NUMERICAL MODELLING OF A CILIUM USING AN INTEGRAL EQUATION

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1 NUEICAL ODELLING OF A CILIU USING AN INTEGAL EQUATION IHAI EBICAN, DANIEL IOAN Key words: Cl, Numercl nlyss, Electromgnetc feld, gnetton. The pper presents fst nd ccurte method to model the mgnetc behvour of n rtfcl clum. The method s bsed on xwell equtons, formulted s ntegrl equtons for the mgnetton. The domn of the clum s dvded n D fnte elements, ech element beng unformly mgneted long the clum drecton. Numercl results, s mgnetc flux densty, torque nd force n ech element of the clum re presented for two test cses. The clum s plced n n unform mgnetc feld.. INTODUCTION cro-fludcs s the scence nd technology of mnpultng nd nlysng flud flow n structures of sub-mletre dmensons. The vlblty of mcrofludcs technology s essentl for the development of dvnced products n vrety of pplcton res, the most mportnt of whch s the bomedcl feld. ny ndustrl reserch groups re studyng wys of mcro-fludc mnpulton. A method of flud mnpulton technology n mcro-fludcs systems, nspred by nture s bsed on cl. Ths pproch enbles effectve locl flud mnpulton, nd the possblty to generte complex flow ptterns. The rtfcl cl re thn fs consstng of polymer mtrx flled wth mgnetc nnoprtcles []. Dependng on the nture of the prtcles, the f cn be super-prmgnetc or ferromgnetc wth remnent mgnetton. The movement of the rtfcl cl cn be ctvely controlled, preferbly usng mgnetc feld or n electrcl feld []. In ths pper, fst nd ccurte method to model the mgnetc behvour of the rtfcl clum s proposed. The method s bsed on the ntegrl equtons of the mgnetc feld, n term of the mgnetton. Poltehnc Unversty of Buchrest, Electrcl Engneerng Deprtment, Numercl ethods Lbortory, Splul Independente 33, 0600 Buchrest, omn, E-ml: mh_r@n.pub.ro, dnel@n.pub.ro ev. oum. Sc. Techn. Électrotechn. et Énerg., 56,, p , Bucrest, 0

2 360 h ebcn, Dnel Ion. INTEGAL EQUATIONS ETHOD Let us consder the clum s lner mgnetc wre wth rectngulr cross secton plced n unform mgnetc feld Fg. ). The mgnetc flux densty B), torque T), force F) n the clum re obtned by mgnetc feld computton. Fg. Computton domn of the clum. Strtng from the xwell equtons for the mgnetosttc regme [3] n the domn Ω Ω Ω Ω s the surroundng r, Ω s the clum), rot H 0, dv B 0, B µ 0 H µ 0, ) wth n pproprte open boundry condton on surfce Ω e.g., B n B 0, where B 0 s the known exctton feld), t s obtned the Posson's equton: vm dv Vm dv vm V m. ) Ω The totl mgnetc feld n the clum s: dv dv H H 0, 3 3) Ω where H 0 s the externl mgnetc feld. Thus, tkng n consderton the equton χh p, the ntegrl equton for the mgnetton s: χ dv dv χh 3 0 p. ) Ω

3 3 Numercl modellng of clum usng n ntegrl equton 36 Let us consder tht the clum s dvded n n segments D fnte elements), nd ech prllelepped element s unformly mgnetsed long the clum drecton Fg. ). In ech mgneted domn D, the volume chrge densty vm s null. Insted, the surfce chrge densty Sm on the seprton surfces S nd S re not ero. Fg. Dscretton of the clum for the mgnetc feld computton. Becuse the thckness of the clum g s very smll compred wth the other dmensons, t cn be neglected further. Thus, the seprton surfce S between two fnte elements s pproxmted wth strght lne perpendculr to the plne XOY Fg. 3). On ech seprton lne S, there s constnt lner chrge densty g Sm. Fg. 3 Approxmted computtonl domn left sde) nd seprton lne S rght sde).

