Mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid: Natural Convection Dominated Regime
 Rama Subba Reddy Gorla^{1}Email author,
 Ali Jawad Chamkha^{2} and
 Ahmed Mohamed Rashad^{3, 4}
https://doi.org/10.1186/1556276X6207
© Gorla et al; licensee Springer. 2011
Received: 18 October 2010
Accepted: 9 March 2011
Published: 9 March 2011
Abstract
A boundary layer analysis is presented for the mixed convection past a vertical wedge in a porous medium saturated with a nano fluid. The governing partial differential equations are transformed into a set of nonsimilar equations and solved numerically by an efficient, implicit, iterative, finitedifference method. A parametric study illustrating the influence of various physical parameters is performed. Numerical results for the velocity, temperature, and nanoparticles volume fraction profiles, as well as the friction factor, surface heat and mass transfer rates have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.
Introduction
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 highperformance. Several authors have tried to establish convective transport models for nanofluids. Nanofluid is a twophase mixture in which the solid phase consists of nanosized particles. In view of the nanoscale size of the particles, it may be questionable whether the theory of conventional twophase flow can be applied in describing the flow characteristics of nanofluid. Nanofluids are also solidliquid composite materials consisting of solid nanoparticles or nanofibers with sizes typically of 1100 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 particleliquid 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 wellestablished theories. Other perplexing results in this rapidly evolving field include a surprisingly strong temperature dependence of the thermal conductivity [3] and a threefold 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 fluidsaturated 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 nonisothermal 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 twodimensional problem. We consider at y = 0, the temperature T and the nanoparticle fraction ϕ take constant values, T_{W} and ϕ_{W}, respectively. The ambient values, as y tends to infinity, of T and ϕ are denoted by T_{∞} and ϕ_{∞}, respectively. The OberbeckBoussinesq 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.
where, ρ_{f}, μ, and β are the density, viscosity, and volumetric volume expansion coefficient of the fluid, while ρ_{p} 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, k_{m}, of the porous medium. The coefficients that appear in Equations 3 and 4 are, respectively, the Brownian diffusion coefficient, D_{B}, and the thermophoretic diffusion coefficient, D_{T}.
Where u_{∞} = cx^{m} and g_{ x }= g cos ϕ represents the xcomponent of the acceleration due to gravity.
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 1315 was solved over the region covered by ξ = 01 to provide the other half of the solution for the entire mixed convection regime. Moreover, it may be remarked that the system of Equations 1315 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].
Numerical Method and Validation
Equations 1315 represent an initialvalue problem with ξ playing the role of time. This general nonlinear problem cannot be solved in closed form and, therefore, a numerical solution is necessary to describe the physics of the problem. The implicit, tridiagonal finitedifference method similar to that discussed by Blottner [14] has proven to be adequate and sufficiently accurate for the solution of this kind of problems. Therefore, it is adopted in the present study. All the firstorder derivatives with respect to ξ are replaced by twopoint backwarddifference formulae when marching in the positive ξ direction. Then, all the secondorder differential equations in η are discretized using threepoint central difference quotients. This discretization process produces a tridiagonal set of algebraic equations at each line of constant ξ which is readily solved by the wellknown Thomas algorithm (see Blottner [14]). During the solution, iteration is employed to deal with the nonlinearity aspect of the governing differential equations. The problem is solved line by line starting with line ξ = 0 where similarity equations are solved to obtain the initial profiles of velocity, temperature and concentration, and marching forward (or backward) in ξ until the desired line of constant ξ is reached. Variable step sizes in the η direction with Δη_{1} = 0.001 and a growth factor G = 1.035 such that Δη_{ n }= G Δη_{ n }_{1} and constant step sizes in the ξ direction with Δξ = 0.01 are employed. These step sizes are arrived at after many numerical experimentations performed to assess grid independence. The convergence criterion employed in this study is based on the difference between the current and the previous iterations. When this difference reached 10^{5} for all the points in the ηdirections, the solution was assumed to be converged, and the iteration process was terminated.
Results and discussion
In this section, a representative set of graphical results for the dimensionless velocity S '(ξ,η), temperature θ(ξ,η), and nanoparticle volume fraction f(ξ,η) as well as the local skinfriction coefficient C_{f}_{ x }= S"(ξ,0) (reciprocal of local friction factor), reduced local Nusselt number Nu_{ x }= θ"(ξ,0) (reciprocal of rate of heat transfer), and the reduced local Sherwood number Sh_{ x }= f"(ξ,0) (reciprocal of rate of mass transfer) is presented and discussed for various parametric conditions. These conditions are intended for various values of buoyancy ratio, Nr, Lewis number Le, thermophoresis parameter Nt, Brownian motion parameter Nb, wedge angle parameter m, and mixed convection parameter ξ, respectively.
Concluding Remarks
In this article, we presented a boundary layer analysis for the mixed convection past a vertical wedge embedded in a porous medium saturated with a nano fluid. Numerical results for friction factor, surface heat transfer rate, and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le. The results indicate that, as Nr and Nt increase, the friction factor increases, whereas the heat transfer rate (Nusselt number) and mass transfer rate (Sherwood number) decrease. As Nb increases, the friction factor and surface mass transfer rates increase, whereas the surface heat transfer rate decreases. As Le increases, the heat transfer rate decreases, whereas the mass transfer rate increases. As the wedge angle increases, the heat and mass transfer rates increase.
Abbreviations
List of symbols
 D _{B} :

