Naseem Ahmad 1 and Kamran Ahmad 2

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1 International Journal of Dynamics of Fluids. ISSN Volume 13, Number 1 (2017), pp Research India Publications Radiation Effects on Unsteady Boundary Layer Flow of nanofluids Cu-water and Ag-water over a Stretching Plate and Heat Transfer with Convective Boundary Condition Naseem Ahmad 1 and Kamran Ahmad 2 1 Department of Mathematics, Jamia Millia Islamia New Delhi , India 2 Department of Mathematics, Jamia Millia Islamia New Delhi , India Abstract In this paper, we deal with unsteady laminar boundary layer flow of nano-fluids Cuwater and Ag-water past a stretching plate with convective surface boundary condition in the presence of thermal radiation. The present flow belongs to the category of boundary layer flow of Skiadis type. The closed form solution has been obtained for convective heat transfer under the given conditions. The main thrust of our study is to read the following: a) effect of radiation parameter on the convective heat transfer, b) the effect of volume fraction of nano-sized particles of Cu in Cu-water and Ag in Ag-water nano-fluids, and c) the time dependence of temperature field and Nusselt number. Keywords: Nano-fluids, heat transfer, boundary layer equations, radiation flux and Nusselt number. INTRODUCTION Due to the number of applications in the industrial manufacturing, the problem of boundary layer flow past a stretching sheet has been considered to study during the last few years. The possible applications of boundary layer flow over a stretching sheet are the formation of boundary layer along liquid film in condensation process, hot rolling, wire drawing, the cooling of metallic plate in a cooling bath, glass fibre

2 2 Naseem Ahmad and Kamran Ahmad production and paper production. The task of heat transfer comes to the problem in the process of drawing the artificial fibre from the polymer solution that emerges from the orifice with a speed which increases from almost zero at the orifice up to a plateau at which it remains constant. In this situation, the moving fibre is of technical interest because it is governed by the rate of cooling at which it is cooled according to the quality of yarn. In recent years, for Proceedings of the National Academy of Sciences of United states of America (PNAS) experiment, Dai and his team used the nanotubes that to consume near-infrared light waves which are slightly longer than visible rays of light and pass carefully through our cells of body without any damage. Stanford university researchers found that if a solution of carbon nanotubes is dropped near-infrared laser beam, the solution would heat up to about F i.e C in two minutes. So, we have been encouraged to analyse the model with convective boundary condition with radiation heat transfer. Crane L.J. [1], investigated the boundary layer flow past a stretching sheet whose velocity is proportional to the distance from the slit. Carragher P. [2], reconsidered the problem of Crane to study the heat transfer and calculated the Nusselt number for the entire range of Prandlt number. In due course of time, the problems related to the boundary layer flow over a stretching sheet were extensibly studied in several variants. Recently the nanotechnology came into the picture and hence we consider the nano-fluids as fluid flowing over the stretching plate. There are number of researchers who follow the pioneer classical work of Skiadis [3], such as F.K.Tsou et al. [4] and Crane [1]. Many scholar like A.Naseem [5], N.Ahmad [6], D. Kelly,K.Vijravelu, L. Andrews [7], N. Ahmad and K. Marwah [8], Sidhdheshr, Mahabaleshwar [9], M.Subhas Abel, P.G. Sidheshwar, Mahantesh M. N [10] and N.Ahmad, M. Mishra [11] have solved unsteady flow past stretching sheet in various variants. In 2011, K.Vajravelu et al. [12] studied convective heat transfer in the flow of viscous Ag-water and Cu-water nano-fluids over a stretching surface. A. Noghrehabadi [13], Nageeb Ah Haroun [14], Y. Yigra [15], Kalidas Das [16], investigated new results relating to the flow of nano-fluids over a stretching surface analytically while Kalidas Das [17] did numerical simulation of nano-fluids flow with convective boundary condition N.M.Sarif [18] found numerical solution of flow and heat transfer problems for stretching sheet with Newtonian heating using Keller box technique. In the present paper, we study the unsteady boundary layer flow of Cu-water and Agwater nano-fluids over stretching plate with convective surface boundary condition to see the effect of radiation on the heat transfer. The flow and heat transfer model have been solved for closed form solution. The related physical constants like skin friction and Nusselt number have been calculated. The effect of radiation parameter, nanoparticle volume fraction and time dependence have been studied with the help of different graphs.

