# Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11242-010-9683-8

- Cite this article as:
- Hamad, M.A.A. & Pop, I. Transp Porous Med (2011) 87: 25. doi:10.1007/s11242-010-9683-8

## Abstract

In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter \({\phi}\) in the range \({0\leq \phi \leq 0.2}\) with the Prandtl number of *Pr* = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter, the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter, and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern.

### Keywords

Porous mediaNanofluidStretching sheetStagnation-point flow### List of Symbols

*a*,*c*Constants

*C*_{f}Skin friction coefficient

*C*_{p}Specific heat at constant pressure

*G*Invariant group of transformations

*k*_{nf}Effective thermal conductivity of the nanofluid

*k*_{f}Thermal conductivity of the fluid

*K*Permeability of the porous medium

*K*_{1}Permeability parameter

*Nu*Local Nusselt number

*Pr*Prandtl number

*q*_{w}Wall heat flux

*Q*_{0}Heat generation/absorption coefficient

*Re*_{x}Local Reynolds number

*S*Suction/injection parameter

*T*Temperature of the nanofluid

*T*_{w}Surface temperature

*T*_{∞}Temperature of the ambient nanofluid fluid

- \({\bar{{u}}, \bar{{v}}}\)
Velocity components along \({\bar{{x}}}\) and \({\bar{{y}}}\) directions

*u*,*v*Dimensionless velocity components

- \({\bar{{u}}_w (\bar{{x}})}\)
Stretching velocity

- \({\bar{{U}}}\)
Free stream velocity of the nanofluid

*U*Dimensionless free stream velocity of the nanofluid

- \({\bar{{v}}_w}\)
Mass flux velocity

- \({\bar{{x}}, \bar{{y}}}\)
Cartesian coordinates along the surface and normal to it

*x*,*y*Dimensionless coordinates

### Greek Letters

*α*_{nf}Effective thermal diffusivity of the nanofluid

*α*_{f}Thermal diffusivity of the fluid

*α*_{i}Constants in (14)

- \({\phi}\)
Solid volume fraction of the nanoparticles

*η*Similarity variable

- λ
Heat generation/absorption parameter

*μ*_{nf}Effective dynamic viscosity of the nanofluid

*μ*_{f}Dynamic viscosity of the fluid

*ν*_{f}Kinematic viscosity of the fluid

*θ*Dimensionless temperature

*ρ*_{nf}Effective density of the nanofluid

*ψ*Stream function

### Subscripts

- nf
Nanofluid

- f
Fluid

- s
Solid

### Superscript

- *
*G*Group variables