Journal of Heat and Mass Transfer Research

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1 Journal o Heat and Mass Transer Research 1 (014) Journal o Heat and Mass Transer Research Journal homepage: Boundary layer lo beneath a uniorm ree stream permeable continuous moving surace in a nanoluid Ioan Pop 1*, Sarkhosh Seddighi,3,4, Noriah Bachok, Fudziah Ismail 1 Department o Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania Department o Mathematics and Institute or Mathematical Research, University Putra Malaysia, UPM Serdang, Selangor, Malaysia 3 Department o Mathematics, Science and Research Branch, Islamic Azad University, Bushehr Branch, Bushehr, Iran 4 Nuclear Science Research School, Nuclear Science and Technology Research Institute (NSTRI), P.O. Box , Tehran, Iran PAPER INFO History: Received 5 February 014 Received in revised orm 4 April 014 Accepted 8 April 014 Keyords: Suction/injection Moving surace Nanoluid Runge-Kutta method Shooting techniques Dual solutions. A B S T R A C T The main purpose o this paper is to introduce a boundary layer analysis or the luid lo and heat transer characteristics o an incompressible nanoluid loing over a permeable isothermal surace moving continuously. The resulting system o non-linear ordinary dierential equations is solved numerically using the ith order Runge Kutta method ith shooting techniques using Matlab and Maple sotares. Numerical results are obtained or the velocity, temperature, and concentration distributions, as ell as the riction actor, local Nusselt number, and local Sherood number or several values o the parameters, namely the velocity ratio parameter, suction/injection parameter, and nanoluid parameters. The obtained results are presented graphically in tabular orms and the physical aspects o the problem are discussed. 014 Published by Semnan University Press. All rights reserved. 1. Introduction Many industrial processes involve the transer o heat by means o a loing luid in either the laminar or turbulent regime as ell as loing or stagnant boiling luids. The processes cover a large range o temperatures and pressures. Many o these applications ould beneit rom a decrease in the thermal resistance o the heat transer luids. This situation ould lead to smaller heat transer systems ith loer capital cost and improved energy eiciencies. An innovative technique, hich uses a mixture o nanoparticles and a base luid, as irst introduced by Choi [1] to develop advanced heat transer luids ith substantially higher conductivities. The resulting mixture o the base luid and nanoparticles having unique physical and chemical properties is reerred to as a nanoluid. Flo and heat in a nanoluid over a stretching/shrinking sheet have become a hot topic and have attracted the interest o many researchers recently. The interest in this ield has been stimulated due to its applications in industrial processes such as in poer generation, chemical processes, and heating or cooling processes. The solid nanoparticles have been suspended into the base luid hich has poor heat transer properties in order to increase its thermal conductivity. Choi et al. [] reported that the thermal conductivity o the base luids increases up to approximately to times ith the addition o a small amount (less than 1% by volume raction) o nanoparticles to the base luids. A good literature on convective lo and applications o nanoluids ere done in the books by Das et al. [3], and Nield and Bejan [4], and in the revie papers by Buongiorno [5-7], Kakaç and Address correspondence to: Ioan Pop, Department o Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania popm.ioan@yahoo.co.uk

