Second Order Slip Flow of Cu-Water Nanofluid Over a Stretching Sheet With Heat Transfer

Transcription

1 Second Order Slip Flow o Cu-Water Nanoluid Over a Stretching Sheet With Heat Transer RAJESH SHARMA AND ANUAR ISHAK School o Mathematical Sciences, Faculty o Science and Technology Universiti Kebangsaan Malaysia UKM Bangi-436, Selangor, Malaysia raj.juit@gmail.com Abstract: - The boundary layer low o Cu-water based nanoluid with heat transer over a stretching sheet is numerically studied. Second order velocity slip low model is considered instead o no-slip at the boundary. The governing partial dierential equations are transormed into ordinary one using similarity transormation, beore being solved numerically. Numerical solutions o these equations are obtained using inite element method (FEM). The variations o the velocity and temperature distribution as well as the skin riction and the heat transer coeicients or some values o the governing parameters, namely, the nanoparticle volume raction and slip parameters are shown graphically and discussed. Comparison with published results or pure luid low case is presented and it is ound to be in excellent agreement. Key-Words: - Nanoluid, Stretching surace, heat transer, FEM, velocity slip condition Introduction The boundary layer low over a stretching sheet plays an important role in aerodynamic, extrusion o plastic sheet, metal-spinning, manuacture o plastic and rubber sheets, paper production etc and thus, remains at the leading edge o technology development. In the industrial operation, metal or more commonly an alloy, is heated until it is molten, whereupon it is poured into a mould or dies which contains a cavity, o required shape. The hot metal issue rom the die is subsequently stretched to achieve the desired product. When the super heated melt issue comes out rom the die, it loses its heat and contract as it cools, this is reerred as liquid state contraction. With urther cooling and loss o latent heat o usion, the atoms o the metal lose energy and become closely bound together in a regular structure. The quality o the inal product greatly depends on the rate o cooling and the process o stretching. In view o such applications, Crane [] initiated the analytical study o boundary layer low due to a stretching sheet. He assumed the velocity o the sheet to vary linearly as the distance rom the slit and obtained an analytical solution. Ater this pioneering work, the low over a stretching surace has drawn considerable attention and a good amount o literature in dierent ield has been generated on this problem [-]. In these studies the no slip o the luid velocity relative to the solid boundary was considered. It is a well-known act that, a viscous luid normally sticks to the boundary. But, there are many luids, e.g. particulate luids, rareied gas etc., where there may be a slip between the luid and the boundary [-]. Wang [3] reported that the partial slip between the luid and the moving surace may occur in particulate luid situations such as emulsions, suspensions, oams and polymer solutions. Fang et al. [4] gave a closed orm solution or slip MHD viscous low over a stretching sheet. Wang [5] investigated the eect o surace slip and suction on viscous low over a stretching sheet. Sajid et al. [6] analyzed the stretching low with general slip condition. Sahoo [7] investigated the low and heat transer solution or third grade luid with partial slip boundary condition. Bhattacharyya et al. [8] analyzed the boundary layer orce convection low and heat transer past a porous plate embedded in the porous medium with irst order velocity and temperature slip eect. Das [9] examined the inluence o partial slip, thermal radiation, chemical reaction and temperature-dependent luid properties on heat and mass transer in hydro-magnetic micropolar luid low over an inclined permeable plate with constant heat lux and non-uniorm heat source/sink. Das [] examined the eects o partial slip, thermal buoyancy and heat E-ISSN: 4-347X 6 Volume 9, 4