4 36 h ebcn, Dnel Ion The mgnetc feld n pont P due to the seprton lne S s: 3, d d C S P r u H, 5) where u. Due to the symmetry, the mgnetc feld s null on the OZ drecton, H 0 Fg. 3). Therefore, the mgnetc feld exsts only n the plne XOY: ) 3, ) d d sn H S P α. 6) The mgnetc feld n the pont P due to the ll seprton lnes s: n P ) u H 7) n n g g s sm P ) dv ) u u H. 8) Fg. shows the seprton surfce between two elements, n whch the mgnettons - nd re constnt. Fg. Seprton surfce between two fnte elements. ) ) cos cos cos dv s β β β, 9)

5 5 Numercl modellng of clum usng n ntegrl equton 363 β β β rccos x ) x x ) y y ) y y ) [ x x ) y y ) ] x x ) y y ) x. [ ] The mgnetc feld n the element D j due to the ll mgnettons n the elements D : 0) H j g n j ) cos β ) ) j, ) x j x j y j y j j x y. ) Outsde the clum the mgnetton s null, thus 0 0 nd n 0. In the element D j the constnt mgnetton s j χh p, where H s the totl mgnetc feld n the element D j, H H j H 0. Thus, n the element D j the equton of the mgnetton s χh χh 0, where p s supposed j j known. Fnlly, ) becomes lner system of n equtons []: n χg )cos β ) χh 0, j, n. j p 3) ) j j Solvng 3), the mgnettons j, j, n n the fnte elements re obtned. oreover, the mgnetc flux densty, the torque T B) nd the force Fgrd B)) n ech element re computed. p 3. TEST CASES The rtfcl clum dopted n ths pper s lner mgnetc wre µ r 00) wth rectngulr cross secton, s s shown n Fg. 5. The clum dmensons re: the length of 00µm, the wdth of 0 µm, nd the thckness of µm. Two test cses re proposed n order to study the mgnetc behvour of the clum: the strght clum test cse ) nd the bended clum test cse ) plced n n unform mgnetc feld B 0 0mT). Fg. 5 shows the confgurtons of the two test cses.

6 36 h ebcn, Dnel Ion 6 Fg. 5 Strght nd bended cl n n unform mgnetc feld cse left sde), cse rght sde).. NUEICAL ESULTS In the test cses consdered, the clum s dvded n 0 fnte elements. The numercl smultons of the proposed method for the two test cses re done by tlb [5]. The computtonl tme s the order of the seconds. Fg. 6 shows the mgnetc flux densty long the strght clum plced n n unform mgnetc feld test cse ). Fg. 6 gnetc flux densty long strght clum stuted n n unform mgnetc feld.

7 7 Numercl modellng of clum usng n ntegrl equton 365 Fgs. 7, 8 show the mgnetc flux densty, torque nd force long the bended clum plced n n unform mgnetc feld test cse ). Fg. 7 gnetc flux densty long bended clum stuted n n unform mgnetc feld. Fg. 8 Torque nd force long bended clum stuted n n unform mgnetc feld. 5. CONCLUSIONS A fst nd ccurte method to model the mgnetc behvour of clum ws proposed. The method s bsed on the ntegrl equton for the mgnetton. The

8 366 h ebcn, Dnel Ion 8 prllelepped domn of the clum ws dvded n D fnte elements, ech element beng unformly mgneted long the clum drecton. The clum ws consdered lner mgnetc wre. Two test cses were dopted n order to study the mgnetc behvour of the clum. A strght clum nd bended clum were plced n n unform mgnetc feld. The mgnetc flux densty, torque nd force n ech element of the clum were computed for the two cses usng the presented method. ACKNOWLEDGEENTS Ths work ws fnncl supported by the Europen Commsson project NP CT Nture-nspred mcro-fludc mnpulton usng rtfcl cl ATIC. eceved on Februry 8, 0 EFEENCES. S. Khder,. Bltussen, P. Anderson, D. Ion, J. den Toonder, P. Onck, Nture-nspred mcrofludc propulson usng mgnetc ctuton, Physcl evew E., 79,, ).. * * *, Artc home pge: 3. B. Thde, Electromgnetc Feld Theory, Upslon Books, Sweden 00).. D. Ion, I.F. Hntl,. ebcn, C. Constntn, FLUXSET Sensor Anlyss bsed on nonlner mgnetc wre model of the core, Electromgnetc Nondestructve Evluton II), IOS Press, Amsterdm, 998, pp * * *, tlb home pge: