Heat source/sink effects of heat and mass transfer of magnetonanofluids over a nonlinear stretching sheet

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1 Available online at.pelagiaresearchlibrary.com Advances in Applied Science Research, 04, 5(3):4-9 ISSN: CODEN (USA): AASRFC Heat source/sink eects o heat and mass transer o magnetonanoluids over a nonlinear stretching sheet Thommaandru Ranga Rao, Kotha Gangadhar, B. Hema Sundar Raju 3 and M. Venkata Subba Rao 3 Department o Mathematics, SVKP College, Markapur, Prakasam, India Department o Mathematics, Acharya Nagarjuna University, Ongole, Andhra Pradesh, India 3 Department o Mathematics, Vignan University, Guntur, Andhra Pradesh, India ABSTRACT This study investigates theoretically the problem o ree convection boundary layer lo o nanoluids over a nonlinear stretching sheet in the presence o MHD and heat source/sink. The governing partial dierential equations are transormed to a system o ordinary dierential equations and solved numerically using ourth order Runge- Kutta method along ith shooting technique. The eects o the magnetic parameter, the Prandtl number, Leis number, poer la velocity parameter, the Bronian motion parameter, thermophoresis parameter, heat source/sink parameter on the luid properties as ell as on the heat and mass transer coeicients are determined and shon graphically. Keyords: nanoluid, MHD, heat source/sink, Bronian motion, thermophoresis, nonlinear stretching sheet. INTRODUCTION The study o convective heat transer in nanoluids is gaining a lot o attention. The nanoluids have many applications in the industry since materials o nanometer size have unique physical and chemical properties. Nanoluids are solid-liquid composite materials consisting o solid nanoparticles or nanoibers ith sizes typically o -00 nm suspended in liquid. Nanoluids have attracted great interest recently because o reports o greatly enhanced thermal properties. For example, a small amount (<% volume raction) o Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is reported to increase the inherently poor thermal conductivity o the liquid by 40% and 50%, respectively [,]. Conventional particle-liquid suspensions require high concentrations (>0%) o particles to achieve such enhancement. Hoever, problems o rheology and stability are ampliied at high concentrations, precluding the idespread use o conventional slurries as heat transer luids. In some cases, the observed enhancement in thermal conductivity o nanoluids is orders o magnitude larger than predicted by ellestablished theories. Other perplexing results in this rapidly evolving ield include a surprisingly strong temperature dependence o the thermal conductivity [3] and a three-old higher critical heat lux compared ith the base luids [4,5]. These enhanced thermal properties are not merely o academic interest. I conirmed and ound consistent, they ould make nanoluids promising or applications in thermal management. Furthermore, suspensions o metal nanoparticles are also being developed or other purposes, such as medical applications including cancer therapy. The interdisciplinary nature o nanoluid research presents a great opportunity or exploration and discovery at the rontiers o nanotechnology. Bachok et al. [6] studied boundary layer lo o a nanoluid past a moving plate in a uniorm ree stream. In another investigation, Bachok et al. [7] discussed heat transer over a permeable stretching sheet or the unsteady boundary layer lo o a nanoluid. Kandasamy et al. [8] studied scaling group transormations or an electrically conducting nanoluid lo over a vertical stretching sheet. Tsou et al. [9] reported both analytical and experimental results or the lo and heat transer in the boundary layer on a continuously 4

2 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 moving surace. Gorla et al. [0] studied the natural convective boundary layer lo over a horizontal plate embedded in a porous medium saturated ith a nanoluid. The magnetohydrodynamic (MHD) lo has attracted a great interest to many researchers during the last several decades oing to the eect o magnetic ield on the boundary layer lo control and applications in many engineering and physical aspects such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. Flo characteristics and heat transer over a stretching sheet have been studied extensively by many researchers in the recent past because o many engineering applications o stretching sheet in manuacturing processes such as hot rolling, ire draing, draing o plastic ilms and artiicial ibers, metal extrusion, crystal groing, continuous casting, glass iber production and paper production. Metallurgy lies in the puriication o molten metal s rom non-metallic inclusions by applying magnetic ield is other application o MHD. Chen [] studied the eects o magnetic ield and suction/injection on convection heat transer o non-netonian poer-la luids past a poer la stretched sheet ith surace heat lux. The magnetohydrodynamic (MHD) orced convection boundary layer lo o nanoluid over a horizontal stretching plate as investigated by Nourazar et al. [] using homotopy perturbation method (HPM). Matin et al. [3] studied entropy analysis in mixed convection MHD lo o nanoluid over a nonlinear stretching sheet. Hayat et al. [4] used homotopy analysis method (HAM) to investigate the MHD boundary layer lo and examine the heat transer analysis or the permeable stretching sheet under varied conditions. In this continuation, Abbas and Hayat [5] studied the stagnation point lo ith slip eects in heat transer analysis or the boundary layer lo over a non-linear stretching sheet. Aman and Ishak [6] presented the hydromagnetic lo and heat transer adjacent to a stretching vertical sheet ith prescribed surace heat lux. The boundary layer los o non-netonian luids over a stretching sheet ith heat and mass transer are important in several areas such as extrusion process, glass ibber, paper production, hot rolling, ire draing, electronic chips, crystal groing, plastic manuactures, and application o paints, ood processing and movement o biological luids (Hayat and Qasim, [7]). Bhargava et al. [8] discussed heat and mass transer o boundary layer lo over a nonlinear stretching sheet under the eects o dierent physical parameters. Khan and Pop [9] investigated the laminar luid lo o a nanoluid rom the stretching lat surace by incorporating the eects o Bronian motion, thermophoresis and reported to be the pioneer or this study o stretching sheet in nanoluid. Pal and Mondal [0] examined the heat and mass transer over a stretching sheet by considering the eects o buoyancy and solutal buoyancy parameters. Anar et al. [] studied the conjugate eects o heat and mass transer o nanoluids over a nonlinear stretching sheet. The heat source/sink eects in thermal convection, are signiicant here there may exist a high temperature dierences beteen the surace (e.g. space crat body) and the ambient luid. Heat generation is also important in the context o exothermic or endothermic chemical reactions. Vajravelu and Hadjinicalaou [] have studied on hydrodynamic convective heat transer rom a stretching surace ith heat generation/absorption. Molla et al. [3] studied natural convection lo along a vertical avy surace ith uniorm surace temperature in presence o heat generation/absorption. MHD heat and mass transer ree convection lo along a vertical stretching sheet in presence o magnetic ield ith heat generation are studied by Samad et al. [4]. Recently, Rahman et al. [5] investigated the thermophoresis eect on MHD orced convection on a luid over a continuous linear stretching sheet in presence o heat generation and Poer-La all temperature Hoever, the interaction o conjugate eects o nanoluids over a nonlinear stretching sheet, has received little attention. Hence, the present study an attempt is made to analyze the steady conjugate eects o heat and mass transer o nanoluid lo over a nonlinear stretching sheet in the presence o MHD and heat source/sink. The governing boundary layer equations have been transormed to a to-point boundary value problem in similarity variables and the resultant problem is solved numerically using the ourth order Runge-Kutta method along ith shooting technique. The eects o various governing parameters on the luid velocity, temperature, concentration, skin riction, reduced Nusselt number and reduced Sherood number are shon in igures and analyzed in detail.. MATHEMATICAL FORMULATION The steady to-dimensional boundary layer lo o a nanoluid past a stretching surace is considered under the assumptions that the external pressure on the plate in x-direction is having diluted nanoparticles. The stretching m velocity is assumed to be U ( x) = U0x, here U 0 is the uniorm velocity and m( m 0) is a constant parameter and x is the coordinate measured along the stretching surace. The lo takes place above the stretching surace at y 0. Here, y is the coordinate axis measured normal to the stretching surace. A uniorm stress leading to equal and opposite orces is applied along the x-axis so that the sheet is stretched, keeping the origin ixed. A uniorm magnetic ield o strength B(x) is applied normal to the Sheet. It is assumed that at the stretching surace, 5