Brownian diffusion coefficient
 D _{T} :

Thermophoretic diffusion coefficient
 f :

Rescaled nanoparticle volume fraction
 g :

Gravitational acceleration vector
 k _{m} :

Effective thermal conductivity of the porous medium
 K :

Permeability of porous medium
 Le :

Lewis number
 Nr :

Buoyancy Ratio
 Nb :

Brownian motion parameter
 Nt :

Thermophoresis parameter
 Nu :

Nusselt number
 P :

Pressure
 q":

Wall heat flux
 Ra _{ x } :

Local Rayleigh number
 r :

Radial coordinate from the center of the wedge
 S :

Dimensionless stream function
 T :

Temperature
 T _{W} :

Wall temperature at vertical wedge
 T _{∞} :

Ambient temperature attained as y tends to infinity
 U :

Reference velocity
 u :

v: Velocity components
 (x:

y): Cartesian coordinates; Greek symbols
 α _{m} :

Thermal diffusivity of porous medium
 β :

Volumetric expansion coefficient of fluid
 ε :

Porosity
 η :

Dimensionless distance
 θ :

Dimensionless temperature
 μ :

Viscosity of fluid
 ρ _{f} :

Fluid density
 ρ _{p} :

Nanoparticle mass density
 (ρc)_{f}:

Heat capacity of the fluid
 (ρc)_{m}:

Effective heat capacity of porous medium
 (ρc)_{p}:

Effective heat capacity of nanoparticle material
 τ:

Parameter defined by equation (13)
 ϕ :

Nanoparticle volume fraction
 ϕ _{W} :

Nanoparticle volume fraction at vertical wedge
 ϕ _{∞} :

Ambient nanoparticle volume fraction attained
 ψ :

Stream function
Declarations
Acknowledgements
The authors are grateful to referees for their excellent comments which helped us to improve the manuscript.
Authors’ Affiliations
References
 Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ: Anomalously increased effective thermal conductivities containing copper nanoparticles. Appl Phys Lett 2001, 78: 718–720. 10.1063/1.1341218View ArticleGoogle Scholar
 Choi SUS, Zhang ZG, Yu W, Lockwood FE, Grulke EA: Anomalous thermal conductivity enhancement on nanotube suspensions. Appl Phys Lett 2001, 79: 2252–2254. 10.1063/1.1408272View ArticleGoogle Scholar
 Patel HE, Das SK, Sundararajan T, Sreekumaran A, George B, Pradeep T: Thermal conductivities of naked and monolayer protected metal nanoparticle based nanofluids: manifestation of anomalous enhancement and chemical effects. Appl Phys Lett 2003, 83: 2931–2933. 10.1063/1.1602578View ArticleGoogle Scholar
 You SM, Kim JH, Kim KH: Effect of nanoparticles on critical heat flux of water in pool boiling heat transfer. Appl Phys Lett 2003, 83: 3374–3376. 10.1063/1.1619206View ArticleGoogle Scholar
 Vassallo P, Kumar R, D'Amico S: Pool boiling heat transfer experiments in silicawater nonofluids. Int J Heat Mass Transf 2004, 47: 407–411. 10.1016/S00179310(03)003612View ArticleGoogle Scholar
 Cheng P, Minkowycz WJ: Free convection about a vertical flat plate embedded in a saturated porous medium with applications to heat transfer from a dike. J Geophys Res 1977, 82: 2040–2044. 10.1029/JB082i014p02040View ArticleGoogle Scholar
 Gorla RSR, Tornabene R: Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium. Transp Porous Media J 1988, 3: 95–106. 10.1007/BF00222688View ArticleGoogle Scholar
 Gorla RSR, Zinolabedini A: Free convection from a vertical plate with nonuniform surface temperature and embedded in a porous medium. Trans ASME J Energy Resour Technol 1987, 109: 26–30. 10.1115/1.3231319View ArticleGoogle Scholar
 Minkowycz WJ, Cheng P, Chang CH: Mixed convection about a nonisothermal cylinder and sphere in a porous medium. Numer Heat Transf 1985, 8: 349–359. 10.1080/01495728508961859View ArticleGoogle Scholar
 Ranganathan P, Viskanta R: Mixed convection boundary layer flow along a vertical surface in a porous medium. Numer Heat Transf 1984, 7: 305–317. 10.1080/01495728408961827View ArticleGoogle Scholar
 Kumari M, Gorla RSR: Combined convection along a nonisothermal wedge in a porous medium. Heat Mass Transf 1997, 32: 393–398. 10.1007/s002310050136View ArticleGoogle Scholar
 Nield DA, Kuznetsov AV: The ChengMinkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 2009, 52: 5792–5795. 10.1016/j.ijheatmasstransfer.2009.07.024View ArticleGoogle Scholar
 Nield DA, Kuznetsov AV: Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transf 52: 5796–5801. 10.1016/j.ijheatmasstransfer.2009.07.023Google Scholar
 Blottner FG: Finitedifference methods of solution of the boundarylayer equations. AIAA J 1970, 8: 193–205. 10.2514/3.5642View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.