation energy in Darcy–Forchheimer three-dimensional flow of nanofluid subject to rotating frame

  • Tasawar Hayat
  • Arsalan AzizEmail author
  • Taseer Muhammad
  • Ahmed Alsaedi


Darcy–Forchheimer three-dimensional rotating flow of nanoliquid in the presence of activation energy and heat generation/absorption is examined. Heat and mass transport via convective process is considered. Buongiorno model has been employed to illustrate thermophoresis and Brownian diffusion effects. Adequate transformation procedure gives rise to system in terms of nonlinear ODE’s. An efficient numerical technique namely NDsolve is used to tackle the governing nonlinear system. The graphical illustrations examine the outcomes of various sundry variables. Heat and mass transfer rates are also computed and examined. Our results indicate that the temperature and concentration distributions are enhanced for larger values of porosity parameter and Forchheimer number.


Rotating frame Nanoparticles Darcy–Forchheimer porous medium Arrhenius activation energy Heat generation/absorption 

List of symbols

u, v, w

Velocity components


Dynamic viscosity


Kinematic viscosity


Drag coefficient


Angular velocity




Hot fluid temperature


Ambient fluid temperature


Thermal diffusivity


Effective heat capacity of nanoparticles


Brownian diffusion coefficient


Heat transfer coefficient


Reaction rate


Activation energy


Surface velocity


Dimensionless variable


Dimensionless temperature


Porosity parameter


Rotation parameter


Schmidt number


Brownian motion parameter


Chemical reaction parameter


Thermal Biot number

Cfx, Cfy

Skin friction coefficients


Local Nusselt number

x, y, z

Coordinate axes


Density of base fluid


Permeability of porous medium


Non-uniform inertia coefficient


Heat generation/absorption coefficient




Hot fluid concentration


Ambient fluid concentration


Thermal conductivity


Heat capacity of fluid


Thermophoretic diffusion coefficient


Mass transfer coefficient


Fitted rate constant


Boltzmann constant


Positive constant

f′, g

Dimensionless velocities


Dimensionless concentration


Forchheimer number


Heat generation/absorption parameter


Prandtl number


Thermophoresis parameter


Dimensionless activation energy


Concentration Biot number


Local Reynolds number


Local Sherwood number


  1. 1.
    Choi SUS, Eastman JA. Enhancing thermal conductivity of fluids with nanoparticles. San Francisco: ASME International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers; 1995.Google Scholar
  2. 2.
    Buongiorno J. Convective transport in nanofluids. J Heat Transf. 2006;128:240–50.CrossRefGoogle Scholar
  3. 3.
    Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluid. Int J Heat Mass Transf. 2007;50:2002–18.CrossRefGoogle Scholar
  4. 4.
    Pantzali MN, Mouza AA, Paras SV. Investigating the efficacy of nanofluids as coolants in plate heat exchangers (PHE). Chem Eng Sci. 2009;64:3290–300.CrossRefGoogle Scholar
  5. 5.
    Kakac S, Pramuanjaroenkij A. Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transf. 2009;52:3187–96.CrossRefGoogle Scholar
  6. 6.
    Abu-Nada E, Oztop HF. Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid. Int J Heat Fluid Flow. 2009;30:669–78.CrossRefGoogle Scholar
  7. 7.
    Turkyilmazoglu M. Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids. Chem Eng Sci. 2012;84:182–7.CrossRefGoogle Scholar
  8. 8.
    Hsiao KL. Nanofluid flow with multimedia physical features for conjugate mixed convection and radiation. Comput Fluids. 2014;104:1–8.CrossRefGoogle Scholar
  9. 9.
    Hayat T, Aziz A, Muhammad T, Ahmad B. Influence of magnetic field in three-dimensional flow of couple stress nanofluid over a nonlinearly stretching surface with convective condition. PLoS ONE. 2015;10:e0145332.CrossRefGoogle Scholar
  10. 10.
    Hayat T, Muhammad T, Alsaedi A, Alhuthali MS. Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation. J Magn Magn Mater. 2015;385:222–9.CrossRefGoogle Scholar
  11. 11.
