1.1. Problem Definition
A steady, constant property, two-dimensional flow of an incompressible nanofluid through a homogenous porous medium with permeability of K, over a stretching surface with linear velocity distribution, i.e., 
is assumed (Figure 1) (Sheikholeslami, Ellahi, Ashorynejad, Domairry and Hayat, 2014).
The fluid is a water based nanofluid containing different types of nanoparticles: Cu, Al2O3, Ag and TiO2. It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermo physical properties of the nanofluid are given in Table 1.
The transport properties of the medium can be considered independent from the temperature when the temperature difference between wall and ambient is not significant (Starov and Zhdanov, 2001). The origin is kept fixed while the wall is stretching and the y-axis is perpendicular to the surface. Using the above-mentioned assumptions, the continuity equation is:
(1) where
u and
v are velocity components in the
x and
y directions, respectively. The Brinkman model
x-momentum equation reads:
(2)(3) where
is the effective viscosity which for simplicity in the present study is considered to be identical to the dynamic viscosity,
. This assumption is reasonable for packed beds of particles (Hooman, Gurgenci and Merrikh, 2007)
The effective density
, the effective dynamic viscosity 
, the heat capacitance 
and the thermal conductivity 
of the nanofluid are given as:
(4)(5)(6)(7)Here, 
is the solid volume fraction.
The hydrodynamic boundary conditions are:
(8) where
is the non-dimensional
x-coordinate and
L is the length of the porous plate.
The following thermal boundary conditions are considered:
(9)(10)The power-law temperature and heat flux distribution, described in Equations (9) and (10), resent a wider range of thermal boundary conditions including isoflux and isothermal cases. For example, by setting n equal to zero, Equations (9) and (10) yield isothermal and isoflux, respectively.
Second law of thermodynamics analysis of porous media is found to be more complicated compared to the clear fluid counterpart due to increased number of variables in governing equations (Nield and Kuznetsov, 2005). In the non-Darcian regime, there are three alternative models for the fluid friction term which are the clear-fluid compatible model, the Darcy model, and the Nield model or the power of drag model. Following the entropy generation function introduced by Nield and Kuznetsov (2005) the volumetric entropy generation rate, 
, reads:
(11)Using boundary layer approximations, Equation (11) reduces to:
(12)Using the stream function, 
, the continuity equation is satisfied:
(13)The hydrodynamic boundary layer thickness scales with 
. This can be found through a scale analysis between the first and the second terms on the right hand side of Equation (2), i.e., the viscous and the Darcy terms. Therefore, instead of the other similarity parameters reported in the literature, the following dimensionless similarity parameter is defined
(14)The u-velocity is assumed to be correlated to 
, a dimensionless similarity function as:
(15) where
is
. Using stream function definition, Equation (15), the stream function and the
-velocity take the following forms:
(16)Substituting from 
and 
into Equations (2) and (4), one will find the following differential equation for the 
-momentum equation:
(17) where
is a parameter having the following form:
(18) where
Re is the Reynolds number. Equation (18) should be solved subjected to the following boundary conditions:
(19)
is the injection parameter. Positive/negative values of 
show suction/injection into/from the porous surface, respectively.
The wall shear stress is the driving force that drags fluid flow along the stretching wall. The wall shear stress term can then be found, in terms of the similarity function, as:
(20)Introducing a similarity function, 
, as:
(21) where
is
and
for the power-law temperature and heat flux boundary conditions, respectively. The thermal energy equation reads:
(22) where
are parameters having the following form:
(23)Which are subjected to the following boundary conditions:
Power-law temperature
(24)
Power-law heat flux
For power-law temperature and heat flux boundary conditions, respectively. Employing the definition of convective heat transfer coefficient, the local Nusselt numbers, become:
(25)Finally, the local volumetric entropy generation rate for the above cases, respectively, reads:
(26) where
is the heat transfer irreversibility due to heat transfer in the direction of finite temperature gradients.
is common in all types of thermal engineering applications.
The last term (
) is the contribution of fluid friction irreversibility to the total entropy generation. Not only the wall and fluid layer shear stress but also the momentum exchange at the solid boundaries (pore level) contributes to 
. In terms of the primitive variables, 
and 
become
(27)(28) where
and
are measured in degrees of Kelvin.
One can also define the Bejan number, 
, as
(29)The Bejan number shows the ratio of entropy generation due to heat transfer irreversibility to the total entropy generation so that a Be value more/less than 0.5 shows that the contribution of 
to the total entropy generation is higher/less than that of 
. The limiting value of 
shows that the only active entropy generation mechanism is HTI while 
represents no 
contribution to the total entropy production.
Figure 1. Schematic theme of the problem geometry