Revolving Ferrofluid Flow due to Rotating Disk

Transcription

1 Poeedings of the Wold Congess on Engineeing 1 Vol III WCE 1, July - 6, 1, London, U.K. Reoling Feofluid Flow due to Rotating isk Paas Ram, Kushal Shama bstat - eoling flow of feofluid oe a otating disk is inestigated by soling the bounday laye equations with bounday onditions by using Neuinge-Rosensweig model. The omponents of eloity and pessue pofile ae alulated numeially. Howee, the solution fo the flow pofile and the hange in bounday laye displaement thikness follows the lines fo odinay isous flow. Expessions ae obtained in the ylindial o-odinate system by onsideing the -axis as axis of otation. Hee, we hae soled non-linea diffeential equations numeially by using powe seies appoximations. Index Tems - xi-symmeti, otating disk, bounday laye, feofluid, magneti-field. F I. INTROUCTION ERROFLUIS 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 aoid thei agglomeation. Beause of the industial appliations of feofluids, the inestigation on them fasinated the eseahes and enginees igoously sine last fie deades. One of the many fasinating featues of the feofluids is the pospet of influening flow by a magneti field and ie-esa [1, ]. Sealing of the otating shafts is the most known appliation of the magneti fluid. Feofluid is widely used in sealing of had dis dies, otating x-ay tubes unde engineeing appliations. The majo appliations of feofluid in eletial field is that 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. Thee ae otationally symmeti flows of the inompessible feofluid in the field of fluid mehanis, haing all thee eloity omponents i. adial, tangential and etial, in spae, diffeent fom eo. In suh type of flows, the aiables ae independent of angula oodinates Coesponding utho: Paas Ram paasam_nit@yahoo.o.in, kushal.nitkk@gmail.om ept. of Mathematis, NIT Kuuksheta, INI Tel. No , Fax No and the angula eloity is unifom at lage distane fom the disk. We onside this type of flow fo an inompessible feofluid when the otating disk is subjeted to the magneti field ( H,,H ) by using Neuinge-Rosensweig model []. This model has been used by Vema [] fo soling paamagneti Couette flow by taking into aount the inteations of extenal magneti field. Rosensweig [5], in his monogaph, has gien an authoitatie intodution to the eseah on magneti liquids and eealed inteesting infomation about the effet of magnetiation. In geneal, magnetiation is a funtion of magneti field, tempeatue and density of the fluid. This leads to onetion of feofluid in the pesene of the magneti field gadient. Kaman [6] otating dis poblem is extended to the ase of flow stated impulsiely fom est, and also the steady state is soled to a highe degee of auay than peiously done by a simple analytial method whih neglets the esembling diffiulties in Cohan s [7] well known solution. The pioneeing study of odinay isous fluid flow, due to the infinite otating dis was aied by Kaman. He intodued the famous tansfomation, whih edued the goening patial diffeential equations into odinay diffeential equations. Cohan obtained asymptoti solutions fo the steady hydodynami poblem fomulated by Kaman. Benton [8] impoed Cohan s solutions and also, soled the unsteady ase. ttia [9] studied the unsteady state in the pesene of an applied unifom magneti field. The effet of the steady flow of odinay isous fluid, due to the otating dis with unifom solution was onsideed by Mithal [1]. ttia [11] disussed about flow due to an infinite disk otating in the pesene of an axial unifom magneti field by taking Hall effet into onsideation. Sunil et al. [1] studied the effet of otation on themosolutal onetion in a feomagneti fluid onsideing a hoiontal laye of an inompessible feomagneti fluid. Venkatasubamanian and Kaloni [1] inestigated the effet of otation on the themo-onetie instability of a hoiontal laye of feofluid heated fom below in the pesene of unifom etial magneti field. as Gupta and Gupta [1] examined the onset onetion in a hoiontal laye of feomagneti fluid heated fom below and otating about a etial axis in the pesene of a unifom magneti field. Ram et al. [15] soled the non-linea diffeential equations unde Neuinge- Rosensweig model by using powe seies appoximations and disussed the effet of magneti field-dependent isosity on eloity ISBN: ISSN: (Pint); ISSN: (Online) WCE 1

2 Poeedings of the Wold Congess on Engineeing 1 Vol III WCE 1, July - 6, 1, London, U.K. omponents and pessue pofile. Futhe, the effet of poosity on eloity omponents and pessue pofile has been studied by Ram et al. [16]. In the pesent ase, we take ylindial oodinates (,, ) whee -axis is nomal to the plane, and this axis is being onsideed the axis of otation. The isous effets ae dominant oe a egion at a small distane fom the disk, if Reynolds numbe is lage whih gies ise to a bounday laye oe the sufae of disk. We hae pesented the bounday laye equations togethe with bounday onditions. These equations along with Maxwell equations ae soled theoetially as well as numeially. lso, it is found that thee is a aiation in the bounday laye displaement thikness as ompaed to the odinay isous flow ase. Hee, the effet of etially applied magneti field in a iula laye of feofluid on a otating disk is studied within the famewok of Neuinge-Rosensweig appoah and aious types of feofluid esponses ae onsideed. This poblem, to the best of ou knowledge, has not been inestigated yet. II. MTHEMTICL FORMULTION N SOLUTION This model onsides the liquid patile in magneti fluid as a mathematial point with only thee degees of feedom. s a omplete set of independent aiables, the following funtions ae hosen: thee salas (density, pessue and tempeatue) and thee etos (eloity, magnetiation and magneti field). Themodynami oeffiients of the magneti fluid ae onsideed as salas. This model onsides the magnetiation M as being paallel to the applied magneti field, thus implying that no inteation of magneti fluid with extenal magneti field though magneti body ouples, and kineti poesses ae onsideed. This model leads to goening equations whih ae onsideed fom the Naie-stokes equation of magnetiation. The system of equations onsists of the following: Equation of ontinuity V. (1) Equation of motion (Momentum equation fo an inompessible feomagneti fluid with onstant isosity in a fame of efeene otating with angula eloity ) V ( V. ) V p ( M. ) H t () f V ( V ) The effet of otation inludes two tems: 1 (a) Centifugal foe gad (b) Coiolis aeleation ( V ) In (), p p is the edued pessue, whee p stands fo fluid pessue. The eloity omponent is less as ompaed to and, and also the flow is steady and axi-symmeti i.e. ( ), ( ). t 1 p M H 1 p 1 M H (5) (6) The appoximate initial and bounday onditions fo the flow due to otation of an infinitely long disk ( ) with onstant angula eloity Hee, ae gien by at ;,,. () at ;,. (7) () does not anish at ; but tends to a finite negatie alue. Fom bounday laye appoximation, we get fo () that 1 p M H On onsideing ey less aiation of magneti field along - dietion and using Kaman tansfomations, E( ), F( ), G( ), p P( ); whee (9) in equations () - (6) with the help of (8), we get a system of non-linea diffeential equations in E, F, G and P as follows: E GE E F F 1 (1) F GF EF E (11) P G GG (1) G E (1) E( ), F() 1, G(), P() P E( ) F( ) (8) (1) The alues of E, F, G and P ae ompaed gaphially with thei oesponding alues in lassial ase. G must tend to a finite limit, say, as tends to infinity. Cohan indiated ISBN: ISSN: (Pint); ISSN: (Online) WCE 1

