On the Revolving Ferrofluid Flow Due to Rotating Disk

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1 ISSN (pint), (online) Intenational Jounal of Nonlinea Siene Vol.13(212) No.3,pp On the Revolving Feofluid Flow Due to Rotating Disk Paas Ram, Kushal Shama Depatment of Mathematis, National Institute of Tehnology, Kuuksheta , Hayana, India (Reeived 19 August 211, aepted 6 Januay 212) Abstat:This pape deals with the theoetial investigation of the effet of otation on feofluid flow due to otating disk by solving bounday laye equations. Hee, we have solved the oupled nonlinea diffeential equations by powe seies appoximations. Expessions fo the omponents of veloity and pessue pofile ae obtained in ylindial o-odinate system by onsideing the z-axis as the axis of otation. It is obseved that thee is signifiant inement in the thikness of the bounday laye ove otating disk in ompaison to the odinay ase of visous fluid flow without otation. The esults fo all above vaiables ae obtained numeially and disussed gaphially. Keywods: Axi-symmeti; otating disk; bounday laye; feofluid; magneti-field 1 Intodution Feofluids ae stable suspensions of olloidal feomagneti patiles of the ode of 1nm in suitable non-magneti aie liquids. These olloidal patiles ae oated with sufatants to avoid thei agglomeation. Beause of the industial appliations of feofluids, sine last five deades, the investigation on them has been fasinating the eseahes and enginees, vigoously. One of the many fasinating featues of the feofluids is the pospet of influening flow by a magneti field and vie-vesa [1, 2]. Sealing of the otating shafts is the most known appliation of the magneti fluids. Feofluids ae widely used in sealing of had disk dives, otating x-ay tubes unde engineeing appliations. The majo appliation of feofluid in eletial field is the ontolling of heat in loudspeakes. Contol on heating makes the life of sound speakes longe and ineases the aoustial powe without any hange in its geometial shape. Magneti fluids ae used in the ontast medium in X-ay examinations and fo positioning tamponade fo etinal detahment epai in eye sugey. Theefoe, feofluids play an impotant ole in the field of bio-medial siene also. Besides, thee ae otationally symmeti flows of the inompessible feofluids in the field of fluid mehanis, having all thee veloity omponents viz. adial, tangential and vetial, in spae, diffeent fom zeo. In suh type of flows, the vaiables ae independent of angula oodinates and the angula veloity is unifom at lage distane fom the otating disk. We onside this type of flow fo an inompessible feofluid when the otating disk is subjeted to the magneti field (H,, H z ). Vema [3, 4, 5] has solved the eseah poblems on paamagneti Couette flow, helial flow with heat and flow though a poous annulus, espetively. Rosensweig [6] has given an authoitative intodution to the eseah on magneti liquids in his monogaph and studied the effet of magnetization esulting in inteesting infomation. The pioneeing study of odinay visous fluid flow due to the infinite otating disk was aied by Kaman [7]. He intodued the famous tansfomation, whih edues the govening patial diffeential equations into odinay diffeential equations. Kaman otating disk poblem is extended to the ase of flow stated impulsively fom est, and also the steady state is solved to a highe degee of auay than peviously done by a simple analytial method whih neglets the esembling diffiulties in Cohan s [8] well known solution. Cohan obtained asymptoti solutions fo the steady hydodynami poblem fomulated by Kaman. Benton [9] impoved Cohan s solutions and also, solved the unsteady ase. Attia [1] studied the unsteady state in the pesene of an applied unifom magneti field. The steady flow of odinay visous fluid due to the otating disk with unifom sution was studied by Mithal [11]. Attia and Aboul-Hasan [12] disussed about flow due to an infinite disk otating in the pesene of an axial unifom magneti field by taking Hall effet into onsideation. Coesponding autho. addess: paasam nit@yahoo.o.in, kushal.nitkk@gmail.om Copyight Wold Aademi Pess, Wold Aademi Union IJNS /611

2 318 Intenational Jounal of Nonlinea Siene, Vol.13(212), No.3, pp In geneal, magnetization is a funtion of magneti field, tempeatue and density of the fluid. This leads to onvetion of feofluid in the pesene of the magneti field gadient. Sunil et al. [13] studied the effet of otation on themosolutal onvetion in a feomagneti fluid onsideing a hoizontal laye of an inompessible feomagneti fluid. Venkatasubamanian and Kaloni [14] investigated the effet of otation on the themo-onvetive instability of a hoizontal laye of feofluid heated fom below in the pesene of unifom vetial magneti field. Das Gupta and Gupta [15] examined the onset onvetion in a hoizontal laye of feomagneti fluid heated fom below and otating about a vetial axis in the pesene of a unifom magneti field. Chauhan and Agawal [16] studied the MHD flow in a paallel-disk hannel patially filled with a poous medium in a otating system inluding Hall uent. Ram et al. [17] solved the non-linea patial diffeential equations using powe seies appoximations and disussed the effet of magneti field-dependent visosity on veloity omponents and pessue pofile. Futhe, the effet of poosity on veloity omponents and pessue pofile has been studied by Ram et al. [18]. Tukyilmazoglu [19] obtained analytial expessions fo the solution of steady, lamina, inompessible, visous fluid of the bounday laye flow due to a otating disk in the pesene of a unifom sution o injetion and homotopy analysis method was employed to obtain the exat solutions.sajid [2] has employed HAM to investigate the slip effets on the two poblems of visous inompessible flows whih oesponds to the plana and axi-symmeti stething of the sheet. In this poblem, we have disussed the solution of the steady lamina flow of an inompessible visous eletially non onduting feofluid ove a otating disk in the pesene of a unifom magneti field and expessions fo omponents of veloity and pessue pofile onsideing vaious types of fluid esponses ae obtained in ylindial o-odinate system with z-axis as the axis of otation. It is found that thee is a vaiation in the bounday laye displaement thikness as ompaed to the odinay visous flow ase. We also have given the expession fo total volume flowing outwads the axis taken ove a ylinde of adius R aound the z axis using the desiption given on page 229 in Shlihting [21]. The pesent study an seve as a theoetial suppot fo expeimental investigations. This poblem, to the best of ou knowledge, has not been investigated yet. 2 Mathematial fomulation and solution The flow in an isotopi medium is onsideed to be steady ( t = ) and Axi-symmeti ( θ = ) exluding the themal effets. The fluid and the feous patiles have the same veloity. The fluid and disk ae assumed to be eletially nononduting. The whole system is otating with angula veloity Ω = (,, Ω) along the vetial axis, whih is taken as z-axis. One additional simplifiation is assumed in momentum equation that the visosity is independent of magneti-field intensity. Basi govening equations of feofluid due to Finlayson [22] ae as follows: The ontinuity equation fo an inompessible feofluid: V = (1) The momentum equation fo an inompessible feomagneti fluid with onstant visosity in a otating fame of efeene: [ ] V ρ + (V )V = p + µ (M. )H + µ f 2 V + 2ρ(Ω V) + ρ t 2 Ω 2 (2) The effet of otation inludes two tems: Centifugal foe (ρ/2) gad Ω 2 and Coiolis aeleation 2ρ(Ω V). In (2), p (ρ/2) Ω 2 = p is the edued pessue, whee p stands fo fluid pessue. Maxwell s equations, simplified fo a non-onduting fluid with no displaement uents: Assumptions H = ; (H + 4πM) = (3) M = χh; χ = µ f µ 1; M H = (4) Attentions have been dawn to the fat that the extenal magneti field inteats with magneti patiles, not only though foes but also though ouples. The veloity omponent v z is less as ompaed to v and v θ. The equation of IJNS fo ontibution: edito@nonlineasiene.og.uk

