MAGNETOHYDRODYNAMIC FLOW OF NANOFLUID WITH HOMOGENEOUS-HETEROGENEOUS REACTIONS AND VELOCITY SLIP

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1 THERMAL SCIENCE, Year 017, Vol. 1, No., pp MAGNETOHYDRODYNAMIC FLOW OF NANOFLUID WITH HOMOGENEOUS-HETEROGENEOUS REACTIONS AND VELOCITY SLIP by Tasawar HAYAT a,b, Maria IMTIAZ a*, and Ahad ALSAEDI b a Departent o Matheatics, Quaid-I-Aza University, Islaabad, Pakistan b NAAM Research Group, Departent o Matheatics, Faculty o Science, King Abdulaziz University, Jeddah, Saudi Arabia Original scientiic paper This article ocuses on the steady agnetohydrodynaic low o viscous nanoluid. The low is caused by a stretching surace with hoogeneous-heterogeneous reactions. An incopressible luid ills the porous space. Copper-water and silverwater nanoluids are investigated in this study. Transoration ethod reduces the non-linear partial dierential equations governing the low into the ordinary dierential equation by siilarity transorations. The obtained equations are then solved or the developent o series solutions. Convergence o the obtained series solutions is explicitly discussed. Eects o dierent paraeters on the velocity, concentration and skin riction coeicient are shown and analyzed through graphs. Keywords: agnetohydrodynaic nanoluid, hoogeneous-heterogeneous reactions, porous ediu, velocity slip condition Introduction The traditional heat transer luids like oil, water, and ethylene glycol ixtures are now recognized as the poor heat transer luids. Addition o solid nanoparticles in traditional heat transer luids enhances the theral conductivity o base luid. The ter nanoluid is credited by Choi [1]. This pioneering experiental research witnessed theral conductivity enhanceent o a nanoluid. He concluded that addition o very sall aount o nanoparticles to traditional heat transer liquids enhanced the theral conductivity o liquid up to two ties. Eastan et al. [] and Choi et al. [3] pointed out that a sall aount (<1% volue raction) o Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil rearkably enhanced the theral conductivity o a luid by 40% and 50%, respectively. Thus the nanoaterials are recognized ore eective in icro/nano electroechanical devices, advanced cooling systes, large scale theral anageent systes via evaporators, heat exchangers and industrial cooling applications. Such luids are very stable with no extra issues o erosion, sedientation, non-newtonian properties and additional pressure drop. This is because o tiny size and low volue raction o nano eleents required or theral conductivity enhanceent. Further the canvas o agnetic ield has iportant applications in edicine, physics and engineering. Many equipent such as MHD generators, pups, bearings and boundary layer control are aected by the interaction between the electrically conducting luid and a * Corresponding author, eail: i_qau@yahoo.co

2 90 THERMAL SCIENCE, Year 017, Vol. 1, No., pp agnetic ield. The behavior o the low strongly depends on the orientation and intensity o the applied agnetic ield. The exerted agnetic ield anipulates the suspended particles and rearranges their concentration in the luid which strongly changes heat transer characteristics o the low. A agnetic nanoluid has both the liquid and agnetic characteristics. Such aterial have ascinating applications like optical odulators, agneto-optical wavelength ilters, non-linear optical aterials, optical switches, optical gratings, etc. Magnetic particles have pivoted role in the construction o loud speakers as sealing aterials and in sink loat separation. Magneto nanoluids are useul to guide the particles up the blood strea to a tuor with agnets. This is due to the act that the agnetic nanoparticles are regarded ore adhesive to tuor cells than non-alignant cells. Such particles absorb ore power than icro particles in alternating current agnetic ields tolerable in huans i. e. or cancer therapy. Nuerous applications involving nanoluids include drug delivery, hypertheria, constrast enhanceent in agnetic resonance iaging and agnetic cell separation. Motivated by all the aoreentioned acts, various scientists and engineers are engaged in the discussion o lows o nanoluids via dierent aspects (see [4-18] and any useul attepts therein). Hoogeneous-heterogeneous reactions occur in any cheically reacting systes such as in cobustion, catalysis, and biocheical systes. The interaction between the hoogeneous reactions in the bulk o the luid and heterogeneous reactions occurring on soe catalytic suraces is generally coplex and is involved in the production and consuption o reactant species at dierent rates both within the luid and on the catalytic suraces. A odel or isotheral hoogeneous-heterogeneous reactions in boundary layer low o viscous luid past a lat plate is studied by Merkin [19]. He presented the hoogeneous reaction by cubic autocatalysis and the heterogeneous reaction by a irst order process and showed that the surace reaction is the doinant echanis near the leading edge o the plate. Chaudhary and Merkin [0] studied the hoogenous-heterogeneous reactions in boundary layer low o viscous luid. They ound the nuerical solution near the leading edge o a lat plate. Bachok et al. [1] ocused on the stagnation-point low towards a stretching sheet with hoogeneous-- heterogeneous reactions eects. Eects o hoogeneous-heterogeneous reactions on the low o viscoelastic luid towards a stretching sheet are investigated by Khan and Pop []. Kaeswaran et al. [3] extended the work o [] or nanoluid over a porous stretching sheet. In general, porous ediu is used or transport and storage o energy. Analysis o low through a porous ediu has becoe the core o several scientiic and engineering applications. These applications include the utilization o geotheral energy, the igration o oisture in ibrous insulation, ood processing, casting and welding in anuacturing processes, the dispersion o cheical containants in dierent industrial processes, the design o nuclear reactors, cheical catalytic reactors, copact heat exchangers, solar power, and any others. Further the use o icro/nano electroechanical systes (MEMS/NEMS) has been increased in any industries. Such systes have association with velocity slip [4-8]. Motivated by such acts, the ain concern in present counication is to exaine the low analysis o nanoluid with hoogeneous-heterogeneous reactions and velocity slip. The relevant probles or velocity and concentration are odeled. The non-linear partial dierential equations are converted into the ordinary dierential equations. Resulting equations are coputed or the series solutions by a odern technique naely the hootopy analysis ethod (HAM) [9-33]. Convergence region o the derived solutions is deterined. Discussion relevant to ebedded paraeters is ade using graphical illustration.

3 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Model developent Let us consider the steady -D low o an incopressible nanoluid over a stretching surace in porous ediu with pereability K. The x-axis is taken along the stretching surace in the direction o otion and y-axis is perpendicular to it. A unior transverse agnetic ield o strength, B 0, is applied parallel to the y-axis. It is assued that the induced agnetic and electric ields eects are negligible, ig. 1. Nanoparticles such as Cu and Ag are considered. Water is treated as a base luid. We have taken a siple hoogeneous-- heterogeneous reaction odel in the ollowing or [16]: A B 3 B rate k ab (1), c while on the catalyst surace we have the single, isotheral, irst order reaction Figure 1. Geoetry o the proble A B, rate ksa () where a and b are the concentrations o the cheical species A and B and k c and k s denote the rate constants. We assue that both reaction processes are isotheral. Under these assuptions, the relevant boundary layer equations are: u v 0 x y u u u n n n 0 u v u B u x y y K a a a u v DA k cab x y y (3) (4) (5) b b b u v DB k cab x y y The subjected boundary conditions are: (6) v u a b u cx 0, v 0, DA ksa, DB ksa at y 0, y y y y0 u 0, a a, b 0 as y (7) 0 where u and are the velocity coponents along the x- and y-directions, respectively, D A and D B are the respective diusion species coeicients o A and B, is the electrical conductivity o luid, the tangential oentu accoodation coeicient, and 0 the olecular ean ree path. The eective density, n, the dynaic viscosity, n, the heat capacitance, (c p ) n, and the theral conductivity, k n, o the nanoluid are given by:

4 904 THERMAL SCIENCE, Year 017, Vol. 1, No., pp ( 1) (8) n s n 5 ( 1 ). (9) ( c ) ( c ) ( 1) ( c ) (10) p n p p s kn ks k ( k ks) k k k ( k k ) s Here is the nanoparticle volue raction, s in subscript is or nanosolid-particles and in subscript is or base luid. Denoting a 0 (a constant) and g(h) and h(h) the diensionless concentration and deining: c y, u cx ( ), v c ( ), a a 0 g ( ), b a 0 h ( ) (1) Equation (3) is satisied autoatically and eqs. (4)-(7) reduce to: s (11) 5 1( ) ( 1 ). Ha 0 (13) 1 g g kgh 0 Sc (14) h h kgh 0 (15) Sc ( 0) 1 ( 0 ), ( 0) 0, ( ) 0 (16) g( 0) K g( 0 ), g( ) 1 (17) s h( 0) Ksg( 0 ), h( ) 0 (18) in which prie indicates the dierentiation with respect to h. Moreover the non-diensional constants in eqs. (13)-(18) are the porosity paraeter,, the Hartan nuber, the Schidt nuber, the easure o the strength o the hoogeneous reaction, k, the easure o the strength o the heterogeneous reaction, K s, the ratio o the diusion coeicient,, and the velocity slip paraeter,. These are deined: 0 c 0 s B v, Ha, Sc, k, Ks,, A A A v B k a k D c 0 (19) ck c D c D c D where 5. s 1 ( 1 ) 1 The diusion coeicients o cheical species A and B are expected to be o a coparable size. This leads to ake a urther assuption that the diusion coeicients D A and D B are equal, i. e. to take = 1 [16]. In this case we have ro eqs. (17) and (18): (0)

5 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Thus eqs. (14) and (15) becoe: subject to the boundary conditions: g ( ) h( ) 1 (1) 1 g g kg( 1 g) 0 () Sc g( 0) K g( 0 ), g( ) 1 (3) s The physical quantity o interest is the skin-riction coeicient, C. It characterizes the surace drag. The shearing stress at the surace o the wall, w, is given by: in which u 1 3 w c x ( 0). n 5 y y0 ( 1 ) The skin riction coeicient is deined: Re C C w (4) w (5) 05. u Rex ( 0) (6) 5. ( 1 ) x uwx/ denotes the local Reynolds nuber. Solutions derivation Zeroth-order deoration probles We choose the initial guesses 0( ) and g0( ) and the linear operators in the ors: 1 1 0( ) [ 1 e ], g K 0( ) 1 e s 1 together with the properties: L and L g (7) L ( ), L ( g) g g (8) g c1 ce c3e L 0, g c4e c5e L 0 (9) where c 1 -c 5 are constants. With eqs. (13) and (), the deinitions o operators N and N g are: 3 ˆ ˆ ˆ ˆ (, p) ˆ (, p) (, p) N (, p), gˆ (, p) 3 1 (, p) ˆ(, p) ˆ 5. (, p) ( 1 ) Ha (30)

6 906 THERMAL SCIENCE, Year 017, Vol. 1, No., pp where Ng[ g ˆ(, p ), ˆ(, p )] 1 gˆ(, p) g p ˆ ˆ(, ) (, p) kgˆ (, p) k[ gˆ (, p)] k[ gˆ (, p)] Sc We construct the zeroth order probles: 0 3 (31) ( 1 p) L [ ˆ(, p) ( )] p N [ ˆ(, p)] (3) ( 1 p) L [ gˆ(, p) g ( )] p N [ gˆ(, p)] (33) g 0 g g ˆ 0 1 ˆ 0 ˆ 0 0 ˆ (, p) (, p), (, p), (, p) 0, gˆ ( 0, p) K gˆ ( 0, p), gˆ (, p) 1 (34) and g are the non-zero auxiliary paraeters and or p = 0 and p = 1 we have: ˆ(, 0 ) ( ), ˆ(, 1 ) ( ), gˆ(, 0 ) g ( ), gˆ(, 1) g( ) (35) 0 0 Note that 0( ) and g0( ) approach ( ) and, g( ), respectively, when p has variation ro 0 to1. According to Taylor series we have: 1 ˆ(, ) 0 ( ) ( ) p p, ( ) 1! 1 gˆ(, p) g0( ) g ( ) p, g ( )! s ˆ(, p ) p gˆ(, p) 1 where the convergence depends upon and g. By proper choice o and g the series (36) converge or p = 1 and so: The th order deoration probles p p0 p0 (36) ( ) ( ) ( ), g( ) g ( ) g ( ) (37) The resulting probles at this order are given by: L (, p) 1 ( ) R, ( ) (38) L g g (, p) g1 ( ) g R g, ( ) (39) s ( 0) ( 0) ( 0) ( ) g ( 0) K g ( 0) g ( ) 0 (40) 0, 1 1, , [ l l l l ] ( ) Ha1 l0 R (4) (41)

7 THERMAL SCIENCE, Year 017, Vol. 1, No., pp R 1 Sc 1 l g, ( ) g1 [ g1l l kg1l gl j g j kg1l gl ] kg1 l0 j0 (43) where the general solutions are: in which and g ( ) ( ) 1 e 3e, ( ) ( ) 4e 5e c c c g g c c (44) denote the special solutions. Analysis o the results Convergence o the derived series solutions Now the solutions o eqs. (13) and () subject to the boundary conditions (16) and (3) are coputed by eans o HAM. We choose auxiliary paraeters and g or the unctions and g, respectively. The convergence o obtained series and rate o the approxiation or HAM strongly depend upon the values o the auxiliary paraeters. For ranges o adissible values o and g, the -curves or 13 th -order o approxiations are plotted in igs. and 5. We can see that the perissible values or and g or Cu-water are and 1. g 0. 3 and or Ag-water are and 1 g 0. 1 Further, the series solutions converge in the whole region o h ( 0 ) when g 1. Figure. The - curve o or Cu-water Figure 3. The - curve o g or Cu-water. Figure 4. The - curve o or Ag-water Figure 5. The - curve o g or Ag-water Table 1 shows the convergence o the series solutions. It is observed that convergence is achieved at 17 th order o approxiations. In tab. soe existing therophysical properties o water and nanoparticles are given.

8 908 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Table 1. Convergence o HAM solutions or dierent order o approxiations when = 0., = 0.4, k = 0.3, = 1, Ha = 0.5, K s = 0.3, and Sc = 0.5 Order o approxiations () 0 g ( 0) Table. Therophysical properties o water and nanoparticles 3 ( kg / ) c ( j / kgk ) k( W / k ) p ( K ) Pure water Copper, Cu Silver, Ag Aluina, Al O Titaniu Oxide, TiO Results and discussion The eects o dierent paraeters on the diensionless low and concentration proiles are investigated and presented graphically in this section. Diensionless velocity proiles Figures 6-9 exhibit the diensionless velocity proiles or dierent values o nanoparticle volue raction,, Hartan nuber, velocity slip paraeter,, and porosity paraeter,. Eects o volue raction o nanoparticles Cu and Ag on the velocity proile can be seen ro ig. 6. Here the velocity proile and boundary layer thickness decrease when volue raction or the nanoparticles increases. The eects o Hartan nuber on the velocity are depicted in ig. 7. We analyzed that the velocity is reduced when we increase the values o Hartan nuber. In act applied agnetic ield has the tendency to slow down the oveent o the luid which leads to a decrease in the velocity and oentu boundary layer thickness. Variations o velocity slip paraeter on velocity proile can be seen in ig. 8. There is a decrease in velocity when velocity slip paraeter is increased. Fro ig. 9, we have seen that larger values o porosity paraeter correspond to the less velocity. Porosity paraeter depends on the pereability paraeter K Increase in porosity paraeter leads to the lower pereability paraeter. This lower pereability paraeter causes a reduction in the luid velocity. Diensionless concentration proiles Eects o the easure o the strength o the hoogeneous reaction, k, the easure o the strength o the heterogeneous reaction, K s, and the Schidt nuber on the concentration proile, g, are shown in igs Eects o k on the concentration are analyzed in ig. 10. It is observed that increasing the easure o the strength o the hoogeneous reaction k

9 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Figure 6. Eects o on Figure 7. Eects o Ha on Figure 8. Eects o on Figure 9. Eects o on Figure 10. Eects o k on g Figure 11. Eects o K s on g decreases the theral boundary layer thickness. Figure 11 illustrates the eects o K s on concentration proile g. There is an increase in concentration g when the easure o the strength o the heterogeneous reaction K s is increased. The behavior o Schidt nuber on the concentration proile is siilar to that o K s (see ig.1). Skin riction coeicient and concentration Figure 13 shows the skin riction coeicient () 0 as a unction o nanoparticle volue raction. The skin riction coeicient enhances with increasing values o.

10 910 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Figure 1. Eects o Sc on g Figure 13. Eects o on skin riction coeicient The results o the skin riction coeicient are exained or both types o nanoluids. We observe that the Ag-water nanoluid gives a higher drag orce opposite to the low when copared with the Cu-water nanoluid. The variation o diensionless concentration or dierent values o K s and k are shown in igs. 14 and 15, respectively. Fro ig. 14 it is observed that concentration at the surace decreases as the strength o the heterogeneous reaction increases or dierent types o nanoluids. One can see ro ig. 15 that g ( 0) decreases with the increase o hoogeneous reaction strength k. Inluence o Sc on g ( 0) or two dierent types o nanoparticles is shown in ig. 16. It is clear that the concentration decreases with an increase o Schidt nuber. In tab. 3 soe nuerical values o skin riction coeicient are given or Cu and Ag nanoparticles. Tabular values show that skin riction coeicient increases by increasing and Ha while it decreases or larger. Table 4 shows that diensionless concentration decreases by increasing K s, Sc, k, and. Figure 14. Eects o K s on the concentration Figure 15. Eects o k on the concentration Figure 16. Eects o Sc on the concentration

11 THERMAL SCIENCE, Year 017, Vol. 1, No., pp Table 3. Nuerical values o skin riction coeicient or Cu and Ag when = 0.4, k = 0.3, K s = 0.3, and Sc =0.5 Ha C Re or Cu C Re or Ag Table 4. Nuerical values o diensionless concentration or Cu and Ag approxiations when = 0., = 0.4, and Ha = 0.5 K s Sc k g ( 0) or Cu g ( 0) or Ag x x Concluding rearks Here MHD low o nanoluid by a stretching sheet in presence o hoogeneous- -heterogeneous reactions is considered. Convergent approxiate solution is constructed. The ollowing observations are ade. An increase in the values o, Ha,, and has siilar eects on the velocity ( ) in a qualitative sense. Concentration proile increases by increasing K s and Sc while it decreases when k is increased. The values o skin riction coeicient are higher or Ag-water when enhances. Higher values o K s, k, and Sc correspond to saller values o diensionless concentration. Noenclature a concentration o cheical specie A, [ ] B 0 unior agnetic ield strength, [kgs A] b concentration o cheical specie B, [ ] C local skin riction coeicient c stretching constant, [s] c p speciic heat, [ s 1 ] D A diusion coeicient o specie A, [ s 1 ] D B diusion coeicient o specie B, [ s 1 ] K pereability, [ ] Ks easure o the strength o the heterogeneous reaction, [ ]

12 91 THERMAL SCIENCE, Year 017, Vol. 1, No., pp k easure o the strength o the hoogeneous reaction, [ ] k c hoogeneous rate constant, [s 1 ] k s heterogeneous rate constant, [s 1 ] L, L g linear operator or velocity and concentration ields, [ ] Ha Hartan nuber, [ ] N, N g non-linear operators [ ] R, R g, th order non-linear operators [ ] Re x local Reynolds nuber Sc Schidt nuber [ ] u, v velocity coponents along x and y axes, respectively [s 1 ], g non-zero auxiliary paraeters, [ ] Greek sybols velocity slip paraeter, [ ] ratio o the diusion coeicient, [ ] porosity paraeter, [ ] 0 olecular ean ree path, [] viscosity, [kg s 1 ] kineatic viscosity, [ s 1 ] density, [kg 3 ] electrical conductivity, [s 3 A kg 1 3 ] tangential oentu accoodation.coeicient, [ ] t w surace shear stress, [kgs 1 3 ] nanoparticle volue raction, [ ] Subscripts base luid n nanoluid s nanosolid-particles Superscripts ` derivative with respective to h Reerence [1] Choi, S. U. S., Enhancing Theral Conductivity o Fluids with Nanoparticle, Proceedings, ASME International Mechanical Engineering Congress and Exposition, 66 (1995), Jan., pp [] Eastan, J. A., et al., Anoalously Increased Eective Theral Conductivity o Ethylene Glycol- Based Nanoluids Containing Copper Nanoparticles, Applied Physics Letters, 78 (001), 6, pp [3] Choi, S. U. S., et al., Anoalous Theral Conductivities Enhanceent on Nanotube Suspension, Applied Physics Letters, 79 (001), 14, pp [4] Turkyilazoglu, M., Nanoluid Flow and Heat Transer Due to a Rotating Disk, Coputers & Fluids, 94 (014), May, pp [5] Turkyilazoglu, M., Unsteady Convection Flow o Soe Nanoluids Past a Moving Vertical Flat Plate with Heat Transer, Journal o Heat Transer, 136 (013), 3, [6] Sheikholeslai, M., Goriji-Bandpy, M., Free Convection o Ferroluid in a Cavity Heated ro Below in the Presence o an External Magnetic Field, Powder Technology, 56 (014), Apr., pp [7] Sheikholeslai, M., et al., Analytical Investigation o MHD Nanoluid Flow in a Sei-Porous Channel, Powder Technology, 46 (013), Sept., pp [8] Sheikholeslai, M., et al., Lattice Boltzann Method or MHD Natural Convection Heat Transer Using Nanoluid, Powder Technology, 54 (014), Mar., pp [9] Xu, H., et al., Flow and Heat Transer in a Nano-Liquid Fil Over an Unsteady Stretching Surace, International Journal o Heat and Mass Transer, 60 (013), May, pp [10] Rashidi, M. M., et al., Entropy Generation in Steady MHD Flow Due to a Rotating Porous Disk in a Nanoluid, International Journal o Heat and Mass Transer, 6 (013), Jul., pp [11] Niu, J., et al., Slip-Flow and Heat Transer o a Non-Newtonian Nanoluid in a Microtube, Plos One, 7 (01), 5, e3774 [1] Khan, J. A., et al., On Model or Three-Diensional Flow o Nanoluid: An Application to Solar Energy, Journal o Molecular Liquids, 194 (014), Jun., pp [13] Farooq, U., et al., Heat and Mass Transer o Two-Layer Flows o Third-Grade Nanoluids in a Vertical Channel, Applied Matheatics and Coputation, 4 (014), Sept., pp [14] Sheikholeslai, M., et al., Magnetic Field Eects on Natural Convection Around a Horizontal Circular Cylinder Inside a Square Enclosure Filled with Nanoluid, International Counications in Heat and Mass Transer, 39 (01), 7, pp [15] Nadee, S., et al., Heat Transer Analysis o Water-Based Nanoluid Over an Exponentially Stretching Sheet, Alexandria Engineering Journal, 53 (014), 1, pp [16] Haq, R. U., et al., Therophysical Eects o Carbon Nanotubes on MHD Flow over a Stretching Surace, Physica E: Low-diensional Systes and Nanostructures, 63 (014), Sept., pp. 15- [17] Haq, R. U., et al., Convective Heat Transer in MHD Slip Flow over a Stretching Surace in the Presence o Carbon Nanotubes, Physica B: Condensed Matter, 457 (015), 15, pp

13 THERMAL SCIENCE, Year 017, Vol. 1, No., pp [18] Zhang, W. M., et al., A Review on Slip Models or Gas Microlows, Journal o Microluid Nanoluid, 13 (01), 6, pp [19] Merkin, J. H., A Model or Isotheral Hoogeneous-Heterogeneous Reactions in Boundary Layer Flow, Matheatical and Coputer Modelling, 4 (1996), 8, pp [0] Chaudhary, M. A., Merkin, J. H., A Siple Isotheral Model or Hoogeneous-Heterogeneous Reactions in Boundary Layer Flow: I. Equal Diusivities, Fluid Dynaics Research, 16 (1995), 6, pp [1] Bachok, N., et al., On the Stagnation-Point Flow Towards a Stretching Sheet with Hoogeneous- Heterogeneous Reactions Eects, Counications in Nonlinear Science and Nuerical Siulation, 16 (011), 11, pp [] Khan, W. A., Pop, I., Eects o Hoogeneous-Heterogeneous Reactions on the Viscoelastic Fluid Towards a Stretching Sheet, Journal o Heat Transer, 134 (01), 6, pp. 1-5 [3] Kaeswaran, P. K., et al., Hoogeneous-Heterogeneous Reactions in a Nanoluid Flow Due to Porous Stretching Sheet, International Journal o Heat and Mass Transer, 57 (013),, pp [4] Rashidi, M. M., et al., Investigation o Entropy Generation in MHD and Slip Flow Over a Rotating Porous Disk with Variable Properties, International Journal o Heat and Mass Transer, 70 (014), Mar., pp [5] Mahoud, M. A. A., Waheed, S. E., MHD Flow and Heat Transer o a Micropolar Fluid Over a Stretching Surace with Heat Generation (Absorption) and Slip Velocity, Journal o the Egyptian Matheatical Society, 0 (01), 1, pp. 0-7 [6] Ibrahi, W., Shankar, B., MHD Boundary Layer Flow and Heat Transer o a Nanoluid Past a Pereable Stretching Sheet with Velocity, Theral and Solutal Slip Boundary Conditions, Coputers & Fluids, 75 (013), Apr., pp [7] Rooholghdos, S. A., Roohi, E., Extension o a Second Order Velocity Slip/Teperature Jup Boundary Condition to Siulate High Speed Micro/Nanolows, Coputers & Matheatics with Applications, 67 (014), 11, pp [8] Malvandi, A., Ganji, D. D., Brownian Motion and Therophoresis Eects on Slip Flow o Aluina/Water Nanoluid Inside a Circular Microchannel in the Presence o a Magnetic Field, International Journal o Theral Sciences, 84 (014), Oct., pp [9] Liao, S. J., On the Relationship Between the Hootopy Analysis Method and Euler Transor, Counications in Nonlinear Science and Nuerical Siulation, 15 (010), 6, pp [30] Arqub, O. A., El-Ajou, A., Solution o the Fractional Epideic Model by Hootopy Analysis Method, Journal o King Saud University, 5 (013), 1, pp [31] Turkyilazoglu, M., A Note on Hootopy Analysis Method, Applied Matheatics Letters, 3 (010), 10, pp [3] Abbasbandy, S., Shivanian, E., Predictor Hootopy Analysis Method and its Application to Soe Nonlinear Probles, Counications in Nonlinear Science and Nuerical Siulation, 16 (011), 6, pp [33] Shehzad S. A., et al., Therophoresis Particle Deposition in Mixed Convection Three-Diensional Radiative Flow o an Oldroyd-B Fluid, Journal o Taiwan Institute o Cheical Engineers, 45 (014), 3, pp Paper subitted: Septeber, 014 Paper revised: April 5, 015 Paper accepted: April 5, Society o Theral 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 ters and conditions

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