Analysis of Heat Transfer over a Stretching Rotating Disk by Using Homotopy Analysis Method

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1 Vol. 5(16) Jul. 15, PP Anlysis of Het Trnsfer over Stretching Rotting Disk by Using Hootopy Anlysis Method M. Khki 1 *, E. Dbirin nd D.D. Gnji 3 1 Deprtent of Mechnicl Engineering, Islic Azd University, Sri Brnch, Sri, Irn. Deprtent of Mechnicl Engineering, Islic Azd University, Sri Brnch, Sri, Irn. 3 Deprtent of Mechnicl Engineering, Bbol University, Bbol, Irn. *Corresponding Author's E-il: [email protected] Abstrct T his work studies the proble of stedy three diensionl flows nd het trnsfer of viscous fluid on rotting disk stretching in rdil direction hs been investigted. The governing equtions, continuity nd oentu for this proble re reduced to n ordinry for nd re solved by Hootopy Anlysis Method (HAM). The ccurcy of HAM is uthenticted by copring with nuericl solution s Boundry Vlue Proble (BVP). It hs been ttepted to show the cpbilities nd wide-rnge pplictions of the HAM in coprison with the nuericl ethod. This ethod led to high ccurte pproprite results for nonliner probles in coprison with nuericl solution. Keywords: Boundry lyer flow, Hootopy Anlysis Method, Stretching rotting disk. 1. Introduction The study of flow field due to rotting disk hs found ny pplictions in different fields of engineering nd industry. A nuber of rel processes cn be undertken using disk rottion such s: fns, turbines, centrifugl pups, rotors, viscoeters, spinning disk rectors nd other rotting bodies. The history of rotting disk flows goes bck to the celebrted pper by Von Krn [1-3] who initited the study of incopressible viscous fluid over n infinite plne disk rotting with unifor ngulr velocity. This odel is further investigted by ny reserchers to provide nlyticl nd nuericl results for better understnding of the fluid behvior due to rotting disks [4-5]. The investigtion on rotting disks fluid flow ws the in purpose of ny pervious reserches [6-8]. In ost cses, these probles do not dit nlyticl solution, so these equtions should be solved using specil techniques. In recent decdes, uch ttention hs been devoted to the newly developed ethods to construct n nlytic solution of eqution; such s Perturbtion techniques which re too strongly dependent upon the so-clled sll preters *9+. Other ny different ethods hve introduced to solve nonliner eqution such s Adoin s decoposition ethod [1], Hootopy Perturbtion Method (HPM) [1 14], Vritionl Itertion Method (VIM) [15 18], Colloction Method [19]. In contrst with the previous nlytic techniques, HAM [-] hs the following benefits: firstly, unlike ll previous nlytic techniques, the HAM provides us with gret freedo to express solutions of given non-liner proble by ens of different bse functions. Secondly, unlike ll previous nlytic techniques, the HAM lwys provides us with fily of solution expressions in the uxiliry preter h, even if non-liner proble hs unique solution. Thirdly, unlike perturbtion techniques, the HAM is independent of ny sll or lrge quntities. So, the HAM cn be pplied no tter if governing equtions nd boundry/initil conditions of given Article History: IJMEC DOI: 64913/115 Received Dte: Mr. 19, 15 Accepted Dte: Jun. 3, 15 Avilble Online: Jul. 7, 15 9

2 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 non-liner proble contin sll or lrge quntities or not [3-6]. The in purpose of this study is to pply HAM to find pproxite solutions of the flow nd het trnsfer over rotting disk tht is stretching in the rdil direction. A cler conclusion cn be drwn fro the nuericl ethod s (NUM) results tht the HAM provides highly ccurte solutions for nonliner differentil equtions.. Proble stteent nd theticl forultion Let us consider three diensionl linr flow of stedy incopressible fluid over rotting disk, which hs constnt ngulr velocity-x. The disk is stretching in rdil direction with velocity uwð~rþ. The governing Nvier Stokes equtions nd energy eqution with the corresponding boundry conditions for n xi-syetric flow nd het trnsfer in cylindricl coordintes re given by [7-31]: 1 ( ru ) w r r z, (1) u w v r z r r r r r z r u u v 1 p u 1 u u u u w v r z r r r r z r v v uv v 1 v v v u w v r z z r r r z w w 1 p w 1 w w u r z r r r z T 1 w T T T T T, z ; u ru ( r / R ), v r ( r / R ), w, T T z ; u, v, T T.,,, w w w () (3) (4) (5) (6) In the bove equtions uv, nd w re the coponents of velocity in r, nd z directions, is the fluid density, ( k/ C ) is the therl diffusivity nd p is the pressure. The preter is T constnt known s disk stretching preter [8]. p Figure 1. Schetic digr of the physicl syste 93 PISSN: , EISSN:

3 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 A prgtic pproch to find boundry lyer equtions is to introduce non-diensionl vribles in the governing Eqs (1) (6). We consider the following non-diensionl vribles for cur-rent proble. r z 1 u v w 1 r, z Re, u, v, w Re, R R R R R (7) p T T p, T, R T Where Re R / is Reynolds nuber, R is the reference length nd T is the reference teperture. It is noteworthy tht the corresponding scles in the xil direction re sller by 1/ fctor R thus iplicitly nticipting tht Re 1[8]. For high Reynolds nuber, i.e. Re, the resulting boundry lyer equtions in diensionless for re obtined s follows 1 ( ru ) w r r z, (8) u w r z r r z u u v p u u r z r z v w v uv v, p, z u w r z Pr z T T 1 T, z ; u ru ( r), v rv ( r), w, T T z ; u, v, T. w w w (9) (1) (11) (1) (13) where Pr / T is the Prndtl nuber. By using Lie group nlysis by Slee Asghr [3-34] nd introducing siilrity vrible nd the siilrity functions re given by: z r ( b )/ w r h( ), T ( ). b / b /, u r f ( ), v r g ( ), ( b )/ (14) where the for of h( ) will be deterined fro the continuity eqution. Fro the continuity eqution, the stre function ( xy, ) is de-fined s ru, rw. z r (15) Eq.(14) together with Eq.(15)give r ( b 3 )/ f ( ). (16) 94 PISSN: , EISSN:

4 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Fro Eqs.(15) nd (16), the for of w is given by: ( b )/ b 3 b w r f f. To ke the two vlues of w [(16) nd (17)] consistent we require b 3 b h( ) f f (17) (18) Using the siilrity trnsfortion (14) nd (18), the continuity eqution (8) is utoticlly stisfied nd the boundry lyer proble (9) (3) is conveniently trnsfored into self-siilr for: 3 b b f ff f g 3 b b g fg f g, 3 b Pr f, f (), f (), g () 1, () 1 f ( ), g( ), ( )., (19) () (1) () 3. Appliction of Hootopy Anlysis Method on proble For HAM solutions, we choose the initil guess nd uxiliry liner opertor in the following for: f, g, L( f ) f f, L( g ) g g, L( ), 1 L ( c1 c c3), L ( c4 c5), L ( c6 c7), c i re constnts. Let where i( 1,..,6) P,1 denotes the ebedding preter nd indictes non zero uxiliry preters. We then construct the following equtions: Zeroth order defortion equtions (3) (4) (5) (1 P) L F( ; p) f ( ) p N F( ; p), F(; p) ; F(; p), F( ; p) 3 d 3 b d b d N [ F ( ; p)] f ( ; p) f ( ; p) f ( ; p) f ( ; p) 3 d d d g( ; ) p (1 P) L G ( ; p) g ( ) p N G ( ; p), g ( ; p) ; g (; p) 1 (6) (7) (8) (9) (3) 95 PISSN: , EISSN:

5 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 d 3 b d d d N [ G ( ; p)] g ( ; p) f ( ; p) g ( ; p) b d g ( ; p) f ( ; p) d (1 P) L ( ; p) ( ) p N ( ; p), ( ; p) ; (; p) 1 d 3 b d d d N [ ( ; p)] ( ; p) Pr f ( ; p) ( ; p), For p nd p 1we hve F( ;) f ( ), F( ;1) f ( ). G( ;) ( ), G( ;1) ( ). ( ;) ( ), ( ;1) ( ). (31) (3) (33) (34) (35) (36) (37) When p increses fro to 1 then F( ; p), G ( ; p) nd ( ; p) vries fro f( ), g( ) nd ( ) to f ( ), g ( ) nd ( ). By Tylor's theore nd using eqution (19), F ( ; p ), G ( ; p ) nd ( ; p) cn be expnded in power series of p s follows: 1 ( f ( ; p)) F ( ; p) f ( ) f ( ) p, F ( ), 1! p p 1 1 ( g( ; p)) G( ; p) g ( ) g ( ) p, G ( ), 1! p p1 1 ( ( ; p)) ( ; p) ( ) ( ), ( ), p 1! p p1 (38) (39) (4) In which is chosen in such wy tht this series is convergent t p 1, therefore we hve through eqution (1) tht F( ) f ( ) f ( ). 1 G( ) g ( ) g ( ). 1 ( ) ( ) ( ). 1 (41) (4) (43) th order defortion equtions 96 PISSN: , EISSN:

6 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 L F ( ) F ( ) R ( ), 1 k f 1 1 f b 3 b R f 1 f 1 k f k f 1 k f k k g 1 k g k M f 1 F (; p) ; F(; p), F( ; p) ' ' ' L G ( ) g ( ) R ( ), g 1 1 g 3 b b R g 1 f 1 k gk g 1 k fk k g (; p) ; g ( ; p). ' ' L ( ) ( ) ( ), R 1 R 3 b 1 1 k 1 k k ' Pr f (; p) ; ( ; p). ' (44) (45) (46) (47) (48) (49) (5) (51) (5), 1, 1, 1 Now we deterine the convergency of the result, the differentil eqution, nd the uxiliry function ccording to the solution expression. So let us ssue: H( ) 1 (54) We hve found the nswer by ple nlytic solution device. For first defortion of the solution re presented below. (53) f b b b b g : 3 4 b b b (55) (56) 97 PISSN: , EISSN:

7 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/ Pr Pr Prb Pr b : Pr Prb 3 (57) The solutions f( ), g( ) nd ( ) were too long to be entioned here, therefore, they re shown grphiclly. 4. Convergence of the HAM solution As pointed out by Lio, the convergence region nd rte of solution series cn be djusted nd controlled by ens of the uxiliry preter ħ*-]. In generl, by ens of the so-clled ħ-curve, it is strightforwrd to choose n pproprite rnge for ħ which ensures the convergence of the solution series. Figure. The - vlidity for 6, 8, 1 nd 11 th order pproxition when n 1, Pr 3, 1, b 1 nd.5 To influence of ħ on the convergence of solution, we plot the so-clled ħ-curve of f (), g () nd () by 6, 8, 1 nd 11 th-order pproxition, s shown in Fig. nd 3. For n 1, Pr 3, 1, b 1nd.5 the rnges for f () equl to 1.3, for g () equl to 1.5, for n 3,Pr.7,3, 1, b 3 nd.1 the rnges 1.5.4, gives suitble vlue of for convergency. Then.9 is suitble vlue which is used for solution. 98 PISSN: , EISSN:

8 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Figure 3. The - vlidity for 6, 8, 1 nd 11 th order pproxition when n 3, Pr.7,3, 1, b 3 nd.1 5. Result nd discussions In this study, the ccurcy nd vlidity of this pproxite solution on stedy three diensionl flows nd het trnsfer of viscous fluid on rotting disk stretching in rdil direction hs been investigted. The results of HAM nd nuericl solution re copred in Figures (4). The nuericl solution is perfored using the lgebr pckge Mple 15., to solve the present cse. The softwre uses second-order difference schee cobined with n order bootstrp technique with eshrefineent strtegies: the difference schee is bsed on either the trpezoid or idpoint rules; the order iproveent/ccurcy enhnceent is either Richrdson extrpoltion or ethod of deferred corrections [34]. By the drwing of -D Figures 4, of nuericl solution nd HAM solution for f ( ), f ( ) nd ( ) it cn be seen in grphicl results the obtined nlyticl solution in coprison with the nuericl ones represents rerkble ccurcy. 99 PISSN: , EISSN:

9 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Figure 4. The coprison between the nuericl nd HAM solution for f( ), g( ) nd ( ) when n 1,3,Pr.7,3,.1, b 1. The effects of controlling preters (disk stretching preter nd power-lw stretching index n ) on the ziuthlly velocity g( ), the rdil velocity f ( ), the verticl velocity f ( ) nd the teperture ( ) re presented in Figs. 3 nd 4. 3 PISSN: , EISSN:

10 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Figure 5. Vrition of () ziuthlly velocity profiles, (b) rdil velocity profiles, (c) verticl velocity profiles nd (d) teperture profiles for different vlues of disk stretching preter when n 1, 1, b n, Pr 3. Figs(7) Shows the coprison of the ziuthlly velocity g( ), the rdil velocity f ( ), the verticl velocity f ( ) nd the teperture ( ) for known vlues of the preters n1, 1, b n, Pr 3 nd different vlue of stretching preter respectively. The results show tht, by incresing vlue of leds to increse rdil nd verticl velocities increse while the ziuthl velocity nd teperture profiles decrese. 31 PISSN: , EISSN:

11 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Figure 6. Vrition of () ziuthlly velocity profiles, (b) rdil velocity profiles, (c) verticl velocity profiles nd (d) teperture profiles for power-lw index n when 1, b n, Pr 3,.1. In ddition, Figs. (6) Shows the 4shows the effects of power-lw stretching index-n of the ziuthlly velocity g( ), the rdil velocity f ( ), the verticl velocity f ( ) nd the teperture ( ) for known vlues of the preters 1, b n, Pr 3,.1. The results show tht, s the vlue of n increses, the teperture nd the two coponents of velocity decrese. Figure 7. Diensionless teperture predicted by HAM nd nuericl ethod (NUM) for different Pr Prndtl nuber t n 3, 1, b 3,.1 Moreover, Fig7 shows the effect of Prndtl nuber on the diensionless teperture profile. As observed, n increse in the Prndtl nuber leds to decrese in the teperture. This is n greeent with the physicl fct tht the therl boundry lyer thickness decreses with incresing Pr. 3 PISSN: , EISSN:

12 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 Conclusion In this pper, we ipleented the Hootopy Anlysis Method (HAM) for finding solutions of flow nd het trnsfer over rotting disk stretching non-linerly in rdil direction. The governing equtions, continuity nd oentu for this proble re reduced to n ordinry single third for by using siilrity trnsfortion. Furtherore, the obtined solutions by HAM re copred with nuericl solution. By Copring between nuericl solution nd Anlyticl Results proves the ccurcy nd vlidity of ethod. In ddition, the effect of stretching preter index n is to decrese the velocity nd the teperture profile. Moreover Increse of stretching preter shows n increse in verticl nd rdil coponents of the velocity nd decrese in ziuthl coponent nd teperture profiles. References [1] A. Aziz, A siilrity solution for linr therl boundry lyer over flt plte with convective surfce boundry condition, Coun. Nonliner Sci. Nuer. Siul. 14 (9), [] Abbsi, M., Hzeh Nv, Gh., Rhiipetroudi, I., Anlytic solution of hydrodynic nd therl boundry lyers over flt plte in unifor stre of fluid with convective surfce boundry condition, Indin J. Sci. Res, 1(14), 1, [3] A.H. Nyfeh, Perturbtion Methods, Wiley, New York, USA,. [4] A. Vhbzdeh, M. Fkour, D. D. Gnji, I. Rhiipetroudi, Anlyticl ccurcy of the one diensionl het trnsfer in geoetry with logrithic vrious surfces, Centrl Europen Journl of Engineering, 4(14), 4, [5] F. White, Viscous Fluid Flow, McGrw, New York, 1991, [6] Gnji, D. D., Abbsi, M., Rhii, J., Gholi, M., Rhiipetroudi, I., On the MHD squeezeflow between two prllel disks with suction or injection vi HAM nd HPM, Front. Mech. Eng. DOI 1.17/s [7] G.W. Blun, S. Kuei, Syetries nd differentil equtions, in: Applied Mtheticl Sciences, vol. 1, Springer, Berlin, New York, [8] H. Schlichting, K. Gersten, Boundry-lyer Theory, Springer, Verlg,. [9] I. Rhii Petroudi, D. D. Gnji, A. B. Shotorbn, M. Khzyi Nejd, E. Rhii, R. Rohollhtbr, F. Therini, Sei nlyticl ethod for solving nonliner eqution rising of nturl convection porous fin, THERMAL SCIENCE, 16 (1), 5, [1] I.V. Shevchuk, Convective Het nd Mss Trnsfer in Rotting Disk Systes, Springer, Berlin, 9. [11] J.H. He, Hootopy perturbtion ethod: new nonliner nlyticl technique, Applied Mthetics nd Coputtions, 135 (3), [1] J.H. He, Hootopy perturbtion ethod for solving boundry vlue probles, Physics Letters A, 35 (6), [13] J. H. He, Vritionl itertion ethod for utonoous ordinry differentil systes, Applied Mthetics nd Coputtion, 114 (), [14] J.H. He, Vritionl itertion ethod soe recent results nd new interprettions, Journl of Coputtionl nd Applied Mthetics, 7 (7),1,3 17. [15] K. Millsps, K. Pohlhusen, Het trnsfer by linr flow fro rotting plte, J. Aerosp. Sci. 19 (195),, [16] Khki, M., Abbsi, M., Rhii Petroudi, I., M. Nghdi Bzneshin, R., Sei- nlyticl investigtion of trnsverse gnetic field on Viscous Flow over Stretching Sheet, MAGNT Reserch Report, (14),, 5-57 [17] L. Rosehed, Linr Boundry Lyer, Oxford University Press, Oxford, [18] M. Abbsi, D. D. Gnji, I. Rhiipetroudi, M. Khki, Coprtive nlysis of MHD boundry-lyer flow of viscoelstic fluid in pereble chnnel with slip boundries by using HAM, VIM, HPM. Wlilk Journl for Science nd Technology, 11(14), 7, [19] M. Awd, Het trnsfer fro rotting disk to fluids for wide rnge of Prndtl nubers using the syptotic odel, J. Het Trnsfer 13 (8), 1. [] M. Abbsi, A. Ahdin CHshi, I. Rhiipetroudi, Kh. Hosseinzdeh, Anlysis of fourth grde fluid flow in chnnel by ppliction of VIM nd HAM, Indin J.Sci.Res. 1(14),, [1] M. Jlil, S. Asghr, Flow of power-lw fluid over stretching surfce: lie group nlysis, Int. J. Non-Liner Mech. 48 (13) [] M. Jlil, S. Asghr, S.M. Irn, Self siilr solutions for the flow nd het trnsfer of Powell Eyring fluid over oving surfce in prllel free stre, Int. J. Het Mss Trnsfer 65 (13) [3] M. Jlil, S. Asghr, M. Mushtq, Lie group nlysis of ixed convection flow with ss trnsfer over stretching surfce with suction or injection, Mth. Prob. Eng. 1 (1). [4] M. Shhbbei, S. Sedodin, M. Soleynibeshei, I. Rhiipetroudi,MHD effect on therl perfornce of cylindricl spin porous fin with teperture dependent het trnsfer coefficient nd eissivity, Interntionl Journl of Energy & Technology, 6 (14), 16, 1 1 [5] N.K. Ibrgiov, CRC Hndbook of Lie Group Anlysis of Differentil qutions, Syetries, Exct Solutions, nd Conservtion Lws, vol. I, CRC Press Llc, PISSN: , EISSN:

13 Mehrn Khki et l. / Vol. 5(16) Jul. 15, pp IJMEC DOI: 64913/115 [6] P. Olver, Appliction of Lie groups to differentil equtions, Grdute Texts in Mthetics, vol. 17, 1986, Springer, New York. [7] Rhiipetroudi, I., Gnji, D.D., Khzyi Nejd, M., Rhii, J., Rhii, E., Rhiifr, A., Trnsverse gnetic field on Jeffery Hel proble with Cu-wter nnofluid between two non-prllel plne wlls by using colloction ethod, Cse Studies in Therl Engineering, 4(14), 193-1, [8] Slee Asghr, Mudssr Jlil, Muhd Hussn, Mustf Turkyilzoglu, Lie group nlysis of flow nd het trnsfer over stretching rotting disk, Interntionl Journl of Het nd Mss Trnsfer 69 (14) [9] SJ. Lio, Beyond perturbtion: introduction to the hootopy nlysis ethod. Boc Rton: Chpn & Hll, CRC Press, 3. [3] SJ. Lio, Hootopy Anlysis Method in Nonliner Differentil Equtions, Springer & Higher Eduction Press, Heidelberg, 1. [31] SJ. Lio, Series solutions of unstedy boundry-lyer flows over stretching flt plte, Studies in Applied Mthetics, 117(6), *3+ S. Moni, S. Abusd, Appliction of He s vritionl itertion ethod to Helholtz eqution, Chos Solitons & Frctls, 7 (6), 5, [33] T.V. Kárán, Über linre und turbulente Reibung, Zeit. Angew. Mth. Phys. 1 (4) (191) [34] T. Von Kárán, C. Lin, On the existence of n exct solution of the equtions of Nvier Stokes, Coun. Pure Appl. Mth. 14 (3) (1961) PISSN: , EISSN: