Numerical contribution to the viscoelastic magnetic lubrication of human joint in periodic motion

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1 ca of Bongnrng an Bomcancs ol. 6 No. 4 Numrcal conrbuon o scolasc magnc lubrcaon of uman jon n proc moon KZYSZTOF C. WIEZCOLSKI Tcncal nrsy of Gańs Faculy of Ocan Engnrng an Sp Tcnology 8-95 Gańs Gyna Marm nrsy Faculy of Marn Engnrng -mal:wrzc@am.gyna Ts papr prsns auor s numrcal conrbuon o unsymmrcal scolasc yroynamc lubrcaon of uman jons w synoal flu n procally cang m an unsay magnc fl. W assum a bon a n uman jon mos n wo rcons namly n crcumfrnc an mran rcons. Basc quaons scrbng flow of synoal flu n uman p jon ar sol analycally an numrcally. Numrcal calculaons ar prform n Maca Profssonal Program ang no accoun mo of fn ffrncs. Ts mo sasfs sably of numrcal soluons of paral ffrnal quaons an alus of capacy forcs occurrng n uman jons. Ky wors: scolasc lubrcaon uman jon proc moon magnc fl. Prlmnars Tr s a numbr of currn sus wos auors a ffrn approacs o suy of jon bomcancs. Lubrcaon of uman jon unr unsay proc conons an for ral scolasc proprs of synoal flu as no bn xamn ro. scolasc lubrcaon of uman jon n unsay proc moon an magnc fl was no consr n paprs [] [4] [7] [] [9]. In prsn suy cangs a occur urng scolasc lubrcaon of uman jons unr aryng proc unsay conons ar xamn. In papr w assum roaonal moon of uman bon proc an unsay flow of scolasc synoal flu proc m-pnn gap g cangabl synoal flu scosy arabl gomry of gap g consan nsy of synoal flu an sormal ncomprssbl flow of synoal flu. W also assum a bon a can ma roaonal moon n wo rcons a wo arous angular locs s fgur. In cas of unsymmrcal flow of synoal flu r componns of s locy cor pn on arabls wl m an prssur funcon p pn on.. T gap g may b

2 6 K. C. WIEZCOLSKI a funcon of arabls an m. T symbol nos co-orna n crcumfrnc rcon s co-orna n gap g rcon sans for a gnrang ln of roaonal bon surfac or co-orna n longunal rcon.. Basc quaons T problm of lubrcaon of uman p jon wll b prsn by mans of consraon of momnum connuy an Maxwll s quaons [] []: DS µ N / µ / wr: S srss nsor synoal flu locy m/s magnc nnsy cor /m w componns N magnzaon cor /m w componns N N N µ magnc prmably coffcn of fr spac mgs lcrc prmably coffcn of synoal flu s 4 m g. W assum a synoal flu s a goo nsulaor.. lcrc conucy coffcn σ. Moror scon-orr approxmaon of gnral consu quaon gn by ln an Ercsn can b wrn n followng form []: S pi β L L T gra a gra a T L T L a L wr: p prssur I un nsor an frs wo ln Ercsn nsors L nsor of gran flu locy cor s L T nsor of ranspos of a marx of gran cor of a bologcal flu s m s a acclraon cor m/s. T symbols: β san for r maral consans of synoal flu wr nos ynamc scosy Pas symbol β rmns psuoscosy coffcn Pas an scrbs frcon forcs bwn scolasc parcls of synoal flu. T acclraon rms a bn nglc. Only m ras of locy componn a bn ran. T angnal an rcal acclraon of jon surfac arabl n m s an no accoun. W also nglc Ψ an Ψ / an cnrfugal forcs wr s raus of curaur of bon surfac. W assum a componns of magnc nnsy cor an componns of magnsaon cor ar consan n g rcons of jon gap. W rqur curlnar orogonal sysm of co-ornas w Lamé coffcns rspcly. From bounary conons of n layr follows a. fr bounary smplfcaons sysm of consraon of momnum connuy an Maxwll s quaons as form of sysm. Equaons of moon ar as follows [9] []: p β µ N N µ

3 Numrcal conrbuon o scolasc lubrcaon of uman jon 6 p 4 µ µ β N N p 5. 6 Trms mulpl by facor β scrb nflunc of scolasc proprs of synoal flu on lubrcaon procss. If coffcn β ns o zro n s of quaons 6 ns o quaons scrbng lubrcaon of uman jons prsn n paprs [] [9]. Maxwll s quaons ar as follows []: gra roro µ E / 7 nc: X E µ ξ ξ 8 Y E µ ξ ξ 9 Z E µ ξ ξ wr:. W no: µ µ < πc < c < b m π/8 π/ b s ar componns of magnc nnsy cor /m ξ B µ N ar componns of magnc nucon cor B n T N χ ar componns of magnsaon cor N /m χ s mnsonlss magnc suscpbly of synoal flu. In orr o r soluons of abo s of quaons an oscllang proc moons wll b scuss.

4 64 K. C. WIEZCOLSKI. T mo of ngraon For locy componns an prssur wou loss gnraly followng approac as bn nrouc [8] [9]: xp p p p xp xp 4 wr: s an angular locy s scrbng proc prurbaons n unsay flow of synoal flu an magnc fl n jon gap an s an magnary un. Gap g as followng form: ~ o xp 5 wr: ~ nos m-pnn prurbaon of gap g caus by unsay wor conons m-npnn coffcn of prurbaons of gap g m-npnn prmary gap g an o oal alu of gap g. Bcaus of lnar form of quaons 7 a sparaon of a say flow from an unsay flow of synoal flu s possbl. W nsr srs 4 no s of quaons 7 an w qua rms of sam uppr nxs n bracs an sam powrs of xp funcons. Equaons of moon for say conons n say magnc fl an Nwonan flu a form [9]: p M 6 p 7 p M 8 9 for ϕ πc < c < b m π/8 π/ b s r.

5 Numrcal conrbuon o scolasc lubrcaon of uman jon 65 T sysm of quaons 6 9 rmns an unnown prssur funcon p an unnown componns of ol locy cor n rcons rspcly. Maxwll s quaons for say conons a form []: ξ Y ξ Z ξ X a < π c < c < b m π /8 π / b s. T sysm of quaons rmns unnown componns of magnc nnsy cor n rcons rspcly. Equaons of moon of sps of corrcon alus for unsay proc moon an conons a form [8] [9]: p M p p M for... < πc < c < b m π/8 π/ b s. T symbol: β 5 nos an apparn scosy Pas. Ts scosy pns on locy formaons caus by scolasc proprs s coffcn β an by angular locy scrbng proc prurbaons. Ts fac ncas a synoal flu as non-nwonan proprs. If coffcn β ns o zro n scolasc an non-nwonan flu proprs ar nglc. T sysm of quaons 4 rmns unnown corrcons p of prssur funcon an unnown corrcons of componns of ol locy cor n rcons rspcly. T Maxwll quaons for sps of corrcons caus by unsay conons a form []: ξ µ E Y ξ E Z µ X µ E 4 ξ 6

6 K. C. WIEZCOLSKI 66 for... πc < c < b m π/8 π/ b s. T sysm of quaons 6 rmns unnown corrcons of componns of magnc nnsy cor n rcons rspcly. Bfor calculaons w mus nsr followng xprssons [] no quaons an 6: ξ 7 for... ϕ πc < c < b m π/8 π/ b s r. T funcons M M n quaons 6 8 an funcons M M for... n quaons a followng forms: n n n n n M χ µ 8 for... ϕ πc < c < b m π/8 π/ b s r. T funcons oban from quaons an funcons for... oban from quaons 6 ar nsr no xprsson 8. Suc funcons ar nsr no quaons 6 8. frwars sysm of quaons 6 9 an sysm of quaons 4 ar sol n orr o rmn unnown funcons p p for Bounary conons Bon a lops angular locs an n rcons an rspcly. cabulum mos n crcumfrnal an mronal rcons. T gap g cangs n m n rcal rcon. Moror s assum a angnal acclraon of bon a surfac ars n m. nc for sysm of

7 Numrcal conrbuon o scolasc lubrcaon of uman jon 67 quaons 6 9 an 4 a ;... bounary conons[8] [9] [] ar as follows: W assum: 9 for for. 4 nc o xp xp 5 for... 6 Tm-npnn arag gap g w prurbaon assums followng form: [ m m ~ ] cos m sn m m wr m s an arag m pro of jon gap prurbaons opraor of a ral par of complx numbr. locs of bon an acabulum surfacs ar procally pnn on m. Toal angnal locs of bon surfac an acabulum surfac n rcons a followng forms [4] [5]: 7 xp for bon 8 xp cons for acabulum

8 68 K. C. WIEZCOLSKI wr ar m-npnn coffcns of angnal locy cangs of bon surfac a... an ar m-npnn consan coffcns of angnal locy cangs of acabulum a locy of synoal flu an prssur 5.. Soluons for saonary flow In frs sp of soluons w assum a saonary flow. Dynamc scosy can b a funcon of an only. T sysm of quaons 6 9 for bounary conons 9 an a as followng soluons [8] [9]: p M s s s 9 p M s s s. 4 W ngra connuy quaon 9 w rspc o arabl. Imposng bounary conon on locy componn n gap g rcon w oban:. 4 Imposng bounary conon for on soluon 4 w a:. 4 W nsr soluons 9 an 4 no 4. nc prssur p for say conons an magnc fl s rmn by followng mof ynols quaon: p M 6 6 wr s / r b m b s. p M [ ] 4

9 Numrcal conrbuon o scolasc lubrcaon of uman jon Corrcons for unsay flow an scolasc proprs Imposng bounary conons 4 on sysm of quaons 4 w oban cangs of componns of synoal flu locs caus by scolasc proprs an unsay moon n followng form: [ ] W Π sn sn sn sn 44 wr: [ ] [ ] W sn sn xp xp 45 M p Π 46 for an... Ingrang connuy quaon 4 w rspc o arabl for bounary conon w oban:. 47 Imposng bounary conon 6 on soluon 47 w arr a:. 48 If w a no accoun rul of ffrnaon of ngrals w arabl lms of ngraon an f w us aonally conons 4 n quaon 48 assums followng form: 49 for... If w nsr soluons 44 for no quaon 49 n afr fnal calculaons n ppnx w oban: M p M p

10 K. C. WIEZCOLSKI 7 [ ] 6 ] [ 6 5 for... πc < c < b m π/8 π/ b s. Mulplyng bo ss of quaon 5 by xprsson xp an quang rms of ral pars of complx numbr on bo ss of quaon w oban followng squnc of mof ynols quaons: M p M p * * sn [ ] sn cos 6 β [ ] sn cos 6 β cos 5 for... < πc < c < b m π/8 π/ b s r an * sn cos xp β β β. 5 Formula 5 sows a oal apparn scosy * of synoal flu pns aonally on m. Ts fac can b xplan only by ru of rologcal proprs of synoal flu. T mof ynols quaon 5 rmns followng prssur funcons: p p p. Ts funcons fn prssur corrcons caus by unsay an scolasc proprs of synoal flu n magnc fl.

5 T m-npnn coffcns of crcumfrnal locs of sprcal

11 Numrcal conrbuon o scolasc lubrcaon of uman jon 7 6. Parcular cass of uman jons In a parcular cas of p jon w sprcal bon a w a sprcal coornas an Lamé coffcns n followng form: ϕ r sn /. 5 T m-npnn coffcns of crcumfrnal locs of sprcal bon a can b xprss as s fgur : / sn ϕ sn / snϕ sn a b c

12 7 K. C. WIEZCOLSKI Fg.. oaonal proc unsay moon of bon a an acabulum n wo rcons: a b p jon c bon a an acabulum for arbrary uman jon n proc moon n normal cor wr ar angular locs of sprcal bon a an r prurbaons n crcumfrnal rcon ϕ an ar angular locs of sprcal bon a an r prurbaons n mronal rcon. Symbol nos n s cas raus of sprcal bon a. Fg.. ang of rgon of lubrcaon on sprcal an yprbolc bon as In yprbolcal coornas for yprbolc bon a n uman jon followng Lamé coffcns ar al: w Λ asc Λ 4 aλ Λ asc an 55

13 Numrcal conrbuon o scolasc lubrcaon of uman jon 7 w a Λ π arccos. b a Λ a w W ma followng noaons: a s smalls raus of bon crossscon a a w s largs raus of bon cross-scon w a a b s jon lng. T rgon of lubrcaon Ω a bon a of sprcal p jon n sprcal co-ornas an rgon of lubrcaon Ω an acabulum of yprbolc p jon n yprbolc coornas unr unsay conons ar sown n fgur. 7. Mof ynols quaons n sprcal co-ornas L us prsn mof ynols quaon for unsay moon n magnc fl bu wou scolasc proprs of synoal flu.. for β. W assum a a cnr of sprcal bon a s a pon O an cnr of sprcal carlag a pon O x y z. Eccnrcy as followng alu: D [ ].5 s fgur. In sprcal coornas w assum n bounary layr us for synoal flu flow w oban: ϕ an Lamé coffcns 5. W also assum m-npnn coffcns of gap g prurbaons n form: / nc by ru of 7 an arag gap g s a sum of nfn srs n followng form []: sn m Γ Γ.8 π m π m 56 m wr: ϕ / cosϕ sn / snϕ sn / cos / {[ cosϕ sn / snϕ sn / cos /] mn D mn }.5 57 sn x π x π x for < x x π. 58

14 K. C. WIEZCOLSKI 74 Fg.. Tm-pnn cangs of gap g unr bounary conons If lf an rg ss of quaons 4 an 5 ar a for... an scolasc proprs of synoal flu ar nglc.. coffcn β ns o zro n w arr a mof ynols quaon: M p M p ϕ ϕ sn cos cos sn sn ϕ ϕ sn sn 6 sn 6 cos sn sn sn ϕ sn cos sn 6 ϕ ϕ sn sn cos sn ] sn [ 6 ϕ 59

15 Numrcal conrbuon o scolasc lubrcaon of uman jon 75 wr M M ϕ M M an sums of nfn srs assum followng forms []: cos sn π π 4 for π < < π for π. W a ϕ π π/8 π/ an gap g as form Γ wras Γ for m π an Γ.7π for m π/6. T mof ynols quaon 59 rmns oal prssur funcon p for unsay conons n magnc fl. 8. Numrcal calculaons T prssur srbuon p for saonary moon an s corrcons p p p... for unsay flow of synoal flu w scolasc proprs can b calcula from quaons 4 5 n lubrcaon rgon Ω nca n fgur. I s a scon of bowl of spr. In s cas on bounary of rgon Ω prssur p as alu of amosprc prssur p a an corrcons of prssurs p for ar qual o zro. W us m-pn scosy 5. Toal prssur p for unsay flow of synoal flu n p jon gap wou scolasc proprs w rmn ns rgon Ω from quaon 59. On bounary of rgon Ω oal prssur p assums alus of amosprc prssur p a. Numrcal calculaons ar prform for rgon Ω : ϕ π π/8 π/ wr Γ for m π an Γ.65 [m].8 [s ].5 [s ]. [s ].9 [s]. [s] [µm].5 [µm] [µm]. [Pas] 8 [g/m ]. Mnmal alu of gap g mn. [µm] an maxmal alu of gap g max 7. [µm] an w a no accoun m pro π/. Numrcal calculaons of prssur srbuons aryng w m ar prsn n fgurs 4 5 an 6. Fgur 4 sows prssur srbuon aryng w m caus by roaon.5 s of bon a n mronal rcon only for normal p jon wc s no affc by magnc fl. W a no accoun angular locy prurbaons. s on sprcal a of bon n mronal rcons... oaon abou bon a n crcumfrnal rcon s no an no accoun. W assum a acabulum s moonlss... W also assum prurbaons of jon gap g n unsay moon a angular locy qual o. s. For m π/ π/ w oban maxmal alus of prssur wc ar.67 6 Pa;.45 6 Pa;.67 6 Pa rspcly.