3 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 the temperature T and the nanoparticle raction C take constant values T and C hereas the ambient values o temperature T and the nanoparticle raction C are attained as y tends to ininity. Under these assumptions along ith the Boussinesq and boundary layer approximations, the system o equations, governing the lo ield are given by Continuity equation u v + = 0 x y Momentum equation x y x y p u u u = µ ρ u + v σ B u ( ) ρ β ( ) ( ρ ρ ) β ( ) + C T T T C C C g Energy equation ˆ T T φ T DT T q u + v = α T + τ DB + T T x y yˆ yˆ + T y ρ cp Species equation C C C D T u + v = DB + x y y T y T ( ) (.) (.) (.3) (.4) The boundary conditions or the velocity, temperature and concentration ields are u = U ( x) = U x m, v = 0, T = T, C = C at y = 0 0 u 0, v 0, T T, C C as ŷ (.5) here u and v are the velocity components along the x and y axes, respectively, g is the acceleration due to gravity, µ is the viscosity, ρ the volumetric thermal expansion, ( ρ c) p is the density o the base luid, p β C ρ is the density o the nanoparticle, T is the coeicient o the volumetric concentration expansion, ( ) β is the coeicient o c ρ and are the heat capacity o the luid and the eective heat capacity o the nanoparticle material respectively, m υ = µ ρ is the kinematic viscosity o the luid, k is the thermal conductivity, B( x) B x / ield o constant strength, here B 0 is constant, α k ( ρc) / = is the magnetic = is the thermal diusivity parameter, D B is the 0 Bronian diusion coeicient, τ is a parameter deined by ( c) / ( c) parameter, here q0 is any constant. ρ ρ, p 0 m q = q x is the heat source/sink The continuity equation (.) is satisied by the Cauchy Riemann equations u = ψ y and v ψ = x (.6) here ψ ( x, y) is the stream unction. In order to transorm the equations (.) to (.5) into a set o ordinary dierential equations, the olloing similarity transormations and dimensionless variables are introduced. 6

4 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 m + m υu 0 x ( m + ) U 0 x T T C C ψ =, = y, θ ( ) =, φ ( ) = m + υ T T C C σ B Gr Gm υ υ M = = = = = ρ 0, λ, δ, Pr, Le 3 / 3 / U 0 Re x Re x α DB τ DB ( C C ) τ DT ( T T ) U ( x) x q0 Nb =, Nt =, Re x =, Q = υ T υ υ U ρ c ρ ρ p ρ ( C ) gn( T T ) ρ gn ( C C ) Gr = ρ, Gm = / υ Re υ Re / x x 0 p (.7) here ( ) is the dimensionless stream unction, θ - the dimensionless temperature, φ - the nanoparticle volume riction, - the similarity variable, Nb - the Bronian motion parameter, Nt - the thermophoresis parameter, Pr - the Prandtl number, Q- heat source/sink parameter, λ- thermal buoyancy parameter, δ-solutal buoyancy parameter, Re x - the local Reynolds number based on the stretching velocity, M - the magnetic parameter, Le - the Leis number, Gr-local thermal Grasho number, Gm-local solutal Grasho number. Here, β and β are proportional to 3 x [6])., that is β β 3 3 T = nx, C = n x, here n and In vie o the equations (.6) and (.7), the equations (.) to (.5)) transorm into n are the constant o proportionality (Makinde and Olenreaju T C m M m + m + ''' + '' ' + ( λθ δφ ') = 0 Pr θ θ θ φ θ m + θ " + ' + Nb ' ' + Nt ' + Q = 0 (.8) (.9) Nt φ " + Le φ ' + θ '' = 0 Nb (.0) The transormed boundary conditions can be ritten as = 0, ' =, θ =, φ = at = 0 ' 0, θ 0, φ 0 as (.) The skin-riction, Nusselt number and Sherood number or the present problem o nanoluid are deined as τ qx qmx C =, Nu =, Sh = ρu k ( T T ) k ( C C ) (.) here T, C q = k qm = DB, τ u = µ y y y at y=0 The associated expressions o dimensionless skin-riction ''(0), reduced Nusselt number θ '(0) and reduced Sherood number φ '(0) deined as 7

5 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 C Nu Sh Cx = Re x, Nur =, Shr = m + m + m + Rex Re x (.3) 3 SOLUTION OF THE PROBLEM The set o non-linear coupled dierential Eqs. (.8)-(.0) subject to the boundary conditions Eq. (.) constitute a to-point boundary value problem. In order to solve these equations numerically e ollo most eicient numerical shooting technique ith ith-order Runge-Kutta-integration scheme. In this method it is most important to choose the appropriate inite values o. To select e begin ith some initial guess value and solve the problem ith some particular set o parameters to obtain '', θ ' and φ '. The solution process is repeated ith another large value o until to successive values o '', θ ' and φ ' dier only ater desired digit signiying the limit o the boundary along. The last value o is chosen as appropriate value o the limit or that particular set o parameters. The three ordinary dierential Eqs. (.8)-(.0) ere irst ormulated as a set o seven irst-order simultaneous equations o seven unknons olloing the method o superposition [7]. Thus, e set, y =, y = ', y = '', y = θ, y = θ ', y = φ, y = φ ' y ' = y, y ' = y y 3 ( ) y ( ) ( 0 ) ( 0 ) ( 0 ) ( ) ( ) = 0, 0 = y ' = m y + ( δ y m + m + + M y λ y ) y y y = δ y ' = y, y 0 = y ' = P r y y + N b y y + N ty + Q y m + y = δ y ' = y, y 0 = N t y ' = L ey y + y N b y δ = 3 ' Eqs. (.8)-(.) then reduced into a system o ordinary dierential equations, i.e., here δ, δ and δ 3 are determined such that it satisies y ( ) 0, y4 ( ) 0 and 6 ( ) 0, δ until the boundary conditions y ( ) 0, y ( ) 0 and ( ) δ δ and 3 y. The shooting method is used to y are satisied. Then guess 4 the resulting dierential equations can be integrated by ith-order Runge-Kutta integration scheme. The above 6 procedure is repeated until e get the results up to the desired degree o accuracy, 0. RESULTS AND DISCUSSION 6 0 In order to get a clear insight o the physical problem, the velocity, temperature and concentration have been discussed by assigning numerical values to the governing parameters encountered in the problem. Numerical computations are shon rom igs.-. The velocity or dierent values o the magnetic ield parameter (M) is shon in ig.. As is no ell knon, the velocity decreases ith increases in the magnetic ield parameter due to an increase in the Lorentz drag orce that opposes the luid motion. Figs. (a) to (c) are shos the eect o the thermal buoyancy parameter on the velocity, temperature and concentration proiles. The thermal buoyancy parameter signiies the relative eect o the thermal 8

6 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 buoyancy orce to the viscous hydrodynamic orce. The lo is accelerated due to the enhancement in buoyancy orce corresponding to an increase in the thermal buoyancy parameter, i.e. ree convection eects. It is noticed that the thermal buoyancy parameter inluence the velocity ield almost in the boundary layer hen compared to ar aay rom the plate. It is seen that as the thermal buoyancy parameter increases, the velocity ield increases. Also noticed that temperature and concentration proiles decreases ith an increasing the thermal buoyancy parameter. Figs. 3(a) to 3(c) are shos the eect o the solutal buoyancy parameter on the velocity, temperature and concentration proiles. It is seen that as the solutal buoyancy parameter increases, the velocity ield decreases. Also observed that temperature and concentration proiles increases ith an increasing the solutal buoyancy parameter..0 ' δ=λ=m=, Q=, Pr=Le=, Nt=Nb= M=0 M= M= M= Fig. Velocity or dierent values o M.0 ' δ=m=, M=Q=, Pr=Le=, Nt=Nb= λ= λ= λ=3 λ= Fig.(a) Velocity or dierent values o λ 9

7 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 δ=m=, M=Q=, Pr=Le=4, Nt=Nb= λ= λ= λ=3 λ=4 θ Fig.(b) Temperature or dierent values o λ.0 δ=m=, M=Q=, Pr=Le=4, Nt=Nb= λ= λ= λ=3 λ=4 φ Fig.(c) Concentration or dierent values o λ 0

8 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 ' λ=m=, M=Q=, Pr=, Le=4, Nt=Nb= δ= δ= δ=3 δ= Fig.3(a) Velocity or dierent values o δ.0 θ λ=m=, M=Q=, Pr=Le=4, Nt=Nb= δ= δ=.0 δ=.5 δ= Fig.3(b) Temperature or dierent values o δ

9 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 λ=m=, M=Q=, Pr=Le=4, Nt=Nb= δ= δ=.0 δ=. δ=.5 φ Fig.3(c) Concentration or dierent values o δ.0 δ=λ=m=, M=Q=, Pr=Le=4, Nb= Nt= Nt= Nt=5 Nt=0.3 θ Fig.4(a) Temperature or dierent values o Nt

10 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 δ=λ=m=, M=Q=, Pr=Le=4, Nb= Nt= Nt= Nt=5 Nt=0.3 φ Fig.4(b) Concentration or dierent values o Nt.0 δ=λ=m=, M=Q=, Pr=Le=4, Nt= Nb= Nb= Nb=5 Nb=0.3 θ Fig.5(a) Temperature or dierent values o Nb 3

11 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 δ=λ=m=, M=Q=, Pr=Le=4, Nt= Nb= Nb= Nb=5 Nb=0.3 φ Fig.5(b) Concentration or dierent values o Nb.0 δ=λ=m=, M=Q=, Le=4, Nt=Nb= Pr= Pr=3 Pr=5 Pr=7 θ Fig.6 Temperature or dierent values o Pr 4

12 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9.0 δ=λ=m=, M=, Pr=Le=4, Nt=Nb= Q=- Q= Q= Q=0.3 θ Fig.7 Temperature or dierent values o Q.0 δ=λ=m=, M=, Pr=4, Nt=Nb=Q= Le= Le=3 Le=5 Le=7 φ Fig.8 Concentration or dierent values o Le 5

13 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 Cx Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M=.0.0 Pr=Le=, λ=δ=, Q= m Fig.9 Eect o Nt, Nb, Q and m on the skin riction Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M=.0 Pr=Le=, λ=δ=, Q= -θ(0) m Fig.0 Eect o Nt, Nb, Q and m on the reduced Nusselt number 6

14 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3): Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M= Nt=, Nb=, M=.0 Pr=Le=, λ=δ=, Q= -φ(0) m Fig. Eect o Nt, Nb, Q and m on the reduced Sherood number The eect o thermophoresis parameter on temperature and concentration ields is shon in igs. 4(a) and 4(b). The thermophoretic orce generated by the temperature gradient creates a ast lo aay rom the stretching surace. In this ay more luid is heated aay rom the surace, and consequently, as Nt increases, the temperature ithin the boundary layer increases. The ast lo rom the stretching sheet carries ith it thermophoretic orce leading to an increase in the concentration boundary layer thickness. The eect o the Bronian motion o the luid on the temperature and concentration is shon in Figs. 5(a) and 5(b). As expected, the increased Bronian motion o the luid carries ith it heat and the thickness o the thermal boundary layer increases. An increase in the Bronian motion o the luid leads to a decrease in the concentration proiles. Fig. 6 sho the temperature proiles or several values o the Prandtl number (Pr). The temperature proiles decrease as the Prandtl number increases since, or high Prandtl numbers, the lo is governed by momentum and viscous diusion rather than thermal diusion. Fig. 7 shos the eect o the heat source/sink parameter (Q) on the temperature proiles. The temperature proiles signiicantly increase as the heat source/sink parameter increases. The various values o Leis number (Le) on concentration is plotted in ig. 8. The concentration proiles signiicantly contract as the Leis number increases. Fig. 9 shos the eects o the thermophoresis parameter, Bronian motion parameter, non linear stretching parameter (m) and the magnetic parameter on the all skin riction. It can be seen that the skin riction coeicient increases hen the thermophoresis parameter, non linear stretching parameter and magnetic parameter increases. Hoever, increasing the Bronian motion parameter leads to a decrease in the skin riction coeicient. Figs. 0 sho the eects o the thermophoresis parameter, Bronian motion parameter, non linear stretching parameter and the magnetic parameter on the reduced Nusselt number. We note a decrease in the reduced Nusselt number hen Nt or Nb or m or M increases. Fig. depicts the eects o the thermophoresis parameter, Bronian motion parameter, non linear stretching parameter and the magnetic parameter on the reduced Sherood number φ '(0). We observe an increase in the φ '(0) hen Nb increases and decrease hen Nt or m or M increases. The variations o skin riction coeicient, reduced Nusselt number and reduced Sherood number or dierent values o λ, δ, Pr, Q and Le are shon in Table. It is observed that Cx is an increasing unction o δ and Pr, hile ith increasing values o λ, Q and Le. Hoever, it is ound that θ '(0) decreases or large values o δ and Q hereas increases or increasing values o λ, Pr and Le. It is urther observed rom this table that φ '(0) is increasing unction o λ, δ, Q and Le hereas φ '(0) decreases ith increasing values o Pr. Table. shos the comparison o reduced Nusselt number and reduced Sherood number o the previous published data. 7