    Pang C, Lee JW, Kang YT. Review on combined heat and mass transfer characteristics in nanofluids. Int J Thermal Sci. 2015;87:49–67.CrossRefGoogle Scholar
  12. 12.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. On magnetohydrodynamic three-dimensional flow of nanofluid over a convectively heated nonlinear stretching surface. Int J Heat Mass Transf. 2016;100:566–72.CrossRefGoogle Scholar
  13. 13.
    Goshayeshi HR, Safaei MR, Goodarzi M, Dahari M. Particle size and type effects on heat transfer enhancement of Ferro-nanofluids in a pulsating heat pipe. Powder Technol. 2016;301:1218–26.CrossRefGoogle Scholar
  14. 14.
    Shehzad N, Zeeshan A, Ellahi R, Vafai K. Convective heat transfer of nanofluid in a wavy channel: Buongiorno’s mathematical model. J Mol Liq. 2016;222:446–55.CrossRefGoogle Scholar
  15. 15.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. Numerical study for nanofluid flow due to a nonlinear curved stretching surface with convective heat and mass conditions. Results Phys. 2017;7:3100–6.CrossRefGoogle Scholar
  16. 16.
    Eid MR, Alsaedi A, Muhammad T, Hayat T. Comprehensive analysis of heat transfer of gold-blood nanofluid (Sisko-model) with thermal radiation. Results Phys. 2017;7:4388–93.CrossRefGoogle Scholar
  17. 17.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. A revised model for Jeffrey nanofluid subject to convective condition and heat generation/absorption. PLoS ONE. 2017;12:e0172518.CrossRefGoogle Scholar
  18. 18.
    Sheikholeslami M, Hayat T, Alsaedi A. Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using lattice Boltzmann method. Int J Heat Mass Transf. 2017;108:1870–83.CrossRefGoogle Scholar
  19. 19.
    Hayat T, Muhammad T, Shehzad SA, Alsaedi A. An analytical solution for magnetohydrodynamic Oldroyd-B nanofluid flow induced by a stretching sheet with heat generation/absorption. Int J Thermal Sci. 2017;111:274–88.CrossRefGoogle Scholar
  20. 20.
    Aziz A, Alsaedi A, Muhammad T, Hayat T. Numerical study for heat generation/absorption in flow of nanofluid by a rotating disk. Results Phys. 2018;8:785–92.CrossRefGoogle Scholar
  21. 21.
    Animasaun IL, Koriko OK, Adegbie KS, Babatunde HA, Ibraheem RO, Sandeep N, Mahanthesh B. Comparative analysis between 36 nm and 47 nm alumina-water nanofluid flows in the presence of Hall effect. J Therm Anal Calorim. 2018. Scholar
  22. 22.
    Mahanthesh B, Gireesha BJ. Scrutinization of thermal radiation, viscous dissipation and Joule heating effects on Marangoni convective two-phase flow of Casson fluid with fluid-particle suspension. Results Phys. 2018;8:869–78.CrossRefGoogle Scholar
  23. 23.
    Mahanthesh B, Gireesha BJ, Shehzad SA, Rauf A, Kumar PBS. 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.CrossRefGoogle Scholar
  24. 24.
    Gireesha BJ, Kumar PBS, Mahanthesh B, Shehzad SA, Abbasi FM. Nonlinear gravitational and radiation aspects in nanoliquid with exponential space dependent heat source and variable viscosity. Microgravity Sci Technol. 2018;30:257–64.CrossRefGoogle Scholar
  25. 25.
    Gireesha BJ, Mahanthesh B, Thammanna GT, Sampathkumar PB. Hall effects on dusty nanofluid two-phase transient flow past a stretching sheet using KVL model. J Mol Liq. 2018;256:139–47.CrossRefGoogle Scholar
  26. 26.
    Sheikholeslami M, Hayat T, Alsaedi A. Numerical simulation for forced convection flow of MHD CuO–H2O nanofluid inside a cavity by means of LBM. J Mol Liq. 2018;249:941–8.CrossRefGoogle Scholar
  27. 27.
    Sheikholeslami M. Numerical modeling of nano enhanced PCM solidification in an enclosure with metallic fin. J Mol Liq. 2018;259:424–38.CrossRefGoogle Scholar
  28. 28.
    Sheikholeslami M. Influence of magnetic field on Al2O3–H2O nanofluid forced convection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM. J Mol Liq. 2018;263:472–88.CrossRefGoogle Scholar
  29. 29.
    Sheikholeslami M. Finite element method for PCM solidification in existence of CuO nanoparticles. J Mol Liq. 2018;265:347–55.CrossRefGoogle Scholar
  30. 30.