3 Poeedings of the Wold Congess on Engineeing 1 Vol III WCE 1, July - 6, 1, London, U.K. that fomal asymptoti expansion (fo lage ) of the system of equations (1) - (1) is a powe seies in exp( ), i.e. tan E ( ) / F ( ) i i e E ( ) (15) i Bi e F ( ) (16) i G ( ) C i e (17) i i e ( P P )( ) (18) Let E ( ) a and F ( ) b. Using this supposition and equations (1) - (1), we get some additional bounday onditions fo the appoximate solution. IV. ISCUSSION The poblem onsideed hee inoles a numbe of paametes, on the basis of whih, a wide ange of numeial esults hae been deied. Of these esults, a small setion is pesented hee fo beity. The numeial esults fo the eloity pofiles, fo,, omponents of the eloity, ommonly known as adial, tangential, etial (axial) eloities, ae shown in figue 1.1, 1., 1. espetiely. III. RESULTS Fist fou oeffiients in the equations (15) - (18) ae alulated with the help of (1) and additional bounday onditions, whih ae as follows: b 1a 1, b 7 7a, b b 8 19a 11a 6 a 1b a 19b B 1, 6 B a 7b a 11b B, 1 B 6 C a 1, 8a C 6 7a C, a C 6 6a 1, 1 1a, 16 19a 11a Using the alues a. 5, b. 6, and. 886 fom Cohan [7], we alulate the alues of the oeffiients 1,,,, B1, B, B, B, C1, C, C, C, 1,,, and. We daw the gaphs of eloity omponents and asymptoti pessue with the dimensionless paamete. The pesent esults gie the good appoximate solution of the aboe system of non-linea diffeential equations. The bounday laye displaement thikness is alulated as d F( )d The fluid is taken to otate at a lage distane fom the wall, the angle beomes In figue 1.1, E shows the adial eloity pofile with the aiation of dimensionless paamete, known as Kaman paamete. Hee, the adial eloity E =.889 is maximum at., afte that it deeases smoothly and fo lage alues of, it oneges to eo. E1shows the adial eloity omponent fo odinay isous fluid ase with peak alue is as 1.. It is notieable that the adial eloity E has ey less peak alue in ompaison to E1 beause of thikening of the feofluid laye due to the otation of the whole system. On the othe hand, the magneti foe edues the pessue on fluid and ineases the adial eloity. The effet of otation is moe ponouned than the foe of magnetiation in the sense of fluid thikening. In othe wods, figues 1.1 and, with the aiation of dimensionless paamete, hae onese behaio to eah othe. In figue 1., F shows the tangential eloity pofile ompaison with Newtonian ase, i.e. F1. In ou ase, if we inease the alue of, the tangential eloity F deeases ontinuously and goes to eo fo lage alue of. It is obseed fom the table, the alue of tangential eloity is.976 at 1, wheeas in Benton s ase fo the odinay isous fluid, the tangential eloity is.68 fo the same alue of. Theefoe, at 1, ou alue ineases 6.7% appoximately in ompaison to the Benton s alue. Fom figue, it is lea that F1 oneges to eo little faste than F. Hee the tangential omponent of eloity follows almost the same tend as that of the odinay isous flow. ISBN: ISSN: (Pint); ISSN: (Online) WCE 1

4 Poeedings of the Wold Congess on Engineeing 1 Vol III WCE 1, July - 6, 1, London, U.K. Figue 1., shows the axial eloity pofile, whih is eo in the beginning and tends to a finite alue in the last. When we inease the alue of, it deeases ontinuously in the negatie egion. Hee the axial eloity oneges to at 1. 8 onwads. Figue, shows the pessue pofile with the initial pessue, P at. The pessue goes to negatie egion fo fist few alues of. t., it goes to maximum negatie alue, whih is fte ontinuously ineasing the alue of, pessue also ineases and at 1., it omes out to.119. When we inease the alue of ontinuously fom 1. onwads, pessue ineases in the positie dietion and it takes peak alue as.7667 at.. fte that, fo lage alue of, pessue beomes P. Compaing figues 1.1 and, we onlude that when adial eloity ineases, the pessue of the feofluid deeases and when adial eloity deeases, feofluid pessue ineases. These figues eeal onese behaio to eah othe. Hee the tangential eloity diminishes slowe than axial