NATURAL CONVECTION BOUNDARY LAYER FLOW ALONG A SPHERE EMBEDDED IN A POROUS MEDIUM FILLED WITH A NANOFLUID

Transcription

1 Latin American Applied Research 44:49-57 (04) NATURAL CONVECTION BOUNDARY LAYER FLOW ALONG A SPHERE EMBEDDED IN A POROUS MEDIUM FILLED WITH A NANOFLUID A.M. RASHAD Department of Mathematics, Facult of Science, Asan Universit, 858, Egpt. am_rashad@ahoo.com Abstract A boundar-laer analsis is presented for the natural convec tion boundar laer flo about a sphere embedded in a porous medium filled ith a nanofluid using Brinkman-Forchheimer- Darc etended model. The model use d for the nanofluid inc orporates the ef fects of Bronian motion and thermophoresis. The governing partial differential equa tions are transformed into a set of nonsimilar equations and solved numericall b an efficient implicit, iterative, finite-difference method. Comparisons ith previousl published ork are performed and ecellent agreement is obtained. A parametric stud of the phsical parameters is conducted and a representative set of numerical results for the veloci t, temperature, and nanoparticles volume fraction profiles as ell as the local skin-friction coefficient, l ocal Nus selt and Sherood numbers is illustrated graphicall to sho interesting features of the solutions. Keords Natural convection; porous medium; sphere; nanofluid; thermophoresis. I. INTRODUCTION Convective heat transfer from fied or rotating bodies embedded in porous media has been etensivel studied b man investigators oing to its man applications in engineering, such as post accidental heat removal in nuclear reactors, solar collectors, dring processes, heat echangers, geothermal energ recover, and crude oil etraction, ground ater pollution, thermal energ storage, building construction, flo through filtering media, modeling of packed sphere beds, ground ater pollution and filtration processes, to name just a fe of these applications. Hoever, representative studies in this area ma be found in the monographs b Vafai (000), Pop and Ingham (00), Ingham and Pop (00) and Nield and Bejan (006) have summarized the importance of both boundar and inertia effects in porous media. Yih (000) eamined the non-darc MHD natural convection flo over a permeable sphere in a porous medium. EL-Hakiem and Rashad (007) investigated the non- Darc natural convection flo over a vertical clinder saturated porous medium. EL-Kabeir et al. (007) studied the natural convection from a sphere embedded in a variable porosit porous medium. EL-Kabeir et al. (008) have studied the effect of thermal radiation on the combined heat and mass transfer on MHD non- Darc free convection about a horizontal clinder embedded in a saturated porous medium. Nanofluids are prepared b dispersing solid nanoparticles in fluids such as ater, oil, or ethlene glcol. These fluids represent an innovative a to increase thermal conductivit and, therefore, heat transfer. Unlike heat transfer in conventional fluids, the eceptionall high thermal conductivit of nanofluids provides for eceptional heat transfer, a unique feature of nanofluids. Advances in device miniaturization have necessitated heat transfer sstems that are small in size, light mass, and high-performance. Several authors have tried to establish convective transport models for nanofluids. An innovative technique for improving heat transfer b using ultra fine solid particles in the fluids has been used etensivel during the last several ears. The term nanofluid refers to these kinds of fluids b suspending nano-scale particles in the base fluid and has been introduced b Choi (995). Use of metallic nanoparticles ith high thermal conductivit ill increase the effective thermal conductivit of these tpes of fluid remarkabl. Buongiorno (006) found that the nanoparticle absolute velocit can be ritten as the sum of the base fluid velocit and the slip velocit. Nield and