BIOCONVECTION HEAT TRANSFER OF A NANOFLUID OVER A STRETCHING SHEET WITH VELOCITY SLIP AND TEMPERATURE JUMP

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1 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp BIOCONVECTION HEAT TRANSFER OF A NANOFLUID OVER A STRETCHING SHEET WITH VELOCITY SLIP AND TEMPERATURE JUMP by Bingyu SHEN a, Liancun ZHENG a*, Chaoli ZHANG a,b, and Xinin ZHANG b a School o Mechanical Engineering, University o Science and Technology Beijing, Beijing, China b School o Mathematics and Physics, University o Science and Technology Beijing, Beijing, China Original scientiic paper This paper presents an investigation or bioconvection heat transer o a nanoluid containing gyrotactic microorganisms over a stretching sheet, in which the eects o radiation, velocity slip, and temperature jump are taken into account. The non-linear governing equations are reduced into our ordinary dierential equations by similarity transormations and solved by homotopy analysis method, which is veriied with numerical results in good agree. Results indicate that the density o motile microorganisms and gyrotactic microorganisms increase with bioconvection Rayleigh number, while decrease with increasing in bioconvection Peclet number and bioconvection Lewis number. It is also ound that the Nusselt number, Sherwood number, and gyrotactic microorganisms density depend strongly on the buoyancy, nanoluids, and bioconvection parameters. Key words: nanoluid, gyrotactic microorganisms, bioconvection, velocity slip, temperature jump Introduction As a new generation o highly eicient heat transer luid, nanoluids have been etensively investigated by many researchers. The term nanoluids was coined by Choi [1] at the ASME Winter Annual Meeting, which reers to a liquid containing a dispersion o submicronic solid particles (nanoparticles) with typical length on the order o 1-50 nm. Nanoluids have the higher heat conductivity eiciency than pure luid due to a volume raction (usually < 5% ) o metal nanoparticles. Heat and mass transer o nanoluids have been widely investigated. Nazar et al. [] studied the unsteady boundary-layer low o a nanoluid over a stretching sheet caused by an impulsive motion or a suddenly stretched surace using the Keller-bo method. Buongiorno [] developed a new correlation eplanation or convective heat transport o nanoluids considering the Brownian diusion. Chamkha and Ismael [4] considered the steady-conjugate natural convection heat transer o three types o nanoluids in a square porous cavity which was heated by a triangular solid wall under a wide range o considered parameter. Xu et al. [5] studied the mied convection low o nanoluids caused by both the eternal pressure and the buoyancy orce in a vertical channel, the eects o the Prandtl number and other parameters were analyzed. * Corresponding author, liancunzheng@16.com

2 48 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp In [6] the eect o a convective surace on the heat transer characteristics o nanoluids over a static or moving wedge in the presence o thermal radiation was investigated with three types o nanoparticles considered. Alam and Hossain [7] considered the eects o viscous dissipation and Joule heating on the heat and mass transer o a -D steady MHD orced convection low o nanoluids over a non-linear stretching sheet and studied numerically. Recently, Kuznetsov and Nield [8] analytically studied the natural convective boundary-layer low o nanoluids past a vertical plate. Unlike the commonly employed constant conditions, a convective heating boundary condition was used in study Makinde and Aziz [9]. Dulal [10] numerically studied the low and heat transer o an incompressible viscous luid past an unsteady stretching permeable sheet. Anbuchezhian et