MIXED CONVECTION FLOW OF A NANOFLUID PAST A NON-LINEARLY STRETCHING WALL

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1 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp MIXED CONVECTION FLOW OF A NANOFLUID PAST A NON-LINEARLY STRETCHING WALL y Adeel AHMAD a,, Saleem ASGHAR a,c, Mudassar JALIL a*, and Ahmed ALSAEDI c a Department o Mathematics, COMSATS Institute o Inormation Technology, Islamaad, Pakistan Laoratory J.-A. Dieudonne, University o Nice Sophia Antipolis, Nice, France c Non-Linear Analysis and Applied Mathematics Research Group, Faculty o Science, King Adulaziz University, Jeddah, Saudi Araia Introduction Original scientiic paper This paper deals ith the oundary-layer mixed convective lo o a viscous nanoluid past a vertical all stretching ith non-linear velocity. The governing equations are transormed into sel similar ordinary dierential equations using appropriate transormation. Using group theoretic method it is shon that the similarity solutions are possile only or the non-linear stretching velocity having speciic orm. Numerical solution o the coupled governing equations is otained using Keller Box method. Correlation expression o reduced Nusselt and Sherood numers are otained y perorming linear regression on the data otained rom numerical results. The authenticity o these results is estalished y calculating the percentage error eteen the numerical results and correlation expression hich is oserved to e less than 5%. Eects o Bronian and thermophoretic diusions and nanoparticles concentration lux on the Nusselt and Sherood numers are discussed. Key ords: mixed convection lo, non-linear stretching, nanoluid, correlation expression The model presented y Buongiorno [1] is considered in this article to examine the mixed convective lo and heat transer o nanoluid past a non-linearly stretching sheet. The prolem o natural convection past a vertical plate as irst studied theoretically and experimentally y Pohlhausen et al. []. Merkin and Pop [3] used a similarity transormation to analyze mixed convection oundary-layer lo over a vertical semi-ininite plate. Steinruck [4] ound a ne similarity solution o mixed convection lo along a vertical plate. Kuznetsov and Nield [5, 6] studied the prolems o natural convection or nanoluid. Considering the impact o stretching and heating/cooling o surace on the quality o the inished product o real processes, the modeling o such processes is undertaken ith the help o dierent stretching velocities and surace temperature distriutions. These stretching are linear [7], polynomial [8], and exponential [9]. A lot o research has een conducted on stretching phenomena in Netonian and non-netonian luids and has een idely and extensively quoted. Recently, the stretching o plate in nanoluid has een a ocus o study due to its applications. Bachok et al. [1] studied the lo o nanoluid over a moving surace. Khan and Pop * Corresponding author, mudassarjalil@yahoo.com

2 568 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp [11] investigated numerically the prolem o laminar lo o nanoluid due to stretching o the surace. Rana and Bhargava [1] investigated the nanoluid lo hich results rom the sheet stretching non-linearly. Matin and Jahangiri [13] presented numerical solution using Keller-ox method or the orced convection MHD lo o nanoluid over a permeale stretching plate ith viscous dissipation. Hayat et al. [14] studied the MHD lo o viscous nanoluid caused y a permeale exponentially stretching surace. In [15] Hayat et al. addresses the mixed convection lo o Casson nanoluid over a stretching surace in presence o thermal radiation, heat source/sink and irst order chemical reaction. Ahmad et al. [16] explores the oundary-layer lo and heat transer o a viscous nanoluid ounded y a hyperolically stretching sheet. Recently, Ahmad [17] studied the lo o a classical non-netonian luid, Reiner-Philippo luid, in the presence o nanoparticles over a non-linear stretching sheet. Why this paper In a recent paper, Xu and Pop [18] solved a prolem o mixed convection lo o a nanoluid over a linearly stretching surace in a uniorm ree stream velocity. They introduced a similarity transormation hich converts the governing equations into a set o sel-similar non-linear ODE (suitale or numerical solution). Using group theoretic method, e sho that that the similarity transormation used in [18] or uniorm ree stream is unique. This means to say that in the asence o ree stream velocity, uniorm or otherise, these equations do not admit any similarity transormation or linear stretching. The question arises; is this condition hold or non-linear stretching as ell. In this paper, e provide an anser to this question and sho that there exists a non-linear stretching o the orm x 1/ or hich the similarity transormations are possile hen there is no ree stream. Ater having ound the similarity transormations y group theoretic method; e solve the prolem o mixed convection lo o nanoluid over a non-linear (x 1/ ) stretching surace in the asence o ree stream. We convert the governing equations into sel-similar non-linear ODE using the similarity so otained. The coupled non-linear resulting equations are solved numerically. The telling points o this study are: (a) the ull range o possile similarity transormations or the lo o nanoluid past linear and non-linear stretching o a vertical plate has een presented y group theoretic method; this ill help the scientists and engineers alike, to discern hat to ocus and hat not to ocus on; hile attempting the ne prolems using similarity transormations, () a sel-similar oundary value prolem or the mixed convection lo o nanoluid past non-linear stretching surace is ormulated and the solution is presented or the irst time, and (c) inding the correlation expression o reduced Nusselt and Sherood numers using linear regression on the numerical results. The prolem ormulation Vertical stretching sheet at y = ith x-axis aligned vertically upard has een considered. The sheet is stretching ith non-linear velocity in x-direction in a viscous ased nanoluid. At the sheet the temperature, T, and the nanoparticle raction, ϕ, take the constant values T and ϕ, respectively. The amient values o T and ϕ are given as T and ϕ. Employing the Oereck Boussinesq approximation and the assumption that the nanoparticle concentration is dilute, the oundary-layer equations or the conservation o total mass, momentum, thermal energy, and nanoparticles are ritten: T T T T φ DT T u + v = α + τ D B + x T (1)

3 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp ρ µ 1 φ ρ β ρ ρ g φ φ x u u u u + v = + g T T p ( ) ( ) ( ) ( ) T T T T φ DT T u + v = α + τ D B + x T φ φ φ D T u + v = DB + x T T y y here u and v are the velocity components in x- and y-direction, respectively, ρ the density o the ase luid, ρ p is the density o particles, and µ, α, and β are the viscosity, thermal conductivity, and volumetric volume expansion coeicient o the nanoluid, respectively. Also, in eqs. (1)-(4), g is the gravitational acceleration, D B the Bronian diusion coeicient, D T the thermophoretic diusion coeicient, and α and τ are deined: k ( ρc) p α =, τ = ( ρc) ( ρc) The appropriate oundary conditions o the prolem are: uxy (, ) = Ux ( ), vxy (, ) =, T= T, φ = φ at y= uxy (, ), T= T, φ = φ as y Introducing the non-dimensional parameters x y ρ UL T T φ φ x =, y =, T =, φ = L L µ T T φ φ u v ρ UL v u = v = v = U U U,, µ µ ρ UL the oundary value prolem o eqs. (1)-(5) and otain the dimensionless equations o the orm: u v + = x () (3) (4) (5) (6) (7) u u u u + v = + λt N rφ x T T 1 T T φ T u + v = + N + Nt x Pr φ φ 1 φ 1 N T u + v = + x Le N t y Le y T T φ φ u = U x v = T = = y = T φ ( ),,, φ at uxy (, ), T=, φ = as y The dimensionless parameters in the previous equations are given: (8) (9) (1) (11)

4 57 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp L T L φ ρp ρ τdb φ λ = ( 1 φ ) g β, Nr = g, N = U U ρ ν τ DT T µ ν ν Nt = N =, ν =, Pr =, Le= νt ρ α D here λ is the Richardson numer, N r the nanoparticles concentration lux parameter, N Bronian motion parameter, N t thermophoresis parameter, Pr the Prandtl s numer, n the kinematic viscosity o the luid, and Le the Leis numer. It is orth mentioning that rather than taking any speciic orm e have assumed that the all stretching velocity is generalized unction o x. Hoever, using group theoretic method (given elo) e reach at the conclusion that similarity transormation or the modeled prolem is possile only i e take stretching velocity U(x) = x 1/. The scaling group o transormations The similarity transormations are otained y introducing the olloing scaling group o transormations [19-1]: εa ε εc εd εe ε p * εq Γ : x = xe, y = ye, u = ue, v = ve, T = Te, φ = φe, U = Ue (1) here ε is a small parameter and a,, c, d, e, p, and q are some constants to e determined. The point transormation, Γ, transorms the co-ordinates (x, y, u, v, T, ϕ, U) to the ne co-ordinates (x *, y *, u *, v *, T *, ϕ *, U * ). Sustituting eq. (1) in eqs. (7)-(11), e get: B u x v + e = * * ε ( a+ + c d) * * (13) u u u u v T N x * * * * ε( a+ + c d) * ε( a+ + c) ε( a+ c e) * ε( a+ c p) * + e = e + λe e * * * r φ * * * * * * * T ε( a+ + c d) * T ε( a+ + c) 1 T ε( a+ + c p) T φ ε( a+ + c e) T u + e v = e + N e e * * * + N * * t * x Pr (15) φ φ 1 φ 1 N T u * * * * * ε( a+ + c d) * ε( a+ + c) ε( a+ + c e+ p) + e v = e + e * * * * x Le Le Nt * * ε( c q) * * * * εe * * ε p * * φ u ( x,) = e U ( x ), v ( x,) =, e T ( x,) = 1, e ( x,) = 1 (17) * * * * * * u ( x, ) =, T ( x, ) =, φ ( x, ) = The previous system o equations ecomes invariant under Γ, hen the parameters deined in eq. (1) satisy the olloing relations: a+ + c d =, a+ + c =, a+ c e=, a+ c p = a+ + c p =, a+ + c e=, a+ + c e+ p = c q =, e=, q = The solution o these equations in terms o a is given: a a = d =, c = (19) 4 (14) (16) (18)

5 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp For these relations o constants, the scaling transormation (1) reduced to the olloing orm: εa εa/4 εa/ εa/4 Γ : x = xe, y = ye, u = ue, v = ve () * ε a/ T = T, φ = φ, U = Ue Expanding the exponentials in () using Taylor series expansion, up to order, ε, e get: εa εa εa x x = ε ax, y y= y, u u = u, v v = v : 4 4 Γ (1) * ε a T T =, φ φ =, U U = U The characteristic equation is otained y denoting the dierences eteen the ne and the original variales as dierentials and equating each term, gives: dx dy du dv dt dφ du = = = = = = ax a a a a () y u v U 4 4 For the simplicity, e take a = 1. No solving the previous system o equations, the olloing similarity transormations are otained: y 1/ 1/4 1/ η=, u= x 1/4 ( η), v= x h( η), T= θη ( ), φ= s( η), U( x) = Cx (3) Cx 1 here C 1 and C are some constants. The equation o continuity gives the actual orm o h(η). It is clear rom eq. (3) that the orm o stretching velocity is non-linear, x 1/. Thus using group theoretic method, e can claim that or the mixed convection lo o nanoluid past a stretching surace the similarity transormation is possile only or U(x) = x 1/. This oservation is given in the literature or the irst time. Here e assume C 1 = 3 1/ / and C = 1, or the sake o simplicity. Using the continuity equation, e can deine stream unction, ψ, such that: ψ ψ u =, v = (4) x Equation (3) together ith eq. (4) gives: ψ = ( η ) (5) 3/4 3 x Note that the to orms or v ill e consistent i: 3 η h( η) = 3 (6) Using the similarity transormations and variales given y eq. (3), the eqs. (7)-(11) are conveniently transormed into ODE: λθ N s = (7) r θ + θ + θ + θ = (8) Pr N s Nt Nt s + Le s + N θ = (9)

6 57 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp suject to the olloing oundary conditions: = 1, =, θ = 1, s= 1 at η = (3), θ, s as η It is important to mentions that all the parameters appearing in the previous equations are ree o independent variales unlike some previous studies on the convective lo o nanoluid. It is also important to mention that recently Xu and Pop [18] has otained the ODE ith constant parameters suject to non-zero ree stream velocity. The results and discussion The physical quantities o practical interest i. e., the local Nusselt numer and the heat lux at the surace q are deined: 1 xq 1 3u 4 o Nu =, q = k( T T ) x θ () k( T T ) ν From this to expressions, conveniently e can rite: 1/ Rex Nu = θ () hich is deined as reduced Nusselt numer and ill denote as Nu r. The value o reduced Nusselt numer is calculated or 75 sets o values o parameters N r, N, and N t and linear regression is perormed on the data hich yield the correlation: Nu r Nu = N.1948N.13118N (31) rest vs. N r vs. N t vs. N N, N t, Nr Figure 1. Reduce Nusselt numer against Bronian motion parameter, thermophoresis parameter, and nano-particles concentration lux parameter or Pr =.5, Le = 1., and λ = 1.. For each curve the other to parameters are ixed to e.1 r t The previous expression or estimated reduced Nusselt numer is otained or Pr =.5, Le = 1., and λ = 1.. This can e seen rom the previous expression that reduced Nusselt numer decrease ith an increase in each parameter N r, N, and N t. The same ehavior o reduced Nusselt numer can e oserved rom ig. 1. For dierent values o Prandtl, Leis, and Richardson numers correlation or the reduced Nusselt numer is given in ta. 1. To estalish the reliaility o correlation o estimated reduced Nusselt numer the maximum percentage error is displayed in ta. 1. The percentage error in all the cases is less than 4% hich conirm the reliaility o these expressions or the practical purposes. Tale 1. Correlation o reduced Nusselt numer (Nu r = C + C Nr N r + C N N + C Nt N t ) and maximum percentage error [1 (Nu Rest Nu)/Nu] hen the values o Bronian, thermophoresis and uoyancy ratio parameters are considered in the interval (,.5) Pr Le λ C C Nr C N C Nt Max. % error

7 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp In similar ashion the estimated expression or the Sherood numer can e ritten. For illustration e are riting one expression or the particular values Pr =.5, Le = 1., and λ = 1.. From previous correlation this can e seen that the reduce Sherood numer increase ith an increase in N and decrease ith an increase N r and N. Expressions (31) and (3) sho that rate o change ith respect to N r is very small. These oservations can also e read rom the numerical results hich are shon in ig.. Conclusions Sh = N + 1.1N.586N (3) Rest r t N, N t, Nr Figure. Reduce Sherood numer against Bronian motion parameter, thermophoresis parameter, and nano-particles concentration lux parameter or Pr =.5, Le =1., and λ = 1.. For each curve the other to parameters are ixed to e.1 Mixed convection-oundary-layer lo o a viscous nanoluid past a vertical all stretching ith non-linear velocity is discussed. Similarity transormations are achieved using group theoretic method. The governing non-linear PDE are transormed into ODE using these transormations. It is shon that similarity solutions are possile only or the non-linear stretching velocity having orm x 1/. Reduced Nusselt and Sherood numers are given through correlation expression y perorming linear regression on the numerical results. It is oserved that oth the reduced Nusselt numer and Sherood numer decrease ith the increase in nano-particles concentration lux, and Bronian motion parameters. Hoever, increasing thermophoresis parameter reduced Nusselt numer is decreased and Sherood numer is increased. Analytical expressions and numerical results ascertain that nano-particles concentration lux parameter plays comparatively role in the variation o reduced Nusselt numer and Sherood numer as compared to other parameters. Sh.6.4. vs. N vs. N r vs. N t Nomenclature a,, c, d, e, p, and q scaling constants, [-] C 1, C similarity constants, [-] c speciic heat, [Jkg 1 K 1 ] D B Bronian diusion coeicient, [m s 1 ] D T thermophoretic diusion coeicient, [m s 1 ] dimensionless stream unction, [-] g gravitational acceleration, [ms ] L reerence length, [m] Le Leis numer, (= n / D B ) [-] N Bronian motion parameter, (= τd B Δϕ / n ) [-] N r nanoparticles concentration lux parameter, [= LΔϕ(ρ p ρ )g / ρ U ], [-] N t thermophoresis parameter, (= τd T ΔT / nt ), [-] Nu local Nusselt numer, [= θ'()], [-] 1/ Nu r reduced Nusselt numer, (= Re x Nu), [-] Pr Prandtl numer, (= n / α), [-] q surace heat lux, [-] Re x local Reynolds numer,(= U x / n ), [-] s dimensionless nanoparticle raction, [= (ϕ ϕ ) / (ϕ ϕ )], [-] Sh Sherood numer, [= s'()], [-] T luid temperature, [K] U stretching velocity, [m s 1 ] u, v velocity components along x- and y-axes, [m s -1 ] U reerence velocity, [ms 1 ] x, y Cartesian co-ordinates, [m] Greek symols α thermal diusivity, (= κ / ρ c ), [m s 1 ]

8 574 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp β volumetric volume expansion coeicient, [K 1 ] ε small perturation parameter, [-] η similarity variale, [= ȳ/ x 1/4 ] [ ] λ Richardson numer, [= LΔT(1 ϕ )gβ / U ], [-] µ dynamic viscosity o the luid, [kgm 1 s 1 ] n kinematic viscosity o the luid, [m s 1 ] ϕ nano-particle raction, [-] ρ density, [kgm 3 ] ψ stream unction, [=(/3 1/ )x 3/4 (η)], [m s 1 ] τ ratio o eective heat capacity o the nanoparticle material to the heat capacity o the luid, [=(ρc) p / (ρc) ], [-] Reerences [1] Buongiorno, J. Convective Transport in Nanoluids. Journal o Heat Transer, 18 (6), 3, pp. 4-5 [] Pohlhausen, E., et al., Der Waermeaustausch zischen esten Koerpern und Fluesigkeiten mit kleiner Reiung und kleiner Waermeleitung. ZAMP-Zeitschrit uer Angeandte Mathematik und Physik,1 (191), pp [3] Merkin, J., Pop, I. The Forced Convection Flo o a Uniorm Stream over a Flat Surace ith a Convective Surace Boundary Condition, Communications in Non-linear Science and Numerical Simulation,16 (11), 9, pp [4] Steinruck, H. Aout the Physical Relevance o Similarity Solutions o the Boundary-Layer Flo Equations Descriing Mixed Convection Flo Along a Vertical Plate, Fluid Dynamics Research, 3 (3), 1-, pp [5] Kuznetsov, A., Nield, D. Natural Convective Boundary-Layer Flo o a Nanoluid past a Vertical Plate, International Journal o Thermal Sciences, 49 (1),, pp [6] Nield, D., Kuznetsov, A., The Cheng-Minkoycz Prolem or the Doule-Diusive Natural Convective Boundary-Layer Flo in a Porous Medium Saturated y a Nanoluid, International Journal o Heat and Mass Transer, 54 (11), 1-3, pp [7] Crane, L. J. Flo past a Stretching Plate, ZAMP-Zeitschrit uer Angeandte Mathematik und Physik, 1 (197), 4, pp [8] Banks, W., Similarity Solutions o the Boundary-Layer Equations or a Stretching Wall, Journal de Mecanique Theorique et Appliquee, (1983), 3, pp [9] Magyari, E., Keller, B. Heat and Mass Transer in the Boundary-Layers on an Exponentially Stretching Continuous Surace, Journal o Physics D: Applied Physics, 3 (1999), 5, pp [1] Bachok, N., et al., Boundary-Layer Flo o Nanoluids over a Moving Surace in a Floing Fluid, International Journal o Thermal Sciences, 49 (1), 9, pp [11] Khan, W., Pop, I. Boundary-Layer Flo o a Nanoluid past a Stretching Sheet, International Journal o Heat and Mass Transer, 53 (1), 11-1, pp [1] Rana, P., Bhargava, R., Flo and Heat Transer o a Nanoluid over a Non-Linearly Stretching Sheet: A Numerical Study, Communications in Non-linear Science and Numerical Simulation, 17 (1), 1, pp. 1-6 [13] Matin, M. H., Jahangiri, P., Forced Convection Boundary-Layer Magnetohydrodynamic Flo o Nano- Fluid over a Permeale Stretching Plate ith Viscous Dissipation, Thermal Science, 18 (14), Supp. 1, pp. S587-S598 [14] Hayat, T., et al., MHD Flo o Nanoluids over an Exponentially Stretching Sheet in a Porous Medium ith Convective Boundary Conditions, Chinese Physics B, 3 (14), 5, 5471 [15] Hayat, T., et al., Mixed Convection Flo o Casson Nanoluid over a Stretching Sheet ith Convectively Heated Chemical Reaction and Heat Source/Sink, Journal o Applied Fluid Mechanics, 8 (15), 4, pp [16] Ahmad, A., et al., Flo and Heat Transer o a Nanoluid over a Hyperolically Stretching Sheet, Chinese Physics B, 3 (14), 7, 7441 [17] Ahmad, A., Flo o Reiner Philippo Based Nano-Fluid past a Stretching Sheet, Journal o Molecular Liquids, 19 (16), Complet, pp θ Superscripts dimensionless temperature, [=(T T ) / (T T )], [-] dierentiation ith respect to η dimensionless quantity * transormed co-ordinates under scaling group o transormation Suscripts p ase luid nano particle condition at the all amient condition

9 THERMAL SCIENCE: Year 18, Vol., No. 1B, pp [18] Xu, H., Pop, I., Mixed Convection Flo o a Nanoluid over a Stretching Surace ith Uniorm Free Stream in the Presence o Both Nanoparticles and Gyrotactic Microorganisms, International Journal o Heat and Mass Transer, 75 (14), Aug., pp [19] Pakdemirli, M., Yurusoy, M., Similarity Transormations or Partial Dierential Equations, SIAM Revie, 4 (1998), 1, pp [] Irahim, F. S., et al., Lie-Group Analysis o Radiative and Magnetic Field Eects on Free Convection and Mass Transer Flo past a Semi-Ininite Vertical Flat Plate, Electronic Journal o Dierential Equations, 39 (5), Apr., pp [1] Jalil, M., et al., Sel Similar Solutions or the Flo and Heat Transer o Poell-Eyring Fluid over a Moving Surace in a Parallel Free Stream, International Journal o Heat and Mass Transer, 65 (13), Oct., pp Paper sumitted: Decemer, 15 Paper revised: Septemer 5, 16 Paper accepted: Septemer 6, Society o Thermal Engineers o Seria Pulished y the Vinča Institute o Nuclear Sciences, Belgrade, Seria. This is an open access article distriuted under the CC BY-NC-ND 4. terms and conditions