Kuznetsov (009) have studied the classical problem of free convection boundar laer flo of a Netonian fluid past a vertical flat plate in a porous medium saturated b a nanofluid. Chamkha et al. (0a) have studied the natural convection past a sphere embedded in a porous medium saturated b a nanofluid. A similarit analsis for the problem of a stead boundar-laer flo of a nanofluid on an isothermal stretching circular clindrical surface is investigated b Gorla et al. (0a). Gorla et al. (0b) have also analzed mied convective boundar laer flo over a vertical edge embedded in a porous medium saturated ith a nanofluid. Chamkha et al. (0b) analzed the effect of melting on unstead hdromagnetic flo of a nanofluid past a stretching sheet. The problem of natural convection boundar laer of a non-netonian fluid about vertical cone embedded in a porous medium saturated ith nanofluid has been performed b Rashad et al. (0). Khan and Aziz (0) have analzed double-diffusive natural convective boundar laer flo over a vertical plate embedded in a porous medium saturated ith a nanofluid. Chamkha et al. (0a, 03a) presented the effect of radiation on mied convection over a edge and cone embedded in a porous medium filled ith a nanofluid. Cheng (0) considered free convection boundar laer flo over a horizontal clinder of elliptic cross section in porous media saturated b a nanofluid. Uddin et al. (0) have considered a numerical solution for free convection boundar laer flo from a heated upard facing horizontal flat plate embedded in 49

2 Latin American Applied Research 44:49-57 (04) a porous medium filled b a nanofluid. Chamkha and Rashad (0) studied the natural convection from a vertical permeable cone in nanofluid saturated porous media for uniform heat and nanoparticles volume fraction flues. Chamkha et al. (0b) have investigated the effect of radiation on boundar-laer flo of a nanofluid on a continuousl moving or fied permeable surface. Chamkha et al. (03b) have analzed the transient natural convection flo of a nanofluid over a vertical clinder. The problem of natural convection boundar-laer flo adjacent to a vertical clinder embedded in a thermall stratified nanofluid saturated non- Darc porous medium is investigated b Rashad et al. (03). The present ork has been undertaken in order to analze the natural convection boundar laer flo about a sphere embedded in a porous medium filled ith a nanofluid using Brinkman-Forchheimer-Darc etended model. The effects of Bronian motion and thermophoresis are included for the nanofluid. Numerical solutions of the boundar laer equations are obtained and discussion is provided for several values of the nanofluid parameters governing the problem. II. MATHEMATICAL FORMULATION Consider stead natural convection boundar-laer flo from along a sphere embedded in non-darc porous medium filled ith a nanofluid. The model used for the nanofluid incorporates the effects of Bronian motion and thermophoresis. The sphere surface is maintained at a constant temperature T and a constant nano-particle volume fraction C and the ambient temperature and nano-particle volume fraction far aa from the surface of the sphere T and C are assumed to be uniform. For T >T and C >C an upard flo is induced as a result of the thermal and nano-particle volume fraction buoanc effects. The schematics of the problem under consideration and the coordinate sstem are shon in Fig.. The porous medium is assumed to be uniform, isotropic and in local thermal equilibrium ith the fluid. All fluid properties are assumed to be constant. Under the Boussinesq and boundar-laer approimations, the governing equations for this problem can be ritten as: The fluid is assumed to Netonian, incompressible and viscous. All fluid properties are assumed to be constant ecept the densit variation in the buoanc force term of the -momentum equation. Upon treating the