Analytical solution of a transient Hartmann flow with Hall current and ion slip using finite Fourier transform

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1 Blgaran Chmcal Commncaons, Volm 6, Nmbr 3pp. 6 65) Analycal solon of a ransn Harmann flow wh Hall crrn and on slp sng fn Forr ransform W. Abd El-Mgd*, H. Aa and M. Elbarawy Dparmn of Engnrng Mahmacs and Physcs, Facly of Engnrng, El-Fayom Unvrs El-Fayom 635, Egyp Rcvd Ags 8, 3; Rvsd Novmbr 8, 3 Th ransn Harmann flow of an lcrcally condcng vscos ncomprssbl fld bond by wo paralll nslang poros plas s sdd sng fn Forr ransform. An xrnal nform magnc fld s appld whl h fld moon s sbjcd o a consan prssr gradn. Th Hall crrn and h on slp ar akn no consdraon n h momnm qaons. Th ffc of h Hall crrn and on slp on h vlocy and dsrbon of h flow s nvsgad. Ky words: Harmann flow; Fn Forr sn ransform FFST); aplac ransform T); condcng fld; Hall crrn; Ion slp. INTRODUCTION Th Harmann flow dfnd as a magnohydrodynamc MHD) flow bwn wo paralll plas s a classcal ss ha has many applcaons n MHD pmps, MHD powr gnraors, acclraors, arodynamcs hang, and prolm ndsry. In rf. [] h nflnc of a ransvrs nform magnc fld on h flow of a condcng fld bwn wo nslad nfn paralll saonary plas s sdd. Thn a lo of rsarch work has bn don concrnng h Harmann flow ndr dffrn physcal ffcs [- ]. In mos cass h Hall and on slp rms wr gnord n applyng Ohm s law as hy hav no markd ffc for small and modra vals of h magnc fld. Howvr, whn an applcaon of MHD rqss a srong magnc fld, h nflnc of lcromagnc forc bcoms nocabl []. Undr hs condons, h Hall crrn and on slp ar mporan as hy wll affc h magnd and drcon of h crrn dnsy and consqnly affc h magnc forc rm. In rf. [6] h Hall ffc on h sady moon of lcrcally condcng and vscos flds n channls was sdd. In [8-9] h ffc on h sady MHD Co flow wh ha ransfr was sdd. Th mprars of h wo plas wr assmd hr o b consan [8] or o vary lnarly along h plas n h drcon of h flow [9]. In [] h ffc of Hall crrn on h sady Harmann flow sbjcd o a nform scon and njcon a h bondng plas was sdd. ar, n [] h problm of h nsady sa wh ha ransfr was xndd akng no consdraon h Hall ffc whl nglcng h on slp. In [3] h on slp was akn no consdraon and h qaons of moon wr solvd analycally sng h aplac ransform T) mhod. Th nrgy qaon was solvd nmrcally sng h fn dffrnc mhod akng no consdraon h Jol and vscos dsspaons. In hs papr, an analycal solon s prsnd for h ransn flow of an ncomprssbl, vscos, lcrcally condcng fld bwn wo nfn nslang horzonal plas wh h consdraon of boh h Hall crrn and on slp. Th fld s acd pon by a consan prssr gradn n h axal drcon, whl a nform magnc fld s appld prpndclar o h plas. Th ndcd magnc fld s nglcd by assmng a vry small magnc Rynolds nmbr. Th momnm qaons ar solvd analycally sng Fn Forr Sn ransform FFST). Th ffc of h magnc fld, h Hall crrn, and h on slp on h vlocy dsrbon s sdd. DESCRIPTION OF THE PROBEM Th wo nslang poros plas ar locad a y h and xndd from x o and from z o as shown n Fg.. Th fld flow bwn h wo plas s * To whom all corrspondnc shold b sn: E-mal: wald.abdlmagd@nvnsys.com; dp nflncd by a consan prssr gradn n dx Blgaran Acadmy of Scncs, Unon of Chmss n Blgara 6

2 W. Abd El-Mgd al.: Analycal solon of a ransn Harmann flow wh Hall crrn and on slp y = h Uppr pla y Man flow ) x z y = -h owr pla Fg.. Schmac dagram of h sysm. h x-drcon. Th whol sysm s sbjc o a nform magnc fld n h posv y-drcon whl h ndcd magnc fld s nglcd. Th xsnc of h Hall rm gvs rs o h z- componn of h vlocy. Ths h vlocy vcor of h fld s gvn as: v w k ) Th fld flow s govrnd by h momnm qaon Dv v P J B o ) D whr and ar h dnsy and h coffcn of vscosy of h fld, rspcvly. If h Hall and on slp rms ar rand, h crrn dnsy J s gvn by B J v Bo J Bo ) J Bo ) Bo Bo 3) whr s h lcrc condcvy of h fld, s h Hall facor and B s h on slp paramr []. Eqn. 3) may b solvd n J o yld Bo J Bo ) w) ) w ) k ) ) whr Bo s h Hall paramr []. Ths, n rms of Eqns. ) and ), h wo componns of Eqn. ) rad dp Bo ) w), dx y ) 5) 6 w w Bo y ) ) w ), 6) Th scond and hrd rms on h rgh-hand sd rprsn h vscos and Jol dsspaons, rspcvly. Th problm s smplfd by wrng h qaons n non-dmnsonal form. Th characrsc lngh s akn o b h, and h h characrsc m s whl h characrsc vlocy s. W dfn h followng nondmnsonal qans: h x y z h hw ˆ, ˆ, ˆ, ˆ, ˆ, ˆ Ph, ˆ x y z w P, h h h h In rms of h abov non-dmnsonal varabls and paramrs, Eqns. 5)-6) ar wrn as h "has" wll b droppd for convnnc): dp Ha w) 7) dx y ) w w Ha y ) w ) 8) whr Ha s h Harmann nmbr sbjcd o h followng nal and bondary condons: w ; 9) w ; y Th man prpos of hs papr s o solv h paral dffrnal Eqns. 7), and 8) by FFST. SOUTION OF THE EQUATIONS OF MOTION USING FFST Th FFST of a fncon s dfnd as:

3 W. Abd El-Mgd al.: Analycal solon of a ransn Harmann flow wh Hall crrn and on slp n y g G n, g y)sn ) dy ; y ; n,,3,.....) Fs s whr h krnl of h ransform s: ny sn ) ) For ransformng h scond drvav w s h opraonal propry d g y) n n Fs g) cos n )g) - ) G n) s dy ) Ths, f h bondary val problm nvolvs a scond-ordr drvav, xnds ovr a fn doman y, and has bondary condons a boh nds n h followng form: g, f, 3) g, f hn FFST can b sd o ransform h ndordr drvav. Dfnng: v w ) Eqns. 7) and 8) can b combnd as: v dp v Ha ) v 5) dx y ) Applyng FFST o Eqn. 5) ylds Vs n dp k) Vs * cos n ) 6) dx n whr Ha k 7) Th solon of h lnar nhomognos paral dffrnal Eqn. 6) ndr h nal and bondary condons gvn n Eqn. 9) s gvn as dp * cos n ) n - k Vs n dx n, ) n - 8) k) Th nvrs ransform of and hr ral and magnary componns ar gvn by dp * cos n ) v dx n - n n k n - k sn n y 9) v, w Imv ) R ) Fgs. a) and b) prsn h volon of h profls of h vlocy componns and w dp vrss y for H a 3; 3 ; 5, and dx for varos vals of m.;.5;.5; 3.). Th fgrs show h parabolc shap of h profls and ndca ha boh and w rach hr sady sa monoonosly wh m. RESUTS AND DISCUSSION As sn n Tabls and, h comparson bwn T as sd by Aa [3] and FFST mhods for solvng h qaon of moon for Harmann flow ndcas ha h ops ar narly qal whch lnds confdnc o h rsls oband hr. Tabl. Comparson bwn T and FFST for calclang Y =.5 =.5 =3 T FFST T FFST T FFST Tabl. Comparson bwn T and FFST for calclang w =.5 =.5 =3 Y T FFST T FFST T FFST In Fg., h vlocy componn rachs h sady sa fasr han w. Fg. 3 prsns h m valaon of and w a y= for varos vals of h Hall paramr, h on slp paramr a Ha. 63

4 W. Abd El-Mgd al.: Analycal solon of a ransn Harmann flow wh Hall crrn and on slp =. =.5 =.5 =3 ) Ha w rm ), h nflnc of on slp ) on w bcomng clarr for hghr vals of For larg h componns and w ovrshoo, xcdng hr sady sa vals and hn go down owards sady sa; h on slp plays a rol n spprssng hs ovrshoos y Fg. a). Varaon of vrss y B=, B= B=, B= B=, B=3 B=3, B= B=3, B= B=3, B=3 Fg. 3a). Effc of Hall Paramr and on slp paramr on h m dvlopmn of a y= Fg. b). Varaon of w vrss y. In Fg. 3a), ncrasng h paramrs and wll ncras bcas h ffcv condcvy ) dcrass wh ncrasng or whch rdcs h magnc dampng forc on. In Fg. 3b), ncrasng wll ncras h vlocy componn w as a rsl of h Hall ffc, b ncrasng wll dcras w for all vals of as a rsl of dcrasng h sorc rm of w Ha ) ) and ncrasng s dampng ) Ha w rm ), h nflnc of on slp ) on w bcomng clarr for hghr vals of In Fg. 3b), ncrasng wll ncras h vlocy componn w as a rsl of h Hall ffc, b ncrasng wll dcras w for all vals of as a rsl of dcrasng h sorc rm of w Ha ) ) and ncrasng s dampng w B=, B= B=, B= B=, B=3 B=3, B= B=3, B= B=3, B=3 Fg. 3b). Effc of Hall Paramr and on slp paramr on h m dvlopmn of w a y=. Fg. prsns, and w a y= for varos vals of h Harmann nmbr Ha and h on slp paramr wh 3. As shown n Fg. a), for small vals of Ha, ncrasng wll slghly dcras as a rsl of ncrasng h dampng facor on ; frhr ncrasng wll ncras h ffcv condcvy and, n rn, wll dcras h dampng facor on whch ncrass ; on h ohr hand, for largr vals of Ha, bcoms small; ncrasng always dcras h ffcv condcvy and consqnl wll ncras, h ffc of on bcomng mor apparn for larg vals of Ha. In Fg. b), ncrasng h on slp paramr wll dcras w for all vals of Ha, s ffc s mor apparn for hghr vals of Ha.

5 W. Abd El-Mgd al.: Analycal solon of a ransn Harmann flow wh Hall crrn and on slp Ha=, B= Ha=, B= Ha=, B=3 Ha=5, B= Ha=5, B= Ha=5, B=3 Fg. a). Effc of Harmann Nmbr Ha and on slp paramr on h m dvlopmn of a y= w Ha=, B= Ha=, B= Ha=, B=3 Ha=5, B= Ha=5, B= Ha=, B=3 Fg. b). Effc of Harmann Nmbr Ha and on slp paramr on h m dvlopmn of w a Y=. CONCUSIONS FFST mhod can b sd o oban an analycal solon for h ransn Harmann flow of an lcrcally condcng, vscos, ncomprssbl fld bond by wo paralll nslang plas wh Hall crrn and on slp. Th comparson of h FFST mhod wh prvosly sd mhods as T shows ha hs chnq s vry smpl and gvs accra rsls for solvng h govrnng momnm qaon for h whol rang of h physcal paramrs sd. REFERENCES. J Harmann and F azars kgl. Dansk Vdnskab. Slskab Ma.-Fys. Mdd ).. N Tao Jornal of Arospac Scnc ). 3. S Ngam and S Sngh Th Qarrly Jornal of Mchancs and Appld Mahmacs ).. K Cramr and S Pa Nw York: McGraw-Hll) 973). 5. G Son and A Shrman Nw York: McGraw-Hll) 965). 6. I Tan Jornal of Arospac Scnc ). 7. H. A. Aa and M. A. M. Abdn Effc of on slp on h m-varyng Harmann flow of a non-nwonan vscolasc fld wh ha ransfr, Jornal of Appld Mchancs and Tchncal Physcs 5 68 March 3) 8. V Sondalgkar, N Vghnsam and H Takhar IEEE Transacons on Plasma Scnc PS ). 9. V Sondalgkar and A Uplkar IEEE Transacons on Plasma Scnc PS ).. M. A. M. Abdn, H. A. Aa, W. Abbas and W. Abd El-Mgd Effcvnss of porosy on ransn gnralzd Co flow wh Hall ffc and varabl proprs ndr xponnal dcayng prssr gradn Indan Jornal of Physcs Ags 3).. E Abo-El-Dahab Effc of Hall Crrns on Som Magno-hydrodynamc Flow Problms Hlwan Unvrs Hlwan: Masr Thss) 993).. H Aa Canadan Jornal of Physcs ). 3. H Aa Physca Scrpa 66 7 ). АНАЛИТИЧНО РЕШЕНИЕ НА ЗАДАЧАТА ЗА ПРЕХОДНО ТЕЧЕНИЕ НА HARTMANN С ТОК НА HA И ЙОННО ПРИПЛЪЗВАНЕ СПОМОЩТА НА КРАЙНА ТРАНСФОРМАЦИЯ НА FOURIER У. Абд Ел-Мегед*, Х. Атиа, М. Елбарауи Департамент по физично и математично инженерство, Инженерен факултет, Университет в Ел-Фаюм, Ел- Фаюм 635, Египет Постъпила на 8 август, 3 г.; коригирана на 8 ноември, 3 г. Резюме) Изследвано е преходното течение на Harmann flow на електропроводящ несвиваем вискозен флуид между две успоредни изолаторни плочи с помощта на крайна Forr ова трансформация. Задачата е решена при прилагане на външно магнитно поле и постоянен градиент на хидравличното налягане. Токът на Hall и приплъзването на йони са отчетени в уравненията на движението. Установено е тяхното влияние върху скоростния профил в течността. 65

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