Rajesh C. Shah 1,*, Dilip B. Patel 2

Transcription

1 Appied Matematics (5): DOI:.593/j.am.5.5 Matematica Modeing of newy Designed Ferrofuid Based Sider Bearing Incuding Effects of Porosity Anisotropic Permeabiity Sip Veocity at bot te Ends Squeeze Veocity Rajes C. Sa * Diip B. Pate Department of Appied Matematics Facuty of Tecnoogy Engineering Te M. S. University of Baroda Vadodara 39 ujarat State India Department of Matematics Sankac Pate Coege of Engineering Visnagar ujarat State India Abstract Te present paper proposes te matematica mode of ferrofuid ubricated sider bearing wit an incined pad surface ( pate or pad ) incuding combined effects of porosity anisotropic permeabiity sip veocity at bot te ends squeeze veocity under an obique magnetic fied. Fro m te Reynods s equation of te above mode ep ressions bearing caracteristics ike oad capacity friction on moving sider coefficient of friction centre of pressure are obtained wic can be soved numericay to eamine te eporation of its possibe effects on te system. Various sizes of te porous matri at bot te ends are aso discussed te possibe optimization of te bearing caracteristics. Keywords Matematica Modeing Ferrofuid Porosity Incined Bearing Anisotropic Permeab iity. Introduction Ferrofuid or Magnetic fuid [] are stabe coo ida suspensions containing fine ferromagnetic partices dispersing in a iquid caed carrier iquid ( in our case water ) in wic a surfactant is added to generate a coating ayer preventing te foccuation of te partices. Wen an eterna magnetic fied is appied ferrofuids eperience magnetic body ces depends upon te magnetization of ferromagnetic partices. Owing to tese features ferrofu ids are usefu in many appications ike in sensors seaing devices fitering apparatus etc.[3]. Wu[4] in an innovative anaysis deat wit te case of squeeze fim beaviour porous annuar disks in wic e sowed tat owing to te fact tat fuid can fow troug te porous materia as we as troug te space between te bounding surfaces te permance of a porous waed squeeze fim can differ substantiay from tat of a soid waed squeeze fim. Later[5] etended te above anaysis[4] by introducing te effect of veocity sip to porous waed squeeze fim wit porous matri appeared in te above pate. Tey found tat te oad capacity decreases due to te effect of porosity sip. Prakas Vij[6] investigated a porous * Corresponding autor: dr_rcsa@yaoo.com (Rajes C. Sa) Pubised onine at ttp://journa.sapub.org/am Copyrigt Scientific & Academic Pubising. A Rigts Reserved incined sider bearing witout te effect of magnetic fuid found tat porosity caused decrease in te oad capacity friction wie it increased te coefficient of friction. upta Bat[7] found tat te oad capacity friction coud be increased by using a transverse magnetic fied on te bearing a conducting ubricant. Wit te advent of ferrofuid Agrawa[8] studied its effects on a porous incined bearing found tat te magnetizing of partices in te ubricant increased its oad capacity witout affecting te friction on te moving sider. Recenty many autors[9-7] ave anaysed effects of ferrofuid as ubricant in teir study found te increase of efficiency of te bearing permance over conventiona fro m d ifferent viewpoints. In a above investigations none of te autors considered bot te porous pates in teir study. Te porous ayer in te bearing is considered because of its advantageous property of sef ubrication. Wit tis motivation te study of beaviour of an incined sider bearing wit te porous matri attaced to bot te pates (tat is upper ower) is proposed ere wit a ferrofuid ubricant under a magnetic fied obique to te ower surface. A so effects of sip veocity anisotropic permeabiity at bot te porous pates as we as squeeze veocity wen te upper pate approaces to ower one are incuded. Te ferrofuid fow mode considered ere is due to R. E. Rosensweig[]. A ferro fu id ubrication mode is derived te above probem te various sizes of upper ower porous

2 77 Appied Matematics (5): matri is considered computation of various bearing caracteristics ike oad capacity friction coefficient of friction center of pressures. Aso above caracteristics ave been computed two different cases of anisotropic permeabiities at upper ower porous matri. Te ferrofuid used in te computations are water based.. Formuation of te Matematica Mode A scematic diagram of te system under study is presented in Fig. consists of a ferrofuid fim of tickness witin an incined pad surface ( stator ) a sider of engt A in te -direct ion widt B in y-d irection A<< B. Te vaue of is at te inet at te outet. Tis fim tickness is given by = ( ) / A. () Te sider stator bot ave attaced porous matri of tickness ( metres ) respectivey. Bot te porous matri are backed by a soid wa. Te sider moves wit a unim veocity U in te -d irect ion. Aso stator moves normay towards te sider wit a unim veocity = d / dt t is time in seconds. Te basic fow equations governing te above penomenon based on R. E. Rosensweig mode are given by[3] q ( ) ρ + t q q = p + η + µ ( ) () q M H q = (3) (4) H = (5) ( H+ M) = (6) ρ p η q µ M H µ are density fim pressure fuid viscosity fuid veocity free space permeabiity te magnetization vector magnetic fied vector magnetic susceptibiity respectivey. By combining above equations () to (6) under te usua assumption of ubrication negecting inertia terms tat te derivatives of veocities across te fim predominate one- dimensiona equation governing te ubricant fow in te fim region yieds u = p H z η (7) u is te fim fuid veocity in te -direction H is te magnetic fied strengt. Soving equation (7) under te sip boundary conditions given by Sparrow et.a.[5] modified by Sa et.a.[4] wit te addition of sider veocity U to[5] u u = + U s z = s η 5 M = µ H z = u u = s z wen wen s z = (8) ; ( i =) m = 5 η m being sip parameter are porosities in te - direction are permeabiities in te - direction in te porous region yieds s i Figure. Sider bearing wit incined pad surface

3 Rajes C. Sa et a.: Matematica Modeing of newy Designed Ferrofuid Based Sider Bearing Incuding 78 Effects of Porosity Anisotropic Permeabiity Sip Veocity at bot te Ends Squeeze Veocity z ssz s ss u = + + p H + + zu (9) η s s s s s s s = s + s + s s. Substituting te above vaue of u in te integra m of continuity equation u dz + w w = w is te aia component of te fuid veocity in te fim one obtains () 3 3 ss s ssu + + p H w w =. () η 6 s s s s s s Using Darcy s aw te veocity components of te fuid in te porous matri are given as foow: For upper porous region: u = P H ( in d irection ) () η z w = P H ( in z d irection ) (3) η z are fuid permeabiities in te upper porous region in z direction respectivey P is te fuid z pressure in te porous region. For ower porous region: u = P H ( in d irection ) (4) η z w = P H ( in z d irection ) (5) η z are fuid permeabiities in te ower porous region in z direction respectivey P is te fuid pressure z in te porous region. Substituting equations () (3) in te continuity equation upper porous region u w + = (6) z yieds η + = η z z P H P H wic on integration wit respect to z across te upper porous matri ( + ) one obtains z= η (7) z P H = p H η z (8) z = + is non-porous. using Morgan-Cameron approimation[6] tat te surface Substituting equations (4) (5) in te continuity equation ower porous region u w + = (9) z yieds z P H + P H = () η η z wic on integration wit respect to z across te ower porous matri ( ) one obtains

4 79 Appied Matematics (5): z P H = p H () η z z= η using Morgan-Cameron approimation[6] tat te surface z = is non-porous. Considering te norma component of veocity across te fim porous interface are continuous so tat w = + w w = w equations (8) () () yieds f g p H = () g = ( 4s 4 s ss) s( ) η s su f = ( + s ) + s H = KA X ( X ). wic is te Reynods s type equation te considered penomenon. Here magnetic fied is considered to be vanis at te inet outet of te bearing incined at an ange φ wit te -ais. Tus H = H ( )(cos φsin φ) φ = φ ( z ) (3) H = K ( A ) (4) K being a quantity cosen to suit te dimensions of bot sides of equation (4). Suc a magnetic fied attains a maimum at te midde of te bearing producing magnetic pressure. On te oter a constant fied cannot produce magnetic pressure because (d / d) H =. Te direction of te magnetic fied is significant since H as to satisfy te equations H = H = (5) so H arises out of a potentia φ satisfies te equation φ φ A cosφ + = z ( A ) (6) wose soution is determined from equations cosec φ = C ( A ) / C( A) = [ C A 4sin( Cz)] (7) C being an arbitrary constant. Introducing te dimensioness quantities j j X = = j = j = i i = A p KA s= s s= s p = µ = η AU ηu A S = ; i= ; j= z. U equations () (4) impies respectivey = a a X a = ( ) X ; Aso equation () transms to d d p µ X( X) de dx = dx dx (8) = ( + 4 s + 4 s + s s) + s( + ) E = 6SsX + 6 s ( + s ). Equation (8) is known as dimensioness Reynods s equation. 3. Epressions of Bearing Caracteristics Since te pressure is negigibe on te boundaries of te sider bearing compared to inside pressure soving equation (8) under boundary conditions p = wen X =. Te dimensioness fim pressure p is obtained as: X E Q p µ X ( X ) dx = + E dx. Q = dx Te oad carrying capacity W friction on te moving sider F coefficient of friction f te - coordinate of te centre of pressure X are epressed respectivey in dimensioness ms as W µ E Q W = = X dx η A BU F s( + s) E Q s s F = = dx dx η ABU + s s Af F f = = W

5 Rajes C. Sa et a.: Matematica Modeing of newy Designed Ferrofuid Based Sider Bearing Incuding 8 Effects of Porosity Anisotropic Permeabiity Sip Veocity at bot te Ends Squeeze Veocity W 4 X µ E Q Y = = X dx BA A u W = p ddy F = η ddy. z BA 4. Cacuation of Resuts z= Te various bearing caracteristics ike oad capacity frictiona ce on te moving sider te coefficient of friction - coordinate of centre of pressure are computed various sizes of upper ower porous matri =.(m ) =.(m ) =.(m ) =.(m ) using Simpson s one tird rue wit step size. te foowing vaues of te parameters : η =.64 m =.8 µ = 4π 7 (kgms - A - ) 9 µ =.5 K = (A m -4 ) A =.5 (m) =.5 (m) η =. ( kgm - s - ) U =. (ms - ) =.5 (ms - ) a =. (m). 5. Discussion of Resuts Te matematica mode of water based ferrofuid ubricated sider bearing wit an incined pad surface incuding combined effects of porosity anisotropic permeabiity sip veocity at bot te ends squeeze veocity is proposed under an obique magnetic fied. Te resuts of various bearing caracteristics ( refer Section 3 ) are obtained te various vaues of te parameters ( refer Section 4 ) are presented grapicay as foows: Fro m te Figs. -9 te foowing observations are made: Dimensioness oad capacity -- -->.6.4. Vaues of 5 4 Figure. Dimensioness oad capacity =. =. 4 6 Vaues of ----> W various vaues of Dimensioness oad capacity ---->.5 Figure 3. Dimensioness oad capacity =. =. Dimensioness friction ce -- --> W various vaues of Figure 4. Dimensioness friction ce on te moving sider various vaues of =. =. Dimensioness friction ce ----> Figure 5. Dimensioness friction ce on te moving sider various vaues of =. =. () In Figs. 3 te dimensioness oad capacity W various vaues of ower porous matri vaues of upper porous matri Vaues of Vaues of ----> Vaues of Vaues of ----> Vaues of Vaues of ----> are dispayed te constant vaues of =. =. =. =. respectivey. F F

6 8 Appied Matematics (5): Dimensioness coefficient of friction ---> 4 Vaues of Vaues of ----> Figure 6. Dimensioness coefficient of friction =. =. Dimensioness coefficient of friction ----> Figure 7. Dimensioness coefficient of friction =. =. Dimensioness centre of pressure --- -> 6 4 various vaues of various vaues of Figure 8. Dimensioness te ordinate of te centre of pressure various vaues of =. =. Vaues of Vaues of ----> Vaues of Vaues of ----> f f Y Dimensioness centre of pressure ----> Figure 9. Dimensioness te ordinate of te centre of pressure various vaues of =. =. It is observed from te Fig. tat te dimensioness oad capacity decreases wit te increase of porous matri tickness as we as.te decrease rate of oad capacity is sow wit respect to. It is observed from Fig.3 tat te dimensioness oad capacity decreases wit respect to but its beaviour is amost remains same wit respect to Vaues of Vaues of ---->. Te ma imu m dimensioness oad capacity is obtained wen = = but wit te disadvantage tat te bearing as no sef ubricating property. Tis beaviour of decrease in oad capacity wit te insertion of porous matri aso agrees wit te concusions of[56]. According to[5] te above trends squeeze fim bearing can be obtained because of te pysica process as under : Te presence of te porous medium provides a pat te fuid to come out easiy m te sider bearing to te environment. Te iger te permeabiity te more readiy does fuid fow troug te porous materia. Tus te presence of te porous materia decreases te resistance to fow in direction as a consequence te oad carrying capacity is reduced. Te effect of veocity sip is to decrease te resistance encountered by te fuid fowing in te gap itsef by tis means to diminis te oad carrying capacity. () In Figs. 4 5 te dimensioness friction ce on te moving sider F various vaues of ower porous matri vaues of upper porous matri are dispayed te constant vaues of =. =. =. =. respectivey. It is observed tat te same beaviour is obtained F as we ave discussed in () W wit respect to. (3) In Figs. 6 7 te dimensioness coefficient of friction f various vaues of ower porous matri Y