al. [11] studied the low o nanoluids caused by buoyancy along a vertical plate in a porous medium. Mushtaq et al. [1] investigated radiation eects to -D stagnation-point low o viscous nanoluid due to solar energy. Malvandi et al. [1] numerically simulated unsteady stagnation point low o nanoluids with a slip boundary condition, the results show dual solution would eist when the unsteadiness parameter was negative. The parameters o thermophoresis, Brownian motion and the velocity slip played a vital role in the transport process o various nanoluids. Behseresht et al. [14] displayed that the heat transer associated with nanoparticles migration was negligible compared with heat conduction and convection on the natural convection heat transer o nanoluids in a saturated porous medium. Noghrehabadi et al. [15] pointed that the Reynolds number or the temperature proile could be signiicantly aected by Prandtl number. Rahman et al. [16] numerically investigated the steady boundary-layer low and heat transer o nanoluids past a permeable eponentially shrinking/stretching surace with second order slip velocity. Bioconvection is induced by swimming o motile microorganisms, leading the increase o the density o the base luid. Few studies eist on nanoluids containing gyrotactic microorganisms over a convectively heated stretching sheet. Kuznetsov [17] studied both non-oscillatory and oscillatory nanoluid biothermal convection in a horizontal layer o inite depth and analyzed the dependence o the thermal Rayleigh number on the nanoparticle Rayleigh number and the bioconvection Rayleigh number. Khan and Makinde [18] investigated MHD low o nanoluids with heat and mass transer along a vertical stretching sheet in the presence o motile gyrotactic microorganisms. Xu and Pop [19] obtained a more physically realistic result using a passively controlled nanoluid model by an analysis o bioconvection low o nanoluids in a horizontal channel. In a recent paper, Khan et al. [0] investigated the eects o both Navier slip and magnetic ield on boundary-layer low o nanoluids containing gyrotactic microorganisms over a vertical plate. Their results show that the bioconvection parameters tend to reduce the local concentration o motile microorganisms. Xu and Pop [1] presented an analysis on the mied convection low o a nanoluid over a stretching surace with uniorm ree stream containing nanoparticles and gyrotactic microorganisms. In the present paper, we investigate the bioconvection and radiation heat transer o a nanoluid containing gyrotactic microorganisms over a stretching sheet, in which the eects o Brownian motion and thermophoresis are considered according to Rosseland s approimation [] as well as velocity slip and temperature jump. We obtain the analytical solutions by using the homotopy analysis method (HAM). The HAM as introduced by Liao [-7], has been adopted to solve the highly non-linear and coupled dierential equations. Many studies have conirmed the eectiveness o this method by contrast. Mathematical ormulation We consider in this paper the steady incompressible viscous luid containing gyrotactic microorganisms near a stagnation-point. In the co-ordinate system, the origin is a luid stagnation