3 Radiationeffectsonunsteadyboundarylayerflow of nanofluids Cu-water and Ag- 3 MATHEMATICAL FORMULATION Considering two dimensional unsteady boundary layer flow of nano-fluids over a stretching plate, we assume a coordinate system as x-axis along the stretching sheet and y-axis normal to the surface of the sheet in the positive direction. The following figure shows the geometry of the problem where the continuous stretching surface is bx governed by U(x) = for a < 1, where a and b are constants. 1 at t BOUNDARY LAYER FLOW PROBLEM The governing equations for unsteady boundary layer flow of nano-fluids Cu-water and Ag-water past a stretching plate are: Continuity equation Momentum equation u x + v y = 0 (1) u t u u + u + v x = μ nf 2 u y ρ nf y 2 (2) where u and v are the velocity components along x and y axes, respectively, μ nf and ρ nf are dynamic viscosity and density of nano-fluid, respectively. The appropriate boundary conditions are: u(x, 0) = U(x) = y, u = 0 Introducing dimensionless variables x = x h, y = y h, u = uh ν nf bx 1 at, v = vh ν nf,, v(x, 0) = 0 (3a) t = tυ nf h 2 where h is the characteristics length and employing the methodology considered by N. Ahmad and Ranivs [19], we get the velocity distribution as: (3b)

4 4 Naseem Ahmad and Kamran Ahmad u = b 0 1 a 0 t xe ry and v = b 0 r(1 a 0 t) (1 e ry ) (4) where r = r(t) = a 0+b 0 1 a 0 t, b 0 = bh2 ν nf and a 0 = al2 υ nf. HEAT TRANSFER PROBLEM The energy equation governing the heat transfer is u T T + v = α 2 T x y nf 1 q r y 2 (ρc p ) nf y (5) The relevant boundary conditions for energy equation are: y = 0, k nf T y = h f(t p T ) y, T T (6a) (6b) where k nf is the thermal conductivity of the nanofluid, α nf is the thermal diffusivity of the nanofluid, T p is temperature of the plate, T is ambient fluid temperature, i.e., the temperature of the fluid far away from the plate and h f is heat transfer coefficient Referring Rosseland S. [20] and Siegel R. and Howell. JR [21], the radiative heat flux may be considered as q r = 4σ 3k T 4 y (7) where σ and k are the Stefan-Bltzmann constant and the mean absorption coefficient, respectively. Here we use the approximation as it is being used by Battler [22, 23], Pal [24], Mukhopadhya and layek [25], Ishak [26] and very recently N. Ahmad and Ravins [19], as T 4 4T 3 T 3T 4 (8) Using the equations (7) and (8), we have 1 q r = 1 (ρc p ) nf y (ρc p ) nf y ( 4σ 3K y (4T 3 T 3T 4 )) = 16σ T3 2 T 3K (ρc p ) nf y (9) Using the equation (9) and dimensionless variables in the equation (5), we have u T T + v = 1 (α x y υ nf + nf 16σ T3 ) 2 T 3K (ρc p ) nf y (10)

5 Radiationeffectsonunsteadyboundarylayerflow of nanofluids Cu-water and Ag- 5 where bar ( ) has been suppressed for our convenience. Defining the dimensionless temperature T by θ(η) = T T T p T and η = ry, and using u and v from equation (4) in (10), we get with boundary conditions: θ + (Pr) nfk 0 b 0 r 2 (1 a 0 t) (1 e η )θ = 0 (11) θ (0) = h f k nf 1 a 0t a 0 +b 0 and θ 0, as η (12) where (Pr) nf = ν nf α nf is the Prandlt number of nano-fluid, K 0 = 3N 3N+4 k nf k 4σ T 3, the radiation parameter. with N = A solution of the equation (11) together with boundary conditions (12) is θ(η) = h f 1 a (Pr) nfk 0b0 0t e k nf a 0 +b 0 a0+b0 ( (Pr) nfk 0 b 0 a 0 +b 0 ) (Pr) nf K0b0 a0+b0 x where γ(a, x)= e t t a 1 dt, is incomplete gamma function. 0 The effective density of nano-fluid is given as γ( (Pr) nfk 0 b 0, (Pr) nfk 0 b 0 e η ) (13) a 0 +b 0 a 0 +b 0 ρ nf = (1 φ)ρ f + φρ s (14) where φ is the solid volume fraction of nanoparticles. Thus thermal diffusivity of the nano-fluid becomes α nf = where the heat capacitance of the nano-fluid is taken as k nf (ρc p ) nf (15) (ρc p ) nf = (1 φ)ρc pf + φρc ps (16) Brinkman [27], effective dynamic viscosity of the nano-fluid is given by μ nf = μ f (1 φ) 2.5 (17) We are having k nf, the thermal conductivity of the nano-fluid given by Maxwell [28] as k nf = k f { k s+2k f 2φ(k f k s ) k s +2k f +φ(k f k s ) } (18)

6 6 Naseem Ahmad and Kamran Ahmad Skin Friction The skin friction coefficient is defined as C f = μ nf ρ f U 2 ( u y ) at y=0 where, Re x = Ux υ f is the local Reynolds number. = (Re x ) 1 2 ν (1 φ) 2.5 f (1 + a 0 ) (19) b 0 Table-1 Variation in Skin friction for different volume fraction φ of nano-particles φ 1/2 C f Re x Nusselt number The coefficient of convectional heat transfer is called Nusselt number Nu and it is defined as Variation in Nusselt number for different volume fraction φ of nano- Table-2 particles ( T y ) at y=0 Nu = = h f (20) T p T k nf φ Nu(Cu) Nu(Ag) DISCUSSION AND RESULTS The nano-fluids Cu-water and Ag-water have been considered for unsteady boundary layer flow past a stretching plate and heat transfer with radiation boundary condition. The exact solutions of boundary layer equation for flow field and temperature field have been obtained. Skin friction and Nusselt number have also been derived. The effect of Radiation parameter N and the volume fraction φ of nano-sized particles have been studied on temperature field through graphs. We summarize the results in the following paragraphs:

7 Radiationeffectsonunsteadyboundarylayerflow of nanofluids Cu-water and Ag- 7 Figure-1 Effect of radiation parameter Non temperature field for volume fraction parameter φ = 0 at the time t = 0.5 In figure 1, we study the effect of thermal radiation parameter on temperature field. It is observed that as thermal radiation parameter increases, the temperature field decreases, i. e. the rate of heat transfer increases. Figure-2 Effect of thermal radiation parameter on temperature field for volume fraction φ = 0.1 at t = 0.5 Figure 2 is the graph of θ versus spatial co-ordinate η for various values of thermal radiation parameter for a volume fraction φ = 0.1 of nano-particles of Cu in Cu-water and Ag in Ag-water nano-fluids, respectively. In this case, we notice that as N increases, the temperature field decreases. In this case, the process of heat transfer has been accelerated, in turn temperature field decreases.

8 8 Naseem Ahmad and Kamran Ahmad Figure-3 Effect of volume fraction φ for constant radiation parametern = 1 We observe in figure 3 that as volume fraction φ increases temperature field increases. In this case, radiation parameter is constant, so the heat transfer rate does not increase and the heating of nanoparticles leaves the heating effect on fluid as a whole. Figure-4 Effect of volume fraction φ on temperature field for constant radiation N = 10 In figure 4, a graph of temperature profile θ versus η has been drawn for t=0 and N=10. As t=0 so this case is time independent as volume fraction φ increases, the temperature profile increases. Thus we conclude that the radiation intensity N=10 contributes to increases the temperature profile.

9 Radiationeffectsonunsteadyboundarylayerflow of nanofluids Cu-water and Ag- 9 Figure-5 Effect of time on temperature field when φ = 0 and N = 10 It is common phenomenon that as time passes, the fluid loses its heat. This common phenomenon is well supported by our analysis through the graphs appeared in the figure 5. Figure-6 Effect of time on temperature field for φ = 0.1 and N = 10 In figure 6, due to volume fraction φ = 0.1, the temperature increases upto 16 in case of Cu-water and upto 18 in case of Ag-water. This difference is in maximum temperature due to metallic characteristics. As time passes, the temperature starts falling in both the fluids Cu-water and Ag-water nano-fluids. This is normal phenomena if there is no internal heat generation.

10 10 Naseem Ahmad and Kamran Ahmad We observe that the skin friction C f = (Re x ) 1 2 ν (1 φ) 2.5 f (1 + a 0 ) is independent of time b 0 and radiative heat transfer. This is influenced by volume fraction parameter φ only because as the volume fraction of nanofluids increases of the density fluid increases, in turn skin friction increases. The table-1 supports the increasing trend of C f with the increase in φ. As thermal conductivity k nf is the function of volume fraction φ and as φ increases, k nf increase because of k f < k s. Hence, Nusselt number decreases as φ increases. Our result is supported by table-2. CONCLUSION We conclude the following: 1. A closed form solution has been obtained for radiation effects on unsteady boundary layer flow of nano-fluids Cu-water and Ag-water over a stretching plate and heat transfer with convective boundary condition 2. The radiation N = 1 is well digested by nanofluids to get increased their temperature. 3. As radiation increases, the rate of heat transfer gets started to increases. 4. Skin friction is independent of time and radiation. As φ increases, skin friction increases. 5. Nusselt number decreases as volume fraction φ increases. REFERENCES [1] Crane LJ. 1970, Flow past a stretching plate Z. Angew.Math. Physik ZAMP 21, pp [2] Carragher P. 1978, Boundary layer flow and heat transfer for the stretching plate Ph.D. thesis, University of Dublin, Chapter 4,41. [3] Sakiadis BC. 1961, Boundary-layer behaviour on continuous solid surfaces. Boundary-layer equations for 2-dimensional and axisymmetric flow, AICHE J. 7, pp [4] Tsou F.K, Sparrow E.M, and Goldstein R. Jh. 1967, Flow and heat transfer in the boundary layer on a continuous moving surface. International Journal of Heat and Mass Transfer 10.2,pp [5] Ahmad N. 1996, Stretching flat plate with suction in a non-participating medium with a radiation boundary condition. International journal of Heat and Technology 14.1,pp [6] Ahmad N. 1995, On temperature distribution in a no-participating medium with radiation boundary condition. International Journal of Heat and Transfer 14.1, pp

