Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Laminar Free Convection Over a Vertical Wavy Surface Embedded in a Porous Medium Saturated with a Nanofluid

  • 360 Accesses

  • 41 Citations


Numerical analysis is performed to examine laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. In this pioneering study, we have considered the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Non-similar transformations are presented for the governing equations and the obtained PDE are then solved numerically employing a fourth order Runge–Kutta method with shooting technique. A detailed parametric study (nanofluid parameters) is performed to access the influence of the various physical parameters on the local Nusselt number and the local Sherwood number. The results of the problem are presented in graphical forms and discussed.

This is a preview of subscription content, log in to check access.


a :

Amplitude of the wavy surface

D :

Brownian diffusion coefficient

\({\bar{{D}}}\) :

Thermophoretic diffusion coefficient

F :

Dimensionless stream function

g :

Acceleration due to gravity

k :

Thermal conductivity

K :

Modified permeability of the porous medium

Le :

Lewis number

N b :

The Brownian motion parameter

N r :

The buoyancy ratio

N t :

The thermophoresis parameter

Nu :

Local Nusselt number

P :


q w :

Heat transfer rate

q m :

Mass transfer rate

Ra :

Rayleigh number

S :

Rescaled nanoparticle volume fraction

Sh :

Local Sherwood number

T :

Temperature of the fluid

u, υ :

Components of velocity of the fluid

x, y :

Coordinate axes

α :

Thermal diffusivity of porous medium

β :

Volumetric coefficient of thermal expansion of fluid

μ :

Viscosity of the fluid

γ :

The ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid


The characteristic length of the wavy surface

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρ c)f :

Heat capacity of the fluid

(ρ c)p :

Effective heat capacity of nanoparticle material

ψ :

Stream function

θ :

Dimensionless temperature

\({\phi }\) :

Nanoparticle volume fraction


Conditions at the wall

∞, •:

Conditions in the free stream


  1. Buongiorno J.: Convective transport in nanofluid. ASME J. Heat Transf. 128, 240 250 (2006)

  2. Chamkha, A.J., Gorla, R.S.R., Ghodeswar, K.: Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid. Transp Porous Med. (2010). doi:10.1007/s11242-010-9601-0

  3. Cheng C.Y.: Natural convection heat and mass transfer near a vertical wavy surface with constant wall temperature and concentration in a porous medium. Int. Commun. Heat Mass Transf. 27, 1143–1154 (2000)

  4. Choi, S.: Enhancing thermal conductivity of fluids with nanoparticle. In: Siginer, D.A., Wang, H.P., (Eds.), Developments and Applications of Non-Newtonian Flows. ASME FED-231, MD-66, 99–105 (1995)

  5. Das S.K., Putra N., Thiesen P., Roetzel W.: Temperature dependence of thermal conductivity enhancement for nanofluids. J. Heat Transf. 125, 567–574 (2003)

  6. Hady F.M., Mohamed R.A., Mahdy A.: MHD free convection flow along a vertical wavy surface with heat generation or absorption effect. Int. Commun. Heat Mass Transf. 33, 1253–1263 (2006)

  7. Hossain M.A., Rees D.A.S.: Combined heat and mass transfer in natural convection flow from a vertical wavy surface. Acta Mech. 136, 133–141 (1999)

  8. Keblinski P., Phillpot S.R., Choi S.U.S., Eastman J.A.: Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids). Int. J. Heat Mass Transf. 45, 855 863 (2002)

  9. Kumar B.V.R., Shalini: Non-Darcy free convection induced by a vertical wavy surface in a thermally stratified porous medium. Int. J. Heat Mass Transf. 47, 2353–2363 (2004)

  10. Kuznetsov, A.V., Nield, D.A.: The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium. Transp Porous Med. (2010). doi:10.1007/s11242-010-9600-1

  11. Mahdy A.: MHD non-Darcian free convection from a vertical wavy surface embedded in porous media in the presence of Soret and Dufour effect. Int. Commun. Heat Mass Transf. 36, 1067–1074 (2009)

  12. Molla M.M., Hossain M.A., Lun S.Y.: Natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation/absorption. Int. J. Therm. Sci. 43, 157–163 (2004)

  13. Nield D.A., Kuznetsov A.V.: The Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int. J. Heat Mass Transf. 52, 5792–5795 (2009)

  14. Putra N., Roetzel W., Das S.K.: Natural convection of nanofluids. Heat Mass Transf. 39, 775–784 (2003)

  15. Wen D., Ding Y.: Natural convective heat transfer of suspensions of titanium dioxide nanoparticles (nanofluids). IEEE Trans. Nanotechnol. 5, 220–227 (2006)

  16. Xuan Y., Yu K., Li Q.: Investigation on flow and heat transfer of nanofluids by the thermal Lattice Boltzmann model. Prog. Comput. Fluid Dyn. 5, 13–19 (2005)

Download references

Author information

Correspondence to Sameh E. Ahmed.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mahdy, A., Ahmed, S.E. Laminar Free Convection Over a Vertical Wavy Surface Embedded in a Porous Medium Saturated with a Nanofluid. Transp Porous Med 91, 423–435 (2012). https://doi.org/10.1007/s11242-011-9852-4

Download citation


  • Wavy surface
  • Nanofluid
  • Porous medium
  • Natural convection