akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 495 CHEICALLY EACTING ON HD BOUNDAY-LAYE FLOW OF NANOFLUIDS OVE A NON-LINEA STETCHING SHEET WITH HEAT SOUCE/SINK AND THEAL ADIATION by Oluwole D. AKINDE a*, Fazle ABOOD b, and ohaed S. IBAHI c, a Faculty o ilitary Science, Stellenbosch University, Saldanha, South Arica b Departent o atheatics, University o Peshawar, Peshawar, Pakistan c Departent o atheatics, GITA University, Visakhapatna, Andhra Pradesh, India Introduction Original scientiic paper https://doi.org/.98/tsci584 In this paper, steady -D HD ree convective boundary-layer lows o an electrically conducting nanoluid over a non-linear stretching sheet taking into account the cheical reaction and heat source/sink are investigated. The governing equations are transored into a syste o non-linear ODE using suitable siilarity transorations. Analytical solution or the diensionless velocity, teperature, concentration, skin riction coeicient, heat and ass transer rates are obtained by using hootopy analysis ethod. The obtained results show that the low ield is substantially inluenced by the presence o cheical reaction, radiation, and agnetic ield. Key words: cheical reaction, hootopy analysis ethod, theral radiation, HD, nanoluids, stretching sheet The concept o a nanoluid has been advanced by Choi [] who showed substantial augentation o heat transported in suspensions o copper or aluinu nanoparticles in water and other liquids. Nanoluids are a new kind o luid. They are dispersions o nanoparticles in liquids that are peranently suspended by Brownian otion. By using dierent solvents and particles we hope to create coposite luids o widely variable and perhaps copletely new properties. The heat transport properties o nanoluids depend on the theral properties, concentration size and shape o suspended nanoparticles. A nanoluid is a ore or less unior dispersion o solid particles with sall diaeters easured in nanoeters. A large nuber o experiental and theoretical studies have been carried out by nuerous researchers on theral conductivity o nanoluids [, ]. A coprehensive survey o convective transport in nanoluids has been eployed by Buongiorno [4], who gave a satisactory explanation or the abnoral increase o the theral conductivity. Buongiorno and Hu [5], studied on the nanoluid coolant in advanced nuclear systes. It is now well accepted act that the ters HD, theral radiation and heat generation extensively appear in various engineering processes. The HD is signiicant in the control o boundary-layer low and etallurgical processes. Again the theral radiation and heat generation possessions ay arise in high teperature ingredients processing operations. Ingredients * Corresponding author, e-ail: akinded@gail.co
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... 496 THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 ay be intelligently designed thereore with judicious ipleentation o radiative heating to produce the desired characteristics. This recurrently occurs in agriculture, engineering, plasa studies, and petroleu industries. Nuerous low coplications under dierent aspects have been considered by the several scholars. Vajravelu and Hadjinicalaou [6] scrutinized the heat transer characteristics over a stretching surace with viscous dissipation in the presence o internal heat generation or absorption. The eect o radiation on convective heat transer probles have been exained by a nuber o researchers using principally algebraic approxiations or the radiative transer siulation. Takhar et al. [7] eployed a dierential non-gray gas approxiation to study nonlinear gas dynaics in a pereable aterial. Seddeek [8] evaluated the eects o radiation and variable viscosity on hydroagnetic convection low with an aligned agnetic ield using a nuerical ethod and a lux approxiation or radiation. Free convection low o a nanoluid over a linearly stretching sheet in presence o agnetic ield has been studied by Haad [9]. Habibi et al. [] analyzed the HD low o nanoluid over a non-linear stretching sheet with eects o viscous dissipation and variable agnetic ield. Very recently a nuber o studies o HD boundary-layer luid low with the eects o theral radiation and nanoluid were reported in the literature [-7]. Heat and ass transer probles with a cheical reaction have received a considerable aount o attention in recent years. In processes such as drying, evaporation, energy transer in a cooling tower and the low in a desert cooler, heat and ass transer occur siultaneously. Anwar et al. [8] studied the conjugate eects o heat and ass transer o nanoluids over a non-linear stretching sheet. Khan and Pop [9] studied boundary-layer heat-ass transer ree convection lows also in porous edia o a nanoluid past a stretched sheet. Shakhaoath et al. [] analyzed the boundary-layer nanoluid low with HD radiative possessions. Chaka [] studied the HD low over a uniorly stretched vertical pereable surace subject to a cheical reaction. Aiy [] analyzed the HD ree convective low and ass transer over a stretching sheet with a hoogeneous cheical reaction o order n (where n was taken to be,, or ). Cheical reaction eects on HD heat and ass transer low o nanoluid near the stagnation point over a pereable stretching surace in presence o heat source was carried out by Gireesha and udrasway []. Olanrewaju and akinde [4] reported a nuerical solution or the cobined eects o theral diusion and thero diusion on cheically reacting HD boundary-layer low with heat and ass transer past a oving pereable surace. Hootopy analysis ethod (HA), proposed by Liao [5], is a very powerul ethod and has been eployed by nuerous researchers in various physical phenoena [6, 7]. In this paper, we extend the results o the odel presented by Poornia and Bhaskar [] or cheical reaction and heat source/sink. We shall apply HA to solve the siilarity equations obtained ro the governing boundary-layer equations with the help o siilarity transorations. atheatical orulation A steady -D boundary-layer low o an incopressible electrically conducting and radiating nanoluid past a stretching surace is considered under the assuptions that the external pressure on the stretching sheet in the x-direction is having diluted nanoparticles. The x-axis is taken along the stretching surace and y-axis noral to it. A unior stress leading to equal and opposite orces is applied along the x-axis so that the sheet is stretched, keeping the origin ixed. The stretching velocity is assued to be U w (x) = U x where U the unior velocity is and ( ) is a constant paraeter. The luid
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 497 is considered to be a gray, absorbing eitting radiation but non-scattering ediu. A unior agnetic ield is applied in the transverse direction to the low. The luid is assued to be slightly conducting, so that the agnetic eynolds nuber is uch less than unity and hence the induced agnetic ield is negligible in coparison with the applied agnetic ield. Also, there is cheical reaction between the diusing species and the luid. Eploying the Oberbeck-Boussinesq approxiation, the governing equations o the low ield can be written in the diensional or as [-]: u ν + = x y ρ u u p u u v ( C ) ( ) ( ) ( ) TgT T p CgC C Bu x y x µ + = + + y ρ β ρ ρ β σ T T T qr Q T C DT T u + v = α + ( T T ) τ D + B + x y y ( ρc) y ( ρc) y y T y C C C DT T u + v = DB + KC x y y T y () () () (4) where u and v are the velocity coponents in the x- and y-directions, respectively, g the acceleration due to gravity, µ the viscosity, ρ the density o the base luid, ρ p the density o the nanoparticle, β T the coeicient o voluetric theral expansion, β c the coeicient o voluetric concentration expansion, T the teperature o the nanoluid, C the concentration o the nanoluid, T w the teperature along the stretching sheet, C w the concentration along the stretching sheet, T the abient teperature o the nanoluid, C the abient concentration o the nanoluid, D B the Brownian diusion coeicient, D T the therophoresis coeicient, B the agnetic induction, q r the radiative heat lux, (ρc) p the heat capacitance o the nanoparticles, (ρc) the heat capacitance o the nanoluid, α = k /(ρc) the theral diusivity paraeter, k the theral conductivity, τ = (ρc) p /(ρc) the ratio between the eective heat capacity o the nanoparticles aterial and heat capacity o the nanoluid, Q the coeicient o heat generation paraeter, and k the rate o cheical reaction paraeter. u= U( x) = Ux, v=, T= T, C= C at y= (5) w w w, u, v, T T, C C at y (6) By using the osseland approxiation [4-7, ], the radiative heat lux, q r, is given by: e e, 4 4σs T 6σsT T q r = k y k y where σ s is the Stephen-Boltzann constant and k e the ean absorption coeicient. It should be noted that by using the osseland approxiation, the present analysis is liited to optically thick luids. The teperature dierence within the low is assued to be suiciently sall, then T 4 in eq. (7) can be easily linearized about T to give T 4 4T T 4. Invoking eqs. (7) and () gets odiied: (7) T T k T 6σ st T Q T C DT T u + v = + + ( T T ) D + τ B + x y ( ρc) y ke( ρc) y ( ρc) y y T y (8)
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... 498 THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 Using the stea unction ψ = ψ(x, y), the velocity coponents u and v are deined as u = ψ/ y, ν = ψ/ x. Assuing that the external pressure on the sheet, in the direction having diluted nanoparticles, to be constant, the siilarity transorations is taken: νux T T C C ( + ) Ux kk 4 ψ= θ = φ = = = = T T C C Nr + e ( ), ( ), ( ),, Nr, + w w ν 4σ st G ν µ τ D ( C C ) ( C ) ρ gn( T T ) Gr ν δ =, Pr =, ν =, =, =, λ =, Le= e e e D γ B w w Nb Gr / / / x α ρ υ ρν x x ku ( C C ) U ( x) x τ D ( ) ( ) ( ) ( ) T Tw T QU w x ρp ρ gn Cw C, Nt =, Q =, Gr = / ν ν νt xν ρν e w w w =, ex = In view o the siilarity transorations, the eqs. ()-(4) are reduce to: + + ( λθ δφ ) = + + ( ) ( ) x B (9) () ( + ) θ + Pr θ + Pr Nbθφ + Pr Nt θ + Pr Qθ = () Nt φ'' + Le φ' + θ γφ = () Nb where λ is the buoyancy paraeter, δ the solute buoyancy paraeter, Pr the Prandtl nuber, Le the Lewis nuber, n the kineatic viscosity o the nanoluid, Nb the Brownian otion paraeter, Nt the therophoresis paraeter, e x the local eynolds nuber based on the stretching velocity, Gr the local theral Grasho nuber, G the local concentration Grasho nuber, the radiation paraeter, Q the heat generation paraeter, γ the cheical reaction paraeter, and, θ, the diensionless strea unctions, teperature, concentration, respectively. Here, β T and β C are proportional to x, that is β T = nx and β C = n x where n and n are the constants o proportionality. The corresponding boundary conditions are: () =, () =, θ() =, φ() =, ( ) =, θ( ) =, φ( ) = () The skin riction coeicient, C, Nusselt nuber, and Sherwood nuber are iportant physical paraeters given by: C ' ( + ) + + / / / = e (), Nu = e θ (), Sh = e φ () x x x Hootopy analysis solution In order to solve eqs. ()-() analytically using HA, we express the solutions by the set o base unctions as { k exp( n): k, n are integers} in the or: (4) a n n= k= θ n= k= φ n= k= k k ( ) = n, exp( ) k k ( ) = bn, exp( n ) k k ( ) = cn, exp( n ) (5)
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 499 where a k,n, b k,n, c k,n are the coeicients. Then ro eqs. ()-(), it is a straightorward atter to choose, () = e, θ () = e, ϕ () = e, as our initial approxiations or (), θ(), and φ(), respectively. The auxiliary linear operators are then chosen: d d L( ) =, d d with the ollowing properties: d θ L( θ) = θ, d d φ L( φ) d ( + + ) =, Lθ ( Ce 4 + Ce 5 ) =, Lφ ( Ce Ce ) L C Ce Ce φ (6) + = (7) 6 7 where C i (i = -7) are arbitrary constants. Let q [,] represent an ebedding paraeter and ћ, ћ θ, and ћ ϕ denote the non-zero auxiliary linear operators and construct the ollowing zeroth order deoration equations: ( ) ( ) ( ql ) ˆ ; q = q N ˆ( ; (8) ( ) ( ) ( ql ) ˆ ; ˆ ˆ θ θ q θ = qθ θn θ θ ( ;, ( ; (9) ( ql ) ˆ( ; ) ˆ ˆ φ φq φ ( ) q N φ φ θ ( ;, ( ; =, () subject to the boundary conditions: ( ; ) =, ( ; ) =, ( ; ) =, ˆ θ( ˆ θ( ˆ q ˆ q ˆ q where the non-linear operators are deined: ( ˆ φ( ; =, ; = ˆ φ ; =, ; = () ˆ( ; ˆ( ) ˆ ˆ ; q ( ; + ( ; + N ˆ ( ; + = ˆ ˆ ( ; ) ( ; ) ˆ q + λθ q δφ ( ; + ˆ θ( ; ˆ ( ) ˆ θ( ; + + Pr ( ; + N ˆ ( ;, ˆ θ θ ( ; = ˆ θ( ; ˆ φ( ; ˆ θ( ; Pr Nb Pr Nt Pr Q ˆ + + + θ ; q ( ) ( ) ( ) ( ) N ˆ ( ;, ˆ φ φ ( ; ˆ φ; q ˆ ˆ ˆ φ; q Nt θ; q = + Le ( ; + γ ˆ φ ( ; Nb setting q = and q =, we obtain ro eqs. (8)-(): ˆ ( ;) = ( ), ˆ ( ;) = ( ), ˆ θ ( ;) = θ( ), ˆ θ ( ;) = θ ( ), ˆ φ ( ;) = φ( ), ˆ φ ( ;) = φ ( ) (5) ( ) Now let: =! ( ; q q=, θ ( ) ( ; θ =! q q=, φ ( ) ( ; φ =! q q= () () (4) (6)
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... 5 THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 ˆ φ ;, by eans o Taylor s theore with respect to q, we obtain: ˆ q; = + + +, ˆ + q θ q; = θ + θ q, ˆ φ q; = φ + φ q and expanding ˆ ( q; ), ˆ θ( q; ), ( q ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (7) thus: = = = The auxiliary paraeters are properly chosen so that series (7) converge at q = and + + θ θ θ ( ) = ( ) + ( ), ( ) = ( ) + ( ), ( ) ( ) ( ) = The resulting probles at the th order deoration are: L ( ) ( ) X = ( ) θ L θ θ( ) θ ( ) = θ ( ) φ L φ φ ( ) X φ ( ) = ( ) + φ φ φ = + (8) = X (9) φ with ( ) + k k k k + k= + k= ( ) = + ( λθ k δφ k) ( ) + () ( + ) θ ( ) + Pr kθ k + Pr Nb θφ k k + θ k= k= ( ) = + Pr Nt θθ k k + Pr Qθ ( ) k = φ Nt, ( ) = φ ( ) + Le kφ k + θ ( ) γφ ( ), k = Nb X =, > () subject to boundary conditions: ( ) ( ) ( ) ( ) ( ) ( ) ( ) =, =, =, θ =, θ =, φ =, φ = () The general solutions o eqs. ()-(5) are expressed: * ( ) = ( ) + + exp( ) + exp( ) C C C * ( ) ( ) exp( ) exp( ) θ = θ + C4 + C5 () * ( ) = ( ) + exp( ) + exp( ) φ φ C6 C7 where * (), θ * (), and ϕ * () are the particular solutions and the constants are to be deterined by the boundary condition at eq. (). Convergence o the HA The convergence o the series solution given by HA depends strongly auxiliary paraeters ћ, ћ θ, and ћ ϕ. In order to choose appropriate values or these auxiliary paraeters, the ћ, ћ θ, and ћ ϕ curves are displayed at th order approxiations as shown in ig.. It is clear ro ig. that the solution or velocity ield converges or.45 ћ., the solution or the teperature proile converges or. ћ θ., and the concentration distribution converge when. ћ ϕ..
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 5 esults and discussion Coputations were carried out with HA or several non-diensional paraeters. Convergence o the series solution up to 5 th order o approxiation is presented in tab.. It is clearly seen that the convergence is obtained ater 5 th order o approxiation. In order to validate the accuracy o the HA technique, we have copared our HA coputation results o reduced Nusselt and Sherwood nubers with those available in open literature in tab., which are in good agreeent. The inluence o the agnetic paraeter on the velocity is shown in ig.. As the agnetic paraeter increases, the velocity decreases. This is because, an application o the agnetic ield within the boundary-layer produces a resistive type orce known as Lorentz orce which opposes the low, and decelerate the luid otion. The eect o the stretching paraeter on the velocity is shown in ig.. It is ound that, the velocity decreases as the stretching paraeter increases...5..5 () θ () ()..5..5. Figurе. The ћ-curves or the th order o approxiation Table. Convergence o HA solutions or dierent order o approxiations when Pr =.6, Le =, =, λ = δ = γ = = Nb = = Nt = = Q =., and ћ = ћ θ = ћ ϕ =.5 Order o approxiation ''() ϕ () θ ().5.765.6967 5.48.5477.65699 5.48.58.79 5.48.58.7 5.48.58.7 Table. Coparison o reduced Nusselt and Sherwood nubers when Pr = Le =, λ = δ = γ = = Q =, Pr = Le = Paraeters Khan and Pop [9] Anwar et al. [8] Poornia and eddy [] Present results Nb Nt θ () ϕ () θ () ϕ () θ () ϕ () θ () ϕ ()...954.94.954.94.9576.99.9576.988...654.55.654.55.6557.55.6558.557...55.688.55.688.554.688.554.688.4.4.495.68.495.68.49465.684.49464.689.5.5.79.57.79.57.79.57.79.5799 Figure shows the eect o the theral buoyancy paraeter on the velocity. Here, the positive buoyancy orce acts like a avorable pressure gradient and hence accelerates the luid in the boundary-layer. This results in higher velocity as theral buoyancy paraeter, λ, increases. The solute buoyancy paraeter eect on the velocity is illustrated in ig.. It is noticed that as the solute buoyancy paraeter increases, the velocity decreases. Figure 4 shows the eect o the radiation paraeter on the diensionless Pr =, = Nb = Nt = γ =. = Nb = Nt= γ =., Q=. λ = δ =.5, Le = =, Q=. λ = δ =.5, Pr = =, Le =.8.8.6.6.4.4....5..6.... 4 6 8 4 6 8 Figure. Eects o and on ()
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... 5 THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 ' ( ).8 = Nb= Nt = γ =. δ =.5, =, Q =. Le =, Pr = = ' ( ).8 = Nb= Nt = γ =. λ=.5, =, Q =. Le =, Pr = = ' ( ).8 Nb = Nt = γ =., λ = δ =.5 Le = =, Q=., Pr = = ' ( ).8 Nb = Nt = γ =., = = Le = =, Q =., λ= δ=.5.6.6.6.6.4. λ...6..4. δ...6..4....6..4. Pr 4 7 4 6 8 4 6 8 Figure. Eects o λ and δ on () 4 6 8 4 6 8 Figure 4. Eects o and Pr on () θ ( ).8.6.4. = Nb = Nt = γ= δ= λ=. Le = = Pr =, Q=. 4 6 8...6. Table. Values o C, Nu and Sh or dierent values o paraeters when Pr = Le =, = and λ = δ =, Nb = Nt =. θ ( ).8.6.4. = Nb = Nt = γ= δ= λ=. Le = Pr =, Q=., = 4 6 8 Figure 5. Eects o and on θ() Q γ () θ () ϕ ().479.6579.5555.5.59.65.4.44499.5848.55568.7498.5958.44..458.4446.49.4.4845.45484.6974.8.4564.56944.47.46649.556.8495.5.5.4788.965.776694..45.7454.464.46.47676.4494..9.6955.47858.5.8866.6988.478...4964.585.8589.5..47889.48.44.5.465.869.49657.47.66.67 velocity. It is observed that as the radiation paraeter increases, the velocity increases. Figure 4 depicts the eect o the Prandtl nuber on the velocity. It is noticed that, an increase in the Prandtl nuber akes the luid to be ore viscous, which leads to decrease in the velocity. Figures 5 and 5 depict teperature proile or agnetic ield paraeter and stretching paraeter. It is observed that in both the cases an increase in agnetic ield paraeter or stretching paraeter leads to increase in luid teperature. The values o skin riction, Nusselt and Sherwood nubers or dierent values o paraeters as illustrated in igs. 4-6 are displayed in tab.. The skin riction increases with and decreases with, Q (>) and γ. Increase in, γ, and Q (>) decrease the Nu while an increase and (Q < ) will enhance the Nu. The Sh increases with Q (>), γ but decreases with and. Fro the igs. 6 and, it is observed that the teperature along the surace decreases or the case o increasing buoyancy paraeter but increases in the case o increase solute buoyancy paraeter. The eect o the radiation paraeter on the teperature is depicted in ig. 7. It is seen that as the radiation paraeter increases, teperature increases. Figure 7 shows the eect o the Prandtl nuber on the teperature. It is noticed that as the
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 5 θ ( ).8 = Nb = Nt = γ= δ=., = Le = = Pr =, Q =. θ ( ).8 = Nb = Nt = γ= λ=., = Le = = Pr =, Q =. θ ( ).8 Nb = Nt= γ= λ= δ=., = Le = = Pr =, Q =. θ ( ).8 Nb = Nt = γ = λ= δ=. Le = =, Q=. = =.6.6.6.6.4....6..4. δ...6.8.4....6..4. Pr 4 7 4 6 8 4 6 8 Figure 6. Eects o λ and δ on θ() 4 6 8 Prandtl nuber increases, the teperature decreases. This is due to the act that or saller values o Prandtl nuber are equivalent to larger values o theral conductivities and thereore heat is able to diuse away ro the stretching sheet. The teperature proiles or various values o heat source paraeter are shown in ig. 8. In ig. 8 it is identiied that the teperature increases, as the heat source paraeter increases. Figure 8 is graphical representation o teperature distributions or dierent values o the cheical reaction paraeter. It is seen that the concentration o the luid decreases with increase o cheical reaction paraeter. The inluence o the Brownian otion paraeter and therophoresis paraeters are illustrated in igs. 9 and. It is clear that the teperature decreases, as the Brownian otion paraeter increases, while teperature increases as the therophoresis paraeter increases. θ ( ).8 = Nb = Nt = γ = λ = δ =. Le = = Pr =, = θ ( ).8 = Nb = Nt = γ =., λ =.5 Le = =, Q =. = δ= θ ( ).8 4 6 8 Figure 7. Eects o and Pr on θ() = Nt = γ = λ = δ =. Le = = Pr = =, Q=. θ ( ).8 = Nb = γ=., λ =.5 Le = = Pr =, Q=. = δ=.6.6.6.6.4. Q..5..5.4. γ....4. Nb..5..5.4. Nt..5..5 4 6 8 4 6 8 Figure 8. Eects o Q and γ on θ() The concentration proiles or dierent values o the agnetic ield paraeter and stretching paraeter are shown in igs. and, respectively. It is ound that the concentration o the low ield increases as the agnetic ield paraeter or stretching paraeter increases. The inluences o the theral and solute buoyancy paraeters on the concentration ield area shown in igs. and, respectively. It is noticed that the concentration decreases as the theral buoyancy paraeter increases, but concentration increase with an increase o sol- 4 6 8 ( ).8.6.4. 4 6 8 Figure 9. Eects o Nb and Nt on θ() = Nb = Nt = γ = δ =., λ = Le = = Pr =, Q=...5.. 4 6 8 ( ) 4 6 Figure. Eects o and on ϕ().8.6.4. = Nb = Nt = γ = δ =., λ = Le = Pr =, Q =., =.5.5
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... 54 THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 ( ).8 = Nb = Nt = γ= δ=. Le = = Pr =, Q=. = ( ).8 = Nb = Nt = λ = γ=. Le = = Pr =, Q=. = ( ).8 = Nt= γ = =., δ=.5 Le = = Pr =, Q=. λ = ( ).8 = Nb = γ = =., δ=.5 Le = = Pr =, Q=. λ =.6.6.6.6.4....6..4. δ...6.8.4. Nb..5..5.4. Nt..5..5 4 6 8 4 6 8 Figure. Eects o λ and δ on ϕ() 4 6 8 4 6 8 Figure. Eects o Nb and Nt on ϕ() ute buoyancy paraeter. Figure shows the eect o the Brownian otion paraeter on the concentration proiles. It is observed that as Brownian otion paraeter increases, the concentration decreases. The eect o the therophoresis paraeter on the concentration o the low ield is presented in ig.. We notice that the concentration increases as the therophoresis paraeter increases. The concentration proiles or dierent values o the cheical reaction paraeter and Lewis nuber are shown in igs. and, respectively. It is ound that the concentration o the low ield decreases as the cheical reaction paraeter and Lewis nuber increases. ( ).8 Nb = Nt =., λ = δ =.5 Le = = Pr =, Q=. = = ( ).8 Nb = Nt = γ =., λ = δ =.5 = Pr =, Q =. = = ().5 = Nb = Nt = δ = γ =. Le = Pr =, Q =. () -.5 = Nb = Nt = λ = γ =. Le = Pr =, Q =..6.4....5. 4 6 8.6.4. Le 4 6 8 Figure. Eects o γ and Le on ϕ() 5 5-4 5-4 5 Figure 4. Eects o,, λ and δ on C The eects o various physical paraeters on skin riction coeicient, Nusselt nuber, and Sherwood nuber are shown in igs. 4-6. The skin riction coeicient increases with an increase agnetic ield intensity but decreases with a cobined increase in radiation paraeter, heat source paraeter, and cheical reaction. This ay be attributed to an increase or decrease in the velocity gradient at the sheet surace as the paraeter varies. The Nusselt nuber decreases with an increase in agnetic ield, heat source and cheical reaction, while -.5 - -.5 - -.5 λ =,.5, - -.5 - -.5 δ =,.5, θ().5.4. λ =.,.,.5 θ().8.6 Nb = Nt = λ= γ=., = Le = =, Q=. δ = ().9.8 Q=., Le = 5, = Pr = = λ = δ = γ= ().5 Q =., Nb = Nt =., Pr = = = λ = δ = γ =.7...4 δ =..6.5 γ = -. Nb = Nt = δ= γ=. Le = Pr =, Q =., = 4 5 4 5 Pr Figure 5. Eects o,, λ, δ,, and Pr on Nu. δ =....5 Nb.. Nt =.,.,..4 4 5 4 5 6 7 8 9 Le Figure 6. Eects o, Nb, Nt, γ,, and Le on Sh.5 =,,
akinde, O. D., et al.: Cheically eacting on HD Boundary-Layer Flow o... THEAL SCIENCE: Year 8, Vol., No. B, pp. 495-56 55 an increase with heat sink and theral radiation will enhance Nusselt nuber. It is interesting, to note that buoyancy paraeter enhances both the heat and ass transer rate at the sheet surace. oreover, the Sherwood nuber increases with an increase in cheical reaction and heat source but decreases with an increase in agnetic ield and theral radiation. Conclusions In this paper, we have analyzed theoretically the eects o theral radiation, cheical reaction on steady HD heat and ass transer low o nanoluids over a non-linear stretching sheet with heat source. The transored two-point non-linear boundary value proble has been solved with HA. Good agreeent o HA solution is observed with those obtained by nuerical ethods. The present solutions have shown that the velocity decreases and teperature increases with agnetic ield paraeter. A rise in the radiation paraeter raises the teperature as well as rate o heat transer and the presence o radiation radiates the heat energy away ro the luid. It is the nanoluids property to enhance the theral conductivity. There is a rise in the teperature with an increase in the heat generation, stretching paraeters and solute buoyancy orce and a all with an increase in theral buoyancy paraeter and Prandtl nuber. Species concentration increases within the agnetic ield, stretching paraeter while the concentration decreases or an increase in the values o Lewis nuber, cheical reaction paraeter, and Brownian otion paraeter. Skin riction coeicient increases with an increase in the agnetic paraeter and decreases with an increase in the radiation paraeter. The increase in cheical reaction paraeter decreases the diensionless concentration in the concentration boundary-layer. Thus ass transer rate increases with cheical reaction paraeter. eerences [] Choi, U. S., Enhancing Theral Conductivity o Fluids with Nanoparticles, in: Developents and Applications o Non-Newtonian Flows, (Eds. D. A. Siginer, H. P. Wang), FED-Vol. /D-66 ASE, New York, USA, 995, pp. 99-5 [] Fan, J., Wang, L., Heat Conduction in Nanoluids: Structure-Property Correlation, Int. J. Heat ass Trans., 54 (), 9-, pp. 449-459 [] Kleinstreuer, C., Feng, Y., Experiental and Theoretical Studies o Nanoluid Theral Conductivity Enhanceent: A eview, Nanoscale es Lett, 6 (),, pp. 9-4 [4] Buongiorno, J., Convective Transport in Nanoluids, J. Heat Trans., 8 (5),, pp. 4-5 [5] Buongiorno, J., Hu, W., Nanoluid Coolants or Advanced Nuclear Power Plants, Proceedings, ICANPP Seoul, Korea, Curran Assoc Inc. 5, pp. 5-9 [6] Vajravelu, K., Hadjinicalaou, A., Heat Transer in a Viscous Fluid over a Stretching Sheet with Viscous Dissipation and Internal Heat Generation, Int. Coun. Heat ass Trans., (99),, pp. 47-4 [7] Takhar, H. S., et al., Coputational Analysis o Coupled adiation Convection Dissipative Flow in a Porous ediu Using the Keller-Box Iplicit Dierence Schee, Int. J. Energy es., (998),, pp. 4-59 [8] Seddeek,. A., Eects o adiation and Variable Viscosity on a HD Free Convection Flow Past a Sei-Ininite Flat Plate with an Aligned agnetic Field in the Case o Unsteady Flow, Int. J. Heat ass Trans., 45 (), 4, pp. 9-95 [9] Haad,. A. A., Analytical Solution o Natural Convection Flow o a Nanoluid over a Linearly Stretching Sheet in the Presence o agnetic Field, Int. Coun. Heat ass Trans., 8 (), 4, pp. 487-49 [] Habibi,.., et al., ixed Convection HD Flow o Nanoluid over a Nonlinear Stretching Sheet with Eects o Viscous Dissipation and Variable agnetic Field, echanika, 8 (), Sept., pp. 45-4 [] Sandeep, N., Sulochana, C., Dual Solutions o adative HD Nanoluid Flow over an Exponentially Stretching Sheet with Heat Generation/Absorptio,. Appl. Nano Sci., 6 (5),, pp. -9 [] ana, P., Bhargava,., Flow and Heat Transer o a Nanoluid over a Non-Linearly Stretching Sheet: A Nuerical Study, Coun. Nonlinear Sci Nuer. Siul., 7 (),, pp. -6 [] Poornia, T., Bhaskar,. N., adiation Eects on HD Free Convective Boundary Layer Flow o Nanoluids over a Non-Linear Stretching Sheet, Adv. Appl. Sci. es., 4 (), Jan., pp. 9-
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