3 P. Ram, K. Shama: On the Revolving Feofluid Flow Due to Rotating Disk 319 motion and equation of ontinuity edue to 1 p ρ + µ ρ M [ 2 H + ν v 2 + [ 2 v θ ν p ρ z + µ ρ M z H + ν ( v ( vθ [ 2 v z 2 ) + 2 v z 2 ) + 2 v θ z ] + 2Ωv θ = v v ] 2Ωv = v v θ v z + 2 v z z 2 ] + v z + v z v z = v + v v z z z v z v2 θ v θ z + v v θ v + v + v z z = (8) The appoximate initial and bounday onditions fo the flow due to otation of an infinitely long disk (z = ) with onstant angula veloity ω ae given by } at z = ; v =, v θ = ω, v z = (9) at z ; v =, v θ = Hee, v z does not vanish fo z but tends to a finite negative value. Consideing bounday laye appoximation 1 p ρ + µ ρ M H = ω2 as used by Ram et al.[17] and vey less vaiation of magneti-field along z-dietion and using tansfomations, v = ωe(α), v θ = ωf (α), v z = νωg(α); whee α = z ω ν ; fom equations (5)-(8), we get a system of non-linea oupled diffeential equations in E, F, G and P, as follows: E GE E 2 + F 2 + 2F 1 = (1) F GF 2EF 2E = (11) P G + GG = (12) (5) (6) (7) G + 2E = (13) Bounday onditions (9) beome E() = G() = F () 1 = P () P = E( ) = F ( ) = } (14) G must tend to a finite limit, say, as α, i.e. G( ) =, ( > ). The values of E, F, G and P ae ompaed gaphially with thei oesponding values in lassial ase. Cohan indiated that fomal asymptoti expansions (fo lage α) of the system of equations (1)-(13)ae the powe seies in exp( α), i.e. E(α) F (α) G(α) G( ) + H(α) Integating the equations (1) and (11) between and, and using the elations A i exp(iα) (15) i=1 B i exp(iα) (16) i=1 C i exp(iα) (17) i=1 D i exp(iα) (18) i=1 GE dα = [GE] GF dα = [GF ] Togethe with E ( ) = and F ( ) =, we get G E dα = 2 G F dα = 2 E 2 dα EF dα IJNS homepage:

4 32 Intenational Jounal of Nonlinea Siene, Vol.13(212), No.3, pp E () = F () = (3E 2 F 2 2F + 1) dα (4EF + 2E) dα Using the supposition E () = a and F () = b in equations (1) - (14), we get the following additional bounday onditions fo the appoximate solution: E () = 2, E () = 4b; F () =, F () = 4a; G () =, G () = 2 a, G (19) () = 4; P () = 2a, P () = 4, P () = 8b. 3 Results and disussion Fist fou oeffiients in eah equations (15) - (18), alulated with the help of (14) and (19) ae as follows: A 1 = ( ) 2b a 2 3, A2 = ( ) 2b a 2 2 ; A3 = ( ) 2b a 2, A4 = ( ) 2b a 2 6 ; B 1 = ( 2a b ), B 2 = ( 2a 19b 3 2 6) ; B 3 = ( 2a + 7b 3 + 4), B 4 = ( 2a 3 11b 3 6 1) ; C 1 = ( 2 3 3a ), C 2 2 = ( 2 + 8a 3 6 ) ; C 2 3 = ( 2 7a ), C 2 4 = ( a 3 ) ; 2 D 1 = ( ) 4b a 2 3, D2 = ( 4b a ) ; D3 = ( 4b a ), D4 = ( ) 4b a 2 3. Using the values a =.54, b =.62 and =.886 fom Cohan [8], we alulate the numeial values of the oeffiients A 1, A 2, A 3, A 4 ; B 1, B 2, B 3, B 4 ; C 1, C 2, C 3, C 4 ; D 1, D 2, D 3 and D 4 ; and daw the gaphs of veloity omponents and asymptoti pessue with the dimensionless paamete α. The pesent numeial esults ae pesented in table 1 and give vey good appoximate solution of the above system of non-linea diffeential equations. Table 1: The steady state veloity field and pessue as funtions of α α E F G P (α) P E F E-5 3.6E E-5 1.3E-5-3.2E The bounday laye displaement thikness is alulated as: d = 1 ω Total volume flowing outwad the z-axis, Q = 2πR z= v θ dz = v dz = 2πR α= F (α) dα = ωe(α) ν/ωdα = πr 2 ων G( ) = R 2 ων = R 2 ν α z Hene fom above, the total volume flowing outwad the z-axis is popotional to the dimensionless paamete α. IJNS fo ontibution: edito@nonlineasiene.og.uk

5 P. Ram, K. Shama: On the Revolving Feofluid Flow Due to Rotating Disk 321 The fluid is taken to otate at a lage distane fom the wall, the angle beomes ( / ) v vθ tan φ = = E () z θ F () = = φ = 41 Patiula Case: If we emove the effet of both, the entifugal foe (ρ/2) gad Ω 2 and Coiolis aeleation 2ρ(Ω V) and bounday laye appoximation 1 p ρ + µ ρ M H = ω2, the poblem edues to Cohan s ase [8] of odinay visous fluid flow without otation. The poblem onsideed hee involves a numbe of paametes, on the basis of whih, a wide ange of numeial esults have been deived. Of these esults, a small setion is pesented hee fo bevity. The numeial esults fo the veloity pofiles, ommonly known as