    Sheikholeslami M. Application of Darcy law for nanofluid flow in a porous cavity under the impact of Lorentz forces. J Mol Liq. 2018;266:495–503.CrossRefGoogle Scholar
  31. 31.
    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.CrossRefGoogle Scholar
  32. 32.
    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.CrossRefGoogle Scholar
  33. 33.
    Rashidi S, Mahian O, Languri EM. Applications of nanofluids in condensing and evaporating systems. J Therm Anal Calorim. 2018;131:2027–39.CrossRefGoogle Scholar
  34. 34.
    Akar S, Rashidi S, Esfahani JA. Second law of thermodynamic analysis for nanofluid turbulent flow around a rotating cylinder. J Therm Anal Calorim. 2018;132:1189–200.CrossRefGoogle Scholar
  35. 35.
    Bestman AR. Natural convection boundary layer with suction and mass transfer in a porous medium. Int J Energy Res. 1990;14:389–96.CrossRefGoogle Scholar
  36. 36.
    Makinde OD, Olanrewaju PO, Charles WM. Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture. Afr Mat. 2011;22:65–78.CrossRefGoogle Scholar
  37. 37.
    Maleque KA. Effects of exothermic/endothermic chemical reactions with Arrhenius activation energy on MHD free convection and mass transfer flow in presence of thermal radiation. J Thermodyn. 2013;2013:692516.Google Scholar
  38. 38.
    Awad FG, Motsa S, Khumalo M. Heat and mass transfer in unsteady rotating fluid flow with binary chemical reaction and activation energy. PLoS ONE. 2014;9:e107622.CrossRefGoogle Scholar
  39. 39.
    Abbas Z, Sheikh M, Motsa SS. Numerical solution of binary chemical reaction on stagnation point flow of Casson fluid over a stretching/shrinking sheet with thermal radiation. Energy. 2016;95:12–20.CrossRefGoogle Scholar
  40. 40.
    Shafique Z, Mustafa M, Mushtaq A. Boundary layer flow of Maxwell fluid in rotating frame with binary chemical reaction and activation energy. Results Phys. 2016;6:627–33.CrossRefGoogle Scholar
  41. 41.
    Anuradha S, Yegammai M. MHD radiative boundary layer flow of nanofluid past a vertical plate with effects of binary chemical reaction and activation energy. Glob J Pure Appl Math. 2017;13:6377–92.Google Scholar
  42. 42.
    Khan MI, Qayyum S, Hayat T, Waqas M, Khan MI, Alsaedi A. Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial. J Mol Liq. 2018;259:274–83.CrossRefGoogle Scholar
  43. 43.
    Darcy H. Les Fontaines Publiques De La Ville De Dijon. Paris: Victor Dalmont; 1856.Google Scholar
  44. 44.
    Forchheimer P. Wasserbewegung durch boden. Z Ver D Ing. 1901;45:1782–8.Google Scholar
  45. 45.
    Muskat M. The flow of homogeneous fluids through porous media. MI: Edwards; 1946.Google Scholar
  46. 46.
    Seddeek MA. Influence of viscous dissipation and thermophoresis on Darcy–Forchheimer mixed convection in a fluid saturated porous media. J Colloid Interface Sci. 2006;293:137–42.CrossRefGoogle Scholar
  47. 47.
    Pal D, Mondal H. Hydromagnetic convective diffusion of species in Darcy–Forchheimer porous medium with non-uniform heat source/sink and variable viscosity. Int Commun Heat Mass Transf. 2012;39:913–7.CrossRefGoogle Scholar
  48. 48.
    Sadiq MA, Hayat T. Darcy–Forchheimer flow of magneto Maxwell liquid bounded by convectively heated sheet. Results Phys. 2016;6:884–90.CrossRefGoogle Scholar
  49. 49.
    Shehzad SA, Abbasi FM, Hayat T, Alsaedi A. Cattaneo–Christov heat flux model for Darcy–Forchheimer flow of an Oldroyd-B fluid with variable conductivity and non-linear convection. J Mol Liq. 2016;224:274–8.CrossRefGoogle Scholar
  50. 50.
    Bakar SA, Arifin NM, Nazar R, Ali FM, Pop I. Forced convection boundary layer stagnation-point flow in Darcy–Forchheimer porous medium past a shrinking sheet. Front Heat Mass Transf. 2016;7:38.Google Scholar
  51. 51.
    Hayat T, Muhammad T, Al-Mezal S, Liao SJ. Darcy–Forchheimer flow with variable thermal conductivity and Cattaneo–Christov heat flux. Int J Numer Methods Heat Fluid Flow. 2016;26:2355–69.CrossRefGoogle Scholar
  52. 52.