16 76 K. C. WIEZCOLSKI Fgur 5 sows prssur srbuon aryng w m caus by roaon.8 s of bon a n crcumfrnal rcon ϕ for normal p jon wc s no affc by magnc fl. W an no accoun angular locy prurbaons.9 s on sprcal a of bon n crcumfrnal rcons. oaon abou bon a n mronal rcon s no an no accoun... cabulum s moonlss... For sa of a br comparson of numrcal rsuls w assum prurbaons of sam gap g a angular locy. s. For ms π/ π/ w oban maxmal alus of prssur wc ar.86 6 Pa;.9 6 Pa;.86 6 Pa rspcly. Fgur 6 sows prssur srbuon aryng w m caus by roaon.8 s of bon a n crcumfrnal rcon ϕ an smulanously by roaon.5 s of bon a n mronal rcon for normal p jon bng no affc by magnc fl. W a no accoun angular locy prurbaons.9 s on sprcal a n crcumfrnal rcon ϕ an smulanously locy prurbaons. s on sprcal a of bon n mronal rcons. W assum sam prurbaons of gap g n unsay moon a angular locy. s. For ms π/ π/ w oban maxmal alus of prssur qual o.77 6 Pa;. 6 Pa;.77 6 Pa rspcly..65 [m]. [Pas]. [/s]. [/s].5 [/s]. [/s]. [/s] p [Pa] p max [s] an π/ o [s] p max.67 6 [Pa] C o [N] Lubrcaon surfac.8 4π/ π/ 6π/ 5π/ π/ 7π/ π/ 8π/ 9π/ z π/8 y r π.7. x 7π/ φ..7 π/ x p a y.65 [m]. [Pas]. [/s]. [/s].5 [/s]. [/s]. [/s] π/ o [s] p max.45 6 [Pa] C o [N] Lubrcaon surfac.8 [cm ]

17 Numrcal conrbuon o scolasc lubrcaon of uman jon p [Pa] p max 6π/ 5π/ 4π/ π/ π/ 7π/ π/ y 9π/ z r 8π/ π.7 π/8. x 7π/ φ..7 π/ x p a y Fg. 4. Prssur srbuons caus by roaon n rcon only wr non-zro alus of angular locy.5 s an non-zro angular locy prurbaons n unsay flow an non-zro angular locy prurbaons of gap g prurbaons ar an no accoun.65 [m]. [Pas].8 [/s].9 [/s]. [/s]. [/s]. [/s] [s] an π/ o [s] p max.86 6 [Pa] C o [N] Lubrcaon surfac.8 [cm ]. 6 p [Pa] p max 6π/ 5π/ 4π/ π/ π/ 7π/ π/ y 9π/ z r 8π/ π.7 π/8. x 7π/ φ..7 π/ x p a y

18 78 K. C. WIEZCOLSKI.65 [m]. [Pas].8 [/s].9 [/s]. [/s]. [/s]. [/s]. 6 p [Pa] p max π/ o [s] p max.9 6 [Pa] C o 67 [N] Lubrcaon surfac.8 [cm ] 5π/ 4π/ π/ 6π/ π/ 7π/ π/ 8π/ 9π/ z π/8 y r π.7. x 7π/ φ..7 π/ x p a y Fg. 5. Prssur srbuons caus by roaon n ϕ rcon only wr non-zro alus of angular locy.8 s an non-zro angular locy prurbaons n unsay flow an non-zro angular locy prurbaons of gap g prurbaons ar an no accoun.65 [m]. [Pas].8 [/s].9 [/s].5 [/s]. [/s]. [/s] an π/ o [s] p max.77 6 [Pa] C o 696 [N] Lubrcaon surfac.8 [cm ]. 6 p [Pa] 9π/ y 8π/ z r π 7π/.7 6π/ p max π/8 5π/. 4π/ x π/ 7π/. π/ φ.7 π/ π/ x p a y

19 Numrcal conrbuon o scolasc lubrcaon of uman jon [m]. [Pas].8 [/s].9 [/s].5 [/s]. [/s]. [/s]. 6 p [Pa] π/ o [s] p max. 6 [Pa] C o 65N] Lubrcaon surfac.8 [cm ] 9π/ y 8π/ z r π 7π/.7 6π/ p max π/8 5π/. 4π/ x π/ 7π/. π/.7 φ π/ π/ x p a y Fg. 6. Prssur srbuons caus by roaon of bon a n crcumfrnal rcon ϕ an smulanously n mronal rcon wr non-zro angular locs.8 s.5 s an non-zro angular locy prurbaons n unsay flow an non-zro angular locy prurbaons of gap g prurbaons ar an no accoun. Symbol C o. nos a oal prssur T frs pcurs n fgurs 4 5 an 6 sow prssur srbuons for nal an fnal ms of pro of prurbaons of moon of uman jon. T scon pcurs n fgurs 4 5 an 6 prsn prssur srbuons for ml m pon of pro of prurbaons of moon. frwars prssur srbuons rurn o srbuons wc ar sown n frs pcurs of fgurs 4 5 an 6.

20 8 K. C. WIEZCOLSKI Capacy Co [N] 5 non zro non zro 5 5 π o π o π o π o 4 π o 5 π Tm o π o Fg. 7. Capacy srbuons rsus m for r assumpons corrsponng o r cass prsn n fgurs 4 5 an 6.. for moon of bon a n mronal rcon crcumfrnal rcon an smulanously n crcumfrnal an mronal rcons rspcly Fgur 7 prsns r curs of oal capacy srbuons rsus m n rang of m of prurbaon pro. Calculaons ar prform for followng ms: [s] π/ [s] π/ [s] π/ [s] 4π/ [s] 5π/ [s] π/ [s] For moon of bon a n mronal rcon an for m π/ π/ π/ 4π/ w oban followng alus of capacs: 859 N N 859N N 859 N rspcly. For moon of bon a n crcumfrnal rcon an for ms π/ π/ π/ 4π/ w oban followng alus of capacs: 865 N 67 N 865 N 67 N 865 N rspcly. For smulanous moon of bon a n crcumfrnal an mronal rcons an for ms π/ π/ π/ 4π/ w oban followng alus of capacs: 696 N 65 N 696 N 65 N 696 N rspcly. I s asy o s a prssur srbuons an capacs for ms: [s] π/ [s] 4π/ [s] a sam alus. T prssur srbuons an capacs for m π/ [s] a 4... a sam alus as wll. 9. Conclusons In prsn papr analycal an numrcal soluons of prssur an locs of synoal flu for any uman jon n curlnar orogonal coornas

21 Numrcal conrbuon o scolasc lubrcaon of uman jon 8 ar prsn. Proc prurbaons of unsay lubrcaon an smulanously of scolasc proprs of flu n magnc fl ar an no accoun. In numrcal calculaons on for prssur an capacy srbuons only prurbaons of moon of uman p jon surfacs ar nclu. nw form of ynols quaon r n s papr ns n parcular cas o a wll-nown form of ynols quaon for say moon bng r n arlr paprs. T rsuls oban ral a oal apparn scosy of synoal flu pns on m an on locy formaons. Toal apparn scosy of synoal flu cangs procally n m. n unsay magnc nucon fl qual o. mt w frquncy of abou 6 z cangs prssur srbuon n uman p jon from o 4 pr cn. ppnx If w subsu soluons 44 for n quaon 49 n w a: W p W p sn ] sn [ ] sn[ ] sn[ ] sn [ sn ] sn [ ] sn [ for... ϕ πc < c < b m π/8 π/ b s r. For furr rucon of quaon s ncssary o calcula followng ngrals: [ ] [ ] sn sn 4 an O W 4 an O p

22 8 K. C. WIEZCOLSKI sn sn an O Ingrals 4 ar nsr no quaon. Tus w arr a quaon cnowlgmn Ts papr was suppor by KBN funs n yars 6 as Scnfc Projc KBN 4E--5. uor ans for coopraon w Cnral Insu Dparmn of Bomcal Engnrng a nrsy of lm n Grmany an for coopraon w Clnc of Gorg ugus nrsy Göngn Dparmn of Jaw Oropcs n Grmany. frncs [] DOWSON D. Bo-Trbology of Naural an placmn Synoal Jons [n:] an Mow C. aclff. Woo S.L-Y. s. Bomcancs of Darroal Jon Sprngr-rlag Nw Yor Brln Lonyn Pars Toyo ong Kong 99 ol. Cap. 9 pp [] KĄCKI E. ównana różnczow cząsow w zagannac fzy cn WNT Warszawa 989. [] KNOPP K.Szrg nsończon PWN 956. [4] MOW.C. OLMES M.. LI W.M. Flu ranspor an mcancal proprs of arcular carlag Journal of Bomcancs [5] MOW.C. TCLIFFE.WOO S. Bomcancs of Darroal Jons Sprngr-rlag Brln lbrg Nw Yor 99. [6] MOW.C. SOSLOWSKY L.J. Frcon lubrcaon an war of arroal jons [n:] Mow.C. ays W. C. s. Basc oropc bomcancs Nw Yor an Prss [7] MOW.C. GILK F. Cll Mcancs an Cllular Engnrng Sprngr-rlag Brln lbrg Nw Yor 994. [8] OETEL. LIEN E. Numrsc Srömungsmcan wg Braunscwg Wsban. [9] TEIPEL I. T Impuls Moon of a Fla Pla n a scolasc Flu ca Mcanca Sprngr- rlag [] TESDELL C. Frs Cours n aonal Connuum Mcancs Marylan Jon opns nrsy Balmor 97. [] NGETÜM M. WINKLE-GNIEWEK W. Trbolog n Mzn Trbolog Scmrungscn [] WIEZCOLSKI K. PYTKO S. Moa wyznaczana paramrów bołożysa smarowango cczą nnwonowsą Trbologa [] WIEZCOLSKI K. NOWOWIEJSKI. MISZCZK. Numrcal nalyss of Synoal Flu Flow n Bobarng Gap Procngs of Inrnaonal Confrnc a Sysm Mollng Conrol Confrnc [4] WIEZCOLSKI K. PYTKO S. nalycal calculaons for xprmnal pnncs bwn sar ra an synoal flu scosy Proc. of Inrna. Trbology Confrnc Japan Yooama 995 ol.. pp

23 Numrcal conrbuon o scolasc lubrcaon of uman jon 8 [5] WIEZCOLSKI K. Ol locy an prssur srbuon n sor journal barng unr ln Ercsn lubrcaon 998 SMS Sysm nalyss Molng an Smulaons OP Orsas Publsrs. ssoc. N.. ol [6] WIEZCOLSKI K. T mo of soluons for yroynamc lubrcaon by synoal flu flow n uman jon gap Conrol an Cybrncs ol. No [7] WIEZCOLSKI K. Capacy of form uman p jon gap n m pnn magnc fl ca of Bongnrng an Bomcancs ol. 5 No. pp [8] WIEZCOLSKI K. Prssur srbuon n uman jon gap for lasc carlag an m pnn magnc fl ussan Journal of Bomcancs Prm ol. 7 No. pp [9] WIEZCOLSKI K. Trbolog für mnsclc Gln Trbolog un Scmrungscn 5 pp. 5. [] Wrzcols K. Trbolog für nsabl blas mnsclc Gln m magnscn Fl Inrnaonal Colloquum of Trbology Tcnsc am Esslngn 4 ol. III pp ; papr prsn aonally a Tcncal nrsy of Karlsru Faculy of Srömungslr July 4.

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm