We consider the steady two-dimensional boundary layer of a non-Newtonian nanofluid over a vertical impermeable wall embedded in a porous medium. The x-coordinate is taken along the plate and y-coordinate is normal to it. Temperature T and nanoparticle fraction C are taken as constant values and , respectively. The ambient values of T and C are denoted by and . The Oberbeck-Boussinesq approximation was employed and homogeneity and local thermal equilibrium in the porous medium was assumed. Using the standard boundary layer approximations, we can write the governing equations as follows:
(1)
(2)
(3)
(4)
where
(5)
The boundary conditions for equations (1)-(4) are given in the form:
(6)
Here u, v are velocity components in vertical (x) and horizontal (y) directions, n is the power law index, K is the modified permeability of the porous medium, g is the gravitational acceleration, β is the volumetric expansion of the base fluid, is the density of the base fluid, μ is the consistency index of the power law fluid, T is the local temperature, C is the nanoparticle volume fraction, is the density of nanoparticle, is the effective thermal diffusivity of the porous medium that can be written as , is the molecular thermal diffusivity, represent the thermal diffusivity, γ is the mechanical thermal dispersion coefficient. This value lies between and (see Kairi et al. [4]), d is the pore diameter, τ is ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the fluid, is the Brownian motion coefficient, is the thermophoretic diffusion coefficient, ε is the porosity of the porous medium, respectively, k is the effective thermal conductivity of the porous medium, is the heat capacity of the nanofluid and is the effective heat capacity of the nanoparticle material.
The modified permeability of the porous medium K of the non-Newtonian power law fluid is defined as
and
For , . Equation (1) is satisfied by introducing a stream function such that
(7)
where , is the dimensionless stream function, and is the local Rayleigh number given by
The velocity components are given by
(8)
The temperature and concentration are presented as
(9)
where and are the dimensionless temperature and dimensionless concentration. On using equations (8) and (9), equations (2), (3), (4) and (6) transform into the following two-point boundary value problem:
(10)
(11)
(12)
subject to the boundary conditions
where the primes denote differentiation with respect to η. The non-dimensional constants in equations (10)-(12) are the buoyancy ratio parameter Nr, the coefficient of thermal dispersion γ, the pore diameter dependent Rayleigh number , the Brownian motion parameter Nb, the thermophoresis parameter Nt and the Lewis number Le. These are defined as
(16)