7 Rajes C. Sa et a.: Matematica Modeing of newy Designed Ferrofuid Based Sider Bearing Incuding 8 Effects of Porosity Anisotropic Permeabiity Sip Veocity at bot te Ends Squeeze Veocity te vaues of upper matri are dispayed te constant vaues of =.. =. respectivey. =. = It is observed tat f increase wit te increase of. (4) In Figs. 8 9 te dimensioness coordinate of te centre of pressure Y various vaues of ower porous matri te vaues of upper porous matri are dispayed te constant vaues of =. =. =. =. respectivey. It is observed tat te position of te centre of pressure does not affect muc wit respect to. Aso one can obtain te foowing cases te specific vaues of te parameters: (a) Wen = = te case of[] is obtained from equation (8) as d d p µ X( X) de dx = dx dx ( 4 4 ) = + s + s + s s + s E= 6 s ( + s). (b) Wen µ = te non- ferrofu id case[6] is obtained from equation (8) as d d p de = dx dx dx ( 4 4 ) ( ) = + s + s + s s + s + 6. Concusions E = 6SsX + 6 s ( + s ). Te present paper proposes te matematica mode of water based ferrofuid ubricated sider bearing wit an incined pad surface incuding combined effects of porosity anisotropic permeabiity sip veocity at bot te ends squeeze veocity under an obique magnetic fied. Te resuts of various bearing caracteristics are presented grapicay. Te porous ayer in te bearing is considered because of its advantageous property of sef ubrication. Wit tis motivation te present study is proposed. Based upon te mu ation in Section Resuts & discussion ( Section 4 5 ) te fo o wing concusions can be made designing sider bearing: () Because of aving te sef ubrication property of te porous pate bearings it is suggested to ave bot te porous pate better sef ubrication. () Better oad capacity is obtained wen te tickness of are sma. (3) Sma tickness of as more infuence on better oad capacity wen = =. =.. (4) Sma tickness of as more infuence on better oad capacity wen = =. =.. (5) It soud be noted tat a constant magnetic fied does not enance te bearing caracteristics in tis mode due to Rosensweig since H / = in equation (7). REFERENCES [] R. E. Rosensweig Ferroydrodynamics Cambridge University Press New York 985. [] N. C. Popa I. Potencz L. Brostean L. Vekas Some appications of inductive transducers wit magnetic fuids Sensors Actuators A 59 (997) 97-. [3] M. odowsky New metods seaing fitering ubricating wit magnetic fuids IEEE transactions on Magnetics Mag.- 6 (98) [4] H. Wu Squeeze-fim beavior porous annuar disks Journa of Lubrication Tecnoogy Trans. ASME Series F 9(4) (97) [5] E. M. Sparrow. S. Beavers I. T. Hwang Effect of veocity sip on porous waed squeeze fims Journa of Lubrication Tecnoogy 94 (97) [6] J. Prakas S. K. Vij Hydrodynamic ubrication of a porous sider Journa of Mecanica Engineering Science 5 (973) [7] J. L. upta M. V. Bat An incined porous sider bearing wit a transverse magnetic fied Wear 55 (979) [8] V. K. Agrawa Magnetic fuid based porous incined sider bearing Wear 7 (986) [9] P. Sina P. Cra D. Kumar Ferrofuid ubrication of cyindrica roers wit cavitation Acta Mecanica 98 (993) [] P.Ram P.D.S.Verma Ferrofuid ubricated in porous incined sider bearing Indian journa of Pure Appied Matematics 3 () (999) [] R. C. Sa M. V. Bat Ferrofuid ubrication in porous sider bearing wit veocity sip Internationa Journa of Mecanica Sciences 44 () [] R. C. Sa M. V. Bat Magnetic fuid based porous incined sider bearing wit veocity sip Internationa journa of Appied Mecanics Engineering 8 (3) [3] R. C. Sa M. V. Bat Anaysis of a porous eponentia sider bearing ubricated wit a ferrofuid considering sip veocity Journa of te Braziian Society of Mecanica Sciences Engineering 5 (3) [4] R. C. Sa M. V. Bat Ferrofuid ubrication equation

8 83 Appied Matematics (5): porous bearing considering anisotropic permeabiity sip veocity Indian Journa of Engineering & Materia Sciences (3) [5] R. C. Sa M. V. Bat Ferrofuid squeeze fim between curved annuar pates incuding rotation of magnetic partices Journa of Engineering Matematics 5 ( 5) [6] H. Urreta Z. Leict A. Sancez A. Agirre P. Kuzir. Magnac Hydrodynamic bearing ubricated wit magnetic fuids Journa of Pysics : Conference Series 49 ( 9 ) 3. [7] R. C. Sa D. B. Pate Squeeze fim based on ferrofuid in curved porous circuar pates wit various porous structure Appied Matematics (4) () -3.