Nanofluids are prepared by dispersing solid nanoparticles in fluids such as water, oil, or ethylene glycol. These fluids represent an innovative way to increase thermal conductivity and, therefore, heat transfer. Unlike heat transfer in conventional fluids, the exceptionally high thermal conductivity of nanofluids provides for enhanced heat transfer rates, a unique feature of nanofluids. Advances in device miniaturization have necessitated heat transfer systems that are small in size, light mass, and high-performance. Several authors have tried to establish convective transport models for nanofluids. Nanofluid is a two-phase mixture in which the solid phase consists of nano-sized particles. In view of the nanoscale size of the particles, it may be questionable whether the theory of conventional two-phase flow can be applied in describing the flow characteristics of nanofluid. Nanofluids are also solid-liquid composite materials consisting of solid nanoparticles or nanofibers with sizes typically of 1-100 nm suspended in liquid. Nanofluids have attracted great interest recently because of reports of greatly enhanced thermal properties. For example, a small amount (<1% volume fraction) of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is reported to increase the inherently poor thermal conductivity of the liquid by 40 and 150%, respectively, as previously shown in [
1,
2]. Conventional particle-liquid suspensions require high concentrations (>10%) of particles to achieve such enhancement. However, problems of rheology and stability are amplified at high concentrations, precluding the widespread use of conventional slurries as heat transfer fluids. In some cases, the observed enhancement in thermal conductivity of nanofluids is orders of magnitude larger than that predicted by well-established theories. Other perplexing results in this rapidly evolving field include a surprisingly strong temperature dependence of the thermal conductivity [
3] and a three-fold higher critical heat flux compared with the base fluids [
4,
5]. These enhanced thermal properties are not merely of academic interest. If confirmed and found consistent, then they would make nanofluids promising for applications in thermal management. Furthermore, suspensions of metal nanoparticles are also being developed for other purposes, such as medical applications including cancer therapy. The interdisciplinary nature of nanofluid research presents a great opportunity for exploration and discovery at the frontiers of nanotechnology. Porous media heat transfer problems have several engineering applications, such as geothermal energy recovery, crude oil extraction, ground water pollution, thermal energy storage, and flow through filtering media. Cheng and Minkowycz [
6] presented similarity solutions for free convective heat transfer from a vertical plate in a fluid-saturated porous medium. Gorla and Tornabene [
7] and Gorla and Zinolabedini [
8] solved the nonsimilar problem of free convective heat transfer from a vertical plate embedded in a saturated porous medium with an arbitrarily varying surface temperature or heat flux. The problem of combined convection from vertical plates in porous media was studied by Minkowycz et al. [
9], and Ranganathan and Viskanta [
10]. Kumari and Gorla [
11] presented an analysis for the combined convection along a non-isothermal wedge in a porous medium. All these studies were concerned with Newtonian fluid flows. The boundary layer flows in nano fluids have been analyzed recently by Nield and Kuznetsov and Kuznetsov [
12] and Nield and Kuznetsov [
13]. A clear picture about the nanofluid boundary layer flows is still to emerge.
This study has been undertaken to analyze the mixed convection past a vertical wedge embedded in a porous medium saturated by a nanofluid. The effects of Brownian motion and thermophoresis are included for the nanofluid. Numerical solutions of the boundary layer equations are obtained and discussion is provided for several values of the nanofluid parameters governing the problem.
Analysis
We consider the steady, free convection boundary layer flow past a vertical wedge placed in a nano-fluid-saturated porous medium. The co-ordinate system is selected such that x-axis is aligned with slant surface of the wedge. The flow model and coordinate system are shown in Figure .
We consider the two-dimensional problem. We consider at y = 0, the temperature T and the nano-particle fraction ϕ take constant values, TW and ϕW, respectively. The ambient values, as y tends to infinity, of T and ϕ are denoted by T∞ and ϕ∞, respectively. The Oberbeck-Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium are assumed. We consider the porous medium whose porosity is denoted by ε, and permeability by K.
We now make the standard boundary layer approximation based on a scale analysis and write the governing equations.
where, ρf, μ, and β are the density, viscosity, and volumetric volume expansion coefficient of the fluid, while ρp is the density of the particles. The gravitational acceleration is denoted by g. We have introduced the effective heat capacity (ρc)m and effective thermal conductivity, km, of the porous medium. The coefficients that appear in Equations 3 and 4 are, respectively, the Brownian diffusion coefficient, DB, and the thermophoretic diffusion coefficient, DT.
The boundary conditions are taken to be
We introduce a stream line function ψ defined by
so that Equation 1 is satisfied identically. We are then left with the following three equations:
Proceeding with the analysis, we introduce the following dimensionless variables:
Where u∞ = cxm and gx = g cos ϕ represents the x-component of the acceleration due to gravity.
Substituting the expressions in Equation 12 into the governing Equations 9-11, we obtain the following transformed equations:
where the parameters are defined as
The transformed boundary conditions are
It is noted that the
ξ parameter here represents the forced flow effect on free convection. The case of
ξ = 0 corresponds to pure free convection, and the limiting case of
ξ = 1 corresponds to pure forced convection. The above system of Equations 13-15 was solved over the region covered by
ξ = 0-1 to provide the other half of the solution for the entire mixed convection regime. Moreover, it may be remarked that the system of Equations 13-15 with the boundary conditions (17) reduces to the equations of combined convection along an isothermal wedge in a porous medium; when (
Nr =
Nb =
Nt = 0), this case has been studied by Kumari and Gorla [
11].
The local friction factor is given by
The heat transfer rate is given by
The heat transfer coefficient is given by
Local Nusselt number is given by
The mass transfer rate is given by
where
hm = mass transfer coefficient,
and Sherwood number is given by
1 Ali Jawad Chamkha,2 and Ahmed Mohamed Rashad3,4