2 56 I. Pop / JHMTR 1 (014) Pramuanjaroenkij [8], Wong and Leon [9], Saidur et al. [10], Wen et al. [11], Mahian et al. [1], and many others. The lo induced by a moving boundary is important in the study o extrusion processes and is a subject o considerable interest in the contemporary literature (Fang et al. [13]). For example, materials hich are manuactured by extrusion processes as ell as heattreated materials travelling beteen a eed roll and a indup roll or on conveyor-belts possess the characteristics o stretching/shrinking suraces. Polymer sheets and ilaments are also manuactured by continuous extrusion rom a die to a indup roller hich is located a inite distance aay (Sparro and Abraham [14]). For both impermeable and permeable shrinking sheets, multiple solutions ere discovered (Liao and Pop [15]). Most o these solutions are based on the boundary layer assumption and thereore do not constitute exact solutions o the Navier Stokes equations (Wang [16]). The inluence o thermal radiation on boundary layer lo over a shrinking sheet in a nanoluid has been studied by Zaimi et al. [17]. The partial dierential equations are transormed to the ODE and are solved by shooting alongside ith sixth order o Runge-Kutta integration technique. It is observed that radiation has dominant eect on the heat transer and the mass transer rates. In general, suction tends to increase the skin riction and heat transer coeicients, hereas injection acts in the opposite manner. Bachok et al. [18, 19] have studied the boundary layer lo over a stretching/shrinking surace in a nanoluid. Finally, e mention the paper by Ibrahim et al. [0] on the MHD stagnation point lo and heat transer due to nanoluid toards a stretching sheet. In this study, ive dierent types o nanoparticles, Silver Ag, Copper Cu, Copper Oxide CuO, Titania TiO and Alumina AlO 3 ere considered in the boundary layer lo over a permeable continuous moving surace ith suction and injection. The governing boundary layer equations have been transormed to a to-point boundary value problem using similarity variables. These have been numerically solved using ourth order Runge Kutta method ith shooting technique. The eects o governing parameters on luid velocity, temperature, and particle concentration have been discussed.. Mathematical ormulation Consider a steady lo o a nanoluid in the region y 0 past a moving semi-ininite permeable lat plate, as shon in Fig. 1, here x and y are the Cartesian coordinates measured along the plate and are normal to it, respectively. It is assumed that the plate moves into or out o the origin at the uniorm speed U, here U is the constant velocity o the external (inviscid) lo and is the constant moving parameter. 0 corresponds to the donstream movement o the plate rom the origin and 0 corresponds to the plate moving into the origin (opposing lo). It is also assumed that the mass lux velocity is v( x ), here v( x) 0 corresponds to the suction and v( x) 0 corresponds to the injection or ithdraal o the luid, respectively. Further, e assume that the uniorm temperature and the uniorm nanoluid volume raction at the surace o the plate are T and C, hile the uniorm temperature and the uniorm nanoluid volume raction ar rom the surace o the plate are T and C, respectively (see Kuznetsov and Nield [1]). Under these assumptions, the basic nanoluid conservation equations are (Buongiorno [5-7] and Tivari and Das []): u v 0 x y u u u 1 p n u u v x y n x n x y u v v 1 p n v v v x y n y n x y T T T T u v n x y x y C T C T DB x x y y D T T T T x y C C C C u v DB x y x y D T T T T x y boundary conditions o these equations are: u u U, v v, T T, C C at y 0 u U U, v 0, T T, C C as y here, u and v are the velocity components along the x and y axes, respectively, T is temperature o the nanoluid, (1) () (3) (4) (5) (6)

3 I. Pop / JHMTR 1 (014) C is the nanoparticle volume raction, p is the pressure, is the density o the nanoluid, D B is the Bronian diusion coeicient and D T is the thermophoretic diusion coeicient, ( C) p /( C), here ( C) is the heat capacity o the luid and ( C) p is the eective heat capacity o the nanoparticle material, respectively. n is the eective thermal diusivity o the nanoluid, n is its eective viscosity o the nanoluid, and n is its eective density o the nanoluid, hich are given by (Khanaer et al. [3] or Oztop and Abu-Nada [4]): k n n,.5 n, (1- ) ( C p) n ( C ) (1 )( C ) ( C ) k ( k k ) ( k k ) k k k k k p n p p s n s s ( s ) ( s) here is the dynamic viscosity o the base luid being proposed by Brinkman [5], k n is the thermal conductivity o the nanoluid, k and k s are the thermal conductivities o the base luid and o the solid particles, respectively. ( C p ) n is the heat capacitance o the nanoluid. Strictly, expressions (7) are restricted to spherical (or near spherical) nanoparticles ith other expressions being required or other shapes o nanoparticles. We deine no the olloing boundary layer variables x 1/ y u x, y Re, u, L L U 1/ v u v v Re, u, v, U U U p p T T C C p,, h U T T C C here L is the characteristic length o the plate and Re U L / is the Reynolds number. Substituting Eq. (9) or Eqs. (1) to (5) and using the boundary layer approximation in hich Re 1, e obtain the olloing dimensionless boundary layer equations or the problem under consideration: (8) (7) n u v x y y h D D T B y y T y T y y h h DB h D u v x y T ith the ne boundary conditions as ollos: u u, v v( x), 1, h 1 at y 0 u 1, 0, h 0 as y (11) (1) (13) We look or a similarity solution to Eqs. (9-1) along ith the boundary conditions (13) o the olloing orm (see Weidman et al. [6]): x ( ), ( ), h h( ), y / x (14) here is the stream unction hich is deined in the usual orm as u / y and v / x. Thus, e have: u '( ), v (1 / x ( ) '( ) (15) here primes denote dierentiation ith respect to. In order that Eqs. (9-1) have similarity solutions; it is necessary that v( x) has the olloing orm: v ( x ) (1 / x ) 0 (16) here 0 is the constant mass lux parameter ith 0 0 or suction and 0 0 or injection, respectively. Substituting variables (14) or Eqs. (9) to (1), e obtain the olloing ordinary (similarity) dierential equations: 1 ''' '' 0.5 (1 ) (1 s / ) 1 kn / k '' Pr.5 (1 ) 1 ( C p) s / ( Cp) ' Nb ' h' Nt ' 0 (17) (18) u v 0 x y u u n u u v x y n y (9) (10) Nt h'' Le h' '' 0 Nb (19) and the boundary conditions (13) become: (0), '(0), (0) 1, h(0) 1 0 '( ) 1, ( ) 0, h( ) 0 as (0)

4 58 I. Pop / JHMTR 1 (014) here Pr is the Prandtl number, Le is the Leis number, Nb is the Bronian parameter, and Nt is the thermophoresis parameter, hich are deined as: DB ( C C) Pr, Le, Nb, D B DT ( T T ) Nt T (1) The quantities o practical interest in this study are the skin riction coeicient C, the local Nusselt number Nu x, and the local Sherood number deined as: Sh x, hich are C x, x q Nu x U k ( T T ), xqm Shx D ( C C ) B () here, q, and q m are the surace shear stress, the surace heat lux, and the surace mass lux ould be: n n y y y 0 y 0 q m u T, q k, C DB y y 0 Using variables (8) and (14), e obtain: 1/ 1 (Re x ) C ''(0),.5 (1 ) 1/ kn ( / Re x ) Nux '(0), k ( / Re ) Sh h'(0) 1/ x x (3) (4) here Re x U x / is the local Reynolds number. It is orth mentioning to this end that or 0 (pure viscous luid), Eq. (17) ith the corresponding boundary conditions (0) or ( ) reduces to Eq. (4a) ith the boundary conditions (4b,c,d) rom the paper by Wedman et al. [6]. 3. Results and discussion Numerical solutions to the nonlinear ordinary dierential equations (17-19) ith the boundary conditions (0) ere obtained using the ith order Runge Kutta [7] ith the shooting technique. We ind Figure 1. Physical model and coordinate system: a) lat plate moving out o the origin; b) lat plate moving into the origin the missing slopes ''(0) and '(0), or some values o the governing parameters, namely, the nanoparticle volume raction, the moving parameter, and the suction/injection parameter 0 using the Maple and Matlab sotares. Five types o nanoparticles ere considered, namely, Ag, Cu, CuO, TiO, and AlO 3. Folloing Oztop and Abu-Nada [4], the value o the Prandtl number Pr is taken as 6. (or ater) and the values o the volume raction parameter is rom 0 to 0. ( ) in hich 0 corresponds to the pure (Netonian) luid. It is orth mentioning that e have used data related to thermophysical properties o the luid and nanoparticles as listed in table 1 to compute each case o the nanoluid. The numerical results are summarized in Table and Figs. to 16. Figs. () to (7) sho the variation o ''(0) (skinriction coeicient) ith respect to or Ag, Cu, CuO, TiO, and AlO3 - ater nanoluids and dierent values o 0 hen Nb 0.3, Nt 0.1, Le 1, Pr 0.1, and 0.1. It is seen that the solution is unique hen 0. There are to solutions (upper and loer branches) hen c 0 (opposite lo), and no solution hen c 0, here c is the critical value o or hich

5 I. Pop / JHMTR 1 (014) the solution exists. The values o ''(0) are positive hen 1, and they become negative hen the value o exceeds 1, or all values o the suction/injection parameter 0. Physically, the positive value o ''(0) means that the luid exerts a drag orce on the plate, and the negative value means the opposite. The zero value o ''(0) hen 1 does not mean separation, but it corresponds to the equal velocity o the plate and the ree stream. A comparison o the obtained values c or several values o 0 ith those reported by Weidman et al. [6] is given in Table. It is seen that the results are in a very good agreement so that e are conident that the present results are accurate. Fig. (8) shos the variation o the reduced skin-riction coeicient ''(0) ith respect to or Ag, Cu, CuO, TiO, and AlO3 -ater nanoparticles hen Nb 0.3, Nt 0.1, Le 1, Pr 0.1, 0 0, and 0.1. The skin-riction ''(0) increases hen the nonoparticles have the order o Ag Cu CuO TiO AlO3. As it is clear rom Fig. (7), the dierence beteen to nonoparticles TiO and AlO 3 is so less compared to other particles. Figs. (9) and (10) sho the variation o '(0) ith respect to or Ag- ater and AlO3 - ater nanoluids and dierent values o the 0 hen Nb 0.3, Nt 0.1, Le 1, Pr 0.1, 0 0,and 0.1. It is seen that the solution is unique hen 0, hile dual solutions are ound to exist hen 0 i.e. hen the plate and the ree stream move in the opposite directions. The values o - h' (0) are positive or all values (positive or negative) o and or all values o the suction/injection parameter 0. The variation o ''(0) and '(0) ith respect to or Ag and AlO3 - ater nanoparticles and dierent values o nanoparticle volume raction ( 0 and 0. ) hen Nb 0.3, Nt 0.0, Le 1, Pr 0.1, and 0 0 (impermeable plate) has been shon in Figs. (11) to (14). The values o ''(0) are positive hen 1, and they become negative hen the value o exceeds 1, or both values o the parameter considered. The values o '(0) are positive or all values o and or both values o nanoparticle volume raction. The values o ''(0) and '(0) increase hen the variable increases rom 0 Figure. Variation o the reduced skin-riction coeicient ''(0) ith or 0 and dierent values o 0 hen Nt = 0.1, Nb=0.3, Le=1, and Pr=6.. Figure 3. Variation o the reduced skin-riction coeicient ''(0) ith or Ag-ater nanoluid and dierent values o 0 hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. to 0.. The variation o c or dierent values o ( 0, 0. ) is very iddling. Figs. () to (14) also sho that or a particular value o 0, the solution exists up to the certain critical value o c 0 or 0. Beyond this value, the boundary layer approximations break don, and thus the numerical

