Numerical Solutions of Dissipative Natural Convective Flow from a Vertical Cone with Heat Absorption, Generation, MHD and Radiated Surface Heat Flux

  • R. M. Kannan
  • Bapuji Pullepu
  • Sabir Ali ShehzadEmail author
Original Paper


The laminar natural convective hydromagnetic viscous fluid flow induced by a cone under aspect of radiated heat flux with thermal radiation, heat absorption and generation is addressed here. The basic equations of conservation of momentum, mass and energy are utilized for the modeling of physical problem. The consequential expressions are worked out by using Crank–Nicholson approach. The implementation of this method leads to conversion of non-dimensional expressions into system of tri-diagonal expressions. The obtain numerical data is visualized for momentum, local-average shear stresses, rate of heat transportation and temperature for various constraints Pr, Δ, M, ε and Rd with the help of graphical sketches. It is reported that the temperature of liquid is boost up with an enhancement in heat generation constraint. The larger Prandtl number corresponds to weaker temperature profiles. The average shear stress coefficient increase for higher radiation constraints and Prandtl number.


Finite difference method MHD Thermal radiation Viscous dissipation Vertical cone 

List of Symbols

\( F_{0}^{''} (0) \)

Shear-stress co-efficient in Ref: [13]


Grashof number


Rate of change of velocity due to gravity


Thermal conductivity


Mean sink co-efficient


Reference span


Magnetic constraint


Local Nusselt number


Dimensionless Local Nusselt numeral

\( \overline{Nu} \)

Dimensionless average Nusselt numeral


Prandtl number


Uniform wall heat flux per unit area


Non-dimensional local radius of the cone


Local radius of the cone




Non-dimensional temperature




Non-dimensional time


Non-dimensional velocity in X-direction


Velocity component in x-direction


Non-dimensional velocity in Y-direction


Rate component in y-direction


Non-dimensional spatial co-ordinate


Spatial coefficient along cone generator


Non-dimensional spatial coefficient along the normal to the cone generator


Spatial coefficient along the normal to the cone generator

Greek Symbols


Thermal diffusivity


Volumetric thermal expansion


Electrical conductivity


Stefan–Boltzmann constant


Non-dimensional heat source/sink constraint


Non-dimensional time step


Non-dimensional finite difference grid size in X-direction


Non-dimensional finite difference grid size in Y-direction


Viscous dissipation parameter


Semi vertical angle of the cone


Dynamic viscosity


Kinematic viscosity




Non-dimensional local skin friction


Non-dimensional local skin friction

\( \bar{\tau } \)

Non-dimensional average skin friction



Condition on the wall

Free stream condition


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Gebhart, B.: Effects of viscous dissipation in natural convection. J. Fluid Mech. 14, 225–232 (1962)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alamigir, M.: Over-all heat transfer from vertical cones in laminar free convection: an approximate method. J. Heat Transf. Trans. ASME 101, 174–176 (1979)CrossRefGoogle Scholar
  3. 3.
    Raptis, A.: Flow through a porous medium in the presence of a magnetic field. Int. J. Energy Res. 10, 97 (1986)CrossRefGoogle Scholar
  4. 4.
    Lin, F.N.: Laminar free convection from a vertical cone with uniform surface heat flux. Lett. Heat Mass Transf. 3, 49–58 (1976)CrossRefGoogle Scholar
  5. 5.
    Chen, C.K., Chen, C.H., Minkowycz, W.J., Gill, U.S.: Non-Darcian effects on mixed convection about a vertical cylinder embedded in a saturated porous medium. Int. J. Heat Mass Transf. 35, 3041–3046 (1992)CrossRefGoogle Scholar
  6. 6.
    Vajravelu, K., Nayfeh, J.: Hydro magnetic convection at a cone and wedge. Int. Commun. Heat Mass Transf. 19, 701–710 (1992)CrossRefGoogle Scholar
  7. 7.
    Hossain, M.A., Paul, S.C.: Free convection from a vertical permeable circular cone with non-uniform surface heat flux. Heat Mass Transf. 37, 167–173 (2001)CrossRefGoogle Scholar
  8. 8.
    Pullepu, B., Ekambavanan, K., Chamkha, A.J.: Unsteady laminar free convection from a vertical cone with uniform surface heat flux. Nonlinear Anal. Model. Control 13, 47–60 (2008)zbMATHGoogle Scholar
  9. 9.
    Pullepu, B., Chamkha, A.J.: Transient laminar MHD free convective flow past a vertical cone with non-uniform surface heat flux. Nonlinear Anal. Model. Control 14, 489–503 (2009)zbMATHGoogle Scholar
  10. 10.
    Kumari, M., Nath, G.: Natural convection from a vertical cone in a porous medium due to the combined effects of heat and mass diffusion with non-uniform wall temperature/concentration or heat/mass flux and suction/injection. Int. J. Heat Mass Transf. 52, 3064–3069 (2009)CrossRefGoogle Scholar
  11. 11.
    Sunitha, S., Prasad, N.R., Reddy, B.: Radiation and mass transfer effects on MHD free convection flow past an impulsively started isothermal vertical plate with dissipation. Therm. Sci. 13, 71–181 (2009)Google Scholar
  12. 12.
    Jordan, J.Z.: Network simulation method applied to radiation and dissipation effects on MHD unsteady free convection over vertical porous plate. Appl. Math. Model. 31, 2019–2033 (2007)CrossRefGoogle Scholar
  13. 13.
    Hossain, M.A., Paul, S.C.: Free convection from a vertical permeable circular cone with non-uniform surface temperature. Acta Mech. 151, 103–114 (2001)CrossRefGoogle Scholar
  14. 14.
    Khan, M.S., et al.: Heat generation, thermal radiation and chemical reaction effects on MHD mixed convection flow over an unsteady stretching permeable surface. Int. J. Basic Appl. Sci. 1, 363–377 (2012)CrossRefGoogle Scholar
  15. 15.
    Khan, M.S., et al.: Non-Newtonian MHD mixed convective power-law fluid flow over a vertical stretching sheet with thermal radiation, heat generation and chemical reaction effects. Acad. Res. Int. 3, 80–92 (2012)Google Scholar
  16. 16.
    Mohamed, R.A., Osman, A.N.A., Abo-Dahab, S.M.: Unsteady MHD double-diffusive convection boundary layer flow past a radiative hot vertical surface in porous media in the presence of chemical reaction and heat sink. Meccanica 48, 931 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hsiao, K.: Stagnation electrical MHD nanofluid mixed convection with slip boundary on a stretching sheet. Appl. Therm. Eng. 98, 850–861 (2016)CrossRefGoogle Scholar
  18. 18.
    Li, J., Zheng, L., Liu, L.: MHD viscoelastic flow and heat transfer over a vertical stretching sheet with Cattaneo-Christov heat flux effects. J. Mol. Liq. 221, 19–25 (2016)CrossRefGoogle Scholar
  19. 19.
    Hsiao, K.: Combined electrical MHD heat transfer thermal extrusion system using Maxwell fluid with radiative and viscous dissipation effects. Appl. Therm. Eng. 112, 1281–1288 (2017)CrossRefGoogle Scholar
  20. 20.
    Zhao, J., Zheng, L., Zhang, X., Liu, F.: Convection heat and mass transfer of fractional MHD Maxwell fluid in a porous medium with Soret and Dufour effects. Int. J. Heat Mass Transf. 103, 203–210 (2016)CrossRefGoogle Scholar
  21. 21.
    Turkyilmazoglu, M.: Mixed convection flow of magnetohydrodynamic micropolar fluid due to a porous heated/cooled deformable plate: exact solutions. Int. J. Heat Mass Transf. 106, 127–134 (2017)CrossRefGoogle Scholar
  22. 22.
    Sheikholeslami, M., Shehzad, S.A.: Non-Darcy free convection of Fe3O4-water nanoliquid in a complex shaped enclosure under impact of uniform Lorentz force. Chin. J. Phys. 56, 270–281 (2018)CrossRefGoogle Scholar
  23. 23.
    Cookey, I.C., Ogulu, A., Omuho-Pepple, V.M.: The influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Int. J. Heat Mass Transf. 46, 2305–2311 (2003)CrossRefGoogle Scholar
  24. 24.
    Chen, C.H.: MHD mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption. Int. J. Nonlinear Mech. 44, 296–603 (2008)Google Scholar
  25. 25.
    Chandrakala, P.: Radiation effects on flow past an impulsively started vertical oscillating plate with uniform heat flux. Int. J. Dyn. Fluids 6, 209–215 (2010)Google Scholar
  26. 26.
    Seth, G.S., Hussain, S.M., Sarkar, S.: Hydro magnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid past an accelerated moving vertical plate with ramped temperature and ramped surface concentration through a porous medium. J. Egypt. Math. Soc. 23, 197–207 (2015)CrossRefGoogle Scholar
  27. 27.
    Elbashbeshy, E.M.A., et al.: Effect of thermal radiation on free convection flow. Therm. Sci. 20, 555–565 (2016)CrossRefGoogle Scholar
  28. 28.
    Pullepu, B., Sambath, P., Viswanathan, K.K.: Effects of chemical reactions on unsteady free convective and mass transfer flow from a vertical cone with heat generation/absorption in the presence of VWT/VWC. Math. Probl. Eng. 2014, 1–20 (2014). (Id 849570) CrossRefGoogle Scholar
  29. 29.
    Sambath, P., Pullepu, B., Hussain, T., Shehzad, S.A.: Radiated chemical reaction impacts on natural convective MHD mass transfer flow induced by a vertical cone. Results Phys. 8, 304–315 (2018)CrossRefGoogle Scholar
  30. 30.
    Soundalgekar, V.M., Ganesan, P.: Finite difference analysis of transient free convection with mass transfer on an isothermal vertical flat plate. Int. J. Eng. Sci. 19, 757–770 (1981)CrossRefGoogle Scholar
  31. 31.
    Ganesan, P., Rani, H.P.: Unsteady free convection on vertical cylinder with variable hat and mass flux. Heat Mass Transf. 35, 259–265 (1999)CrossRefGoogle Scholar
  32. 32.
    Ganesan, P., Palani, G.: Finite difference analysis of unsteady natural convection MHD flow past an inclined plate with variable surface heat and mass flux. Int. J. Heat Mass Transf. 47, 4449–4457 (2004)CrossRefGoogle Scholar
  33. 33.
    Palani, G., Kim, K.Y.: Influence of magnetic field and thermal radiation by natural convection past vertical cone subjected to variable surface heat flux. Appl. Math. Mech. 33, 605–620 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pop, I., Watanabe, T.: Free convection with uniform suction or injection from vertical cone for constant wall flux. Int. Commun. Heat Mass Transf. 19, 275–283 (1992)CrossRefGoogle Scholar
  35. 35.
    Na, T.Y., Chiou, J.P.: Laminar natural convection over a frustum of a cone. Appl. Sci. Res. 35, 409–421 (1979)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Mahanthesh, B., Gireesha, B.J., Reddy Gorla, R.S., Abbasi, F.M., Shehzad, S.A.: Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary. J. Magn. Magn. Mater. 417, 189–196 (2016)CrossRefGoogle Scholar
  37. 37.
    Gireesha, B.J., Mahanthesh, B., Reddy Gorla, R.S., Manjunatha, P.T.: Thermal radiation and Hall effects on boundary layer flow past a non-isothermal stretching surface embedded in porous medium with non-uniform heat source/sink and fluid particle suspension. Heat Mass Transf. 52(4), 897–911 (2016)CrossRefGoogle Scholar
  38. 38.
    Sampath Kumar, P.B., Gireesha, B.J., Mahanthesh, B., Gorla, R.S.R.: Radiative non-linear 3D flow of ferrofluid with Joule heating, convective condition and coriolis force. Therm. Sci. Eng. Prog. 3, 88–94 (2017)CrossRefGoogle Scholar
  39. 39.
    Mahanthesh, B., Gireesha, B.J.: 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. 8, 869–878 (2018)CrossRefGoogle Scholar
  40. 40.
    Mahanthesh, B., Gireesha, B.J., Shehzad, S.A., Rauf, A., Sampath Kumar, P.B.: Nonlinear radiated MHD flow of nanofluids due to a rotating disk with irregular heat source and heat flux condition. Physica B Condens. Matter 537, 98–104 (2018)CrossRefGoogle Scholar
  41. 41.
    Mahanthesh, B., Gireesha, B.J., Sheikholeslami, M., Shehzad, S.A.: Nonlinear Radiative flow of Casson nanofluid a cone and wedge with magnetic dipole: mathematical model of renewable energy. J. Nanofluids 7(6), 1089–1100 (2018)CrossRefGoogle Scholar
  42. 42.
    Gireesha, B.J., Sambath Kumar, P.B., Mahanthesh, B., Shehzad, S.A., Abbasi, F.M.: Nonlinar gravitational and radiation aspects in nanofluid with exponential space dependent heat source and variable viscosity. Microgravity Sci. Technol. 30(3), 257–264 (2018)CrossRefGoogle Scholar
  43. 43.
    Makinde, O.D., Mahanthesh, B., Gireesha, B.J., Shashikumar, N.S., Monaledi, R.L., Tshehla, M.S.: MHD Nanofluid flow past a rotating disk with thermal radiation in the presence of aluminium and titanium alloy nanoparticles. Defect Diffus. 384, 69–79 (2018)CrossRefGoogle Scholar
  44. 44.
    Prakash, J., Gouse Mohiddin, S., Vijaya Kumar Varma, S.: Free convective MHD flow past a vertical cone with variable heat and mass flux. J. Fluids 2013, 404985 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  • R. M. Kannan
    • 1
  • Bapuji Pullepu
    • 1
  • Sabir Ali Shehzad
    • 2
    Email author
  1. 1.Department of MathematicsSRMISTKattankulathurIndia
  2. 2.Department of MathematicsCOMSATS University IslamabadSahiwalPakistan

Personalised recommendations