Physical aspects of Darcy–Forchheimer flow and dissipative heat transfer of Reiner–Philippoff fluid

  • M. Gnaneswara Reddy
  • M. V. V. N. L. Sudharani
  • K. Ganesh KumarEmail author
  • Ali. J. Chamkha
  • G. Lorenzini


The main focus of the present research work is to elaborate the Reiner–Philippoff fluid flow over a stretching sheet along with thermal radiation effect. A Darcy–Forchheimer medium was imposed and a linear stretching surface was used to generate the flow. Application of appropriate transformation yields nonlinear ordinary differential equation through nonlinear Navier–Stokes equations and solved by Runge–Kutta–Fehlberg shooting technique. Importance of influential variables such as velocity and temperature was elaborated graphically. It is envisaging that the boost up values of γ declines the both velocity and temperature profiles.


Reiner–Philippoff fluid Darcy–Forchheimer Thermal radiation Stretching sheet 

List of symbols



\(u_{\text{w}} \left( x \right) = ax^{1/3}\)

Stretched velocity


Permeability of porous medium

\(F = \frac{{C_{\text{b}} }}{{K^{*1/2} }}\)

Non-uniform inertia coefficient of porous medium


Drag coefficient


Forchheimer number


Porosity parameter


Fluid temperature


Wall temperature


Temperature outside the surface


Prandtl number


Heat flux from the sheet


Thermal conductivity


Nusselt number

\(Re_{\text{x}} = \frac{{u_{\text{w}} x}}{\nu }\)

Reynolds number

Greek letter


Reiner–Philippoff fluid parameter


Bingham number


Electrical conductivity,


Fluid density


Thermal diffusivity


Wall shear stress



  1. 1.
    Hayat T, Khan MI, Qayyum S, Alsaedi A, Khan MI. New thermodynamics of entropy generation minimization with nonlinear thermal radiation and nanomaterial. Phys Lett A. 2018;382:749–60.CrossRefGoogle Scholar
  2. 2.
    Rashid MM, Ijaz K, Hayat T, Khan MI, Alsaedi A. Entropy generation in flow of ferromagnetic liquid with nonlinear radiation and slip condition. J Mol Liq. 2019;276:441–52.CrossRefGoogle Scholar
  3. 3.
    Waqas M, Jabeen S, Hayat T, Khan MI, Alsaedi A. Modeling and analysis for magnetic dipole impact in nonlinear thermally radiating carreau nanofluid flow subject to heat generation. J Magn Magn Mater. 2019;485:197–204.CrossRefGoogle Scholar
  4. 4.
    Kumar R, Kumar R, Sheikholeslami M, Chamkha AJ. Irreversibility analysis of the three dimensional flow of carbon nanotubes due to nonlinear thermal radiation and quartic chemical reactions. J Mol Liq. 2019;274:379–92.CrossRefGoogle Scholar
  5. 5.
    Souayeh B, Kumar KG, Reddy MG, Rani S, Hdhiri N. Slip flow and radiative heat transfer behavior of Titanium alloy and ferromagnetic nanoparticles along with suspension of dusty fluid. J Mol Liq. 2019;111–223.Google Scholar
  6. 6.
    Kumar KG. Scrutinization of 3D flow and nonlinear radiative heat transfer of non-Newtonian nanoparticles over an exponentially sheet. Int J Numer Methods Heat Fluid Flow. 2019. Scholar
  7. 7.
    Lokesh HJ, Gireesha BJ, Kumar KG. Characterization of chemical reaction on magnetohydrodynamics flow and nonlinear radiative heat transfer of Casson nanoparticles over an exponentially sheet. J Nanofluids. 2019;8(6):1260–6.CrossRefGoogle Scholar
  8. 8.
    Gireesha BJ, Krishnamurthy MR, Kumar KG. Nonlinear radiative heat transfer and boundary layer flow of Maxwell nanofluid past stretching sheet. J Nanofluids. 2019;8(5):1093–102.CrossRefGoogle Scholar
  9. 9.
    Kumar KG, Gireesha BJ, Krishnamurthy MR, Rudraswamy NG. An unsteady squeezed flow of a tangent hyperbolic fluid over a sensor surface in the presence of variable thermal conductivity. Results Phys. 2017;7:3031–6.CrossRefGoogle Scholar
  10. 10.
    Kapur JN, Gupta RC. Two dimensional flow of Reiner–Philipp off fluids in the inlet length of a straight channel. Appl Sci Res. 1965;14:13–24.CrossRefGoogle Scholar
  11. 11.
    Ghoshal S. Dispersion of solutes in non-Newtonian flows through a circular tube. Chem Eng Sci. 1971;26:185–8.CrossRefGoogle Scholar
  12. 12.
    Na TY. Boundary layer flow of Reiner–Philipp off fluids. Int J Non-Linear Mech. 1994;29:871–7.CrossRefGoogle Scholar
  13. 13.
    Yam KS, Harris SD, Ingham DB, Pop I. Boundary-layer flow of Reiner–Philipp off fluids past a stretching wedge. Int J Non-Linear Mech. 2009;44:1056–62.CrossRefGoogle Scholar
  14. 14.
    Dat VD, Alsarraf J, Moradikazerouni A, Afrand M. Numerical investigation of γ-AlOOH nano-fluid convection performance in a wavy channel considering various shapes of nano additive. Powder Technol. 2019;345:649–65.CrossRefGoogle Scholar
  15. 15.
    Gandhi CVY, Vishal C, Saha P, Rao R. Functionalized multi-walled carbon nanotubes based newtonian nano fluids for medium temperature heat transfer applications. Thermal Sci Eng Progress. 2019;12:13–23.CrossRefGoogle Scholar
  16. 16.
    Abdullah AAA, Al-Rashed A, Shahsavar O, Rasooli MA. Numerical assessment into the hydrothermal and entropy generation characteristics of biological water-silver nano-fluid in a wavy walled micro channel heat sink. Int Commun Heat Mass Transf. 2019;104:118–26.CrossRefGoogle Scholar
  17. 17.
    Ajam H, Jafari SS, Freidoonimehr N. Analytical approximation of MHD nano-fluid flow induced by a stretching permeable surface using Buongiorno’s model. Ain Shams Eng J. 2018;9:525–36.CrossRefGoogle Scholar
  18. 18.
    Xu J, Bandyopadhyay K, Jung D. Experimental investigation on the correlation between nano-fluid characteristics and thermal properties of Al2O3 nano-particles dispersed in ethylene glycol–water mixture. Int J Heat Mass Transf. 2016;94:262–8.CrossRefGoogle Scholar
  19. 19.
    Sheikholeslami M, Sadoughi MK. Simulation of CuO-water nanofluid heat transfer enhancement in presence of melting surface. Int J Heat Mass Transf. 2018;116:909–19.CrossRefGoogle Scholar
  20. 20.
    Sheikholeslami M, Rokni HB. Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force. Comput Methods Appl Mech Eng. 2017;317:419–30.CrossRefGoogle Scholar
  21. 21.
    Sheikholeslami M, Sajjadi H, Delouei AA, Atashafrooz M, Zhixiong L. Magnetic force and radiation influences on nanofluid transportation through a permeable media considering Al2O3 nanoparticles. J Thermal Anal Calorim. 2019;136(6):2477–85.CrossRefGoogle Scholar
  22. 22.
    Jafaryar M, Sheikholeslami M, Li Z, Moradi R. Nanofluid turbulent flow in a pipe under the effect of twisted tape with alternate axis. J Therm Anal Calorim. 2019;135(1):305–23.CrossRefGoogle Scholar
  23. 23.
    Sheikholeslami M, Jafaryar M, Shafee A, Li Z. Nanofluid heat transfer and entropy generation through a heat exchanger considering a new turbulator and CuO nanoparticles. J Therm Anal Calorim. 2019;134(3):2295–303.CrossRefGoogle Scholar
  24. 24.
    Sheikholeslami M, Shehzad SA. Numerical analysis of Fe3O4–H2O nanofluid flow in permeable media under the effect of external magnetic source. Int J Heat Mass Transf. 2018;118:182–92.CrossRefGoogle Scholar
  25. 25.
    Sheikholeslami M, Gerdroodbary MB, Moradi R, Shafee A, Li Z. Application of neural network for estimation of heat transfer treatment of Al2O3–H2O nanofluid through a channel. Comput Methods Appl Mech Eng. 2019;344:1–12.CrossRefGoogle Scholar
  26. 26.
    Nakayama A. A unified treatment of Darcy–Forchheimer boundary-layer flows. Transp Phenomena Porous Media. 1998;1:179–204.CrossRefGoogle Scholar
  27. 27.
    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(1):137–42.CrossRefGoogle Scholar
  28. 28.
    Kishan N, Srinivas M. Thermophoresis and viscous dissipation effects on Darcy–Forchheimer MHD mixed convection in a fluid saturated porous media. Adv Appl Sci Res. 2012;3(1):60–74.Google Scholar
  29. 29.
    Sadiq MA, Hayat T. Darcy–Forchheimer flow of magneto Maxwell liquid bounded by convectively heated sheet. Results Phys. 2016;6:884–90.CrossRefGoogle Scholar
  30. 30.
    Makinde OD, Ogulu A. The effect of thermal radiation on the heat and mass transfer flow of a variable viscosity fluid past a vertical porous plate permeated by a transverse magnetic field. Chem Eng Commun. 2008;195(12):1575–84.CrossRefGoogle Scholar
  31. 31.
    Ganesh NV, Hakeem AKA, Ganga B. Darcy–Forchheimer flow of hydro magnetic nanofluid over a stretching/shrinking sheet in a thermally stratified porous medium with second order slip viscous and Ohmic dissipations effects. Ain Shams Eng J. 2018;9:939–51.CrossRefGoogle Scholar
  32. 32.
    Raju SS, Kumar KG, Rahimi-Gorji M, Khan I. Darcy–Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle in the presence of viscous dissipation. Microsyst Technol. 2019;25(9):3399–405.CrossRefGoogle Scholar
  33. 33.
    Kumar KG, Shehzad SA, Ambreen T, Anwar MI. Heat transfer augmentation in water-based TiO2 nanoparticles through a converging/diverging channel by considering Darcy–Forchheimer porosity. Revista Mexicana de Física. 2019;65:373–81.CrossRefGoogle Scholar
  34. 34.
    Kumar KG, Chamkha AJ. Darcy–Forchheimer flow and heat transfer of water-based Cu nanoparticles in convergent/divergent channel subjected to particle shape effect. Eur Phys J Plus. 2019;134(3):107.CrossRefGoogle Scholar
  35. 35.
    Kumar KG, Rahimi-Gorji M, Reddy MG, Chamkha AJ, Alarifi IM. Enhancement of heat transfer in a convergent/divergent channel by using carbon nanotubes in the presence of a Darcy–Forchheimer medium. In: Microsystem technologies, 1–10, 2019.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsSJM Institute of TechnologyChitredurgaIndia
  2. 2.Department of MathematicsAcharya Nagarjuna University CampusOngoleIndia
  3. 3.Mechanical Engineering Department, Prince Mohammad Endowment for Nanoscience and TechnologyPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  4. 4.Department of Engineering and ArchitectureUniversity of ParmaParmaItaly

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