fluidsaturated porous medium as a continuum (see, Vafai and Tien, 98), including the non-darcian boundar and inertia effects, and assuming that the Boussinesq approimation is valid, the boundar-laer form of mass, momentum, temperature and nanoparticles volume fraction can be ritten as (see, Huang and Chen, 987; and Nazar and Amin 00) ( RU ) ( RV ) + = 0, () X Y U U X ρf U + V = g sin X Y a g a o T,C Fig.. Flo model and coordinate sstem R Porous Medium Y T,C (( C) ρf β( T T) ( ρp ρf )( C C) ) U μ + μ U K U Y K U + V = α X Y Y T T T *, () C T DT T + τ D B +, (3) Y Y T Y C C C D T T U + V = DB +. (4) ϕ X Y Y T Y The boundar conditions for this problem are defined as follos: Y = 0 : U = 0 V = 0 T = T C = C, Y : U = 0 T = T C = C, (5) here R(i)=a sin(x/a) is the radial distance from smmetric ais to surface of the sphere, a is the radius of the sphere, U and V are the velocit components along X and Y aes, respectivel, T is the temperature, C is the nanoparticle concentration, g is the acceleration due to gravit, K is the permeabilit of porous medium, K * is the empirical constant associated ith the Forchheimer porous inertia term, φ is the porosit, α= k/(ρc) f is the thermal diffusivit of the fluid and τ=(ρc) p /(ρc) f. Further, ρf is the densit of the base fluid and ρ, μ., k and β are the densit, viscosit, thermal conductivit and volumetric thermal epansion coefficient of the nanofluid, hile ρ p is the densit of the nanoparticles, (ρc) f is the heat capacit of the fluid and (ρc) p is the effective heat capacit of the nanoparticle material. The coefficients that appear in Eqs. (3)-(4) are the Bronian diffusion coefficient D B, the thermophoretic diffusion coefficient D T. For, details of the derivation of Eqs. ()-(5), one can refer the papers b Buongiorno (006) and Nield and Kuznetsov (009). The above equations are further non-dimensionalised using the ne variables: 50

3 A. M. RASHAD X Y / 4 a a =, = Gr, u = U, v = V, / / 4 a a υgr υgr 3 ( C) ρ gβ( T ) f T a T T Gr =, θ =, μ T T C C φ = C C, r = asin( / a), (6) here v= μ./ρf is the kinematic viscosit coefficient and Gr is the Grashof number, θ and φ are the nondimensional functions of temperature and nanoparticle concentration, respectivel. Substituting the variables (6) into Eqs. ()-(4) lead to the folloing nondimensional equations: ( ru ) ( rv ) + = 0, (7) u u u sin u + v = + ( θ Nrφ) u Γu Da θ θ θ φ θ θ + = + + Pr, (8) u v Nb Nt, (9) Nt u φ v φ φ + = + θ. (0) Le Le Nb The boundar conditions for this problem are defined as follos: = 0 : u = 0 v = 0 θ = φ =, : u = 0 θ = 0 φ = 0, () here Da and Γ are the Darc and Forchheimer numbers, respectivel (reflect the permeabilit of porous medium and inertia effects), Nr is the buoanc ratio parameter, Nb is the Bronian motion parameter, Nt is the thermophoresis parameter, Pr is the Prandtl number and Le is the Leis number, hich are defined respectivel as; / KGr ( ρ * p ρf )( C C) Da =, Γ= K a, Nr =, a ρf β( T T)( C) ϕρ ( c) pdb( C C ) Nb =, ( ρc ) f υ ϕρ ( c) pdt ( T T ) υ υ Nt =, Pr =, Le =. () ( ρc) f υt α ϕdb To solve Eqs. (7)-(9), subject to the boundar conditions (0), e assume the folloing transformations: ψ = r( f ) (, ), θ = θ (, ), φ = φ(, ), (3) here ψ is the stream function defined in the usual a as u = ( / r) ψ / and v = ( / r) ψ /, therefore the continuit equation is identicall satisfied. In addition the velocities components are Substituting Eqs. (3) into Eqs. (7)-() the transformed equations take the folloing form: sin f + + ff + ( θ Nrφ) f tan Da f f ( + Γ )( f ) = f f, (4) θ Nbφθ f θ + Ntθ = Pr tan θ f f θ, (5) φ Nt + Le + f tan + Nb = φ f Le f φ. (6) along ith the boundar conditions; = 0 : f = 0 f = 0 θ = φ = : f = 0 θ = 0 φ=0, (7) here the primes indicate partial differentiation ith respect to. It can be seen that near the loer stagnation point of the sphere, i.e., hen 0, Eqs. (4)-(6) reduce to the folloing ordinar differential equations: f + ff + ( θ Nrφ) f ( f ) = 0, Da (8) θ + Nbφθ + f θ + Ntθ = 0, Pr (9) Nt φ + Lef φ + θ = 0 Nb (0) subject to the boundar conditions = 0 : f = 0 f = 0 θ = φ =, : f = 0 θ = 0 φ=0. () In practical applications, the phsical quantities of principal interest are the shearing stress and the rate of heat transfer in terms of the skin-friction coefficient C f, Nusselt number Nu and Sherood number respectivel, hich can be ritten as; 3l 4 / 4 Gr a agr q C f = τ, Nu = υ k T T, here U τ = Y Y = 0 agr Sh = D C, q / 4 ( C ) T = m Y Y = 0, m ( ), () C = Y Y = 0. (3) Using the non-dimensional variables (6), and (3) and the boundar conditions (7) into Eqs. () and (3), one can obtain the folloing: Cf ( ) = f (,0), Nu( ) = θ (,0), Sh( ) = φ (,0). (4) III. NUMERICAL METHOD The non-similar Eqs. (4)-(6) are linearized and then descritized using three points central difference quotients ith variable step sizes in the direction and using to-point backard difference formulas in the direction ith a constant step size. The resulting equations form a tri-diagonal sstem of algebraic equations that can be solved b the ell-knon Thomas algorithm (see Blottmer, 977). The solution process starts at =0, here Eqs. (8)-(0) are solved, and then marches for- 5

4 Latin American Applied Research 44:49-57 (04) ard using the solution at the previous line of constant until it reaches the desired value of (=0 o in this case). Due to the nonlinearities of the equations, an iterative solution ith successive over- or underrelaation techniques is required. The convergence criterion required that the maimum absolute error beteen to successive iterations be 0-6. The computational domain as made of 96 grids in the direction and,000 grids in the direction. A starting step size of 0 in the direction ith an increase of 375 times the previous step size and a constant step size in the direction of 0 ere found to give ver accurate results. The maimum value of, hich represented the ambient conditions, as assumed to be 35. The step sizes emploed ere arrived at after performing numerical eperimentations to assess grid independence and ensure accurac of the results. The accurac of the aforementioned numerical method as validated b direct comparison ith the numerical results reported earlier b Huang and Chen (987) and Nazar and Amin (00) for various values of, at Nr = Nb = Nt = 0 and Γ=Da - =0 (in the absence of the nanofluid parameters, permeabilit of porous medium and inertia effects) at Pr=. Table presents the results of this comparison. It can be seen from this table that ecellent agreement beteen the results eists. This favorable comparison lends confidence in the numerical results to be reported in the net section. IV. RESULTS AND DISCUSSION In this section, a representative set of graphical results for the dimensionless velocit f (,), temperature θ(,), and nanoparticles volume fraction φ(,) profiles, as ell as the local skin-friction coefficient C f (), local Nusselt number Nu() (reciprocal of rate of heat transfer), and the local Sherood number Sh() (reciprocal of rate of mass transfer)is presented and discussed for various values namel; Darc number Da, Forchheimer number Γ, nanofluid buoanc ratio Nr, Bronian motion parameter Nb, thermophoresis parameter Nt, Leis number Le and the circumferential position parameter. Figures -4 illustrate the effects of the Darc number Da and Leis number Le on velocit f (,), temperature θ(,), and nanoparticles volume fraction φ(,) profiles, respectivel. Phsicall, the presence of a po- Table. Comparison of values of Nu at Pr= ith Huang and Chen (987) and Nazar and Amin (00) for Da - =, Γ=, Nr=0, Nb= and Nt=. Huang and Nazar and Amin Chen (987) (00) Present results 0 o o o o o o o o o o f'(,) Le= Pr= =90 o Fig. : Effects of Da and Le on the velocit profiles θ (,) Le= 0 0. Pr= =90 o Fig. 3: Effects of Da and Le on the temperature profiles φ (,) Le= 0 0. Pr= =90 o Fig. 4: Effects of Da and Le on the volume fraction profiles rous medium in the flo presents resistance to flo. Thus, the resulting resistive force tends to retard the motion of the fluid along the sphere surface and causes increases in its temperature and volume fraction. This is shon in Figs. -4, b the increases in the velocit profiles and decreases in the values of both the temperature and concentration as the Darc number Da increases. On other hand, it is noticed that an increase in the values of Le results in acceleration of the flo represented b increases in the velocit profiles because of the reduction in the nanoparticle buoanc effect. Moreover, as Le increases both the nanofluid temperature and volume fraction profiles and its thermal and nanoparticle concentration boundar laer thickness decrease. 5

5 A. M. RASHAD C f () Le= 0,,,,Pr= 0 o 0 o 40 o 60 o 80 o 00 o 0 o decrease at ever circumferential position ξ. This behavior results in increases in both the local Nusselt and Sherood numbers according to Eqs. (8) and (9). Also, as Da increases, the all slope of the velocit profile f (ξ,0) increases and, based on Eq. (7), its value increases. Hoever, both C f () and Sh() are enhanced, hereas Nu() is reduced as the Leis number Le increases. This is associated ith the decreases in the nanoparticle concentration boundar laer as Le increases as discussed earlier. Thus, the influence of the Leis number is to reduce the heat transfer rate and enhance either of skin friction and nanoparticle mass transfer rate. Fig. 5: Effects of Da and Le on the local skin-friction coefficient Γ=,5.0,0 Nr= Le= Nu() f'(,) 0.0 Da= 5 Pr= =90 o ,,,,Pr= 0 o 0 o 40 o 60 o 80 o 00 o 0 o Fig. 8: Effects of Γ and Nr on the velocit profiles Fig. 6: Effects of Da and Le on the local Nusselt number Nr= ,,,,Pr= Le= 0 Γ=,5.0,0 Sh() θ (,) Da= 0. Pr= =90 o o 0 o 40 o 60 o 80 o 00 o 0 o Fig. 9: Effects of Γ and Nr on the temperature profiles Fig. 7: Effects of Da and Le on the local Sherood number Figures 5-7 displa the variations of the local skinfriction coefficient C f () (reciprocal of shear stress), local Nusselt number Nu() (reciprocal of rate of heat transfer), and the local Sherood number Sh() (reciprocal of rate of mass transfer) due to changes in the values of Da and Le, respectivel. In general, as the circumferential position parameter increases, the skinfriction coefficient rises considerabl hereas the opposite behavior happens ith the local Nusselt and Sherood numbers. Also, from the above observation, as seen from Figs. 3 and 4, for a given value of the Leis number Le, increasing the Darc number Da causes the temperature and volume fraction at the sphere surface to φ (,) Γ=,5.0,0 Nr= Da= 0. Pr= =90 o Fig. 0: Effects of Γ and Nr on the volume fraction profiles 53

6 Latin American Applied Research 44:49-57 (04) C f () Nr= Γ=,5.0,0 Da=,,,Pr=, 0 o 0 o 40 o 60 o 80 o 00 o 0 o Fig. : Effects of Γ and Nr on the local the skin-friction coefficient Nu() Da=,,,Pr=, o 0 o 40 o 60 o 80 o 00 o 0 o Γ=,5.0,0 Nr= Fig. : Effects of Γ and Nr on the local Nusselt number Sh() Γ=,5.0,0 Da=,,,Pr=, Nr= 0 o 0 o 40 o 60 o 80 o 00 o 0 o Fig. 3: Effects of Γ and Nr on the local Sherood number Figures 8-0 sho the effects of Forchheimer number Γ and nanoparticle buoanc ratio Nr on the velocit f (,), temperature θ(,) and nanoparticles volume fraction φ(,) profiles, respectivel. In these figures, it is predicted that Forchheimer number Γ is found to characterize the fluid inertia effect hich presents an obstacle to flo causing the flo to move sloer and both the nanofluid temperature and volume fraction to increase. Hence, an increase in Γ, causes an increase in the thermal and nanoparticle concentration boundar laer thickness and a decrease in the momentum boundar laer thickness. Hoever, It is shon that the velocit profiles decreases near the all and shoing the opposite trend far aa from the all ith the increase of nanoparticle buoanc ratio Nr. On contrast, both the temperature and nanoparticle volume fraction profiles increase ith increasing values of Nr. These increases in both the temperature and volume fraction values are folloed b corresponding slight increases in both thermal and nanoparticle concentration boundar laer thickness. These behaviors are clearl shon in Figs In Figs. -3, the effects of Γ and Nr on the local skin-friction coefficient C f (), local Nusselt number Nu(), and local Sherood number Sh(), respectivel. For a given buoanc ratio Nr, increasing the Forchheimer number Γ is found to have significant effect on both the skin-friction coefficient and local Nusselt number. As Γ increases, the fluid motion in the boundar laer is decelerated causing enhancement in the momentum boundar laer and hence, in the thermal and nanoparticle concentration boundar laer thickness. Consequentl, the velocit gradient and hence, the skinfriction coefficient decrease ith increasing values of Γ. Consequentl, all the local skin-friction coefficient, local Nusselt and Sherood numbers are reduced ith increasing values of Γ. In addition, a similar behavior is noticed here all of these phsical parameters increase as buoanc ratio Nr. This is due to the fact that the increase of buoanc ratio Nr has a tendenc to decelerate the flo along the sphere surface hich filled ith nanofluids. This behavior in the flo velocit is accompanied b slight increases in the nanofluid temperature and volume fraction as Nr increases. Figures 4-6 depict the effects of the Bronian motion parameter Nb and thermophoresis parameter Nt on velocit f (,), temperature θ(,), and nanoparticles volume fraction φ(,) profiles, respectivel. It is seen that the increasing the Bronian motion parameter Nb produces increases in either of the velocit or temperature profiles ith significant decrease on the volume fraction profiles, hile increasing the thermophoresis parameter Nt leads to increase the velocit, temperature and volume fraction profiles. Figures 7-9 sho the influences of the Bronian motion parameter Nb and the thermophoresis parameter Nt on the local skin friction coefficient C f (), local Nusselt number Nu() and local Sherood number Sh(), respectivel. In nanofluid sstems, oing to the size of the nanoparticles, Bronian motion takes place, and this can enhance the heat transfer properties. This is due to the fact that the Bronian diffusion promotes heat conduction. The nanoparticles increase the sphere surface area for heat transfer. A nanofluid is a to phase fluid here the nanoparticles move randoml and raise the energ echange rates. Hoever, the Bronian motion decreases nanoparticles diffusion. The enhancement in the local Sherood number as Bronian motion parameter Nb changes is relativel small. Therefore, as noted before that, the effect of the Bronian motion parameter Nb is to increase the velocit and temperature profiles hile its volume fraction decreases. This ields 54

7 A. M. RASHAD Nt= Nt= 0.5 f'(,) 0.0 Da= 5 Nb=0.,0.3, Pr= =90 o C f () Da=,,,Pr=, 0 o 0 o 40 o 60 o 80 o 00 o 0 o Nb=0.,0.3, Fig. 4: Effects of Nb and Nt on the velocit profiles Fig. 7: Effects of Nb and Nt on the local skin-friction coefficient Nt= 5 0 Nb=0.,0.3, Nt= 0.35 θ (,) Da= 0. Pr= Nb=0.,0.3, =90 o Fig. 5: Effects of Nb and Nt on the temperature profiles Nu() Da=,,,Pr=, o 0 o 40 o 60 o 80 o 00 o 0 o Fig. 8: Effects of Nb and Nt on the local Nusselt number Nt=.4.3 Nt=.. φ (,) Nb=0.,0.3, Da= 0. Pr= =90 o Fig. 6: Effects of Nb and Nt on the volume fraction profiles reduction in the local Nusselt number and enhancement in both the skin friction coefficient and local Sherood number. On the other hand, it can be seen that the thermophoresis parameter Nt appears in thermal and nanoparticles concentration boundar laer equations. This is consistence ith the ell knon fact that it is coupled ith the temperature function and plas a strong role in determining the diffusion of heat and nanoparticles concentration in the boundar laer. Thus, an increasing in the values of the thermophoresis parameter Nt has a tendenc to decrease in the local Nusselt number hereas this trend is reversed in both the local skin friction coefficient and local Sherood number. Sh() 0.9 Nb=0.,0.3, Da=,,,Pr=, 0 o 0 o 40 o 60 o 80 o 00 o 0 o Fig. 8: Effects of Nb and Nt on the local Sherood number V. CONCLUSIONS This ork considered the natural convection laminar boundar laer flo along a sphere embedded in a porous medium filled ith a nanofluid using the Brinkman-Forchheimer-Darc etended model ere studied. The model used for the nanofluid incorporates the effects of Bronian motion and thermophoresis. The governing equations ere formulated and transformed into a set of non-similar equations hich are solved numericall b an accurate, implicit, iterative, finite-difference method. Comparisons ith previousl published ork ere performed and ecellent agreement as obtained. Calculations ere carried out for a ide range of values 55

8 Latin American Applied Research 44:49-57 (04) of the pertinent parameters to eamine the results from this method. Graphical results for the velocit, temperature and nanoparticles volume fraction profiles as ell as the local skin-friction coefficient and local Nusselt and Sherood numbers are presented and discussed for various parametric conditions. It as found that both the local Nusselt and Sherood numbers raised due to increases in Darc number (reflect the permeabilit of porous medium). Hoever, the both reduced due to increases in either the Forchheimer number (reflect inertia effects), buoanc ratio or the circumferential position. Also, increases in the values of either of Bronian motion parameter, thermophoresis parameter or Leis number produced decreases in the local Nusselt number and increases in the local Sherood number. Finall, the skin-friction coefficient as enhanced as either of Darc number, Leis number, Bronian motion parameter, thermophoresis parameter or the circumferential position increased, hile, the opposite behavior as predicted as the buoanc ratio or Forchheimer number as increased. It is hoped that the present ork ill serve as a motivation for future eperimental ork hich seems to be lacking at the present time. ACKNOWLEDGEMENTS The author is ver thankful to the revieers for their constructive comments hich have improved significantl the qualit of the paper. REFERENCES Blottner, F.G., Finite-difference methods of solution of the boundar-laer equations, AIAA Journal, 8, (970). Buongiorno, J., Convective transport in nanofluids, ASME J. Heat Transfer, 8, (006). Chamkha, A.J. and A.M. Rashad, Natural convection from a vertical permeable cone in nanofluid saturated porous media for uniform heat and nanoparticles volume fraction flues, Int. J. Numerical Methods for Heat and Fluid Flo,, (0). Chamkha, A.J., R.S.R. Gorla and K. Ghodesar, Nonsimilar solution for natural convective boundar laer flo over a sphere embedded in a porous medium saturated ith a nanofluid, Transport in Porous Media, 86, 3- (0a). Chamkha, A.J., A.M. Rashad and E. Al-Meshaiei, Melting effect on unstead hdromagnetic flo of a nanofluid past a stretching sheet, Int. J. Chemical Reactor Engineering, 9, -3 (0b). Chamkha, A.J., S. Abbasband, A.M. Rashad and K. Vajravelu, Radiation effects on mied convection over a edge embedded in a porous medium filled ith a nanofluid, Transport in Porous Medium, 9, 6-79 (0a). Chamkha, A.J., M. Modather, S.M.M. EL-Kabeir and A.M. Rashad, Radiative effects on boundar-laer flo of a nanofluid on a continuousl moving or fied permeable surface, Recent Patents on Mechanical Engineering, 5, (0b). Chamkha, A.J., S. Abbasband, A.M. Rashad