adial, tangential, vetial (axial) veloities, ae shown in figues 1.1, 1.2, 1.3 espetively. Also, we have alulated the angle between the wall and the evolving feofluid, whih is 41. In figue 1.1, the uve E 2 epesents the effets of otation on the adial veloity in ase of feofluid flow due to otating disk. Wheeas, E 1 indiates the adial veloity pofile of the Cohan s study of odinay visous fluid flow. Due to otation, the adial veloity eahes its maximum value nea the sufae of the disk with magnitude.8894 at α =.4, wheeas, in Cohan s ase, the maximum value.181 of adial veloity is attained at ompaatively distant point α =.9. Hee, it is notied that the adial veloity E 2 has vey less peak value in ompaison to E 1 beause of thikening of the fluid laye due to the otation of the whole system. Thus, the effet of otation is moe ponouned than the foe of magnetization in the sense of fluid thikening. Also it is quite inteesting to see that in ase of otation befoe onveging to zeo, the adial veloity one beomes negative. Figue 1.2, is the gaphial ompaison of the tangential veloity of feofluid flow with otation (F 2 ) to that of Cohan s odinay visous fluid flow (F 1 ). In ou ase, if we inease the value of α, the tangential veloity F 2 deeases ontinuously and tends to zeo fo lage value of α. It is obseved fom the table 1, the value of tangential veloity is at α = 1, wheeas fo the odinay visous fluid, the tangential veloity is.468 fo the same value of α. Theefoe, at α = 1, thee is an appoximate inement of 6.7% in the value of tangential veloity in ompaison to that of Cohan s value. Also fom the figue, it is lea that F 1 onveges to zeo little faste than F 2, howeve, both the uves have simila tends. Figue 1.3 shows the axial veloity pofile, whih is zeo in the beginning and tends to a finite value.886 fo α = 14.8 onwads. When we inease the value of α, it deeases ontinuously in the negative egion. Ou axial veloity value is at α = 1, wheeas fo Cohan, the axial veloity is.266 fo the same value of α. Meaning theeby, fo the same value of α, the axial veloity omponent aquies lage value than the value in odinay ase. The pessue pofile P (α) P with the initial pessue, P at α = is shown in figue 2. Hee, pessue goes to negative egion nea the sufae of the disk, and at α =.4, it goes to maximum negative value Onwads to α =.4, pessue stats ineasing with ineasing the value of α, and at α = 1.3, it entes in the positive egion and attains maximum value.7667 at α = 2.2. Finally P (α) P onveges to zeo i.e. P (α) onveges to P. Compaing figues 1.1 and 2, we onlude that when adial veloity ineases, the pessue of the feofluid deeases and when adial veloity deeases, feofluid pessue ineases i.e. they ae onvese in onvegene behaviou. Also, the tangential veloity diminishes slowe than axial veloity omponent. The hange in the uve of adial veloity is faste due to the effet of extenal magneti field esulting in eduing the time equied fo veloity pofile to eah thei onvegene level. In figues 3 and 4, deivatives of adial and tangential omponents of veloities ae shown. In nut shell, the otation of the disk along with evolution of the feofluid esults in an ineased displaement thikness moe than that of Benton [9]. 4 Conlusion Fom these esults, we onlude that magnetization foe i.e., µ (M )H edues the pessue. Also, it has been obseved that magneti field intensity ineases the adial veloity; wheeas, the fluid otation has evese effet. The effet of otation is moe ponouned than the foe of magnetization due to whih the adial veloity takes vey less peak value in ompaison to odinay visous fluid flow ase. Due to the otation, the etadation of the adial veloity ineases the thikness of the magneti fluid laye. Aknowledgement The authos ae extemely thankful to the eviewes fo thei itial omments and valuable suggestions leading to an impovement in the pape. IJNS homepage:

6 322 Intenational Jounal of Nonlinea Siene, Vol.13(212), No.3, pp Refeenes [1] Feynman, R.P., Leighton, R.B. and Sands, M. Letues on Physis. Addison-Wesley, MA, 1963, 1. [2] Shliomis, M.I..Feofluids as themal athets. Physial Review Lettes, 92(24):1889. [3] Vema, P.D.S.and Vedan, M.J..Steady otation of a sphee in a paamagneti fluid. Wea, 52(1979): [4] Vema, P.D.S. and Vedan, M.J..Helial flow of feofluid with heat ondution. J. Math. Phy. Si., 12(1978): [5] Vema, P.D.S. and Singh, M..Magneti fluid flow though poous annulus. Int. J. Non-Linea Mehanis, 16(1981): [6] Rosensweig, R.E..Feohydodynamis. Cambidge Univesity Pess, [7] Kaman, V..Ube laminae und tubulente Reibung. Z. Angew. Math. Meh., I(1921): [8] Cohan, W.G..The flow due to a otating dis. Po. Camb. Phil. Sot., 3(1934): [9] Benton, E.R..On the flow due to a otating disk. J. Fluid Mehanis, 24(1966): [1] Attia, H.A..Unsteady MHD flow nea a otating poous disk with unifom sution o injetion. J. Fluid Dynamis Reseah, 23(1998): [11] Mithal, K.G..On the effets of unifom high sution on the steady flow of a non-newtonian liquid due to a otating disk. Quat J. Meh. Appl. Math, XIV(1961): [12] Attia, H.A. and Aboul-Hassan, A.L..On hydomagneti flow due to a otating disk. Applied Mathematial Modelling, 28(24): [13] Sunil, Bhati, P.K., Shama, D. and Shama, R.C..The effet of otation on themosolutal onvetion in a feomagneti fluid. Int. J. Applied Mehanis and Engineeing, 1(25): [14] Venkatasubamanian, S. and Kaloni, P.N..Effets of otation on the themo-onvetive instability of a hoizontal laye of feofluids. Int. J. Engg. Si., 32(1994): [15] Das Gupta, M. and Gupta, A.S..Convetive instability of a laye of a feomagneti fluid otating about a vetial axis. Int. J. Engg. Si.. 17(1979): [16] Chauhan, D.S. and Agawal, R..Effet of Hall uent on MHD flow in a otating hannel patially filled with a poous medium.chem. Eng. Comm., 197(21): [17] Ram, P., Bhandai, A. and Shama, K..Effet of magneti field-dependent visosity on evolving feofluid. J. Magnetism and Magneti Mateials, 322(21)(21): [18] Ram, P., Shama, K. and Bhandai, A..Effet of poosity on feofluid flow with otating disk. Int. J. Applied Mathematis and Mehanis, 6(16)(21): [19] Tukyilmazoglu, M..On Analyti appoximate solutions of otating disk bounday laye flow subjet to a unifom sution o injetion. Int. J. Mehanial Sienes, 52 (21): [2] Sajid, M..Homotopy analysis of stething flows with patial slip. Intenational Jounal of Nonlinea Siene, 8(29): [21] Shlihting, H..Bounday Laye Theoy. MGaw-Hill Book Company, New Yok, 196. [22] Finlayson, B.A..Convetive instability of feomagneti fluids. J. Fluid Mehanis, 4(4)(197): IJNS fo ontibution: edito@nonlineasiene.og.uk

7 P. Ram, K. Shama: On the Revolving Feofluid Flow Due to Rotating Disk 323 Nomenlatue V Veloity of feofluid (m/s) p Fluid pessue (kg /m s 2 ) M Magnetization (A/m) H Magneti field intensity (A/m) Radial dietion (m) z Axial dietion (m) v Radial veloity (m/s) v θ Tangential veloity (ad/s) v z Axial veloity (m/s) E Dimensionless omponent of adial veloity F Dimensionless omponent of tangential veloity G Dimensionless omponent of axial veloity P Kaman s dimensionless pessue P Initial pessue (absolute value) d Thikness of the feofluid laye (m) Q Total volume flowing outwad the z-axis (m 3 ) Geek Symbols Ω Angula veloity of whole system (ad/s) χ Magneti suseptibility θ Tangential dietion (ad) ω Angula veloity of the disk (ad/s) Gadient opeato (m 1 ) ρ Fluid density (kg/m 3 ) µ Magneti pemeability of fee spae (H/m) µ f Refeene visosity of fluid (kg/ms) χ Magneti suseptibility ν Kinemati visosity (m 2 /s) α Kaman s paamete (dimensionless) Angle of otation (deg) φ E E( ) E 1 E 2 = Case of otation Figue 1.1: Effet of otation on adial veloity pofile E E F F( ) F 1 F 2 = Case of otation Figue 1.2: Effet of otation on tangential veloity pofile F F IJNS homepage:

8 324 Intenational Jounal of Nonlinea Siene, Vol.13(212), No.3, pp G( ) P( )-P G G 1 G 2 = Case of otation G G P( )-P Figue 1.3: Effet of otation on axial veloity pofile -.2 Figue 2: Effet of otation on pessue pofile E' E'( ) Figue 3: Deivative of adial veloity F' F'( ) Figue 4: Deivative of tangenial veloity IJNS fo ontibution: edito@nonlineasiene.og.uk