ijpam.eu Effects of thermal radiation on an unsteady nanofluid flow past over an vertical plate

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Inter national Journal of Pure and Applied Mathematics Volume 113 No. 9 2017, 38 46 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.

1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Effects of thermal radiation on an unsteady nanofluid flow past over an vertical plate E.Geetha 1, R.Muthucumaraswamy 2, M.Kothandapani 3 1 Department of Mathematics Sri Chandrasekharendra Saraswathi Viswa MahaVidyalaya Enathur, Kanchipuram , India ID: geethamuthu06@gmail.com 2 Department of Applied Mathematics Sri Venkateswara College of Engineering Irungattukottai , India 3 Department of Mathematics University college of Engineering ( A Constituent College of Anna University, Chennai) Arni , India Abstract This paper analyzes the characteristics of heat transfer on the unsteady nanofluid flow past on an vertical plate in the presence of thermal radiation has been considered. The exact analytical solutions are derived for different water-based nanofluids containing Cu, Al 2O 3, and T io 2. Solutions for temperature and velocity are obtained and the effects of various different parameters like Prandtl number, Radiation parameter, Thermal grashof number and time on temperature and velocity of the plate are depicted graphically and discussed in a detailed manner. AMS Subject Classification: 74F05,76R10,76D05,82D80 Keywords: Vertical plate, heat transfer, nanofluid, thermal radiation 1 Introduction Conventional heat transfer fluids, for example oil, water, and ethylene glycol mixtures, are fluids whose heat transfer is poor because of their poor thermal conductivity. Application of these fluids as a cooling tool enhances manufacturing and operating costs. Many attempts have been taken by many researchers to intensify the thermal conductivity of these fluids by suspending nano/micro particles in liquids [3], [9]. Nanofluids are made of ultrafine nanoparticles (100 nm) enervated in a base fluid, which can be water or an organic solvent [10]. Nanofluids are found to exhibit higher conductive, minimum clogging, boiling, and convective heat transfer performances compared to conventional fluids [5], [21], [28]. There are several industrial and engineering applications of nanofluids such as chemical production, solar and power plant cooling, cooling of transformer oil, production of microelectronics, automotive and cooling of air conditioning, advanced nuclear systems, nano-drug delivery, micro fluidics, transportation, biomedicine, solid state lighting and manufacturing [7]. The free convective flow of a nanofluid over a vertical plate under different boundary condition has been investigated by several researchers [1], [2], [4], [13], [15], [17], [18], [19], [22]. Ho et al. [15] studied natural convective flow of a nanofluid under various flow configurations. The recent book by Das et al [11] and more recent review paper by Kakac and Pramuanjaroenkij [16] examined an excellent aggregation of the study done on nanofluids. Oztop and Abu-Nada [23] ijpam.eu

2 analyzed the numerical study of free convection in partially heated rectangular enclosures filled with nanofluids using different types of nanoparticles. Mohammed J. Uddin et al[20] examined hydrodynamic Free Convective Boundary Layer Flow of a Nanofluid over a Flat Vertical Plate with Newtonian Heating Boundary Condition. Khan and Pop [18] have proposed the problem of laminar fluid flow over the stretching surface in a nanofluid and they investigated it numerically. Anjali Devi and Julie Andrews [6] studied the streamline boundary layer flow of nanofluid over a flat plate and it was found out that the suspended nanoparticles enhance the heat transfer capacity of the fluids. The convective flow and heat transfer of an incompressible viscid nanofluid past a semi-infinite vertical stretching sheet in the presence of a magnetic field was examined by Hamad [14]. Tzou [26],[27] presented the thermal instability of nanofluids in natural convection. Recently, Bachok et al. [8] studied the flow in a boundary layer of a nanofluid past a moving surface in a flowing fluid. Polidori et al. [24] studied the natural convection heat transfer of Newtonian nanofluids in streamline boundary layers using the integral approach, for the case of γ Al 2 O 3 water nanofluids whose Newtonian behaviour with particle volume fractions was less than 4 percentage. They showed that natural convection heat transfer is not only defined by the nanofluids effective thermal conductivity but also that the sensitivity to the viscosity model exerts a major role in the heat transfer behaviour. Yacob [29] demonstrated the mixed convection of fluid flow and heat transfer over a stretching vertical surface immersed in a nanofluid. Srikanth et al [25] investigated theoretically the MHD flow of a nanofluid past an inclined permeable plate with constant heat source and thermal radiation. To the authors knowledge, no study has been communicated on a unsteady natural convective flow of a nanofluid with thermal radiation on a infinite vertical plate. In this study, a free convective flow of a nanofluid on an infinite vertical plate with thermal radiation has been considered. The coupled nonlinear partial differential equations are converted into a non dimensional form and the solutions are derived by using the laplace transform. Three types of water based nanofluids containing nanoparticles of copper (Cu), aluminum oxide (Al 2 O 3 ) and titanium dioxide (T io 2 ) have been considered in the present work. The governing equations are solved analytically and presented in closed form.the effects on temperature of the plate and the velocity of the plate by the various parameters like thermal radiation parameter, grashof number, Prandtl number and time are shown graphically and discussed in a detailed manner. 2 Mathematical analysis The viscous flow of a incompressible nanofluid past an uniformly accelerated isothermal vertical infinite plate in the presence of thermal radiation has been considered. Further it is an unsteady flow of a viscous incompressible fluid which is initially at rest and surrounds an infinite vertical plate maintaining temperature T. The x-axis is considered along the plate in the vertically upward direction and the y-axis is assigned normal to the plate. At time t 0, the plate and fluid are at the same temperature T. At time t > 0, the plate is accelerated with a velocity its own plane against gravitational field and the temperature from the plate is raised to T w. It is also assumed that a radiative heat flux q r is applied in the normal direction to the plate. The fluid is a water based nanofluid containing three types nanoparticles Cu, Al 2 O 3 and T io 2. It is further assumed that the base fluid and the suspended nanoparticles of the nanofluids are in thermal equilibrium. The thermophysical properties of the nanofluids are given in Table 1. Table 1 Thermo physical properties of water and nanoparticles [12] ijpam.eu

3 Physical Properties Water/Base fluid Cu Al 2 O 3 T io 2 ρ(kg/m 3 ) c p K(W/mK) βx10 5 (K 1 ) φ σ(s/m) 5.5x x x x10 6 Then under usual Boussinesq s approximation the unsteady flow is governed by the following equations: ρ nf u t (ρc p ) nf T t = g(ρβ) nf (T T ) + µ 2 u nf y 2 (1) = k nf 2 T y qr 2 y (2) where u is the velocity components along the x-direction, T the temperature of the nanofluid,µ nf the dynamic viscosity of the nanofluid, β nf the thermal expansion coefficient of the nanofluid, ρ nf the density of the nanofluid, k nf the thermal conductivity of the nanofluid, g the acceleration due to gravity, q r the radiative heat flux and (ρc p ) nf the heat capacitance of the nanofluid which are given by µ nf = µ f (1 φ) 2.5, ρ nf = (1 φ)ρ f + φρ s (ρc p ) nf = (1 φ)(ρc p ) f + φ(ρc p ) s (ρβ) nf = (1 φ)(ρβ) f + φ(ρβ) s (3) where φ is the solid volume fraction of the nanoparticle, ρ f the density of the base fluid, ρ s the density of the nanoparticle, µ f the viscosity of the base fluid,(ρc p ) f the heat capacitance of the base fluid and (ρc p ) s the heat capacitance of the nanoparticle. It is worth mentioning that the expressions (1) are restricted to spherical nanoparticles, where it does not count for other shapes of nanoparticles. The effective thermal conductivity of the nanofluid given by Hamilton and Crosser model followed by Kakac and Pramuanjaroenkij [18], and Oztop and Abu-Nada [19] is given by k nf = k f [k s+2k f 2φ(k f k s)] [k s+2k f +2φ(k f k s)] (4) where k f is the thermal conductivity of the base fluid and k s the thermal conductivity of the nanoparticle. In Eqs. (1) - (4), the subscripts nf, f and s denote the thermophysical properties of the nanofluid, base fluid and nanoparticles, respectively. The initial and boundary conditions of the proposed problem are given by: u = 0, T = T forally, t 0 t > 0 : u = u3 0 ν f t, T = T w aty = 0 u 0 T T asy (5) On introducing the following non dimensional quantities are: U = u u 0, t = t (u 0) 2 ν f, Y = tu0 ν f, P r = µcp k f ijpam.eu

4 θ = T T T w T, Gr = gν f β(t w T ) (u 0) 3, R = 16a σ(t ) 3 k f ( ν2 f u 2 0 ) (6) The local radiant for the case of an optically thin gray gas is expressed by q r y = 4a σ(t 4 T 4 ) (7) It is assume that the temperature differences within the flow are sufficiently small such thatt 4 may be expressed as a linear function of the temperature. This is accomplished by expanding T 4 in a Taylor series about T and neglecting higher-order terms, thus T 4 = 4T 3 T 3T 4 (8) By using equations (6) and (7), equation (2) reduces to (ρc p ) nf T t = k nf 2 T y a σt 3 (T T ) (9) By using equations (3),(4) and (6), equations (1) and (9) leads to, U L 1 T = L 3 2 U Y + L 2 2 Grθ θ L 5 t = L θ P r Y R 2 P r θ (10) Where L 1 = (1 φ) + φ( ρs ρ f ), L 2 = (1 φ) + φ( (ρβ)s 1 (ρβ) f ), L 3 = (1 φ) 2.5 L 4 = (1 φ) + φ( (ρcp)s (ρc p) f ), L 6 = k f [k s+2k f 2φ(k f k s)] [k s+2k f +2φ(k f k s)] (11) Where R is the radiation parameter, Pr is the Prandtl number, Gr is the thermal grashof number, Gr approximates the ratio of the buoyancy force to the viscous force acting. Large R signifies a large radiation effect while R 0 corresponds to that there is no radiaton effect. The corresponding initial and boundary conditions are, U = 0, θ = 1 forally, t 0 t > 0 : U = gt, θ = 1 aty = 0 U 0 θ 0 asy (12) Where g = L1 L 3 3 Solution Procedure The solutions are in terms of exponential and complementary error function. The relation connecting error function and its complementary error function is as follows: erfc(x) = 1 erf(x) (13) The dimensionless governing equations (10), subject to the initial and boundary conditions (12), are solved by the usual Laplace-transform technique and the solutions are derived as follows: θ = 1 2 [exp(2η abt)erfc(η a + bt) + exp( 2η abt)erfc(η a bt)] (14) U = gt[(1 + 2η 2 g)erfc(η g) 2η g π exp( η 2 g)] + c d erfc(η g) 1 c 2 + cexp(dt) 2d d [exp(2η abt)erfc(η a + bt) + exp( 2η abt)erfc(η a + bt)] [exp(2η gdt)erfc(η g + dt) + exp( 2η gdt)erfc(η g dt)] [exp(2η a(b + d)t)erfc(η a + (b + d)t) + exp( 2η a(b + d)t) erfc(η a (b + d)t)] (15) cexp(dt) 2d ijpam.eu

5 Wherea = L5P r l 6, b = R L 5P r, c = L2Gr L 1 L 3a d = L3ab L 1 L 3a,, g = L1 L 3 4 Results and Discussion In order to get a clear sight on the physical of the problem, extensive numerical computations are conducted for various values of the thermo physical parameters are depicted in graphs. We consider three different types of nanofluids containing aluminium oxide, copper and titanium oxide nanoparticles with water as a base fluid. The values of volume fraction of nanoparticles re considered to be in the range 0 φ 02. The case φ = 0 corresponds to the regular fluid.i.e. nanoscale characteristics are eliminated. 4.1 Effects of parameters on velocity profiles The velocity profiles for different types of nanoparticles (Cu, Al 2 O 3 andt io 2 ) and constant solid volume fraction φ = 0.1, Gr=5, Pr=6.2, R= 0.5 and t=0.2 are taken and graph is plotted which is shown in Fig.1. It explains that the velocity distributions for Al 2 O 3 - water and T io 2 - water are almost the same as their densities are near to each other, but due to high density of Cu, for Cu W ater the dynamic viscosity enhances more and leads to a thinner boundary layer than other particles. The velocity profiles of Cu W ater with coordinate η for different values of,thermal grashof number, nanoparticle solid volume fraction and time are depicted in graph which are shown in Fig. 2, 3, 4 when Pr = 6.2, Gr = 5, φ = 0.1, R = 0.5 and t=0.2 is kept at constant. In Fig.2, it explains that the velocity U increases with an decreasing values of thermal grashof number Gr. Fig.3 shows that the effect of solid volume fraction φ of nanoparticles on the fluid velocity. The fluid velocity U increases for increasing values of φ, results that in the increase of the momentum boundary layer thickness. Fig.4 reveals that the fluid velocity decreases with an increasing value of time. 4.2 Effects of parameters on temperature profiles Fig.5 shows that the fluid temperature variations for the three types of water-based nanofluids Cu - water, Al 2 O 3 - Water and T io 2 - Water when Pr =6.2, t =0.2, R = 0.5 and φ = 0.1. Due to higher thermal conductivity of Cu -Water nanofluids, the temperature of Cu - water nanofluid is found to be higher than Al 2 O 3 - water and T io 2 - water nanofluids. It is also seen that the thermal boundary layer thickness is more for Cu -water thanal 2 O 3 -water and T io 2 -water nanofluids. The temperature profiles for different values of radiation parameter is shown in Fig.6 when Pr =6.2, t =0.2 and φ = 0.1 for Cu - Water. It is found that the temperature of Cu - Water nanofluid decreases with increasing values of R. because decrease in the values of R for given k nf and T means a decrease in the Rosseland radiation absorptivity a*. Since divergence of the radiative heat flux qr y increases, a* decreases which in turn causes to increase the rate of radiative heat transfer to the fluid and hence the fluid temperature increases. Fig.7 depicts the variation of temperature of the fluid for prandtl number Pr when t =0.2, R = 0.5 and φ = 0.1. The temperature profiles exhibit that the fluid temperature decreases as Pr increases. This is due to the fact that a higher Prandtl number fluid has relatively low thermal conductivity, which falling off the conduction and it results temperature decreases. ijpam.eu

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7 5 Conclusion The purpose of this study is to get the exact solution for the unsteady free convective nanofluid flow over a vertical plate in the presence of thermal radiation. Three different types of nanofluids are taken into account and the expression for temperature of the fluid and velocity of the plate has been obtained in closed form by using the laplace transform. The effects of various parameters on velocity and temperature are depicted graphically and the most important remarks can be concluded as follows: An increase in the radiation parameter as well as Prandtl number leads to the decrease in the temperature of the fluid. An increase in the volume solid fraction and time leads to an increase in the velocity of the plate but the trend is reversed in the case of thermal grashof number. 6 References 1. E. Abu-Nada, Effect of nanofluid variable properties on natural convection in enclosures, Int J Therm Sci.,49 (2010), E. Abu-Nada, A. Chamkha, Effect of nanofluid variable properties on natural convection in enclosures filled with CuOEGwater nanofluid, Int J Therm Sci., 49 (2010), E. Abu-Nada, F. Hakan, Oztop, I. Pop, Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection, Superlattices and Microstructures, 51 (2012), S. M. Aminossadati, B. Ghasemi, Natural convection cooling of a localized heat source at the bottom of a nanofluid-filled enclosure, Eur J Mech B/Fluids,28 (2009), A. Akbarinia, M. Abdolzadeh, R. Laur, Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes, Appl Therm Eng., 31 (2011), S.P. Anjali Devi,Julie Andrews, Laminar Boundary Layer Flow of Nanofluid Over A Flat Plate, International Journal of Applied Mathematics and Mechanics, 7 (2011), S. P. Anjali Devi and P. Suriyakumar, Hydromagnetic Mixed Convective Nanofluid Slip Flow past an Inclined Stretching Plate in the Presence of Internal Heat Absorption and Suction, Journal of Applied Fluid Mechanics, 9 (2016), N. Bachok, A. Ishak, I. Pop, Boundary layer flow of nanofluid over a moving surface in a flowing fluid, Int. J. Therm. Sci.,49 (2010), S. U. S. Choi, Enhancing Thermal Conductivity of Fluids with Nanoparticles, Developments and Applications of Non-Newtonian Flows, FED-231/MD, 66 (1995), S.U.S. Choi, Nanofluids: from vision to reality through research, J of Heat Transf.,131 (2009), S. K. Das, S.U.S. Choi, W. Yu and T. Pradeep, Nanofluids: Science and Technology, John Wiley and Sons, Inc.,Hoboken, New Jersey, USA, (2007). 12. S. Das, R.N. Jana, Natural convective magneto-nanofluid flow and radiative heat transfer past a moving vertical plate, Alexandria Engineering Journal, 54 (2015), B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int J Therm Sci 49 (2010), M. A. A. Hamad, Analytical Solution of Natural Convection Flow of a Nanofluid Over a Linearly Stretching Sheet in the Presence of Magnetic Field, International communications in Heat and Mass Transfer, 38 (2011), C.H. Ho, M. W. Chen, Z. W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int J Heat ijpam.eu