3 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp point and the -ais is the lat direction, that is y = 0. The low region is conined to y > 0and the stretching surace temperature, nanoparticle volume raction, and microorganisms raction are deined to have constant values T w, C w, and n w, respectively, while at a large value o y, temperature, nanoparticle volume raction, and microorganisms raction have constant values T, C, and n, respectively. Flow induced by bioconvection only take place in a dilute suspension o nanoparticles so that the nanoparticle volume raction, C, is lower than 1. The boundary-layer governing equations considering thermal radiation and bioconvection are given: u v + = 0 (1) y u u u u (1 C ) ρ β( T T )g u + v = µ µ + y y k ρ () [( ρp ρ )( C C ) + ( n n ) γ( ρm ρ )]g ρ q u v D C C C DT T u + v = DB + y y T y r T T T T C DT T y + = α + τ B + y y y y T y ρcp n n n c + = n y y Cw C y u v D bw nc y The slip boundary conditions or the governing equations are: σ v u u = a + λ 0 y= 0, v = 0, σ v y σt r λ0 T T = Tw+ y= 0, C = Cw, n= nw, y = 0, (6) σt r+ 1 Pr y u = 0, v= 0, T T, C C, n n when y where a is a positive constant, σ v and σ T are the tangential momentum accommodation coeicient and the thermal accommodation coeicient, respectively, [], λ 0 is the molecular mean ree path, and r the speciic heat ratio [8]. The T is the temperature, C and n are the densities o nanoparticle and motile microorganisms, respectively, τ = ( ρcp) p/( ρ cp) is the ratio o eective heat capacity o the nanoparticle material to the heat capacity o the luid, ρ p the density o nanoparticles, ρ m the microorganism density, ρ the base luid density, γ the average volume o a microorganism, D B the Brownian diusion coeicient, D T the thermophoretic diusion coeicient, D n the diusivity o microorganisms, b the chemota constant, and W c the maimum cell swim-ming speed. The qr/ y = ( 16 σ qt / k) T/ y is obtained according to Rosseland s approimation and the Taylor epansion, σ q and k are Stean-Boltzmann constant and absorption coeicient, respectively. Now eq. () reduces to: () (4) (5)

4 50 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp T T 16σ qt T T C DT T u + v = α + + τ D + B y ρck p y y y T y The physical low model and co-ordinate system is shown in ig. 1. (7) n C T U Similarity transormations Introducing the stream unction and similarity transormation: y Ra η =, ψ = α Ra ( η), T T C C θη ( ) =, φη ( ) =, T T C C w w n n (1 C ) βg Tw χη ( ) =, Ra = n n ρ αµ where α = α + 16 σqt /ρ ck p, ψ the stream unction satisying u = ψ / y and v= ψ /, η the similarity variable, the dimensionless stream unction, θ the dimensionless temperature unction, and φ and χ are dimensionless nanoparticle raction and microorganisms raction unctions, respectively. Equations (1)-(5) and (7) can be reduced into the ollowing similarity equations: N + θ Nrφ Rbχ A = 0 (8) 4 Pr θ + θ + Nbθφ + Nt( θ ) = 0 (9) 4 φ + Le φ + Nt / Nbθ = 0 (10) 4 χ + Lb χ Pe( φ χ + φ χ + σφ ) = 0 (11) 4 and the reduced boundary conditions are: (0) = λ + λ (0), (0) = 0, θ (0) = 1 + δθ (0), φ(0) = χ (0) = 1 (1) 1 = 0, θ 0, φ 0, χ 0 as η (1) 1 14 where λ1 = a Ra /( α + N) the stretching velocity parameter, λ = λ 0 ( σ v)ra / σ v the velocity slip parameter, δ = rλ0 ( σt ) Ra /[ σ TPr( r + 1)] the temperature jump parameter, Pr = µα / the Prandtl number, N= 16 σ T /µρ ck p the thermal radiation parameter, Nr = ( ρp ρ ) Cw/ ρ β(1 C ) Tw the buoyancy ratio parameter or Cw = Cw C, Tw = Tw T, Rb = γ( ρm ρ ) nw/ ρ β Tw(1 C ) the bioconvection Rayleigh number 1 ( nw = nw n ), A= Ra / k the permeability parameter, Nb = τ D Pr/ B Cw µ and Nt = τ D Pr/ T Tw T µ the Brownian motion parameter and the thermophoresis parameter, respectively, Le = µ /PrDB the Lewis number, Lb = µ /PrD the bioconvection Lewis number, Pe bw / D the bioconvection Peclet number, and σ = n / n the bioconvection constant. = c n n C T U 0 U w T w C w n w Figure 1. Flow coniguration and co-ordinate system y w w

