Nanofluids Slip Mechanisms on Hydromagnetic Flow of Nanofluids over a Nonlinearly Stretching Sheet under Nonlinear Thermal Radiation

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1 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. Nanoluids Slip Mechanisms on Hydromagnetic Flo o Nanoluids over a Nonlinearly Stretching Sheet under Nonlinear Thermal Radiation 1 S.P.Anjali Devi and Meala Selvaraj 1 Former Proessor & Head, Department o Applied Mathematics, Bharathiar University, Coimbatore-46 Tamilnadu, India. Research Scholar,Department o Mathematics, Bharathiar University, Coimbatore Tamilnadu, India. Orcid Id: Abstract In this paper, Heat transer characteristics o to dimensional, steady hydromagnetic boundary layer lo o ater based nanoluids containing metallic nanoparticles such as copper (Cu) and Silver (Ag) over a nonlinearly stretching surace taing into account the eects o nonlinear thermal radiation and viscous dissipation has been investigated numerically. The model used or the nanoluids incorporates the eects o Bronian motion and thermophoresis. The governing nonlinear partial dierential equations ere transormed into nonlinear ordinary dierential equations using similarity transormations and then are solved numerically subject to the transormed boundary conditions by most eicient Nachtsheim- Sigert shooting iteration scheme or satisaction o asymptotic boundary conditions along ith ourth order Runge-Kutta Integration method. Numerical computations are carried out or distributions o velocity, temperature and nanoparticles volume raction by means o graphs or dierent values o physical parameters such as magnetic interaction parameter, nonlinear stretching parameter, Ecert number, temperature ratio parameter, radiation parameter, Prandtl number, Bronian motion parameter, thermophoresis parameter and Leis number. The numerical results o the problem are validated by comparing ith previously published results in the literature. Numerical values o sin riction coeicient and Nusselt number at the all are also obtained and given in tabular orm. Sherood number is vanished due to ne mass lu condition. Key ords: Nanoluid, Stretching Sheet, MHD, Radiation. Nomenclature c stretching coeicient B 0 magnetic induction nanoparticle volume raction ambient nanoparticle volume raction D B Bronian diusion coeicient D T thermophoretic diusion coeicient dimensionless stream unction Ec Ecert number * Rosseland mean absorption coeicient Le Leis number M magnetic ield parameter n nonlinear stretching parameter Nb Bronian motion parameter Nt thermophoresis parameter Nu local Nusselt number Pr Prandtl number q r radiative heat lu Re local Reynolds number T temperature o the nanoluid ithin the boundary layer T temperature at the surace o the sheet T temperature o the ambient nanoluid u velocity along the surace o the sheet v velocity normal to the surace o the sheet (, y) Cartesian coordinates Gree symbols nthermal diusivity o the nanoluid ρ n density o the nanoluid (c p) n heat capacity o the nanoluid μ n viscosity o the nanoluid υ n inematic viscosity o the nanoluid ψ stream unction η similarity variable θ dimensionless temperature θ surace all temperature φ dimensionless rescaled nanoparticle volume raction κ n thermal conductivity o the nanoluid τ nanoparticle heat capacity ratio σ magnetic permeability σ * Stean-Boltzmann constant Subscripts surace conditions conditions ar aay rom the surace Superscripts dierentiation ith respect to η 1440

2 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. INTRODUCTION Ultrahigh perormance cooling is one o the most vital needs o many industrial technologies. Nanoluids hich ehibit ultra high perormance cooling are engineered by suspending nanoparticles ith average size belo 100nm in traditional heat transer luids such as ater, oil and ethylene glycol. Nanoluid is the term coined by Choi (1995) [5] to describe the ne class o nanotechnology based heat transer luids that ehibit thermal properties superior to those o their host luids or conventional particle luid suspensions. A comprehensive study on the nanoluids characteristics is documented by Das et al.