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1 Engineering Conferences International ECI Digital Archives 5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry Refereed Proceedings Suer Effect of teperature dependent viscosity on natural convective boundary layer flow over a horizontal plate ebedded in a nanofluid saturated porous ediu Shoba Bagai University of Delhi Chandrashekhar Nishad University of Delhi Follow this and additional works at: Part of the Materials Science and Engineering Coons Recoended Citation Shoba Bagai and Chandrashekhar Nishad, "Effect of teperature dependent viscosity on natural convective boundary layer flow over a horizontal plate ebedded in a nanofluid saturated porous ediu" in "5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry", Prof. Kabiz Vafai, University of California, Riverside; Prof. Adrian Bejan, Duke University; Prof. Akira Nakayaa, Shizuoka University; Prof. Oronzio Manca, Seconda Università degli Studi Napoli Eds, ECI Syposiu Series, (204). This Conference Proceeding is brought to you for free and open access by the Refereed Proceedings at ECI Digital Archives. It has been accepted for inclusion in 5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry by an authorized adinistrator of ECI Digital Archives. For ore inforation, please contact franco@bepress.co.

2 Proceedings of the 5th International Conference on Porous Media and its Applications in Science and Engineering ICPM5 June 22-27, 204, Kona, Hawaii EFFECT OF TEMPERATURE DEPENDENT VISCOSITY ON NATURAL CONVECTIVE BOUNDARY LAYER FLOW OVER A HORIZONTAL PLATE EMBEDDED IN A NANO- FLUID SATURATED POROUS MEDIUM Shobha Bagai Cluster Innovation Centre, University of Delhi, New Delhi, India-0007 Chandrashekhar Nishad Departent of Matheatics, University of Delhi, New Delhi, India-0007 ABSTRACT The effect of teperature dependent viscosity on natural convective boundary-layer flow of a nano-fluid over an isotheral horizontal plate is investigated nuerically. The viscosity of the fluid is assued to have an exponentially decaying dependence on teperature ν = ν 0 e b(t T ). A siilarity analysis is presented for field equations ebodying conservation of total ass, oentu, theral energy and nano-particles. The analysis shows that velocity, teperature and nano-particle volue fraction profiles in the respective boundary layers depend on the viscosity paraeter γ besides the pertinent paraeters such as buoyancy ratio Nr, Brownian otion Nb, therophoresis Nt and Lewis nuber Le. Fro physical considerations one expects that an increase in the viscosity ust result in lower heat transfer rates. Results in accordance with the physical considerations are obtained in this study. The study also investigates the effect of the presence of an internal heat source in the porous ediu by considering the internal heat generation ter in the energy equation. In the presence of the heat source the heat transfer rate is lower whereas the nanoparticle volue fraction rate is higher than those obtained in the absence of the source ter. INTRODUCTION Materials of nanoeter size have unique cheical and physical properties. As a result nano-fluids finds its applications in industries such as electronics and autootive and also has applications in storage of nuclear waste aterial. Nanotechnology represents the ost relevant technology that is being currently explored to enhance heat transfer. The coining of the ter "nano-fluid" is credited to Choi []. The characteristic feature of nano-fluids is theral conductivity enhanceent, a phenoenon observed by Masuda et al [0]. Buongiorno [2] gave a detailed explanation for the abnoral increase of theral conductivity and viscosity. He also focused on heat transfer enhanceent in convective situation in nano-fluids. Buongiorno proposed a odel incorporating the effects of Brownian diffusion and the therophoresis. Detail explanations on Benard proble (the onset of convection in a horizontal layer unifor heated fro below) for a nano-fluid flow in a porous ediu (the Horton- Rogers-Lapwood proble) are given by Kuznetsov and Nield [6] [8], Nield and Kuznetsov [] [6] and Tzou [7] [8]. Recently Kuznetsov and Nield [9] gave a revised odel on Cheng Minkowycz proble of natural convection over a vertical plate in a porous ediu saturated by nano-fluid. Abu-Nada et al [] presented heat transfer enhanceent in a differentially heated enclosure using variable theral conductivity and variable viscosity of nano-fluid ( Al 2 O - water and CuO - water). Experiental and theoretical study on the effective theral conductivity and viscosity of nanofluids was given by Murshed et al. []. They reported that the theral conductivity and viscosity of nano-fluids increases with the nano-particle volue fraction. Gorla and Chakha [4] gave the siilarity solution for the natural convection past an isotheral horizontal plate in porous ediu saturated with nano-fluids. They gave the nuerical results for friction factor, surface heat transfer rate and ass transfer rate was presented for paraetric variations of the Brownian otion, buoyancy ratio, therophoresis and Lewis nuber on nano-fluid. Uddin et al. [9] extended this work for theral convective boundary condition on the boundary layer flow of a nano-fluid over a upward facing pereable horizontal plate. All the above entioned research work was done without internal heat generation which inspire us to work on this diension. In the present article we study the effect of relevant paraeters such as therophoresis, Brownian otion, buoyancy ratio, Lewis nuber and variable teperature dependent viscosity on free convective heat transfer rates (Nusselt nuber), ass transfer rate (Sherwood nuber) and non-diensional velocity in the presence of internal heat generation for an isotheral horizontal plate ebedded in porous ediu saturated with nano-fluid. Siilarity solutions are obtained for exponentially decaying viscosity of the saturating fluid. NOMENCLATURE c = specific heat at constant pressure D B = Brownian diffusion coefficient D = ass diffusivity of porous ediu D T = therophoretic diffusion coefficient

3 f = diensionless strea function g = acceleration due to gravity K = pereability of porous ediu k = theral conductivity of porous edia Le = Lewis nuber n = power law fluid index Nb = Brownian otion paraeter Nr = buoyancy ratio paraeter Nt = therophoresis paraeter = Nusselt nuber q''' = internal heat generation per unit volue q = heat flux = local Rayleigh nuber Sh = Sherwood nuber T = teperature u, v = velocity coponent in x- and y-direction x, y = Cartesian coordinates along the plate and noral to it respectively Greek Sybols α = effective theral diffusivity β = coefficient of theral expansion γ = viscosity paraeter φ = nanoparticle volue fraction η = siilarity variable µ = dynaic viscosity ν = kineatic viscosity θ = diensionless teperature ρ = density ω = rescale nanoparticle volue fraction ψ = strea function Subscripts f = physical property related to fluid p = physical property related to porous ediu w = wall condition = abient conditions Forulation of the proble We consider the proble of free convection boundary layer flow past a horizontal plate placed in a nano-fluid saturated porous ediu in the presence of internal heat generation. We select a coordinate frae in which the x axis is in the horizontal direction (along the plate) and y axis is noral to the plate. At the surface (y = 0) the teperature T and the nano-particle fraction φ take constant values T w and φ w respectively. At the abient (y ) values of teperature and nano-particle volue fraction denoted by T and φ respectively, with T w > T and φ w > φ. The Oberbeck-Boussinesq approxiation is eployed and the hoogeneity and local theral equilibriu in the porous ediu are assued. We consider a porous ediu whose porosity is denoted by ε and pereability by K. The Darcy velocity is denoted by v r. The following four field equations ebody the conservation of total ass, oentu, theral energy and nani-particles respectively. The field variables are the Darcy velocity v r, the teperature T and the nano-particle volue fraction φ (Khan and Pop [5], Nield and Bejan [2])..v r = 0 () ρ f v ρ ε t = P µ K vρ + (2) fρ p + ( f) ρ f ( β( T T { )} g T ( ρc) t + ( ρc ) f v ρ. T = k 2 T + q'''+ ( ) p D B f. T + D T ε ρc T. T T φ t + ε vr. φ = D B 2 φ + D T T 2 T (4) We consider a steady state flow and assue that the flow is slow so that the advective ter and the Forchheier quadratic ter do not appear in the oentu equation.. In keeping with Oberbeck- Boussinesq approxiation and an assuption that the nano-particle concentration is dilute, the oentu equation (2) can be written as (Kuznetsov, Nield [9] and Gorla, Chakha [4]) 0 = P µ K vr + (5) ( r p r f )f ( f )+ ( f )r f β( T T ) g where dynaic viscosity µ is written as µ = ρ f ν, ν is the kineatic viscosity. Making the standard boundary layer approxiation based on scale analysis, we have the governing equations. u x + v y = 0 (6) P x = ρ ν f K u (7) P y = ( φ )ρ φ βg( T T ) ( ρ p ρ φ )g ( φ φ ) (8) ()

4 u T x + v T y = α 2 T y + 2 q''' ρc f T τ D B y y + ( ) f + D T T 2 T y ε u φ x + v φ y = D 2 φ B y + D T 2 T (0) 2 T y 2 where α = τ = ε ρc ρc (9) k is the theral diffusivity of the fluid and ( ρc) f ( ) p is a paraeter. Eliinating P fro the ( ) f equations [7], [8] we have ν( T) u ( ) y + u ν T y = ( φ )βgk T x + ( ρ p ρ φ )gk φ ρ φ x The appropriate boundary conditions for the proble are () v = 0, T = T w, φ = φ w at y = 0 (2) u 0, T T, φ φ as y () The continuity equation (6) satisfied by introducing the strea function ψ defined as u = ψ ψ, v = ψ x We introduce the following siilarity transforations (4) ψ = α Rα x f ( η) (5) η = y x Ra x (6) T T = T w θ( η) (7) φ φ = φ w w( η) ν( T)= ν 0 e b ( T T ) (8) (9) where T w = T w T, φ w = φ w φ, b is a diensional ( constant and = φ )βgk T w x is the Rayleigh ν 0 a nuber. Using transforations (5) (9) in equations (), (9) and (0), we obtain the following syste of non linear ordinary differential equations f ''( η)= γθ '( η)f ' η ( )+ 2 ( ηeγθ η) θ ' ( η ) Nrω '( η) (20) ( )+ Nt( θ '( η ) 2 (2) θ ''( η)+ f ( η )θ '( η)+ Nbω '( η)θ ' η + e η = 0 ω ''( η)+ Le f ( η )ω '( η)+ Nt Nb θ '' ( η )= 0 2 (22) where q''' = k T w Ra x 2 x e η. The transfored boundary conditions associated with the syste (20) (22) are At η = 0, θ =, ω =, f = 0 (2) As η, θ 0, ω 0, f ' 0 (24) The pries denote differentiations with respect to η. The paraeters Nr (buoyancy ratio), Nb (Brownian otion), Nt (therophoresis), Le (Lewis nuber), γ (diensionless viscosity) are defined as ( Nr = r r p f ) f w, Nβ = ( τ B f f )r f β T w α w Nt = t D T α T DT w, Le = α e D B, γ = bdt w The heat transfer rate at the surface q w and the local Nusselt nuber are defined as. q w = k T y y=0 (25) = q x w = q '( 0)Ra T w k x (26) The ass transfer rate at the surface N w and Sherwood nuber Sh are defined as φ N w = D (27) y y=0 Sh = N x w = w '( 0)Ra φ w x (28) where D is ass diffusivity of the porous ediu. 2 Results and discussion The syste of non-linear ordinary differential equations (20)-(22) with boundary conditions (2)-(24) are solved nuerically for different values of Nr, Nb, Nt, Le

5 and γ using shooting ethod. Table gives the results for the heat transfer rate volue friction rate Sh, and the nano-particle at the leading edge x = 0 for Nb = Nr = Nt = 0.5 in the presence of internal heat generation (IHG) source and without internal heat generation (WIHG). It is observed that in both the cases the heat transfer rate and the nano-particle volue friction rate increases if the viscosity of the base fluid decreases that is as the viscosity paraeter γ increases. Table Effect of γ and Le on and Sh for Nr = Nb = Nt = 0.5 in the presence of internal heat generation (IHG) and without internal heat generation (WIHG). The results of the heat transfer rate and the at the ( ) are copared with the results nano-particle volue fraction rate Sh constant viscosity γ = 0 given by Uddin et al.[9]. Table 2 shows that the obtained results in this study are in good agreeent with the known result for each values of Nr, Nb, Nt in the absence of internal heat generation. The effect of the viscosity paraeter on diensionless velocity f '( η), ( ) and nano-particle volue fraction ( ) for a typical case for Nr = Nb = Nt = 0.5 in the teperature θ η ω ' η presence or absence of internal heat generation, are illustrated in figures l Sh Le IHG WIHG IHG WIHG Table 2 Coparison of results with Uddin et al. [9] in the absence of internal heat generation ter. PRESENT UDDIN ET AL. (202) Sh Sh Nr Nb = 0., Nt = 0., Le = 0, γ = Nt Nb = 0., Nr = 0.5, Le = 0, γ = Nb Nt = 0., Nr = 0.5, Le = 0, γ =

6 Figure : Effect of viscosity paraeter γ on velocity profile for Nr = Nb = Nt = 0.5 and Le = 50 in the presence of internal heat generation Figure 4: Effect of viscosity paraeter γ on teperature profile for Nr = Nb = Nt = 0.5 and Le = 50 in the absence of internal heat generation Figure 2: Effect of viscosity paraeter γ on velocity profile for Nr = Nb = Nt = 0.5 and Le = 50 in the absence of internal heat generation Figure 5: Effect of viscosity paraeter γ on nano particle volue fraction profile for Nr = Nb = Nt = 0.5 and Le = 50 in the presence of internal heat generation Figure : Effect of viscosity paraeter γ on teperature profile for Nr = Nb = Nt = 0.5 and Le = 50 in the presence of internal heat generation Figure 6: Effect of viscosity paraeter γ on nano particle volue fraction profile for Nr = Nb = Nt = 0.5 and Le = 50 in the absence of internal heat generation

7 CONCLUSIONS We have exained the influence of viscosity on natural convection boundary layer flow over a horizontal plate in a porous ediu saturated with nano-fluid. Nuerical results for non-diensional velocity, heat transfer rate and ass transfer rate have been presented for paraetric variations of the non-diensional viscosity paraeter γ, buoyancy ratio Nr, Brownian otion paraeter Nb, therophoresis paraeter Nt and Lewis nuber Le. We have assued teperature and the nano-particle volue fraction are constant along the wall. The heat transfer rate (Nusselt nuber) and ass transfer rate (Sherwood nuber) decreases when Lewis nuber increases. It is also observed that the heat transfer rate and ass transfer rate is ore as the viscosity paraeter increases or in other words the viscosity of the fluid decreases. REFERENCES [] Abu-Nada E., Masoud Z., Oztop H. F., Capo A., Effect of nano-fluid variable properties on natural convection in enclosures.,int. J. Ther. Sci.49, (200). [2] Buongiorno, J.: Convective transport in nano-fluids. ASME J. Heat Transf. 28, (2006). [] Choi, S.U.S.: Enhancing theral conductivity of fluids with nanoparticles. In: Siginer, D.A.,Wang, H.P. (eds.) Developent and Applications of Non-newtonian Flows,Vol. 2, pp , ASME, NewYork (995). [4] Gorla R. S. R., Chakha A., Natural convective boundary layer flow over a horizontal plate ebedded in a porous ediu saturated with a nano-fluid. J. Modern Physics 2, 62-7 (20). [5] Khan, W.A., Pop, I., Free convection boundary layer flow past a horizontal flat plate ebedded in a porous ediu filled with a nano-fluid. ASME J. Heat Trans. (20). [6] Kuznetsov A.V., Nield D.A., Theral instability in a porous ediu layer saturated by a nano-fluid: Brinkan Model. Transp Porous Med 8, (200a). [7] Kuznetsov A.V., Nield D.A., Effect of local theral nonequilibriu on the onset of convection in a porous ediu layer saturated by a nano-fluid. Transp Porous Med 8, (200b). [8] Kuznetsov A.V., Nield D.A., The onset of doublediffusive nano-fluid convection in a layer of a saturated porous ediu. Transp Porous Med 85, (200c). [9] Kuznetsov A.V., Nield D.A., The Cheng-Minkowycz proble for natural convective boundary layer flow in a porous ediu saturated by a nano-fluid: A revised odel. Int. J. Heat and Mass Transfer, 65, (20). [0] Masuda, H., Ebata, A., Teraae, K., Hishinua, N.: Alteration of theral conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, (99). [] Murshed S.M.S., Leong K.C., Yang C. Investigations of theral conductivity and viscosity of nanofuids. Int. J. Theral Sciences 47, (2008). [2] Nield, D.A., Bejan, A.: Convection in Porous Media, rd ed. Springer, New York (2006). [] Nield, D.A., Kuznetsov, A.V., Theral instability in a porous ediu layer saturated by a nano-fluid. Int. J. Heat Mass Transfer 52, (2009). [4] Nield, D.A., Kuznetsov, A.V., The effect of local theral non-equilibriu on the onset of convection in a nano-fluid. Int. J. Ther. Sci. (2009c). [5] Nield, D.A., Kuznetsov, A.V., The onset of convection in a horizontal nano-fluid layer of finite depth. European Journal of Mechanics B/Fluids 29, (200). [6] Nield, D.A., Kuznetsov, A.V., The effect of vertical throughflow on theral instability in a porous ediu layer saturated by a nano-fluid. Transp Porous Med 87, (20). [7] Tzou, D.Y., Instability of nano-fluids in natural convection. ASME J. Heat Transf. 0, (2008a). [8] Tzou, D.Y.: Theral instability of nano-fluids in natural convection. Int. J. Heat Mass Transf. 5, (2008b). [9] Uddin Md. J., Khan W. A., Isail A. I. Md., Free convection boundary layer flow fro a heated upward facing horizontal flat plate ebedded in a porous ediu filled by a nano-fluid with convective boundary condition. Transp Porous Med 92, (202)