    Hayat T, Haider F, Muhammad T, Alsaedi A. On Darcy–Forchheimer flow of viscoelastic nanofluids: a comparative study. J Mol Liq. 2017;233:278–87.CrossRefGoogle Scholar
  53. 53.
    Umavathi JC, Ojjela O, Vajravelu K. Numerical analysis of natural convective flow and heat transfer of nanofluids in a vertical rectangular duct using Darcy–Forchheimer–Brinkman model. Int J Thermal Sci. 2017;111:511–24.CrossRefGoogle Scholar
  54. 54.
    Muhammad T, Alsaedi A, Shehzad SA, Hayat T. A revised model for Darcy–Forchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin J Phys. 2017;55:963–76.CrossRefGoogle Scholar
  55. 55.
    Sheikholeslami M. Influence of Lorentz forces on nanofluid flow in a porous cavity by means of non-Darcy model. Eng Comput. 2017;34:2651–67.CrossRefGoogle Scholar
  56. 56.
    Muhammad T, Alsaedi A, Hayat T, Shehzad SA. A revised model for Darcy–Forchheimer three-dimensional flow of nanofluid subject to convective boundary condition. Results Phys. 2017;7:2791–7.CrossRefGoogle Scholar
  57. 57.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. Darcy–Forchheimer three-dimensional flow of Williamson nanofluid over a convectively heated nonlinear stretching surface. Commun Theor Phys. 2017;68:387–94.CrossRefGoogle Scholar
  58. 58.
    Hayat T, Haider F, Muhammad T, Alsaedi A. Darcy–Forchheimer squeezed flow of carbon nanotubes with thermal radiation. J Phys Chem Solids. 2018;120:79–86.CrossRefGoogle Scholar
  59. 59.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. An optimal analysis for Darcy–Forchheimer 3D flow of Carreau nanofluid with convectively heated surface. Results Phys. 2018;9:598–608.CrossRefGoogle Scholar
  60. 60.
    Hayat T, Aziz A, Muhammad T, Alsaedi A. An optimal analysis for Darcy–Forchheimer 3D flow of nanofluid with convective condition and homogeneous–heterogeneous reactions. Phys Lett A. 2018;382:2846–55.CrossRefGoogle Scholar
  61. 61.
    Wang CY. Stretching a surface in a rotating fluid. Z Angew Math Phys. 1988;39:177–85.CrossRefGoogle Scholar
  62. 62.
    Takhar HS, Chamkha AJ, Nath G. Flow and heat transfer on a stretching surface in a rotating fluid with a magnetic field. Int J Therm Sci. 2003;42:23–31.CrossRefGoogle Scholar
  63. 63.
    Nazar R, Amin N, Pop I. Unsteady boundary layer flow due to a stretching surface in a rotating fluid. Mech Res Commun. 2004;31:121–8.CrossRefGoogle Scholar
  64. 64.
    Javed T, Sajid M, Abbas Z, Ali N. Non-similar solution for rotating flow over an exponentially stretching surface. Int J Numer Methods Heat Fluid Flow. 2011;21:903–8.CrossRefGoogle Scholar
  65. 65.
    Zaimi K, Ishak A, Pop I. Stretching surface in rotating viscoelastic fluid. Appl Math Mech Engl Ed. 2013;34:945–52.CrossRefGoogle Scholar
  66. 66.
    Rosali H, Ishak A, Nazar R, Pop I. Rotating flow over an exponentially shrinking sheet with suction. J Mol Liq. 2015;211:965–9.CrossRefGoogle Scholar
  67. 67.
    Mustafa M, Hayat T, Alsaedi A. Rotating flow of Maxwell fluid with variable thermal conductivity: an application to non-Fourier heat flux theory. Int J Heat Mass Transf. 2017;106:142–8.CrossRefGoogle Scholar
  68. 68.
    Hayat T, Muhammad T, Mustafa M, Alsaedi A. An optimal study for three dimensional flow of Maxwell nanofluid subject to rotating frame. J Mol Liq. 2017;229:541–7.CrossRefGoogle Scholar
  69. 69.
    Hayat T, Haider F, Muhammad T, Alsaedi A. Three-dimensional rotating flow of carbon nanotubes with Darcy–Forchheimer porous medium. PLoS ONE. 2017;12:e0179576.CrossRefGoogle Scholar
  70. 70.
    Maqsood N, Mustafa M, Khan JA. Numerical tackling for viscoelastic fluid flow in rotating frame considering homogeneous–heterogeneous reactions. Results Phys. 2017;7:3475–81.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • Tasawar Hayat
    • 1
    • 2
  • Arsalan Aziz
    • 1
    Email author
  • Taseer Muhammad
    • 3
  • Ahmed Alsaedi
    • 2
  1. 1.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  2. 2.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of MathematicsGovernment College Women UniversitySialkotPakistan

Personalised recommendations