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10.539Lecture13IntroductiontoBoundary-LayerTheory

Prof.DeanWang

1. Boundary-LayerTheoryBoundary-layertheoryisacollectionofperturbativemethodsforobtaininganasymptoticapproximationtothesolution𝑦 𝑥 ofadifferentialequationwhosehighestderivativeismultipliedbyasmallparameter𝜀.Solutionstosuchequationsusuallydevelopregionsofrapidvariationsas𝜀 → 0.Ifthethicknessoftheseregionsapproaches0as𝜀 → 0,theyarecalledboundarylayers,andboundary-layertheorymaybeusedtoapproximate𝑦 𝑥 .Example1.Exactlysolubleboundary-layerproblem.Considerthedifferentialequation

𝜀𝑦!! + 1+ 𝜀 𝑦! + 𝑦 = 0𝑦 0 = 0 𝑦 1 = 1

(1)

Solution:Theexactsolutionofthisequationis 𝑦 𝑥 = !!!!!!!/!

!!!!!!!/! (2)

Inthelimit𝜀 → 0+,thissolutionbecomesdiscontinuousat𝑥 = 0,asshowninthefollowingfigure.

Therearetwostandardapproximationsthatonemakesinboundary-layertheory.Intheouterregion(awayfromaboundarylayer)isslowlyvarying,soitisvalidtoneglectanyderivativeof𝑦 𝑥 whicharemultipliedby𝜀.Insideaboundarylayerthederivativeof𝑦 𝑥 arelarge,buttheboundarylayerissonarrowthatwemayapproximatethecoefficientfunctionsofthedifferentialequationbyconstants.Thus,wecanreplaceasingledifferentialequationbyasequenceofmuchsimplerapproximateequationsintheinnerandouterregions.

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

0

0.5

1

1.5

2

2.5

30 = 0.10 = 0.00250 = 0.0005

x0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

y

0

0.5

1

1.5

2

2.5

30 = 0.10 = 0.00250 = 0.0005

K!

E=*<93)!K'!L6$742%$0)$!+%+36+)*$!:%/+0*$123*1)$!9$%:3)<'!!! ! !" !! ! !" ! !!!

! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! GMH!

L$%<!4567!6+646*32;*3/)!9$%:3)<D!?)!?675!4%!0)4)$<6+)!*!3)*06+-2%$0)$!9)$4/$:*46;)!*99$%=6<*46%+!4%!! ! !*7!! ! !!'!N345%/-5!4567!67!%+31!*!&6$742%$0)$!06&&)$)+46*3!)>/*46%+D!64!67!+%+36+)*$!*+0!67!</85!4%%!06&&68/34!4%!7%3;)!6+!83%7)0!&%$<'!!@%3/46%+I!@4)9!"I!O%?);)$D!6+!$)-6%+7!?5)$)!!!*+0!!!!*$)!+%4!3*$-)!G7/85!$)-6%+7!*$)!8*33)0!%/4)$!$)-6%+7H'!A4!67!;*360!4%!+)-3)84!!""!!8%<9*$)0!?645!!!! '!J5/7D!6+!%/4)$!$)-6%+7!?)!*99$%=6<*4)!45)!7%3/46%+!4%!GMH!:1!45)!7%3/46%+!4%!45)!%/4)$!)>/*46%+!

!!!"#! ! !!!"# ! !!!!! GPH!@4)9!KI!GPH!67!)*71!4%!7%3;)!:)8*/7)!64!67!36+)*$'!J5)!7%3/46%+!?5685!7*467&6)7!!!!" ! !!!!!67!

!!"# ! ! !! !"! !!!! ! GQH!@4)9!M'!A+!45)!:%/+0*$1!3*1)$!G45)!6++)$!$)-6%+HD!!!67!7<*33!7%!64!67!;*360!4%!*99$%=6<*4)!!!!!:1!"'!L/$45)$<%$)D!76+8)!!!;*$6)7!$*96031!6+!45)!+*$$%?!:%/+0*$1!3*1)$D!?)!<*1!+)-3)84!!"!8%<9*$)0!?645!!"!'!O)+8)D!6+!45)!6++)$!$)-6%+!?)!*99$%=6<*4)!45)!7%3/46%+!4%!GMH!:1!45)!7%3/46%+!4%!45)!6++)$!$)-6%+

! ! !!!" !!"! ! !! GRH!!@4)9!P'!GRH!67!*!36+)*$!)>/*46%+!6&!?)!$)-*$0!!!*7!45)!0)9)+0)+4!;*$6*:3)'!A47!7%3/46%+!67!

! ! ! !!" ! ! ! !!!!" !! GSH!@4)9!Q'!F!67!0)4)$<6+)0!:1!*71<94%468*331!<*4856+-!45)!%/4)$!*+0!6++)$!7%3/46%+7!GQH!*+0!GSH'!

!!"# ! ! !! !"! !!!! !! !!"# ! !!! !"!! &%$!!!67!7<*33!!GTH!! ! ! !!" ! ! ! !!!!" ! !! !!!!!!" !!! &%$!! ! !!! GUH!

@4)9!R'!GTH!*+0!GUH!-6;)7!! ! !