2 generation/absorption on the low and heat transer o nanoluids over a permeable stretching surace. Noghrehabadi et al. [] analyzed the eect o partial slip on low and heat transer o nanoluids past a stretching sheet. Zheng et al. [] analysed the eect o velocity slip with temperature jump on MHD low and heat transer over a porous shrinking sheet. However, in all o these papers, only the irst order Maxwell slip condition was considered. Recently, Wu [3] proposed a new second order slip velocity model. Fang et al. [4] analyzed the eect o second order slip on viscous luid low over a shrinking sheet. Fang and Aziz [5] studied the low o a viscous luid with a second order slip over a stretching sheet without considering the heat transer aspect. Nandeppanavar et al. [6] studied the second order slip low and heat transer over a stretching sheet. To the best o the authors knowledge, no inormation available on the eect o second order slip on the low and heat transer o a nanoluid past a stretching sheet. Thereore, in the present paper, we investigate the eect o second order slip on the low and heat transer over a stretching sheet immersed in a Cu-water nanoluid. The nanoluid model proposed by Tiwari and Das [7], which analyze the behavior o nanoluids taking into account the solid volume raction is employed. The mathematical model o the problem is highly non-linear whose analytical solution is very hard to ind out, so the only choice let is approximate numerical solution. Thereore, in this study, inite element method is used as a tool or the numerical simulation. Numerical results o the local skin riction coeicient and the local Nusselt number as well as the velocity and temperature proiles are presented or dierent values o the physical parameters. Problem ormulation Consider the two-dimensional low over a lat sheet with heat transer in a water based nanoluid containing Cu nanoparticles. We assume that the sheet coincides with the plane y = and the low is conined to y >. Two equal and opposite orces are applied along the x - axis so that the wall is stretched keeping the origin ixed. It is assumed that the sheet is stretched with velocity uw = cx, where c > is the stretching rate. It is also assumed that the base luid (i.e. water) and the nanoparticles are in thermal equilibrium. The thermo-physical properties o the water and Cu are given in table. Assuming that the nanoluid is viscous and incompressible, and using the nanoluid model as proposed by Tiwari and Das [7], the governing boundary layer equations o mass, momentum and thermal energy or nanoluids can be written as (see Tibari and Das [7]), u v + = () x y ρ n u u + v u = µ u n () x y y T T T ( ρ c p) n u + v = kn (3) x y y where y is the coordinate measured in the direction normal to the sheet, u and v are the velocity components along the x axes and y axes, T is the nanoluid temperature, ρ n is the eective density o the nanoluid, µ n is the eective dynamic viscosity o nanoluid, k n is the thermal conductivity and ( ρ c p ) n is the heat capacity o the nanoluid, which are given by µ ρn = ( φ) ρ + φρ s, µ n =.5 ( φ) ( ρc ) = ( φ)( ρc ) + φρ ( c ) (4) p n p p s kn ( ks + k ) φ( k ks ) = k ( k + k ) + φ( k k ) s s Table. Thermo-physical properties o water and Cu φ ρ c p k 3 ( kg m ) ( J Kg K ( Wm K ) Pure water (H O) Copper (Cu) where φ is the solid volume raction o the nanoluid, ρ is the density o the base luid, ρ s is the density o the nanoparticle, µ is the dynamic viscosity o the base luid, ( ρ c p) is the heat capacity o the base luid, ( ρ c ) is the heat capacity p s E-ISSN: 4-347X 7 Volume 9, 4

3 o the nanoparticle, k is the thermal conductivity o the base luid and k s is the thermal conductivity o the solid nanoparticle. Eqs. () - (3) are subjected to the ollowing boundary conditions: x u = uw + Uslip, v=, T = Tw = T + C at y = l (5) u =, T = T as y where U slip is the slip velocity at the wall. The Wu s slip velocity model (valid or arbitrary Kundsen numbers, K n ) is used in this paper and is given as ollows [3] 3 3 αl 3 l u U slip = λ 3 α Kn y 4 u + ( ) 4 l l λ (6) Kn y u u = A + B y y Following Nandeppanavar et al. [6], we introduce the ollowing similarity transormation: / u = cx ( ), v = ( cν ) ( ), = y c / ν, θ ( ) = ( T T )/( Tw T ) (7) where primes denote dierentiation with respect to. Using transormation (7), Eq. () is automatically satisied, while Eqs. () and (3) respectively reduce to the ollowing nonlinear ordinary dierential equations: µ n s + ( φ) + φ ρ ( ) = µ ρ (8) k ( ) n ρcp s θ + Pr ( φ) + φ ( θ θ) = k ( ) ρcp (9) subject to the boundary conditions () =, () = + γ () + δ (), θ() = ( ), θ ( ) () where γ = A c/ ν ( > ) is the irst order velocity slip parameter, δ = Bc ( / ν ) ( < ) is the second order velocity slip parameter and Pr = µ / k is the Prandtl number. Physical quantities o interest are the skin riction coeicient C and the local Nusselt number Nu, which are deined as τ w, qw C = Nu = () ρ uw k ( Tw T ) where τ w is the surace shear stress and q w is the surace heat lux, which are given by τ w = µ u T n, qw = kn y y= y () y= Using the similarity variables (5), we obtain / k / n Re x C = (), Re = ().5 x Nu ( ) k θ (3) φ where Re = u x/ ν is the local Reynolds number. x w 3 Method o Solution The set o ordinary dierential equations (8)-(9) are highly non-linear, and cannot be solved analytically. Thereore, the inite element method [8-3] is implemented to solve this system numerically. For computational purposes, the dimensionless spatial coordinate is discretized by uniorm elements o step size h =.. Care has been taken in choosing or a given set o parameters because or a ixed value o (where corresponds to ) or all calculations may produce inaccurate results. The Gauss quadrature ormula has been used to calculate the integrals. Owing to the nonlinearity o the system o equations an iterative scheme has been used to solve it. An initial guess is taken at each node point. The system o equations is then linearized by incorporating the unctions, which are assumed to be known values o the unctions and θ. Ater applying the given boundary conditions, the remaining system o equations has been solved using Gausselimination method. This gives us new values o unknowns. This process continues till the absolute dierences o two successive iterate value o unknowns is less than the accuracy o.. 4 Code veriication In order to veriy the accuracy o the applied numerical scheme, comparisons o the present results corresponding to the values o heat transer coeicient or γ =, δ = and φ = or prescribed surace temperature case are made with the available results o Grubka and Bobba [3] and Chen [4], as E-ISSN: 4-347X 8 Volume 9, 4

4 presented in Table. The results are ound in excellent agreement, and thus gives conidence that the numerical results in our case are accurate. Table. Comparison with previous studies or θ () γ = δ = φ =. Pr { } Grubka & Babba [3] Chen [4] Present Table 3. Reduced skin riction coeicient and reduced Nusselt number or various value o φ φ Skin riction Nusselt Number Table 4. Reduced skin riction coeicient or various value o γ and δ γ δ =.5 δ =. δ =. δ = Table 5. Reduced Nusselt number or various value o γ and δ γ δ =.5 δ =. δ =. δ = Results and discussion A systematic study o selected control parameters governing the low regime i.e. nanoparticle concentration ( φ ), irst order slip parameter ( γ ) and second order slip parameter ( δ ) has been conducted and the results are depicted in Figs. - and Tables 3-5. In the present computations the ollowing deault parameter values have been prescribed: φ =., γ =., δ = -., Pr = 6. (water). Variations o the reduced skin riction coeicient and the reduced Nusselt number with respect to nanoparticle concentration ( φ ), irst order slip parameter ( γ ) and second order slip parameter ( δ ) are shown in Tables 3-5. It is observed that the reduced skin riction coeicient is an increasing unction o nanoparticle concentration ( φ ), whereas it a decreasing unction o irst order slip parameter ( γ ) and second order slip parameter ( δ ). The heat transer rate is also an increasing unction o nano particle concentration ( φ ) and a decreasing unction o irst order slip parameter ( γ ) and second order slip parameter ( δ ). The positive value o the reduced Nusselt number shows that the heat is transerring rom the plate to the luid i.e. cooling o the plate. Thus, it can be concluded that the nanoluid can be eectively used or the ast cooling o the plate, while slip eect slow down the heat transer rate. Figs. - depict the eect o nanoparticle concentration ( φ ) on the variation o velocity and temperature in the boundary layer. On observing these igures, we see that velocity decreases and temperature increases in the boundary layer region with the increase o nanoparticle concentration. It has been ound that when the volume raction o the nanoparticle increases rom to., the thickness o the thermal boundary layer increases (See Fig. ). Since nanoparticle enhanced the thermal conductivity o the luid, and higher values o thermal conductivity are accompanied by higher values o thermal diusivity. The high value o thermal diusivity causes a drop in the temperature gradients and accordingly increases the boundary layer thickness as demonstrated in Fig.. As temperature gradient decreases with the increase o nanoparticle raction, so Nusselt number i.e. rate o heat transer should also reduce with the increase o nanoparticle raction rom to.. But in case o nanoluid, the Nusselt number is a multiplication o temperature gradient and the thermal conductivity ratio (conductivity o the nanoluid to the conductivity o the base luid). The reduction in temperature gradient E-ISSN: 4-347X 9 Volume 9, 4

5 due to the presence o nanoparticles is much smaller than thermal conductivity ratio, which accompanied the enhancement o Nusselt number by increasing the volume raction o nanoparticles, as it can be seen rom Table 3. '() φ=. φ=. φ=. Fig. Velocity distribution or various value o φ θ() φ=. φ=. φ=. Fig. Temperature distribution or various value o φ Figs. 3-6 show the eect o irst order slip parameter on velocity, temperature, shear stress and eective temperature gradient in low ield. It is seen that or increased irst order slip the lateral velocity decreases near the surace but increases at large distance. Thus, irst order slip o luid on the stretching surace causes decrease in low velocity. Fig. 4 shows that temperature o low ield increases with the increase o irst order slip parameter. These results are in very good agreement with reported results in partial slip case by Mahmoud [9] and Noghrehabadi []. It is observed rom Figs. 5-6 that the magnitude o the wall shear stress i.e..5 ( / ( φ ) ) () decreases with the increase o irst order slip parameter and the value o the eective temperature gradient at the wall i.e. ( k / k ) θ () is also decreases with the increase n o irst order slip parameter, which is in agreement with the results presented in Table 4-5. '() Fig. 3 Velocity distribution or various value o γ θ() γ=.5 γ=. γ=. γ= γ=.5 γ=. γ=. γ=3. Fig. 4 Temperature distribution or various value o γ (-φ) -.5 ''() γ=.5 γ=. γ=. γ=3. Fig. 5 Shear stress distribution or various value o γ Figs. 7- depict the inluence o second order slip parameter on the velocity, temperature, shear stress and eective temperature gradient proile in boundary layer region. From these igures, we observe that velocity decreases and temperature increases with the increase o second order slip E-ISSN: 4-347X 3 Volume 9, 4

6 parameter δ. The magnitude o the wall shear stress and eective temperature gradient at the wall decrease with the increasing value o δ. -(K n /K )θ'() γ=.5 γ=. γ=. γ=3. In this paper, the problem o two-dimensional low o a viscous and incompressible Cu-water nanoluid over a stretching lat sheet with second order slip conditions is studied. The governing partial dierential equations or mass, momentum and energy are transormed into ordinary dierential equations using a similarity transormation. These equations were solved numerically using inite element method. We ound that irst order and second order slip parameters reduce the skin riction as well as the rate o heat transer. The results also indicate that with the increase o nanoparticle volume raction, skin riction coeicient as well as heat transer rate increases Fig. 6 Temperature gradient distribution or various value o γ.3.5. δ=-.5 δ=-. δ=-. δ=-3. (-φ) -.5 ''() δ=.5 δ=. δ=. δ=3. '() Fig. 9 Shear stress distribution or various value o δ Fig. 7 Velocity distribution or various value o δ δ=-.5 δ=-. δ=-. δ=-3. -(K n /K )θ'().5.5 δ=.5 δ=. δ=. δ=3. θ() Fig. Temperature gradient distribution or various value o δ Fig. 8 Temperature distribution or various value o δ 6 Conclusions 7 Acknowledgements The inancial support received rom the Universiti Kebangsaan Malaysia (Project Code: DIP--3) is grateully acknowledged. E-ISSN: 4-347X 3 Volume 9, 4

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8 Navier boundary condition, Int. J. Thermal Sci., Vol. 58,, pp R.K. Tiwari, M.K. Das, Heat transer augmentation in a two-sided lid-driven dierentially heated square cavity utilizing nanoluids, Int. J. Heat Mass Trans., Vol. 5, 7, pp J.N. Reddy, An introduction to the inite element method, Mc Graw-Hill Int. Editions, New York, R. Bhargava, R. Sharma, O.A. Beg, Oscillatory chemically-reacting MHD ree convection heat and mass transer in a porous medium with Soret and Duour eects: inite element modeling, Int. J. Appl. Math. Mech., Vol. 5, No. 6, 9, pp R. Sharma, R. Bhargava, P. Bhargava, A numerical solution o unsteady MHD convection heat and mass transer past a semi-ininite vertical porous moving plate using element ree Galerkin method, Comp. Material Sci., Vol. 48, No. 3,, pp E-ISSN: 4-347X 33 Volume 9, 4