(007)[8]. Kauui V. Wong and Omar De Leon (010)[11] presented the ide range o applications o nanoluids in current and uture such as nuclear reactors, transportation, electronics cooling, biomedicine and ood. Ahmad et al. (011)[] presented a numerical study o the Blasious and Saiadis los in nanoluids under isothermal condition. Their results revealed that solid volume raction aects the luid lo and heat transer characteristics o nanoluids. An analytical derivation o eective thermal conductivity o nanoluids hich incorporates the contribution o interacial layer as ell as the Bronian motion as solved by Ritu Pasrija and Sunita Srivastava (013)[]. Sandeep Pal et al.(014)[4] has presented a revie on enhanced thermal conductivity o colloidal suspension o nanosized particles (nanoluids).the recent literature o nanoluids as revieed by Mohameed Saad Kamel et al.(016)[13]. Steady boundary layer lo o incompressible luids over a stretching sheet has considerable bearing on various technological processes. The lo over a stretching plate as irst considered by Crane (1970)[7] ho ound a closed orm analytic solution o the sel-similar equation or steady boundary layer lo o a Netonian luid. MHD as initially non in the ield o astrophysics and geophysics and later becomes very important in engineering and industrial processes. Pavlov (1974)[16] gave an eact similarity solution o the MHD boundary layer equations or the steady to-dimensional lo o an electrically conducting luid due to the stretching o a plane elastic surace in the presence o a uniorm transverse magnetic ield. Anjali Devi and Thiyagarajan (006)[9] solved the problem o steady nonlinear MHD lo o an incompressible, viscous and electrically conducting luid ith heat transer over a surace o variable temperature stretching ith a poer la velocity in the presence o variable transverse magnetic ield. The role o thermal radiation is o major importance in some industrial applications such as glass production, urnace design, nuclear poer plants space technology such as in comical light aerodynamics rocet, propulsion systems, plasma physics and space crat reentry aerodynamics hich operates high temperatures. The eect o thermal radiation on the boundary layer lo has been investigated by Raael Cortell (008)[6]. Viscous dissipation plays an important role in changing the temperature distribution hich aects the heat transer rates considerably. The thermal radiation and viscous dissipation eects on the laminar boundary layer about a lat plate in a uniorm stream o luid (Blasius lo), and about a moving plate in a quiescent ambient luid (Saiadis lo) both under convective boundary condition is presented by Olanreaju.P.O et al. (011)[14].Similarity solutions to boundary layer lo and heat transer o nanoluid over nonlinearly stretching sheet ith viscous dissipation eects as studied by Hamad and M.Ferdos (01)[10]. The eect o variable viscosity on the lo and heat transer o a viscous Ag- ater and Cu-ater nanoluids as investigated by Vajravelu (01)[8].Convective-radiation eects on stagnation point lo o nanoluids over a stretching/shrining surace ith viscous dissipation as studied by Pal et al.(014)[15]. The radiating and electrically conducting luid over a porous stretching surace ith the eect o viscous dissipation as researched by Sreenivasalu et al. (016)[6]. Buongiorno (006)[4] proposed a mathematical nanoluid model by taing into account the Bronian motion and thermophoresis eects on lo and heat transer ields.in his or he has considered seven slip mechanisms those aect nanoluid lo such as inertia, Bronian diusion, thermophoresis, diusiophoresis, Magnus eect, luid drainage and gravity. He indicated that o those seven only Bronian diusion and thermophoresis are important slip mechanisms in nanoluids. Reza Azizian et al. (01)[1] has investigated the eect o nanoconvection caused by Bronian motion on the enhancement o thermal conductivity in nanoluids. The non-linear stretching o a lat surace in a nanoluid ith Bronian motion and thermophoresis eects as investigated by Rana and Bhargava (01)[18]. The temperature-dependent thermo-physical properties on the boundary layer lo and heat transer o a nanoluid past a moving semi-ininite horizontal lat plate in a uniorm ree stream ith the eects o Bronian motion, thermophoresis and viscous dissipation due to rictional heating are analyzed by Vajravelu and Prasad (01)[9].The eects o thermal radiation and viscous dissipation on magnetohydrodynamic (MHD) stagnation point lo and heat transer o nanoluids toards a stretching sheet are investigated by Yohannes Yirga and Bandari Shanar (013)[30]. The problem o laminar luid lo hich results rom a permeable stretching o a lat surace in a nanoluid ith the eects o heat radiation, magnetic ield, velocity slip, bronian motion and thermophoresis parameters and convective boundary conditions have been eamined by Reddy (014)[0].Sumalatha et al.(016)[7] published the 1441

3 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. mied convection lo o nanoluids past a nonlinear stretching sheet in the eistence o nanoluids important slip mechanisms ith MHD, variable surace temperature and volume raction. Motivated by the above discussed investigations and applications, in this present or mainly concentrate on the eects o nonlinear thermal radiation, viscous dissipation and variable magnetic ield on heat transer lo o nanoluids (Cu Water nanoluid and Ag Water nanoluid) over a nonlinearly stretching sheet ith variable surace temperature. And also the model includes the eects o Bronian motion and Thermophoresis eects. MATHEMATICAL FORMULATION Consider to-dimensional, hydromagnetic lo over a nonlinearly stretching sheet ith convective heat transer in ater based nanoluids containing copper (Cu) and Silver (Ag) nanoparticles and the Cartesian coordinates such as - ais runs along the direction o the continuous stretching surace and the y - ais is measured normal to the surace o the sheet. It is also considered that the sheet is stretching ith velocity U = c n, here c > 0.Let us assume, the base luid (ater) and the nanoparticles are in equilibrium and the nanoluids is viscous and incompressible.(see Fig. i). Figure i: Physical model o the problem Taing into account the eects o Bronian motion and thermophoresis and based on model developed by Buongiorno [4]. The basic steady boundary-layer equations in the presence o variable magnetic ield, nonlinear thermal radiation and viscous dissipation are given by u v 0 y u u u B ( ) u v n u y y D c u v c D n y p n n T T T T q T T r u p s B n y y y T y y y (1) () (3) D T u v D T B y y T y (4) The boundary conditions are given by u = u () = c n, v =0,T = T () = T + b m, DT T DB 0 at y = 0 y T y u=0,t T,, as y (5) In the above boundary conditions, assume m = n is a surace temperature parameter and the nanoparticle mass lu due to the Bronian motion and thermophoresis eects tends to zero at the boundary(y=0)[a.v.kuznetsov and D.A.Nield [1]]. here the symbols are as deined in the nomenclature. The variable magnetic ield B() = B 0 (n-1)/ (Azal 1993)[1] is applied in the transverse direction. The magnetic Reynolds number is assumed to be small so that the induced magnetic ield is negligible in comparison ith the applied magnetic ield. Since the induced magnetic ield is neglected and B 0 is independent o time, curl E 0. Also, dive 0 in the absence o surace charge density. Hence E 0. The Rosseland approimation [Rosseland (1936)[3],Raptis (1998)[19], Sparro and Cess(1978)[5], Brester (199)[3]] is used to describe the radiative heat lu hich is negligible in direction in comparison to that in y direction. Full radiation term has been taen into account. Employing the Rosseland diusion approimation, the radiative heat lu is modeled as q r Hence * 16σ * 3 T 3 T y * T 3 T 3 * 16 qr T T y 3 y y here σ * is the Stean Boltzmann constant, * is the Rosseland mean absorption coeicient. The nonlinear governing equations (1) to (4) ith the boundary conditions (5) are solved by employing the similarity transormations hich are given belo. c n 1 T T T T n1 c n1, n1, y, (6) (7) (8) Where is the similarity space variable and dimensionless stream unction. is the 144

4 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. Using the Stream unction and v u y The velocity components are epressed as ollos u c n, cn1 n1-1 n v - n 1 (9) Using the similarity transormations (9), equation o continuity (1) is automatically satisied and the equations (7),(8) and (9), the nonlinear partial dierential equation (), (3) and (4) ith boundary conditions (5) are reduced to the olloing nonlinear ordinary dierential equations n b c M 0 n 1 (10) N N R n R n 4n Pr. Ec Pr. c Nb Nt n 1 b1 n (11) Nt Pr Le 0 Nb (1) Here, b 1, c 1 and c are constants hose values are given in Appendi. The appropriate boundary conditions are 0 0, 0 1, 0 1 Nb 0 Nt 0 0 at =0, 0, 0 0 as (13) The nondimensional parameters appeared in Equations (10) to (1) are deined as ollos B0 is the M cn magnetic interaction parameter, radiation parameter cp number, T T Ec cp T T u p N R 1 * is the 4 * T 3 Pr is the Prandtl is the Temperature ratio parameter, is the Ecert number, ( c ) D p s B is the Bronian motion parameter, Nb ( c ) ( c ) D T T p s T Nt ( c ) T p is the thermophoresis parameter and Le D Sin-riction coeicient B is the Leis number. The sin riction coeicient (rate o shear stress) is deined as C U, here u n y y0 Substituting equations (8) and (9) into equation (14), C 1/ n 1 Re = Nusselt number The Nusselt number (rate o heat transer) is deined as Nu q T T (14), here surace heat lu is 16 3 T n 3 y y0 q T (15) Using equations (8) and (9), equation (15) can be ritten as Nu n 4 3 n 1 1 (0) Re n 3N Re Here, c R n1 Due to the eects o Bronian motion and thermophoresis at the boundary, the Sherood number vanishes because hich characteristics the mass lu is zero at y=0. Numerical Solutions In this or, steady, to dimensional, hydromagnetic boundary layer lo o nonlinearly stretching surace over to types o nanoluids namely Cu Water nanoluid and Ag Water nanoluid in the presence o viscous dissipation and nonlinear thermal radiation and also the eects o Bronian motion and thermophoresis has been investigated. The governing nonlinear partial dierential equations are converted to nonlinear ordinary dierential equations by similarity transormations incorporating the necessary similarity variables. The resulting nonlinear ordinary dierential equations (10) to (1) along ith the relevant boundary conditions (13) constitute a nonlinear boundary value problem hich is diicult to solve analytically. Hence, these equations are solved using the most eicient shooting method such as the Nachtsheim-Sigert shooting iteration scheme or satisaction o the asymptotic boundary conditions along ith the Fourth-order Runge Kutta integration method. The diiculty lies in guessing the values 1443

5 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. or ''(0), (0) and (0) properly to get the Ec = convergence and solution. The level o accuracy or convergence is chosen as Pr = 6. N R = Le = Nt = Nb = n = 1 RESULTS AND DISCUSSIONS ' () The numerical and graphical results or to types o ater based nanoluids such as Cu-ater nanoluid and Ag-ater nanoluid are presented. The value o the Prandtl number or the base luid (ater) is ept to be the constant Pr = 6.. In order to veriy the accuracy o the present method, e have compared our results ith those o Cortell [17] and Hamad et al.[10] or the Sin riction coeicient (0) and nondimensional rate o heat transer -0 in the absence o nanoparticles ( = 0), Magnetic interaction parameter and viscous dissipation parameter and ithout thermal radiation parameter ( N R ), Bronian motion and thermophoresis hich is shon in Table and Table 3. It is clearly note that our results are good agreement ith that o Cortell and Hamad et al. Table : Comparison o results or (0) hen = 0 and M = n Cortell Hamad et al. Present or Table 3: Comparison o results or θ(0) hen = 0, Pr = 5.0,Ec = and N R n Cortell Hamad et al. Present or M =,, 1.5, Figure 1: Velocity proiles or various values o M Figure : Eect o M on Temperature proiles 0. M =,, 1.5,.0 M =,, 1.5,.0 Ec = Pr = 6. N R = Le = Nt = Nb = n = 1 Ec = Pr = 6. N R = Le = Nt = Nb = n = 1 Fig.1 to Fig.13 demonstrate the inluence o Magnetic interaction parameter, nonlinear stretching parameter, viscous dissipation parameter, surace temperature parameter, radiation parameter, Leis number, Bronian motion and thermophoresis parameter respectively on velocity distribution, temperature distribution and nanoparticle volume raction o to types o nanoluids such as copper ater nanoluid and silver ater nanoluid Figure 3: Nanoparticle volume action or various values o M 1444

6 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. ' () n =,.0, 3.0, 1 M = Pr = 6. N R = Ec = Nb = Nt = Le = = 0 M = Pr = 6. N R = Ec = Nt = Nb = Le = n =,.0, 3.0, 4.0 Figure 4: Velocity and volume raction proiles or n or Cu ater nanoluid Fig.1 shos the plot o dimensionless velocity or dierent values o magnetic interaction parameter. It is noted that as magnetic interaction parameter increases, decreases, elucidating the act that the eect o magnetic ield is to decelerate the velocity. This result qualitatively agrees ith the epectation since the Lorentz orce hich opposes the lo ield increases as M increases and leads to enhanced deceleration o the lo. Further the eect o magnetic ield is to reduce the boundary layer thicness. Fig. represents the graph o dimensionless temperature or dierent values o magnetic interaction parameter. Increase in M hich enhances the dimensionless temperature distribution. The inluence o magnetic interaction parameter on the dimensionless volume raction is plotted in Fig.3.The igure reveals that the volume raction o the nanoluids boosts or increasing values o M. Fig.4 and ig.5 respectively is a graphical representation o dimensionless velocity, volume raction or Cu ater nanoluid and temperature o both nanoluids or various values o nonlinear stretching parameter. It is noted that as the nonlinear stretching parameter increases, (),and diminishes. Consequently the eect o nonlinear stretching parameter over momentum boundary layer thicness becomes signiicantly less, or cu - ater nanoluid Figure 5:Dimensionless Temperature proiles or n In Fig.6, the eect o Ecert number on temperature distribution is displayed. It implied that the Ecert number enhances temperature and contributes to the thicening o thermal boundary layer thicness. 0. Ec =,,, M = Pr = 6. N R = Nt = Nb = Le = n = 1 Figure 6:Dimensionless temperature distribution at dierent values o Ec 1445

7 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. =, 5,, M = Pr = 6. Ec = N R = Nt = Nb = Le = n = 1 M = Pr = 6. N R = Ec = Nt = Nb = = n = 1 Le =,,, Figure 7:Dimensionless Temperature Proiles or N R =,, 1.5, Figure 8: Radiation parameter eect on Dimensionless Temperature proiles M = Pr = 6. Ec = Nt = Nb = Le = n = 1 Fig.7 depicts the eect o changing temperature ratio parameter on temperature distribution. The thermal boundary layer thicness increases ith increasing surace temperature. This can be eplained by the statement the eect o temperature ratio parameter is to increase the rate o energy transport to the nanoluid and accordingly increase the temperature. An increase in the radiation parameter causes a decrease in the temperature and the thermal boundary layer thicness as displayed in Fig.8.The values o radiation parameter ill cause no change in the velocity proiles o the nanoluids because the transormed momentum equation (10) is uncoupled rom the energy equation (1). Fig.9 shos the eect o Leis number on the volume raction proiles. It illustrates that the volume raction decreases as the Leis number increases. This is because as the values o Leis number gets larger the molecular diusivity gets smaller thereby causes a decrease in the volume raction ield. Figure 9: Dimensionless nanoparticle volume raction or Leis number Figure 10: Temperature proiles or various values 0. Nb = 0.,,, o Bronian motion parameter Nb = 0.,,, Figure 11: Eect o Bronian motion parameter on volume raction distribution M = Pr = 6. N R = Ec = n = 1 Nt = Le = The eect o Bronian motion parameter on temperature and volume raction is shon in Fig.10 and Fig.11. The temperature in the boundary layer has the less result due to the inluence o Bronian motion parameter hereas the volume raction decreases ith the increasing values o M = Pr = 6. N R = Ec = Nt = Le = = n =

8 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. Bronian motion parameter. Bronian motion serves to just arm the boundary layer. Figure 1: Temperature proiles or dierent values o Thermophoresis parameter Thermophoresis parameter plays an important ey role in temperature distribution and nanoparticle volume raction hich is demonstrated through Fig.1 & Fig.13 respectively. It is noticed that the dimensionless temperature as ell as the dimensionless volume raction increases by the increase o the values o the thermophoresis parameter. Increase in Nt causes the increment in the thermophoresis orce hich tends to move nanoparticles rom hot to cold areas and consequently it enhances the magnitude or temperature and nanoparticle volume raction proiles. 0. Nt = 0.,,, Nt = 0.,,, M = Pr = 6. N R = Ec = n = 1 Nb = Le = M = Pr = 6. N R = Ec = n = 1 Nb = Le = The numerical results o the sin riction co eicient and nondimensional rate o heat transer are presented in table 4 and table 5 or both cu - ater nanoluid and silver ater nanoluid. In Table 4, sin riction coeicient increases due to the inluence o magnetic interaction parameter and nonlinear stretching parameter in magnitude. Table 5 illustrates the eect o all the physical parameters on nondimensional rate o heat transer. For increasing values o nonlinear stretching parameter and surace temperature ratio parameter, the nondimensional rate o heat transer enhances meanhile the physical parameters such as magnetic interaction parameter, radiation parameter, thermophoresis parameter, Bronian motion parameter and Ecert number diminishes the nondimensional rate o heat transer. Table 4: Sin riction coeicient or dierent values o M n M and n Cu - ater n n Figure 13: Eect o Thermophoresis parameter on volume raction distribution 1447

9 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. Table 5: Nondimensional Heat transer rate or dierent values o M, n, N R, Nb, Nt, Ec and hen =, Le = and Pr = 6. Cu Water Ag Water M n N R Nb Nt Ec n 4 3 n 4 3 n 1 1 (0) n 1 1 (0) n 3N n 3N R R CONCLUSION A role o Bronian motion and thermophoresis eects on hydromagnetic lo o nanoluids past a nonlinearly stretching sheet under consideration o viscous dissipation and nonlinear thermal radiation have been investigated in this or or to types o nanoluid Cu ater nanoluid and silver ater nanoluid. Using similarity transormations the governing equations o the problem are transormed into nonlinear ordinary dierential equations and solved numerically by using most eicient Nachtsheim- Sigert shooting iteration scheme or satisaction o asymptotic boundary conditions along ith ourth order Runge-Kutta Integration method (FORTRAN pacage). Numerical solutions o the problem are obtained or various physical parameters. From the obtained numerical results and discussion presented in the previous section, the olloing conclusions are dran An increase in magnetic interaction parameter and nonlinear stretching parameter decreases the nanoluid velocity but opposite trend is occurred in sin riction coeicient. A rise in the magnetic interaction parameter, thermophoresis parameter, temperature ratio parameter and viscous dissipation parameter raises the temperature distribution. In the mean hile nonlinear stretching parameter and radiation parameter decreases the temperature distribution. Also the temperature has very less eect due to Bronian motion parameter. Nanoparticle volume raction decelerates ith an increase in the values nonlinear stretching parameter, Bronian motion parameter and Leis number. 1448

10 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. Nanoparticle volume raction accelerates or the increasing values o thermophoresis parameter and magnetic interaction parameter. Nondimensional heat transer rate enhances by means o rise in the values o nonlinear stretching parameter and surace temperature parameter but the nondimensional rate o heat transer decelerates ith an increasing value o magnetic interaction parameter and radiation parameter, thermophoresis parameter, Bronian motion parameter and Ecert number. Sherood number vanishes or nanoluids to phase model ith ne type o boundary condition. Finally, the numerical values o nondimensional rate o heat transer o Ag - ater nanoluid is higher than the Cu - ater nanoluid. Appendi The epressions or the physical quantities and n, c p n [Ahmad et al. (011)],,, n n, n are given through the olloing lines 1, n s n n s s, s s n n, The constants values are as ollos, c 1 n cp n 1 cp cp 1 s, b 1 c 1 cp cp.5 s 1, Table 1:Thermo-physical properties o luid and nanoparticles at 5C Physical properties Water luid Cu Ag C P , K REFERENCES [1] N.Azal, Heat transer rom a stretching surace, Int. J. Heat Mass Transer, vol.36,pp , [] S.Ahmad,A.M.Rohni, I.Pop, Blasious and Saiadis s problems in nanoluids,acta vol.18,pp , 011. Mechanica, [3] M.Q.Brester, Thermal Radiative Transer and Properties John Wiley and sons Inc, 199. [4] J.Buongiorna, Convective transport in nanoluids, Jl. o heat transer,vol. 18(3), pp.40-50,006. [5] S.Choi, Enhancing thermal conductivity o luids ith nanoparticles. I sidiner DA, Wang HP (eds) Developments and applications o non-netonian los,asmefed, 31/MD, pp ,1995. [6] R.Cortell, Eects o viscous dissipation on and radiation on the thermal boundary layer over a nonlinearly stretching sheet, Physics Letters A, vol.37(5), pp , 008. [7] L.J.Crane, Flo past a stretching plate Zeitschrit ür Angeandte Mathemati und Physi, vol.1(4),pp , [8] S.K.Das,S.Choi,W.Yu&T.Pradet, Nanoluids: science and Technology, Wiley, Ne Jersey, 007. [9] S.P.A.Devi, M.Thiyagarajan, Steady nonlinear hydromagnetic lo and heat transer over a stretching surace o variable temperature,heat Mass Transer,vol.4,pp ,006. [10] M.A.A.Hamad,M.Ferdos, Similarity solutions to viscous lo and heat transer o nanoluid over nonlinearly stretching sheet, Appl. Math. Mech. Engl. Ed., vol.33(7), pp , 01. [11] V.Kauui, Wong and Omar De Leon, Applications o nanoluids:current and Future, Advances in mechanical engineering,pp.1-11,010. [1] A.V.Kuznetsov,D.A.Nield, Natural convective boundary layer lo o a nanoluid past a vertical plate: A revised model, Int. J. o thermal sciences, vol.77, 16-19,014. [13] Mohammed Saad Kamel, Raheem Abed Syeal,Abdulameer Amdulhussein, Heat transer enhancement using nanoluid: A revie o the recent literature, American Jour. o Nano Research and Applications, vol.4(1), pp.1-5,016. [14] P.O.Olanreaju,J.A.Gbadeyan,O.O.Agboola, and S.O.Abah, Radiation and viscous dissipation eects or the Blasius and Saiadis los ith a convective surace boundary condition, International Journal o Advances in Science and Technology,vol.(4),011. [15] D.Pal, G.Mandal, and K.Vajravelu, Convective- Radiation Eects on Stagnation Point Flo o Nanoluids Over a Stretching/Shrining Surace ith Viscous Dissipation,Journal o Mechanics, vol.8,pp.1-9,

11 International Journal o Applied Engineering Research ISSN Volume 1, Number (017) pp Research India Publications. [16] K.B.Pavlov, Magnetohydrodynamic lo o an incompressible viscous luid caused by the deormation o a plane surace. Magnytnaya Gidrodinamia, vol.4, pp , [17] Raael Cortell, Viscous lo and heat transer over a nonlinearly stretching sheet, Applied Mathematics and Computation,vol.184,pp ,007. [18] P.Rana,R.Bhargava, Flo and heat transer o a nanoluid over a nonlinearly stretching sheet: A numerical study,commun Nonlinear Sci Numer Simulat,vol.17,1 pp.6,01. [19] A.Raptis. Flo o a micripolarluid past a continuously moving plate by the presence o Radiation, Int. Jour. o Heat & Mass transer,vol.41(18),pp ,1998. [0] M.G.Reddy, Inluence o Magnetohydrodynamic and Thermal Radiation Boundary Layer Flo o a Nanoluid Past a Stretching Sheet,J. Sci. Res.Vol.6(), pp.57-7,014. [1] Reza Azizian, Elham Doroodchi, and Behdad Moghtaderi, Eect o nanoconvection caused by bronian motion on the enhancement o thermal conductivity in nanoluids, Ind. Eng. Chem. Res.,vol.51, pp ,01. [] Ritu Pasrija and Sunita Srivastava, On the Eective Thermal Conductivity o metallic and oide Nanoluids,Int. Jour. o NanoScience and Nanotechnology,vol.4,pp ,013. [3] S.Rosseland, Theoretical Astrophysics, Clarendon Press, Oord, [4] Sandeep Pal, Tiamchand Soni Ariti Agraala and Deepa Sharma, Revie on Enhanced Thermal Conductivity o Colloidal Suspension o Nanosized Particles (Nanoluids),Int. Jour o Advanced Mechanical Engineering,vol.4,pp ,014. [5] E,M,Sparro, R.D.Cess, Radiation heat transer hemisphere, Washington(Chaps. 7 & 10), [6] P.Sreenivasulu,T.Poornima, Bhasar N.Reddy., Thermal radian eects on MHD boundary layer slip lo past a permeable eponential stretching sheet in the presence o Joule heating and viscous dissipation, JAFM, vol.9(1), pp.67-78,016. [7] Sumalatha, Chenna Shaner, Bandari, MHD Mied Convection Flo o a Nanoluid Over a Nonlinear Stretching Sheet ith Variable Wall Temperature and Volume raction,journal o Nanoluids,vol.5(5), pp ,016. [8] K.Vajravelu, The eect o variable viscosity on the lo and heat transer o a viscous Ag- ater and Cuater nanoluids, Journal o Hydrodynamics,vol.5, pp.1-9,01. [9] K.Vajravelu and K.V.Prasad, Heat transer phenomena in a moving nanoluid over a horizontal surace, Journal o Mechanics, vol.8, pp ,01. [30] Yohannes Yirga and Bandari Shanar, Eects o thermal radiation and viscous dissipation on magnetohydrodynamic stagnation point lo and heat transer o nanoluid toards a stretching sheet, Journal o Nanoluids, vol., pp.83 91,