eloity omponents. The hange in the ue of adial eloity is faste due to effet of extenal magneti field, esulting in eduing the time equied fo eloity pofile to eah thei onegene leel. In ou poblem, we hae alulated the displaement thikness, numeially. Hee, the disk is otating along with otation of the feofluid, due to whih thikness is ineasing. The displaement thikness in ou ase is 1.56, wheeas in Benton s ase is In nut shell, we onlude that if we otate the plate with otation of the feofluid, thee is an inement in the thikness of bounday laye. Hee, we hae also alulated angle between wall and feofluid, whih is 1 V. CONCLUSION Fom these esults, we onlude that magnetiation foe i.e., M. H edues the pessue. lso, it has been obseed that magneti field intensity ineases the adial eloity; wheeas, the fluid otation has eese effet. The effet of otation is moe ponouned than the foe of magnetiation due to whih the adial eloity takes ey less peak alue in ompaison to the odinay isous flow ase. ue to the otation, etadation of the adial eloity ineases the thikness of the magneti fluid laye. Conduting feofluids flow with otating disk hae the patial appliations in many aeas suh as otating mahiney, lubiation, oeanogaphy, ompute stoage deies, and isomety and ystal gowth poesses. REFERENCES [1] R.P. Feynman, R.B. Leighton, and M. Sands, Letues on Physis, ddison- Wesley Reading M, Vol [] M.I.Shliomis, Feofluids as Themal Rathets, Physial Reiew Lettes, Vol. 9, No. 18, pp. 1889,. [] J.L. Neuinge, R.E. Rosensweig, Magneti Fluids, Physis of Fluids, Vol. 7, pp , 196. ISBN: ISSN: (Pint); ISSN: (Online) WCE 1

5 Poeedings of the Wold Congess on Engineeing 1 Vol III WCE 1, July - 6, 1, London, U.K. [] P..S. Vema, M.J. Vedan, Steady Rotation of a Sphee in a Paamagneti Fluid, Wea, Vol. 5, pp. 1-18, [5] R.E. Rosensweig, Feohydodynamis, Cambidge Uniesity Pess, [6] V. Kaman, Ube Laminae und Tubulente Reibung, Z. ngew. Math. Meh. Vol. 1, pp. - 5, 191. [7] W. G. Cohan, The Flow due to a Rotating is, Po. Camb. Phil. Sot., Vol., pp , 19. [8] E.R. Benton, On the Flow due to a Rotating isk, J. Fluid Meh, Vol., No., pp , [9] H.. ttia, Unsteady MH Flow nea a Rotating Poous isk with Unifom Sution o Injetion, Jounal of Fluid ynamis Reseah, Vol., pp. 8-9, [1] K.G. Mithal, On the Effets of Unifom High Sution on the Steady Flow of a Non-Newtonian Liquid due to a Rotating isk, Quat J. Meh. ppl. Math, Vol. XIV, pp. 1 1, [11] H.. ttia,.l. boul-hassan, On Hydomagneti Flow due to a Rotating isk, pplied Mathematial Modelling, Vol. 8, pp ,. [1] Sunil, P.K. Bhati,. Shama and R.C. Shama, The Effet of Rotation on Themosolutal Conetion in a Feomagneti Fluid, Int. J. of pplied Mehanis and Engineeing, Vol. 1, No., pp. 71-7, 5. [1] S. Venkatasubamanian, P.N. Kaloni, Effets of otation on the themo-onetie instability of a hoiontal laye of feofluids, Int. Jou. Engg. Si., Vol., No., pp. 7-56, 199. [1] M. as Gupta,.S. Gupta, Conetie instability of a laye of a feomagneti fluid otating about a etial axis, Int. J. Engg. Si., Vol. 17, pp , [15] P. Ram,. Bhandai, K. Shama, Effet of magneti field-dependent isosity on eoling feofluid, Jounal of Magnetism and Magneti Mateials, Vol., No. 1, pp. 76-8, 1. [16] P. Ram, K. Shama,. Bhandai, Effet of poosity on feofluid flow with otating disk, Int. Jounal of pplied Mathematis and Mehanis, Vol. 6, No. 16, pp , 1. d Magneti pemeability of fee spae Fluid density Gadient opeato imensionless Kaman s paamete isplaement thikness of the feofluid laye ngle of otation ngula eloity of the disk Radial eloity Tangential eloity xial eloity ngula eloity of whole system H M p p P P V NOMENCLTURE Magneti field intensity Magnetiation Fluid pessue Redued pessue Kaman pessue Initial pessue (absolute alue) Veloity of feofluid Kinemati isosity f Refeene isosity of fluid ISBN: ISSN: (Pint); ISSN: (Online) WCE 1