15 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 Table Comparison o reduced Nusselt number and reduced Sherood number or Pr=0, Le=0, m=, δ=λ=m=q=0 Nt 0.3 Nb 0.3 Present study Khan and Pop θ '(0) φ '(0) θ '(0) φ '(0) Table : Variation o Cx, θ '(0) & φ '(0) at the all ith Nb, Nt, Pr, Le, λ, δ, m, Q and M Nb Nt Pr Le λ δ m Q M Cx θ '(0) φ '(0) CONCLUSION In the present numerical study, conjugate eects o heat and mass transer o nanoluids over a nonlinear stretching sheet in the presence o MHD and heat generation or absorption. We determined the eects o various parameters on the luid properties as ell as on the shin riction, heat and mass transer rates. We have shon that increasing the magnetic ield parameter M tends to retard the luid lo ithin the boundary layer. The eects o the Prandtl number, the Leis number, the Bronian motion parameter, the thermophoresis parameter, heat source/sink parameter on the skin riction, heat and mass transer coeicients and luid lo characteristics have been studied. We have observed that: The luid velocity decreases hen increasing the magnetic parameter. The thermal and mass boundary layer thickness increases ith the thermophoresis parameter. Increasing the Leis number and solutal buoyancy parameter reduces the heat and mass transer coeicient. Increasing the Bronian motion parameter and heat source/sink parameters enhances the thermal boundary layer thickness. Non linear stretching parameter enhances the shin-riction. REFERENCES [] Eastman, J. A., Choi, S. U. S., Yu, S. Li, W., and Thompson, L. J., (00), Applied Physics Letters, Vol. 78, No. 6, pp [] Choi, S. U. S., Zhang, Z. G., Lockood, W. Yu, F. E., and Grulke, E. A., (00), Applied Physics Letters, Vol. 79, No. 4, pp [3] Patel, H. E., Das, S. K., Sundararajan, T., Sreekumaran, A., George, B., and Pradeep, T., (003), Applied Physics Letters, Vol. 4, No. 83, pp [4] You, S. M., Kim, J. H., and Kim, K. H., (003), Applied Physics Letters, Vol. 83, No. 6, pp [5] Vassallo, P., Kumar, R., and D Amico, S., (004), International Journal o Heat and Mass Transer, Vol. 47, No., pp [6] Bachok, N., Ishak, A., Pop, I., (00), Int. J. Therm. Sci., Vol.49, pp [7] Bachok, N., Ishak, A., and Pop, I., (0), Int. J. Heat Mass Trans., Vol.55, pp [8] Kandasamy, R., Loganathan, P., Arasu, P.P., (0), Nucl. Eng. Des., Vol.4, pp [9] Tsou, F.K., Sparro, E.M., and Goldstien, R.J., (967), Int. Heat Mass Trans., Vol.0, pp.9-3. [0] Rama Subba Reddy Gorla, and Ali Chamkha, (0), Journal o Modern Physics, Vol., pp.6-7. [] Chen, C.H., (008), Int. J. Thermal Sci., Vol.47, pp [] Nourazar, S.S., Matin, M.H., and Simiari, M., (0), J. Appl. Math., doi:55/0/ [3] Matin, M.H., Heirani, M.R., Nobari, Jahangiri, P., (0), J. Therm. Sci. Technol., Vol.7, pp [4] Hayat, T., Qasim, M., Mesloub, S., (0), Int. J. Numer. Methods Fluids, Vol.66, pp [5] Abbas, Z., and Hayat, T., (0), Numer. Meth. Part Dier. Equat., Vol.7, pp

16 Thommaandru Ranga Rao et al Adv. Appl. Sci. Res., 04, 5(3):4-9 [6] Aman, F., and Ishak, A., (00), Heat Mass Trans., Vol.46, pp [7] Hayat, T.,and Qasim, M., (0), Int. J. Numer. Meth. Fluids, Vol.66, pp [8] Bhargava, R., Sharma, S., Takhar, H.S., Beg, O.A., and Bhargava, P., (007), Nonlinear Anal. Model. Cont., Vol., pp [9] Khan, W.A., and Pop, I., (00), Int. J. Heat Mass Trans., Vol.53, pp [0] Pal, D., and Mondal, H., (00), Physica B., Vol.85, pp [] Anar, M. I., Khan, I., Sharidan, S., and Salleh, M. Z., (0), International Journal o Physical Sciences, Vol. 7(6), pp [] Vajravelu, K., and Hadjinicolaou, A., (993), Int. Comm. Heat Mass Trans., Vol.0, pp [3] Molla, M.M., Hossain, M.A., and Yao, L.S., (004), Int. J. Thermal Sci., Vol.43, pp [4] Samad, M.A., and Mohebujjaman, M., (009), Research J. o Applied Sci., Eng. and Tech., Vol.(3), pp [5] Mohammed Abdur Rahman, Alim, M. A. and Jahurul Islam, Md., (03), Annals o Pure and Applied Mathematics, Vol. 4, No., pp.9-04 [6] Makinde, O.D., and Olanreaju, P.O., (00), J. Fluids Eng. Trans., ASME 3: [7] Na, T.Y., (979), Computational Methods in Engineering Boundary Value Problems, Academic Press, Ne York. 9