5 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp The reduced local Nusselt number, local Sherwood number, and local density number o the motile microorganisms may be ound in terms o the dimensionless temperature at the sheet surace, concentration o nanoparticle, and microorganisms at the sheet surace, respectively. Nur = Ra Nu = θ (0) (14) Shr = Ra Sh = φ (0) (15) Nnr = Ra Nn = χ (0) (16) The HAM solution The coupled non-linear boundary value problems eqs. (8)-(1) are solved by using the HAM [-7]. The initial approimations are selected: λ1 [ 1 ep( η) ] ep( η ) 0( η) =, θ0( η) = (17) 1+ λ 1+ δ φ0 ( η) = ep( η ), χ0 ( η) = ep( η ) (18) The auiliary linear operators are chosen as ollows, respectively: 1 [ ] = +, [ θ] = θ + θ, [ φ] = φ + φ, [ χ] = χ + χ (19) 4 The non-linear operators are given by: ( η; q) 1 ( η; q) N1 [ ( η; q), θη ( ; q), φη ( ; q), χη ( ; q) ] = + + ( ; ) + N ηq 4 Pr ( η; q) + θη ( ; q) Nrφη ( ; q) Rbχη ( ; q) A (0) θη ( ; q) θη ( ; q) N [ ( η; q), θη ( ; q), φη ( ; q), χη ( ; q) ] = + ( η; q) + 4 θη ( ; q) φη ( ; q) θη ( ; q) + Nb + Nt (1) φη ( ; q) φη ( ; q) Nt θη ( ; q) N [ ( η; q), θη ( ; q), φη ( ; q), χη ( ; q) ] = + Le ( η; q) + () 4 Nb χη ( ; q) χη ( ; q) N4 [ ( η; q), θη ( ; q), φη ( ; q), χη ( ; q) ] = + Lb( η; q) 4 φη ( ; q) φη ( ; q) χη ( ; q) φη ( ; q) Peσ Pe + ( ; ) χηq () with the boundary conditions: ( η; q) ( η; q) = λ1 + λ, ( η; q) η = 0 = 0 η = 0 η = 0 θη ( ; q) θη ( ; q) = 1 + δ, φη ( ; q) = 1, χη ( ; q) = 1 η= 0 η= 0 η= 0 η = 0 (4)

6 5 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp ( η; q) ( q) ( q) ( q) = 0, θη; = 0, φη; = 0, χη; = 0 η= η= η= η η = where q [0,1] is the embedding parameter. The auiliary unctions are chosen: Results and discussion (5) H ( η) = H ( η) = H ( η) = H ( η ) = 1 (6) θ φ χ Liao [-7] pointed out that the auiliary parameter, h, played a vital role in the convergence o the HAM solutions. By means o h-curves o (0), θ (0), φ (0), and χ (0) in ig. obtained by the 1 th approimation or Pr = 6, Le = 5, N =, λ 1 = 1, Pe = Nb = Nt = 0.5, Nr = Lb = δ = 0.5, λ = σ = 0., Rb = 0.1, and A = 1, it is straightorward to choose a proper value o h to ensure the convergence o the solution series. Unless special indicated, the values o the 6 parameters in this paper are used as the previous (0) values. The reliability o analytical results is veriied with numerical solutions obtained by inite di- 4 θ'[0] erence method using MAPLE The asymptotic boundary conditions at η were replaced by φ'(0) χ'(0) those at η = 6. 0 For illustrations o the results, solutions are plotted or special parameters. In order to validate h the present results, a comparison o numerical solutions with the analytical results obtained by Figure. The h-curves o (0), θ (0), HAM is presented in igs. and 4. The proiles indicate that the two results are in good agreement. φ (0), and χ (0) o the HAM solution The eects o velocity slip λ on the dimensionless velocity and the proiles o temperature or dierent values o temperature jump parameter δ are shown. The velocity value decreases with the velocity slip parameter and the rising in temperature jump parameter leads to the decrease o the surace temperature and thickness o the thermal boundary-layer. In ig. 5, the eects o stretching velocity parameter λ 1 on the dimensionless velocity o nanoluids are shown. As the stretching velocity increases, the velocity value decreases. The proiles o temperature distribution or dierent values o thermal radiation parameter N or Le = 1, Nb = 0. and Nt = 0.1 are shown in ig. 6. It indicates that the rising in thermal radiation parameter leads to the increase o the surace temperature and thickness o the thermal boundary-layer. The variation o volume raction o nanoparticles with transverse distance in the concentration boundary-layer is shown in ig. 7 or dierent values o Lewis number. It is ound that an increase in Lewis number results in reduction o the volume raction o nanoparticles and concentration boundary-layer thickness. This is because Brownian motion coeicient decreases with increasing transverse distance and as a result the rescaled nanoparticle volume raction decreases rapidly or large Lewis number. Figure 8 illustrates the eects o the bioconvection Peclet number and the bioconvection constant, σ, on the density o motile microorganisms in nanoluids. It can be seen that the motile microorganism boundary-layer thickness decreases with the increasing in σ and Peclet number which is like bioconvection Lewis number. The eects o bioconvection Lewis number, Lb, on the density o motile microorganisms in nanoluids are showed in ig. 9. The density o motile microorganisms decreases as Lb

7 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp '(η) λ = 0, 0., 0.5, 1 Numerical solution HAM solution θ(η) λ = 0, 0.5, 1, Numerical solution HAM solution Figure. Eects o velocity slip parameter, λ, on velocity 0 4 η η 6 Figure 4. Eects o temperature jump parameter, δ, on temperature '(η) 1.5 θ(η) λ1 = 0.5, 1, 1.5, N = 0, 0.5, 1, η η 5 6 Figure 5. Eects o the stretching velocity parameter, λ 1, on velocity Figure 6. Eects o the thermal radiation parameter, N, on temperature increases. Simultaneously, the motile microorganism boundary-layer thickness decreases. Like the bioconvection Peclet number, the bioconvection Lewis number plays the same role as regular Lewis number. The variation o local Nusselt number with dierent velocity slip parameters and bioconvection Rayleigh numbers or Nb = Lb = 0.1 is shown in ig. 10. It shows that Nusselt number decreases with an increase in the thermophoresis parameter, Nt. This is due to that thermophoresis increases the temperature in boundary-layer, leading to a rise in the thermal boundary-layer thickness. φ(η) χ(η) Le = 0.1, 1,.5 Pe = 0, 5, 10 (σ = 0.) σ = 0.1, 0., 1 (Pe = 5) η 5 6 Figure 7. Eects o Lewis number on volume raction o nanoparticles proiles 0 1 η 4 5 Figure 8. Eects o Peclet number and σ on volume raction o gyrotactic microorganism

8 54 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp χ(η) Lb = 0, 0.5, 1, Nur 0.5 Rb = 0, 1, (λ = 0.) Rb = 0, 1, (λ = 1.5) η 5 6 Figure 9. Eect o Lb on volume raction o gyrotactic microorganism Nt Figure 10. Eects o λ and Rb on Nusselt number Figure 11 displays the variation o the reduced Sherwood numbers or dierent values o Lewis number, thermophoresis parameter, and bioconvection Peclet number or Rb = λ = 0.5 and Nb = Lb = 0.1. As epected, the Sherwood number increases with Lewis number while decreases with increasing thermophoresis parameter, Nt. This is due to the act that the nanoparticle volume raction increases with this parameter including the concentration boundary-layer, which can be attributed to the act that when the Lewis number and bioconvection Peclet number are high, the nanoparticle concentration is low. Thus mass transer rom the plate to the luid since the concentration at the plate surace is higher than that o the luid. It can be seen rom ig. 1 that the gyrotactic microorganisms density number, Nnr, increases with the bioconvection Peclet number, bioconvection constant, σ, parameter and bioconvection Lewis number Lb or λ = 0.5. Shr Nt = 0.1, 0., (Le = 5) Nnr 1. Pe = 0.1, 0., (σ = 0) Pe = 0.1, 0., (σ = 1) Nt = 0.1, 0., (Le = 0) Conclusions 0. Pe Figure 11. Eects o Lewis number and Nt o Sherwood number 0. Lb Figure 1. Eects o Peclet nuber and σ on density number o the motile microorganisms This paper presents an investigation or bioconvection heat transer o a nanoluid containing gyrotactic microorganisms over a stretching sheet, in which the eects o and radiation, velocity slip and temperature jump are taken into account. The main results can be classiied as ollows. y Velocity proiles decrease with the velocity slip parameter and the stretching velocity parameter. y Temperature decreases with the temperature jump parameter and increases with the thermal radiation parameter inside thermal boundary-layer.

9 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp y Density o nanoparticles decreases with increasing Lewis number inside the boundary-layer. y Density o motile microorganisms decreases with increasing bioconvection Peclet number, bioconvection constant and bioconvection Lewis number inside the boundary-layer. y Nusselt number decreases with an increase in slip parameter, bioconvection Rayleigh number, and thermophoresis parameter. Sherwood number increases with Lewis number, whereas decreases with thermophoresis parameter. The gyrotactic microorganisms density number increases with bioconvection Peclet number, bioconvection constant parameter and bioconvection Lewis number. Acknowledgment The work is supported by the National Natural Science Foundations o China (Nos , ). Nomenclature A permeability parameter,[ ] a positive constant, [s 1 ] b chemota constant, [m] C nanoparticle raction, [ ] C w, C nanoparticle volume raction at stretching surace and at y ininity, [ ] D B Brownian diusion coeicient, [ ] D n diusivity o microorganisms, [ ] D T thermophoretic diusion coeicient, [ ] g gravitational ield, [ms ] k absorption coeicient, [ ] Lb bioconvection Lewis number, [ ] Le Lewis number, [ ] N thermal radiation parameter, [ ] Nb Brownian motion parameter, [ ] Nr buoyancy ratio parameter, [ ] Nt thermophoresis parameter, [ ] n microorganisms raction, [ ] n w, n microorganisms raction at stretching surace and at y ininity, [ ] Pe bioconvection Peclet number, [ ] Pr Prandtl number, [ ] Ra local Rayleigh number, [ ] T temperature, [K] T w, T temperature at stretching surace and at y ininity, [K] uv, luid velocity component in - and y-direction, [ms 1 ] y, streamwise and cross-stream co-ordinate, [m] W c maimum cell swim-ming speed, [ms 1 ] Greek symbols α eective thermal diusivity, [m s 1 ] β volumetric epansion coeicient, [ ] γ average volume o a microorganism, [m ] δ temperature jump parameter, [ ] η similarity variable, [ ] λ 1 stretching velocity parameter, [ ] λ velocity slip parameter, [ ] µ dynamic viscosity, [Pa s] ρ density, [kgm ] ρ c p heat capacity, [Jm K 1 ] σ bioconvection constant, [ ] σ q Stean Boltzmann constant, [ ] σ T thermal accommodation coeicient, [ ] σ v tangential momentum accommodation coeicient, [ ] ψ stream unction, [ ] Subscripts w surace large value o y nanoluid p nanoparticle Reerences [1] Choi, S. U. S., Developments and Applications o Non-Newtonian Flows, ASME Press, New York, USA, 1995 [] Nazar, R., et al., Unsteady Boundary Layer Flow in the Region o the Stagnation-Point on a Stretching Sheet, International Journal o Engineering Science, 4 (004), 11-1, pp [] Buongiorno, J., Convective Transport in Nanoluids, ASME Journal o Heat Transer, 18 (006),, pp [4] Chamkha, A. J., Ismael, M. A., Conjugate Heat Transer in a Porous Cavity Filled with Nanoluids and Heated by a Triangular Thick Wall, International Journal o Thermal Sciences, 67 (01), May, pp [5] Xu, H., et al., Analysis o Mied Convection Flow o a Nanoluid in a Vertical Channel with the Buongiorno Mathematical Model, International Communications in Heat and Mass Transer, 44 (01), May, pp. 15-

10 56 THERMAL SCIENCE: Year 017, Vol. 1, No. 6A, pp [6] Rahman, M. M., et al., The Role o a Convective Surace in Models o the Radiative Heat Transer in Nanoluids, Nuclear Engineering and Design, 75 (014), Aug., pp. 8-9 [7] Alam, M. S., Hossain, S. C., Eects o Viscous Dissipation and Joule Heating on Hydromagnetic Forced Convective Heat and Mass Transer Flow o a Nanoluid along a Nonlinear Stretching Surace with Convective Boundary Condition, Journal o Engineering e-transaction, 8 (01), 1, pp. 1-9 [8] Kuznetsov, A. V., Nield, D. A., Natural Convective Boundary-Layer Flow o a Nanoluid Past a Vertical Plate, International Journal o Thermal Science, 49 (010),, pp [9] Makinde, O. D., Aziz, A., Boundary Layer Flow o a Nanoluid Past a Stretching Sheet with a Convective Boundary Condition, International Journal o Thermal Sciences, 50 (011), 7, pp [10] Dulal, P., Combined Eects o Non-Uniorm Heat Source/Sink and Thermal Radiation on Heat Transer over an Unsteady Stretching Permeable Surace, Communications in Nonlinear Science and Numerical Simulation, 16 (011), 4, pp [11] Anbuchezhian, N., et al., Thermophoresis and Brownian Motion Eects on Boundary-Layer Flow o a Nanoluid in the Presence o Thermal Stratiication Due to Solar Energy, Applied Mathematics and Mechanics, (01), 6, pp [1] Mushtaq, A., et al., Nonlinear Radiative Heat Transer in the Flow o Nanoluid Due to Solar Energy: A Numerical Study, Journal o the Taiwan Institute o Chemical Engineers, 45 (014), 4, pp [1] Malvandi, A., et al., Slip Eects on Unsteady Stagnation Point Flow o a Nanoluid over a Stretching Sheet, Powder Technology, 5 (014), Feb., pp [14] Behseresht, A., et al., Natural-Convection Heat and Mass Transer rom a Vertical Cone in Porous Media Filled with Nanoluids Using the Practical Ranges o Nanoluids Thermo-Physical Properties, Chemical Engineering Research and Design, 9 (014),, pp [15] Noghrehabadi, A., et al., Analyze o Fluid Flow and Heat Transer o Nanoluids over a Stretching Sheet near the Etrusion Slit, Computers & Fluids, 100 (014), Sept., pp. 7-6 [16] Rahman, M. M., et al., Boundary Layer Flow o a Nanoluid past a Permeable Eponentially Shrinking/ Stretching Surace with Second Order Slip Using Buongiorno s Model, International Journal o Heat and Mass Transer, 77 (014), Oct., pp [17] Kuznetsov, A. V., Non-Oscillatory and Oscillatory Nanoluid Bio-Thermal Convection in a Horizontal Layer o Finite Depth, European Journal o Mechanics-B/Fluids, 0 (011),, pp [18] Khan, W. A., Makinde, O. D., MHD Nanoluid Bioconvection Due to Gyrotactic Microorganisms over a Convectively Heat Stretching Sheet, International Journal o Thermal Sciences, 81 (014), July, pp [19] Xu, H., Pop, I., Fully Developed Mied Convection Flow in a Horizontal Channel Filled by a Nanoluid Containing Both Nanoparticles and Gyrotactic Microorganisms, European Journal o Mechanics B/Fluids, 46 (014), July-Aug., pp [0] Khan, W. A., et al., MHD Boundary Layer Flow o a Nanoluid Containing Gyrotactic Microorganisms Past a Vertical Plate with Navier Slip, International Journal o Heat and Mass Transer, 74 (014), July, pp [1] Xu, H., Pop, I., Mied Convection Flow o a Nanoluid over a Stretching Surace with Uniorm Free Stream in the Presence o both Nanoparticles and Gyrotactic Microorganisms, International Journal o Heat and Mass Transer, 75 (014), Aug., pp [] Zheng, L. C., et al., MHD Flow and Heat Transer over a Porous Shrinking Surace with Velocity Slip and Temperature Jump, Mathematical and Computer Modelling, 56 (01), 5-6, pp [] Liao, S. J., An Approimate Solution Technique not Depending on Small Parameters: A Special Eample, International Journal o Non-Linear Mechanics, 0 (1995),, pp [4] Liao, S. J., Homotopy Analysis Method: A New Analytic Method or Nonlinear Problems, Applied Mathematics and Mechanics, 19 (1998), 10, pp [5] Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method [M], CRC Press, Boca Raton, Fla., USA, 00 [6] Liao, S. J., On the Homotopy Analysis Method or Nonlinear Problems, Applied Mathematics and Computation, 147 (004),, pp [7] Liao, S. J., Notes on the Homotopy Analysis Method: Some Deinitions and Theorems, Communications in Nonlinear Science and Numerical Simulation, 14 (009), 4, pp [8] Zheng, L. C., et al., Flow and Radiation Heat Transer o a Nanoluid over a Stretching Sheet with Velocity Slip and Temperature Jump in Porous Medium, Journal o the Franklin Institute, 50 (01), 5, pp Paper submitted: April 4, 015 Paper revised: August 7, 015 Paper accepted: August 7, Society o Thermal Engineers o Serbia Published by the Vinča Institute o Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditions