Abstract
This study employs a numerical solution approach to examine the impacts of viscous dissipative, MHD, among various nanofluid flows across an up straight cone with iso-thermal surface velocity and temperature. After being converted into dimensionless form, the flow field equation is numerically evaluated using an outstanding finite difference approach. Our experiment used different types of nanofluids ( Cu, Al2O3, TiO2 and Ag ) and volume fractions (0, 0.01, 0.02, 0.03, 0.04) to detect differences in heat transfer occurrence depending on temperature and velocity. Graphs have been used to show a parametric analysis for various flow field features.
Similar content being viewed by others
Data availability
The datasets supporting the conclusions of this article are included within the article.
Abbreviations
- \(B_{0}\) :
-
Magnetic field induction (W m−2)
- C\(_{p}\) :
-
Specific heat at constant pressure (J kg−1 K−1)
- Gr\(_{L}\) :
-
Thermal Grashof number (−)
- g:
-
Acceleration due to gravity (m s−2)
- k:
-
Thermal conductivity (W K−1 m −1)
- L:
-
Reference length (m)
- \({\text{Nu}} _{X}\) :
-
Non-dimensional local Nusselt number
- \({\overline{Nu}}\) :
-
Non-dimensional average Nusselt number
- Pr:
-
Prandtl number (−)
- R:
-
Dimensionless local radius
- T′:
-
Temperature (k)
- T:
-
Dimensionless temperature
- t′:
-
Time (s)
- t:
-
Dimensionless time
- U, V:
-
Dimensionless velocity in X, Y direction
- u,v:
-
Dimensional velocity in the direction of x, y axes (ms−1)
- x:
-
Spatial co-ordinate alongside the cone generator (dimensional) (m)
- y:
-
Spatial co-ordinate perpendicular to cone generator (dimensional) (m)
- X:
-
Spatial co-ordinate alongside the cone generator (non-dimensional)
- Y:
-
Spatial co-ordinate perpendicular to cone generator (non-dimensional)
- \(\beta\) :
-
Volumetric thermal expansion coefficient with temperature (K−1)
- \(\epsilon\) :
-
Viscous dissipation parameter (−)
- \(\phi\) :
-
Nanoparticles volume fraction (−)
- μ:
-
Dynamic viscosity (kg m−1 s−1)
- \(\mu _{nf}\) :
-
Nanofluid of the dynamic viscosity (kg m−1 s−1)
- \(\nu\) :
-
Kinematic viscosity (m−1 s−1)
- \(\nu _{f}\) :
-
Base fluid of the kinematic viscosity (m−1 s−1)
- \(\omega\) :
-
Cone apex half-angle (degree)
- \(\rho\) :
-
Density (kg m−3)
- \(\tau _{X}\) :
-
Dimensionless local skin friction
- \({\overline{\tau }}\) :
-
Dimensionless average skin friction
- \(\hbox {f}\) :
-
Base fluid condition
- \(\hbox {nf}\) :
-
Nanofluid condition
- \(\hbox {s}\) :
-
Nanoparticle condition
- \(\hbox {w}\) :
-
Wall condition
- \(\infty\) :
-
Ambient condition
References
F.N. Lin, Laminar free convection from a vertical cone with uniform surface heat flux. Lett. Heat Mass Transf. 3, 49–58 (1976)
T.Y. Na, J.P. Chiou, Laminar natural convection over a slender vertical frustrum of a cone with constant wall heat flux. Wärme-und Stoffübertragung 13(1), 73–78 (1980)
I. Pop, T. Watanabe, Free convection with uniform suction or injection from a vertical cone for constant wall heat flux. Int. Commun. Heat Mass Transf. 19(2), 275–283 (1992)
S.U.S. Choi, Z.G. Zhang, W. Lockwood, F.E. Yu, F.E. Lockwood, E.A. Grulke, Anomalous thermal conductivity enhancement in nanotube suspensions. Appl. Phys. Lett. 79(14), 2252–2254 (2001)
M.A. Hossain, S.C. Paul, Free convection from a vertical permeable circular cone with non-uniform surface heat flux. Heat Mass Transf. 37(2), 167–173 (2001)
M.A. Hossain, S.C. Paul, A.C. Mandal, Natural convection flow along a vertical circular cone with uniform surface temperature and surface heat flux in a thermally stratified medium. Int. J. Numer. Methods Heat Fluid Flow 2, 5586 (2002)
P. Ganesan, G. Palani, 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(19–20), 4449–4457 (2004)
M. Kumari, G. Nath, 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(13–14), 3064–3069 (2009)
S. Mohiddin, S. Gouse, V.K. Varma, N. Iyengar, Unsteady viscoelastic free convection boundary layerflow past a vertical cone with uniform heat and mass flux’’. J. Comput. Math. Sci. 1(2), 103–273 (2010)
C.-Y. Cheng, Soret and Dufour effects on natural convection boundary layer flow over a vertical cone in a porous medium with constant wall heat and mass fluxes. Int. Commun. Heat Mass Transf. 38(1), 44–48 (2011)
G. Palani, K.Y. Kim, Influence of magnetic field and thermal radiation by natural convection past vertical cone subjected to variable surface heat flux. Appl. Math. Mech. 33(5), 605–620 (2012)
G. Palani, A.R. Ragavan, E. Thandapani, Effect of viscous dissipation on an MHD free convective flow past a semi-infinite vertical cone with a variable surface heat flux. J. Appl. Mech. Tech. Phys. 54(6), 960–970 (2013)
C. Ali, S. Abbasbandy, A.M. Rashad, Non-Darcy natural convection flow for non-Newtonian nanofluid over cone saturated in porous medium with uniform heat and volume fraction fluxes. Int. J. Numer. Methods Heat Fluid Flow 2, 558 (2015)
M. Narahari, Unsteady free convection flow past a semi-infinite vertical plate with constant heat flux in water based nanofluids. IOP Conf. Ser. Mater. Sci. Eng. 342(1), 012085 (2018)
P. Sambath, B. Pullepu, T. Hussain, S.A. Shehzad, Radiated chemical reaction impacts on natural convective MHD mass transfer flow induced by a vertical cone. Results Phys. 8, 304–315 (2018)
P. Sambath, D.S. Sankar, K.K. Viswanathan, A numerical study of dissipative chemically reactive radiative MHD flow past a vertical cone with nonuniform mass flux. Int. J. Appl. Mech. Eng. 25(1), 5598 (2020)
Author information
Authors and Affiliations
Contributions
ERK, PS and AJC conceived and designed the study. ERK conducted the literature search and drafted the manuscript. PS and AJC were involved in the analysis and interpretation of data. The study was supervised by PS. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no competing interests.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ragulkumar, E., Sambath, P. & Chamkha, A.J. Free convection nanofluid flow past a vertical isothermal cone surface in the presence of viscous dissipation and MHD with heat flux. Eur. Phys. J. Plus 137, 894 (2022). https://doi.org/10.1140/epjp/s13360-022-03115-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-022-03115-6