8 Mass Transf, 47 (2008), S Kakac, A Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, International Journal of Heat and Mass Transfer, 52 (2009), W.A. Khan, A. Aziz, Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux, Int J Therm Sci, 50 (2011), W. A. Khan, I. Pop, Boundary Layer Flow of a Nanofluid Past a Stretching Sheet, International Journal of Heat and Mass Transfer, 53 (2010), A.V. Kuznetsov, D.A. Nield, Natural convective boundary- layer flow of a nanofluid past a vertical plate, Int J Therm Sci, 49 (2010), Mohammed J. Uddin, Waqar A. Khan, Ahmed I. Ismail, MHD Free Convective Boundary Layer Flow of a Nanofluid past a Flat Vertical Plate with Newtonian Heating Boundary Condition, Plos one, 7 (2012), e S. M. S. Murshed, et al, A review of boiling and convective heat transfer with nanofluids, Renewable and Sustainable Energy Reviews, 15 (2011), E. B. Ogut, Natural convection of water-based nanofluid in an inclined enclosure with a heat source, Int J Therm Sci.48 (2009), H. F. Oztop and E. Abual Nada, Numerical Study of Natural Convection in Partially Heated Rectangular Enclosures Filled With Nanofluids, International Journal of Heat and Fluid Flow,29 (2008), G. Polidori, S. Fohanno, C.T. Nguyen, A note on heat transfer modelling of Newtonian nanofluids in laminar free convection, Int. J. Therm. Sci., 31 (2007) G. V. P. N. Srikanth, G. Srinivas, B. R. K. Reddy, MHD Convective Heat Transfer of a Nanofluid Flow Past an Inclined Permeable Plate With Heat Source and Radiation, International Journal of Physics and Mathematical Sciences, 3 (2013), D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Transfer, 51 (2008), D.Y. Tzou, Instability of nanofluids in natural convection, ASME J. Heat Transfer, 130 (2008), W. Yu, D. M. France, J. L. Routbort, S. U. S. Choi, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transfer Engineering,29 (2008), N. A. Yacob, A. Ishak, R. Nazar, I. Pop, Mixed Convection Flow Adjacent to a Stretching Vertical Sheet in a Nanofluid, Journal of Applied Mathematics, (2013), Article ID ijpam.eu

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Effects of Suction and Internal Heat Generation on Hydromagnetic Mixed Convective Nanofluid Flow over an Inclined Stretching Plate

Available online www.ejaet.com European Journal of Advances in Engineering and Technology, 5, (3): 5-58 Research Article ISSN: 394-658X Effects of Suction and Internal Heat Generation on Hydromagnetic