CYLINDRICAL POLAR SYMMETRY

THE HEAT EQUATION LAPLACE's EQUATION

The Laplacian in 3D in cylindricalIpolercoordinates is

D 8 as tf fr 7 t oo th

iI

In 2D Poler

D ffrfr2 t HoRuhThepointhereisthat if youused separation

of

variables on thesecylindrically synetricproblems

you donot obtain a SEPARATIONofVARIABLESsolutionThere are avariety of differentcoordinatesystems thatlead to separability 1Depending on PDE and B C

x Heat on a Dish strictlyradialsymmetry

let us aCr tPDE 3 du s v f Kur r

to o02 r ab

v herr t trU r

fi

I C u tr O ft Ocr e b

SeparationofVaiables Solutionlet us 4 It 4k

substitute into PDE B c

ME f 149 4 Gr t fCf or

to 9rrtqt rs.li separable

B 4 s tiny Rrr t fCf th4 0Cllr b so

Cllr o isbounded

D hasHe familiar exponentially decayingsolution 4nexpth.int

B Has the solution

far AJolter t BYollerB is a S L problem with weight w r

Ruh Jo is Bessel function order 0 1st landYo is Bessel fucku order0 2ndland

I yx n f O

v

rwsxe JnIx n O I are boundedJok EI 9 Juco son o

HI Yn one unbounded atx O

Since we require war t bonded

including afr O then 13 0

Cf A Jolterso

Apply Brc at r b

Cllr b A Jockb so

knbsdnn.badu are Heteros of JolxI Jolx

VAN xd K Lykn Ibn n 1,2

ucr.tt EanenkitJocknrAs I

Apply I Co

Wr t o ft Ean JocknrNsl

To find an

thHipf B S of by r 51km r andintegrate

rbfinJolkmr dr

A

an Chem Jochmr r d rN s 1

Since Jo1km7Jm are orthogonal wrt

r b

am frfatJoKmr d rm o th s l 2 u

2ne f JilkmDdr a numberLAPLACE'S EQUATION WITH RADIAL and7

DEPENDENCE

I

ex FindSteadyTemperature Distributionu see fr z

plHt stoked

u attheoriginbonded

uir.es sou fr o D

PDE D w s O s Un t f U r t Uzz O

f

it iSeparate ofvariablesU Htt R r andsubstitute into PDE B C

PDE Rrr t fRr tfAzz I a instant

Thechoiceofsignofthecustatleeds to Shin 2

that is

Haz tilt OHeo HCl so

r2RrrtrRrtrfrRs0Tlesoluhwto

HnssinmgIn.bz

E n 1,2

ThesolutmetBO

Rec Jo r tEYofn

sinceUkDisbanded CEOAs

u Eansinn Jotterhsi

Apply B C o

u ca z s To2 lit ansinn Jo misn s I

AnsFozH z sinneddz

Qin s Jo Hf a number

If we perform theintegral weget

an s 4 1 l C D nd 2Quel n3its

ex Circular Annulus SteadyTemperature Tlro

Poe FT s 3 tf It soB e Tlr o f lo TM O f lo

ay

EloX

f lo

T Rcr H O substitute intoPDE BC

R H t fR'HttpRH O

t's t's so

qq.HR

t rR't Yg hi lduiagnessl.no

r R tr R ER 0 Euler EgnatiaH t WH O

hotBessel

The solution for the Rcr equation

Rns Anr't Bnr n n 40Ro hot Bologr n O

Hn lots CnasnotBnsinnOHolo s CotDoo

Tho otbologr thot dologr OA

ECanrntbnr.nl Innnfhere Ken an integer

because Resolutinisperiodicino

similarly periodicity dictates thatI a do oi Tlr O saotbologr

A

anrntbnr.tn wsnOh I

CD

cnrntdnr.tn sinn0Ns

Apply B C

fplots aotbologrA

anrntbnr.tn wsnO1,2 1,2h I

Coo

cnn.n.ztdnr.in sinn0Ms

i Gotbology do d0

anr itbnrii f f cosnodo

cnn.itbnrii.it if do7sinnodOA

Sipser s O

ra s a

bosbn dn soto avoid a singular solution

In this case

TT O r sa

For specificity e.gTle O E f lo 3 cos20

u Tcr o Do t Ia Antosnot BnsnnO

Aos tf Hold0 0 for our case

An too as nodo

Bn flo smnOdO 0 for our case

Ans 0 except n s 2 An 3

Tlr O 3 E cos 20 for our case

Spee ri es his knife

This is an infinite plate with a hole

centered at r O

To avoid unbounded solution

to An Cu s O nd t

84 0Tlr 07 110

Tlr O Dot F BncosnotDnsinno

do Ho do

21

Bns flo comodoO

Dn s t 1 9 SinnOdofld is prescribed all of tleweHicients are known