6 60 I. Pop / JHMTR 1 (014) Figure 4. Variation o the reduced skin-riction coeicient ''(0) ith or Cu-ater nanoluid and dierent values o 0 hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. Figure 6. Variation o the reduced skin-riction coeicient ''(0) ith or TiO -ater nanoluid and dierent values o 0 hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. Figure 5. Variation o the reduced skin-riction coeicient ''(0) ith or CuO-ater nanoluid and dierent values o 0 hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. solution cannot be obtained. The boundary layer separates rom the surace at c 0. Based on our computations, the critical values o c are presented in Table (), hich sho that or all nanoparticles considered, the values o c increase as 0 increases. Figure 7. Variation o the reduced skin-riction coeicient ''(0) ith or AlO3 -ater nanoluid and dierent values o 0 hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. hence, suction delays the boundary layer separation, hile injection accelerates it.

7 I. Pop / JHMTR 1 (014) Figure 8. Variation o the reduced skin-riction coeicient ''(0) ith or Ag, Cu, CuO, TiO,AlO3 -ater nanoluid hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. Figure 10. Variation o '(0) ith or AlO3 -ater nanoluid hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0.1. Figure 9. Variation o '(0) ith or Ag -ater nanoluid hen Nt= 0.1, Nb=0.3, Le=1, Pr=6., and 0.1. Finally, Figs. (15) and (16) present the velocity '( ) and the temperature ( ) proiles or Ag, Cu, CuO, TiO, and AlO3 - ater nanoluids hen Figure 11. Variation o the reduced skin-riction coeicient ''(0) ith or Ag -ater nanoluid and dierent values o hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0 0. Nb 0.3, Nt 0.1, Le 1, Pr 0.1, 0 0, 0.1, and 0.3. It can be seen that all these proiles asymptoticaly satisied all asymptotically the ar ield

8 6 I. Pop / JHMTR 1 (014) Figure 1. Variation o the reduced skin-riction coeicient ''(0) ith or AlO3 -ater nanoluid and dierent values o hen Nt= 0.1, Nb=0.3, Le=1, Pr= 6., and 0 0. Figure 14. Variation o '(0) ith or AlO3 -ater nanoluid and dierent values o hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0 0. Figure 15. Velocity proile or Ag, Cu, CuO, TiO,AlO3 - ater nanoluids hen Nt=0.1, Nb=0.3, Le=1, Pr=6., 0.1, 0.1, and 0 0. Figure 13. Variation o '(0) ith or Ag -ater nanoluid and dierent values o hen Nt=0.1, Nb=0.3, Le=1, Pr=6., and 0 0. boundary conditions equation (19). In these igures the solid lines and the dash lines are or the upper and loer branch solutions, respectively. These velocity and temperature proiles support the existence o dual nature o solutions presented in Figs. () up to (9). The velocity proiles or the upper and loer branch solutions hen 0.3 in Fig. (15) sho that the velocity gradient at the surace is positive, hich produces positive value o the skin riction coeicient. The temperature gradient at the surace as shon in Figs. (14) and (15) is in agreement ith the curves shon in Figs. () up to (9).

9 I. Pop / JHMTR 1 (014) Figure 16. Temperature proiles or Ag, Cu, CuO, TiO,AlO3 - ater nanoluids hen Nt=0.1, Nb=0.3, Le=1, Pr=6., 0.1, 0.1, and 0 0. Table 1.Thermophysical properties o luid and nanoparticles (Oztop and Abu-Nada []) Physical properties Fluid phase (ater) Ag Cu CuO AlO 3 TiO C p ( J/kg K ) (kg/m ) k (W /mk) Table. Comparison o the values o c or various 0 hen 0 (pure luid) 0 c Weidman et al. [4 ] c Present Conclusion This paper have been theoretically the existence o dual similarity solutions in boundary layer lo over a moving surace immersed in a nanoluid ith suction and injection eects have been theoretically studied. The governing boundary layer equations ere solved numerically using the ith order Runge Kutta method ith shooting

10 64 I. Pop / JHMTR 1 (014) technique using the Matlab 1a sotare. Discussion ere carried out or the eects o nanoparticle volume raction, suction/injection parameter 0, and the moving parameter on the skin riction coeicient the local Nusselt number ''(0) and '(0). It as ound that dual solutions exist hen the plate and the ree stream move in the opposite directions. It as also shon that introducing the suction increases the range o or hich the solution exists, and in consequence delays the boundary layer separation, hile it as ound that the injection acts in the opposite manner. Acknoledgements The authors ish to express their thanks to the anonymous Revieers or their valuable comments and suggestions. The inancial supports received rom the Ministry o Higher Education, Malaysia (Project codes: FRGS/1/01/SG04/UPM/03/1) and the Research University Grant (RUGS) rom the University Putra Malaysia are grateully acknoledged. Nomenclature c p C k Nu x Pr q Re T T T x u, v U U u, v x, y Greek letters Speciic heat capacity skin riction coeicient dimensionless stream unction thermal conductivity local Nusselt number Prandtl number surace heat lux local Reynolds number plate temperature luid temperature ambient temperature velocity components along the and directions, respectively plate velocity ree stream velocity Components o velocity Cartesian coordinates along and normal to the surace, respectively Thermal diusivity nanoparticle volume raction Dynamic viscosity Kinematics viscosity Subscript s n Superscript ' Reerences Density Dimensionless temperature velocity ratio parameter surace shear stress stream unction similarity variable solid Fluid nanoluid ambient condition condition at the surace o the plate dierentiation ith respect to [1]. S.U.S. Choi, Enhancing thermal conductivity o luids ith nanoparticles. In: Developments and Applications o Nonnetonian Flos (D. A. Singer and H. P. Wang, Eds.), American Society o Mechanical Engineers, Ne York, NY, USA, 31, (1995). []. S.U.S Choi, Z.G. Zhang, W. Yu, F.E. Lockood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Appl. Phys. Lett., 79, 5-54 (001). [3]. S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanoluids: Science and Technology, Wiley, Ne Jersey, (007). [4]. D.A. Nield, A. Bejan, Convection in Porous Media (4 th edition), Springer, Ne York, (013). [5]. J.Buongiorno, Convective transport in nanoluids., ASME J. Heat Transer, 18, (006). [6]. M. J. Maghrebi M. Nazari T. Armaghani, Forced Convection Heat Transer o Nanoluids in a Porous Channel, Transp Porous Med, 93, (01). [7]. T. Armaghani, M.J. Maghrebi, A.J. Chamkha, M, Nazari, Eects o Particle Migration on Nanoluid Forced Convection Heat Transer in a Local Thermal Non- Equilibrium Porous Channel, 3, (014). [8]. S. Kakaç, A. Pramuanjaroenkij, Revie o convective heat transer enhancement ith nanoluids, Int. J. Heat Mass Transer, 5, (009). [9]. K.V. Wong, O.D. Leon, Applications o nanoluids: current and uture, Adv. Mech. Eng., Article ID , 1-11 (010). [10]. R. Saidur, K.Y. Leong, H.A. Mohammad, A revie on applications and challenges o nanoluids, Reneable and Sustainable Energy Revies, 15, (011). [11]. D. Wen, G. Lin, S. Vaai, K. Zhang, Revie o nanoluids or heat transer applications, Particuology, 7, (011). [1]. O. Mahian, A. Kianiar, S.A. Kalogirou, I. Pop, S. Wongises, A revie o the applications o nanoluids in solar energy, Int. J. Heat Mass Transer, 57, (013). [13]. T. Fang, S. Yao, J. Zhang, A. Aziz, Viscous lo over a shrinking sheet ith a second order slip lo model, Commun. Nonlinear Sci. Numer. Simulat, 15, (010). [14]. E.M. Sparro, J.P. Abraham, Universal solutions or the streamise variation o the temperature o a moving sheet in the presence o a moving luid, Int. J. Heat Mass Transer, 48, (005).

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