11 Radiationeffectsonunsteadyboundarylayerflow of nanofluids Cu-water and Ag- 11 [7] Kelly D, Vajravelu, and Andrews L. 1999, Analysis of heat and mass transfer of a viscoelastic, electrically conducting fluid past a continuous stretching sheet. Nonlinear Analysis: Theory, Methods & Applications 36.6, pp [8] Ahmad N and Marwah K. 2000, Visco-elastic boundary layer flow past a stretching plate with suction and heat transfer with variable conductivity. IJEMS, Vol. 7, pp [9] Siddheshwar P. G and Mahabaleswar U. S. 2005, Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet. International Journal of Non-Linear Mechanics 40.6, pp [10] Abel S.M, Siddheshwar P. G., and Mahantesh M. N. 2007, Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source. International Journal of Heat and Mass Transfer 50.5, pp [11] Ahmad N and Mishra M. 2010, Unsteady boundary layer flow and heat transfer over a stretching sheet. Proc. of UIT: 28th UIT Congress on Heat Transfer, Universitadegli di Brescia, Italy, pp [12] Vajravelu K, et al. 2011, Convective heat transfer in the flow of viscous Ag water and Cu water nanofluids over a stretching surface. International Journal of Thermal Sciences 50.5, pp [13] Noghrehabadi A, Salamat P, and M. Ghalambaz., 2015, Integral treatment for forced convection heat and mass transfer of nanofluids over linear stretching sheet. Applied Mathematics and Mechanics 36.3, pp [14] Nageeb AH, Haroun, et al. 2015, Heat and mass transfer of nanofluid through an impulsively vertical stretching surface using the spectral relaxation method. Boundary Value Problems, pp [15] Yirga Y and Shankar B. 2015, MHD Flow and Heat Transfer of Nano-fluids through a Porous Media Due to a Stretching Sheet with Viscous Dissipation and Chemical Reaction Effects. International Journal for Computational Methods in Engineering Science and Mechanics, 16.5, pp [16] Das K. 2015, Nanofluid flow over a non-linear permeable stretching sheet with partial slip. Journal of the Egyptian Mathematical Society [17] Das K, Pinaki R.D and Prabir K.K. 2015, Numerical simulation of nanofluid flow with convective boundary condition. Journal of the Egyptian Mathematical Society 23.2, pp [18] Sarif N. M., Salleh, M. Z, and Nazar R. 2013, Numerical solution of flow and heat transfer over a stretching sheet with Newtonian heating using the Keller box method. Procedia Engineering 53, pp [19] N.Ahmad and Ravins. 2016, Unsteady visco-elastic boundary layer flow past a stretching plate and heat transfer Russian journal of Mathematics Research Series A, Vol(4), pp [20] Rosseland S. 1936, Theoretical astrophysics. Oxford, The Clarendon press.

12 12 Naseem Ahmad and Kamran Ahmad [21] Siegal R and Howell JR. 1936, Thermal radiation, heat transfer, 3 rd edn. Hemisphere, Washington [22] Bataller and Rafael C. 2008, Similarity solutions for boundary layer flow and heat transfer of a FENE-P fluid with thermal radiation. Physics Letters A , pp [23] Bataller and Rafael C. 2008, Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. Journal of materials processing technology 203.1, pp [24] Pal D. 2009, "Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation. Meccanica 44.2, pp [25] Mukhopadhyay S and Layek G.C. 2009, Radiation effect on forced convective flow and heat transfer over a porous plate in a porous medium. Meccanica44.5, pp [26] Anuar I. 2010, Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect. Meccanica 45.3, pp [27] Brinkman H.C "The viscosity of concentrated suspensions and solutions."the Journal of Chemical Physics 20.4,pp [28] Maxwell and James C. 1881, A treatise on electricity and magnetism. Vol.1. Clarendon press.

MHD Flow and Heat Transfer over an. Exponentially Stretching Sheet with Viscous. Dissipation and Radiation Effects

Applied Mathematical Sciences, Vol. 7, 3, no. 4, 67-8 MHD Flow and Heat Transfer over an Exponentially Stretching Sheet with Viscous Dissipation and Radiation Effects R. N. Jat and Gopi Chand Department