Outline. Review Radial Diffusion. Review Separation of Variables. Review General Solution v(r,t)

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1 aplace s Eqaio Feba 9 9 aplace s s Eqaio a aeo Mechaical Egieeig 5 Seia i Egieeig Aalsis Feba 9 9 Olie eview las class Used essel fcios o solve adial diffsio eqaio aplace s eqaio i wo-diesioal ecagla coodiaes Solio b sepaaio of vaiables esl has sie cosie ad hpebolic sie ad cosie Sie ad cosie eigefcio epasio eview adial Diffsio Goves diffsio hea codcio i clide fo ad is epeae species coceaio Iiial codiio is fiie ad v has v Diffeeial eqaio fo boh ad v v is S-ioville 3 eview Sepaaio of Vaiables Asse v P [ P P Divide b P [ P P P P 4 eview Geeal Solio v ave epoeial solio i ie ad essel s eqaio fo adial fcio d Ae d d dp his is fo of P d d essel s eqaio v P whee eqaio is P ad ae ode zeo essel fcios of fis ad secod kid especivel eview Plo essel Fcios of he Fis Kid fo Iege Odes ME 5 Egieeig Aalsis

2 aplace s Eqaio Feba eview Plo essel Fcios of he Secod Kid of Iege Ode becoes ifiie as appoaches zeo eview adial Solio adial diffsio eqaio fo P asfoed essel eqaio d dz dp P d p d z ν k z dz z d d Solio o secod geeal ν eqaio is A ν kz ν kz adial diffsio wih ν ad p is P A 8 eview adial Solio II v P whee is epoeial deca P A Solio applies i egio o Ms have P fiie a Sice - as we s have fo P o be fiie a odiio P eqies Eigevale solios ae whee Zeos of essel Fcios of he Fis Kid fo Iege Odes -.4 which ae zeos of eview adial Solio III Solio is s of all eigefcios eview adial Solio IV osa iiial codiio U e Iiial codiio v gives b eigefcio epasio [ d d [ d U [ d U U p U e U f ME 5 Egieeig Aalsis

3 aplace s Eqaio Feba 9 9 ME 5 Egieeig Aalsis 3 adial Diffsio a. a.4 a. a.6 a. a. a.4 a.6 a.8 a. a. a.3 a.4 a.5 a.6 a.8 a / [ - / U - 4 eview Diffsio Eqaio adial ecagla fiie A 5 eview Sepaaio of Vaiables adial ecagla P P Ae Ae 6 eview Zeo odaies adial ecagla fo fo P cos si fo fiie si P 7 eview Geeal Solio adial ecagla e si si e 8 eview Eqaios adial ecagla d d d si si si d d d

4 aplace s Eqaio Feba 9 9 aplace s Eqaio Used o epess eqilibi fields of egieeig vaiable like epeae species coceaio elecosaic poeial ad ideal flid flow Wie i geeal coodiaes as aesia lidical Spheical z θ z si coφ φ θ φ φ 9 wo-diesioal Poble Appl o ecagla egio ad Siple boda codiios: is zeo a hee sfaces ad specified a ohe Sepaaio of Vaiables Asse [ [ Divide b Solve ODEs o Ge wo ODEs wih kow geeal solios d d d d Asi cos sih D cosh [ Asi cos [ sih D cosh odaies a ad Sice we s have fo Sice a ad is he solio o a S-ioville poble Asi cos Asi cos > Asi Ms have a iege Eigefcio is si/ Solio wiho oda codiio a gives sih Dcosh D Wih ad D solio is Asi sih wih / Geeal solio is s of each eigefcio liplied b a cosa si sih 3 4 ME 5 Egieeig Aalsis 4

5 aplace s Eqaio Feba 9 9 ME 5 Egieeig Aalsis 5 5 Sepaaio osa: o ave wo choices i sepaaio he fis choice gives as sies ad cosies; he secod choice gives as sies ad cosies Make choice o ge sies ad cosies i coodiae wih wo zeo bodaies 6 Wh Use sih? wo eqivale solios fo he eqaio d /d A sih cosh e D e - Show eqivalece fo defiiios of sih z e z e -z /; cosh z e z e -z / A / ad D A/ Fo sih ad cosh solio gives ol oe fcio sih 7 oda a Mlipl b si/ ad iegae Use eigefcio epasio o ge boda codiio a sih si d d d d si sih si si sih sih si si si 8 Ohogoal Eigefcios Solve fo si si si d d δ δ Uses ohogoali fo eigefcios sih si si sih si d d d 9 oda odiio Gives Siples case is U a cosa sih cos sih si si sih si U d U d d cos fo eve ad fo odd Wie seies sig o ge ol odd es 3 osa oda odiio Solio fo is sih sih si 4 U /U depeds o / / ad / Plo o e page fo /

6 aplace s Eqaio Feba 9 9 ME 5 Egieeig Aalsis Siila o Diffsio Eqaio aplace Diffsio i ope boda 33 Sepaaio of Vaiables aplace Diffsio Ae cosh sih D A 34 oda odiios aplace Diffsio fo fo cos si si cos si fo fo si 35 oda odiio aplace Diffsio cosh sih D have Ms D A A sih o eqivale codiio becase of ope boda i ie 36 Geeal Solio; Fied odiio aplace Diffsio e si si si sih sih si

7 aplace s Eqaio Feba 9 9 Eigevale Epasio fo Diffsio si d si d si d aplace si d sih si d si d sih Gadies as Fles aplace ad diffsio eqaio ae based o cosevaio of fles which ae egaive gadies of poeial aplace eqaio gives eqilibi whee e fl fo egio shold be zeo Deails a ed of peseaio show he e oflow is zeo fo he solio hee e oflow d d d d Gadie oda odiio oside he followig poble 39 Gadie odiio II Appl ew boda codiios o he pevios sepaaio of vaiables esl [ Asi cos [ sih Dcosh ' [ Acos si sih D cosh [ 4 odaies a ad Fo a ad a we have so is he solio o a S-ioville poble Asi cos Acos si Acos si: A cos Ms have / a odd iege; se Eigefcio is cos[/ 4 Solio wiho oda codiio a gives sih Dcosh D Wih A ad D solio is cos sih wih // Geeal solio is s of each eigefcio liplied b a cosa cos sih 4 ME 5 Egieeig Aalsis 7

8 aplace s Eqaio Feba 9 9 oda a Eigefcio epasio cos sih Usal solio pocess gives sih cos cos d d cos sih d 43 oda odiio Gives Siples case is U a cosa cos cos U d U d si si sih sih sih si si[/ fo eve ad fo odd which is U sih sih 44 osa oda Solio Sbsie ad / io geeal solio fo cos sih sih /U depeds o / / ad / Plo o e page fo / Sa eview adial diffsio eqaio Solio i essel fcios Eigefcio epasio of essel fcios o ach iiial codiio a hadle ohe boda codiios aplace eqaio also solved b sepaaio of vaiables hee zeo bodaies ad o ozeo boda Zeo e fl io egio fo oe case Also did zeo fl boda codiio 47 Deails of Zeo e Fl Fo hea fl q k/ q k/ e hea fl leavig egio is iegaed hea fl a all fo bodaies e op k d k d k d k eaiig chas evalae e op ake deivaives of solio fo as evalae a ad ; iegae ad add as show above 48 d ME 5 Egieeig Aalsis 8

9 aplace s Eqaio Feba 9 9 -Diecio Gadies Iegaed -Diecio Iflow si sih sih cos sih sih cos ad cos[ so sih sih Sae iflow a ad as epeced b se of solio sih sih 49 sih sih d d d sih sih sih d sih cosh sih 5 cosh sih -Diecio Gadies si sih sih cosh ad ah sih / cosh si cosh sih si si sih ah 5 e -Diecio Oflow cos si d d sih sih si cos d d d ah d ah sih ah 5 oal e Oflow Is Zeo d d sih ah cosh d d sih S is zeo ah Eqal ad opposie sih 53 ME 5 Egieeig Aalsis 9