!! G"CH!!

!

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Example3.Second-orderlinearboundary-valueproblem.Letusfindanapproximatesolutiontotheboundary-valueproblem

𝜀𝑦!! 𝑥 + 𝑎 𝑥 𝑦! 𝑥 + 𝑏 𝑥 𝑦 𝑥 = 0 0 ≤ 𝑥 ≤ 1𝑦 0 = 𝐴 𝑦 1 = 𝐵

(11)

Solution:Step1:Intheouterregionagoodapproximationto(11)isthefirst-orderlinearequation 𝑎 𝑥 𝑦!"#! 𝑥 + 𝑏 𝑥 𝑦!"# 𝑥 = 0 (12)Step2:Thesolutionto(12)is

𝑦!"# 𝑥 = 𝑐𝑒! !! ! !"!! , (13)

where,𝑐isanintegrationconstant.Step3:Requiringthat𝑦!"# 1 = 𝐵gives𝑐 = 𝐵,sowehave

𝑦!"# 𝑥 = 𝐵𝑒! !! ! !"!! , (14)

Step4:Todeterminethebehaviorof𝑦 𝑥 when𝑥 → 0+,wemayapproximatethefunctionsof𝑎 𝑥 and𝑏 𝑥 by

𝑎 0 = 𝛼 ≠ 0𝑏 0 = 𝛽 (15)

Step5:Intheboundarylayer,𝑦ismuchsmallerthan𝑦′because𝑦israpidlyvarying.Sowemayneglect𝑦comparedwith𝑦′.Sotheinnerapproximationto(11)istheconstantcoefficientdifferentialequation 𝜀𝑦!"!! + 𝛼𝑦!"! = 0 (16)Step6:thegeneralsolutionto(16)is 𝑦!" 𝑥 = 𝐶! + 𝐶!𝑒!!"/! (17)Step7:Applying𝑦!" 0 = 𝑦 0 = 𝐴gives 𝑦!" 𝑥 = 𝐴 + 𝐶! 𝑒!!"/! − 1 (18)Step8:Since𝑦!" 𝑥 in(18)variesrapidlywhen𝑥 = 𝑂 𝜀 ,weconcludethattheboundary-layerthickness𝛿isoforder𝜀.Theasymptoticmatchoftheinnerandoutersolutionstakesplacebetweentherightmostedgeoftheinnerregionandtheleftmostedgeoftheouterregion,sayforvaluesof𝑥 = 𝑂 𝜀!/! .Forsuchvaluesof𝑥,

𝑦!" 𝑥 = 𝐴 − 𝐶!

𝑦!"# 𝑥 ~𝑦!"# 0 = 𝐵𝑒! !! ! !"!!

𝜀 → 0+ (19)

Step9:Byequatingthetwoequationsin(19),wehave

𝐶! = 𝐴 − 𝐵𝑒! !! ! !"!! (20)

Step10:Tosummarize,theboundary-layerapproximationis

𝑦!" 𝑥 ~𝐴𝑒!! ! !/! + 𝐵 1− 𝑒!! ! !/! 𝑒

! !! ! !"!! , 𝑥 = 𝑂 𝜀 , 𝜀 → 0+

𝑦!"# 𝑥 ~𝐵𝑒! !! ! !"!! , 0 < 𝑥 ≤ 1, 𝜀 → 0+

(21)

Step11:Wemayproceedfurtherbycombiningtheabovethetwoexpressionsintoasingle,uniformapproximation𝑦!"#$ ,validforall0 ≤ 𝑥 ≤ 1.Asuitableexpressionis 𝑦!"#$ 𝑥 = 𝑦!" 𝑥 + 𝑦!"# 𝑥 − 𝑦!"#$! 𝑥 , (22)

where,𝑦!"#$! 𝑥 = 𝐴 − 𝐶! = 𝐵𝑒! !! ! !"!!

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Step12:Finally,wehave𝑦!"#$ 𝑥 as

𝑦!"#$ 𝑥 = 𝐵𝑒! !! ! !"!! + 𝐴 − 𝐵𝑒

! !! ! !"!! 𝑒!! ! !/! (24)

Exercise1.Wewishtoobtainanapproximatesolutiontotheboundary-valueproblem

𝜀𝑦!! 𝑥 + 1+ 𝑥 𝑦! 𝑥 + 𝑦 𝑥 = 0 0 ≤ 𝑥 ≤ 1𝑦 0 = 1 𝑦 1 = 1

(25)

Solution:𝑎 𝑥 = 1+ 𝑥and𝑏 𝑥 = 1.Thesolutionto(25)is 𝑦!"#$ 𝑥 = !

!!!− 𝑒!!/! (26)

Thefollowingfiguresshowcomparisonswiththeexactsolutionsfordifferentvaluesof𝜀.Itcanbeseenthattheagreementbecomesbetteras𝜀becomessmaller.

x0 0.2 0.4 0.6 0.8 1

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.80 = 0.05

exactyunif

x0 0.2 0.4 0.6 0.8 1

y

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

20 = 0.01

exactyunif

x0 0.2 0.4 0.6 0.8 1

y

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

20 = 0.001

exactyunif