and K. Vajravelu, Radiation effects on mied convection about a cone embedded in a porous medium filled ith a nanofluid, Meccanica, 48, (03a). Chamkha, A.J., A.M. Rashad and A.M. Al, Transient natural convection flo of a nanofluid over a vertical clinder, Meccanica, 48, 7-8 (03b). Cheng, C.Y., Free convection boundar laer flo over a horizontal clinder of elliptic cross section in porous media saturated b a nanofluid, Int. Comm. Heat Mass Transfer, 39, (0). Choi, S.U.S., Enhancing thermal conductivit of fluids ith nanoparticle, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of Non- Netonian Flos, ASME FED, 3/MD 66, (995). EL-Hakiem, M.A. and A.M. Rashad, Effect of radiation on non-darc free convection from a vertical clinder embedded in a fluid-saturated porous medium ith a temperature-dependent viscosit, J. Porous Media, 0, 09-8 (007). EL-Kabeir, S.M.M., M.A. EL-Hakiem and A.M. Rashad, Group Method Analsis Of Combined Heat And Mass Transfer B MHD Non-Darc Non-Netonian Natural Convection Adjacent To Horizontal Clinder In A Saturated Porous Medium, Applied Mathematical Modelling, 3, (008). EL-Kabeir, S.M.M., M.A. El-Hakiem and A.M. Rashad, Natural convection from a permeable sphere embedded in a variable porosit porous medium due to thermal dispersion, Nonlinear Analsis: Modelling and Control,, (007). Gorla, R.S.R., A.J. Chamkha and A.M. Rashad, Mied convective boundar laer flo over a vertical edge embedded in a porous medium saturated ith a nanofluid: Natural convection dominated regime, Journal of Nanoscale Research Letters, 6, -9 (0a). Gorla, R.S.R., S.M.M. EL-Kabeir and A.M. Rashad, Heat transfer in the boundar laer on a stretching circular clinder in a nanofluid, Journal of Thermophsics and Heat Transfer, 5, (0b). Huang, M. J. and C.K. Chen, Laminar free convection from a sphere ith bloing and suction, J. Heat Transfer, 09, (987). Ingham, D.B. and Pop, I. (Eds.), Transport Phenomena in Porous Media, Pergamon, Oford (00). Khan, W.A. and A. Aziz, Double-diffusive natural convective boundar laer flo in a porous medium saturated ith a nanofluid over a vertical plate: Prescribed surface heat, solute and nanoparticle flues, Int. J. Therm. Sci., 50, (0). Nazar, R. and N. Amin, Free convection boundar laer on an isothermal sphere in a micropolar flu- 56

9 A. M. RASHAD id, Int. Commun. Heat Mass Transfer, 9, (00). Nield, D.A. and A. Bejan, Convection in Porous Media, third Ed., Springer, Ne York (006). Nield, D.A. and A.V. Kuznetsov, The Cheng- Minkocz problem for natural convective boundar laer flo in a porous medium saturated b a nanofluid, Int. J. Heat Mass Transfer, 5, (009). Pop, I. and D.B. Ingham, Convective Heat Transfer: mathematical and computational modelling of viscous fluids and porous media, Pergamon, Oford (00). Rashad, A.M., M.A. EL-Hakiem and M.M.M. Abdou, Natural convection boundar laer of a non- Netonian fluid about a permeable vertical cone embedded in a porous medium saturated ith a nanofluid, Computers and Mathematics ith Applications, 6, (0). Rashad, A.M., S. Abbasband and A.J. Chamkha, Non-Darc natural convection from a vertical clinder embedded in a thermall stratified and nanofluid-saturated porous media, ASME Journal of Heat Transfer, doi: 0.5/ , In press (03). Uddin, M.J., W.A. Khan and A.I.M. Ishmail, Free convection boundar laer flo from a heated upard facing horizontal flat plate embedded in a porous medium filled b a nanofluid ith convective boundar condition, Transp. Porous Media, 9, (0). Vafai, K. and C.L. Tien, Boundar and inertia effects on flo and heat transfer in porous media, Int. J. Heat Mass Transfer, 4, (98). Vafai, K., (Ed.), Handbook of Porous Media, Marcel Dekker, Marcel (000). Yih, K.A., Viscous and Joule heating effects on non- Darc MHD natural convection flo over a permeable sphere in porous media ith internal heat generation, Int. Commun. Heat Mass Transfer, 7, (000). 57