Lecture 7 Diffusion. Our fluid equations that we developed before are: v t v mn t
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1 Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 tu 7 Dffuo Ou flud quato that w dvlopd bfo a: f ( )+ v v m + v v M m v f P+ q E+ v B momtum lot va ollo momtum hag va patl ga/ lo ( ) W hav lookd ollo ad otz fo tm th th lat to two to. Now w wll look at th adom ad dv flow. Oft, w a appoxmat th olloal momtum lo a bg popotoal to th vloty. H w wll aum that mν v M m v f m momtum momtum hag lot va va patl ga/ lo ollo wh ν m th fftv momtum taf ollo fquy. (Th gatv g th bau w a aumg that w a log gy to th oth p.) Thu, v m + v v mνv P+ q( E+ v B) Aumg that all of th aoutd fo fo bala. H, th lft had d of th quato zo. Th gv 0 mνv P+ q( E+ v B) mνv kt + q( E+ v B) o kt q v + ( E+ v B) - fo ow w wll lt B 0 kt q + E D + E wh kt D mνm q - ot that I hav ud a lghtly dfft dfto fo mνm a th dffuo ad th moblty ptvly. Th a taght-fowad quato to gt to but t ha a lot of mplato. Ft, w hav aumd that th vloty wa ot a futo of poto o tm. Th ma that th flud flowg fom o pot to aoth put that th alo flud movg th oth dto om ot of adom walk typ of faho. W fd ot upgly that th ma that f w wh to b abl to pat om of th patl th p a ta aa, w would th patd patl dft o dffu away fom that aa. Th dffuo qut atual. Th od thg that w that th vloty of hagd Pag 1
2 Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 patl ud th flu lmtd. Th aga ot too upg thk of tmal vloty fo a po who ky dvg (~186 mph). What l a w dov? Ft, w a dtm th flux of a gv p. Γ v D + E - fo p If th o lt fld w gt what kow a Fk aw Γ v D Th mply ou adom walk. Ambpola Dffuo If o w to tu off a plama, mot of thm wll day va a po kow a ambpola dffuo. Ambpola om fom two wod. Ambh whh at fo o both d o oud ad pola (Duh?). I t ma that both potv ad gatv hagd dffu at th am at. Suh a qumt ot patulaly upg a w hav qud ad xpt that th dty of th potv ad gatv p mut b about th am. Wthout a appld lt fld w laly fd that Γ D >> Γ D Th mpl that w mut hav a lt fld whh puh th o out ad v to ta th lto. Thu, Γ D + E Γ D + E W a ow olv fo th E fld. 0 } } ( D D) ( ) E ( D D) 0 E ( ) 0 Plag th to ou ommo dffuo quato gv ( D D) Γ a D + ( ) 0 0 ( D D ) 0 ( ) 0 wh ( D D ) ( ) th ambpola dffuo offt. W a dtm appoxmatly what th ambpola dffuo offt by makg om mpl tmat. Pag
3 Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 q, Ft, m, ν. Now ν σ m v. Aumg that both th o ad th lto hav th m am gy, th th vloty of th lto wll b o th od of v M. (Th v m atually a ud tmat a oft th lto gy hgh tha th o gy.) Futh, w wll aum that th ollo o to a o th am od of magtud. Th ot a bad aumpto a oulomb ollo wll b vy mla a wll a ollo wth utal. Thu, w xpt M m Th D D It ut o happ that th ato of q D kt So that kt D kt D T T D - oft 1 T T D Pluggg th to th abov quato fo D a gv D D T 1 + T D ( to 10) D Thu w that th ambpola dffuo td to th low dffuo at but whh oft gatly xd that low at. Fally, w hav to th pot god th otuty quato. f ( )+ v Igog th olloal tm, ad ug th dfto of th flux, t ay to that Pag 3
4 Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy ( v ) but Γ v D + E ( D + ) E t D E - lttg E 0 D Dffuo a Slab Th lat quato allow u to look lo at th day of a plama Ft lt lt (. t) T() t S( ) Th ug th tadad paato of vaabl thqu fo olvg PDE w gt D TS D TS S T TD S 1 T 1 T S D S Cot bau w hav to vay t ad dpdtly W wll all that Cot 1 τ Th T T t / τ - T T 0 τ ad S S S Dτ Λ + / Λ / Λ S S0+ + S0 o S S0Ao ( / Λ)+ S0B ( / Λ) Aumg that th plama ha a lgth of, wth a dty of 0 at ± /, w fd Pag 4
5 Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 S 0 S S Ao / Λ B / Λ / SAo ( / Λ) SB ( / Λ) (fom - / ) SB ( / Λ) (fom + / ) SB 0 ad ± Λ Thu, 1 Λ ( 1± ) Dτ τ ( 1± ) D ad ( ± ) S SA o 1 Thfo (. t) T() t S( ) t D xp ( + ) o ( + ) Fom τ ( ± ) 1 t adly appat that th hgh od mod wll day fat tha D low od mod. Thu a a plama hut off, w would xpt to a o pofl to th dty dtbuto. ± ( )+ ( ) Pag 5
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