View of An improved CVBEM for plane hydrodynamics

À{¿\'l'I {li },I¡\'1' t(ì,\ ^I tji ^\: f 'Ii I )'ÂNAI.\'S fi Ni l\,il;; 1¡ 1 O, ^tr,,

jj'f I)fi ^'f ttliOl.l IIÌ l)li T,,r\l)l)lìO-\ ti\L,\'f tON

l,"INÂLYStì NIi,UÍì¡ìl{JtrÐ

I1'l]

t,,t lt'HtìotÌII t)tì l,';I Pn{)xtÈ,Iltît0È[

ll'or¡rc

lfi. N'9.

19f17.

y¡t.149-lS7

A-r\ Ir![PnOY]rÐ C,l¡1],ilÂ[ ]iolì .t,L,r\]iE IIYD.IìOI)YNÀ1TIICS

'l'I'ttr¡i _{l,Ji'l Iì It.À}

((llrr j-Nlpoer)

illlrt:

aitlr of tìrc

pr'(ìscrtl, u.or'li

ix to givc

1,hc tount.labions

o{ a

lrow rìutìr(ìr'ictìl 1,ecìtrricl rrtr

-

^a, ()onrllltrx Varialtle l3oundar-y lìloLnen1,

l'lctltod -

itr dc.tcl'tttitting holontolpltic functions

fulfilling

soìno givcrr ct¡lrclitions rc:la-

tcrl

Lo

tht'

corrclote ])r'ohlrrrììs

uf

pla,rre lrvrh,orlr.rriu¡tic¡r.

I1 is

n'trll lcnott'lr 1,hat tl.rc trtilirt stcp

of

¿l llItrNI consists

irr t,hc

con- ,st,t'u(:liotì

oÏ

^¿¡rt

itttcglnl

r'opre.sentiùtion

joint to tho

ltorirrtlzl,r'"y

lllolllcnr

¿lntl oli

the

con'cspoucliug inbcglzr,I ctl urìlliorì

on

1,hc ltourrrlalr.. fJut

it

the ltrttttttlar'¡'1rr.rtìrlcttL is folrrrulal,t¡tl

in the

langua,gc

of

an unltno\\'rì [ìol()rÌror'-

¡thir,, tuttci,iott, tho

tlilect ust'of

tltcr Clnuch). fornrula giycs,imrncrrliatel¡, an int,cgt'¿ll t'opr'eseniiìtlion ¿tttacltcrl

to t,he

consid(rt'e(l ltountkrr-v _1x'obltrrn r.ìricJr, jrL (l¿utch)' sìrrrr-ttlaÌit)' a,ttr[itio

r, -

lc'âr-ls

_{on tìrt¡}

autolna,t,icall¡' ìlourrd¿rr'.r'.

to an

integr,¿rl equation

- ^s.itlr

l\lolcxrvcr', usiltsl'

a

(ioì'll't.in sysi]on'r oli intorpolaLing funct,iolrs

of

^l,hc

ttttJ(tìotvn

fultcl;iolt, tlte

sclll'irrg

of thc

bountl¿r,rv i¡11¡rglal trcllLation ^(roLìl(i

lto

¡rclfoltrrctl

u'ithout

rìr)'zrpl)r'oxirtrtl;ion o1'thc boundzrr-v or'¿ìn)'nurure-

licnl

t¡uir.rh'¿r,1,nte.

lrtì1 /(;) lte ^atr

liolotuor'plti<:

fulrcl,ion into tho

sirnple colrncctcrl tlorrtuitt D 1,lie outsitlo

of

¿r lcc'tifialllc..1 ortlarì cnt'yo C.

It is

assurnctl

that /(:) ^is

cotttintultts

on

C tvliclc

it is

hlr<livtr

cililrcl its

roa.l

or its

irr-r¿lgintì,ry

palt, or' ¿l cornl¡i tr¿i,tiorr

of thc

^tr.r.t¡.

Lct

_./(ø)

-

^*o

I !t' ) l-

^o!:

-l

bcr

the

tlcvoloptnt-'nt

of tltc

zz"

cotLsi

tlcltltl lluttt'tiolt il

l,hc noighbourhootl

of infirrit"v.

_'Ilho¡r

thc

Cj¿i,r¿clty"

firrtnulil

for'/(a) a,t'Lr[ l.he

tlonrailr l)[I()

could

l¡c

w,rittcn âs

is

lnÌo$'tr, .l'Gt)

-

_2nt

E /tt'^ ^{rl( -l} ø0,

^{z e}

^tt

-)

lllhis lolnruln,

n,lrich

is jn

_{['¿lct thcr}

jnl,cglal

r'o]ìr'orJerrLâ,tion aftachetl tro

tho

^ploposctl boutrtlar'¡' ploblc'trt, allciws

tts to

tlct,eltninc./(a) orrco tho

1:¡f111's ol'.lT-() on

lhc

bountlar'¡'C zìl'cr

linoryl.

[]u1, thcsevalucs of

./(() rn 0

alsrr s:r,tist.r'2l singulâr int,c:¡¡r'a,l etluation ol¡ta,inetl

-

pcrrforurirìs'z

+ (*

e

6 in

thci irbovo

fornula -

^{antl whic,h.} l'oprcsonts

thc bountla,r)'intoglal

equa,-

'hiol of

1,hc

il'oct'tltto (O\rlll¡lMf. Urrfoltrurately, thc

llroblrrrtr oT solvirrg

TITUS PE1IRILÀ 2

150 AN IMPROVED ^{LCVBEM 1i-r 1}

t¡is inlcg¡al

eqìraLioll cven nLltrl('r'icall;'

is

rrsuall¡'a,

vcr)'rlifficull

one.

In

)âss

tlÌe

sholtcotnings conrcctc'cl

l'itlr.

ata

of oul

problem

in thc

^algorithtn

.,îi, l',

_{J' !;f} ^,

:), :' T:i ì,;"1"ïi ^ili"ä;

the culvo

^C

into

bounclaly elernents C¡

(j:1,ù,

s,het't) C7

is the

sirnple

arc linking the

1loin1,s z¡-y àrrLI z¡' T.,ct non, 1,hc

follos'ing

approximation

i () ^of the ttnknon'n function

.f(()

be

clefinecl b¡'

t7

i(0 - I

^;1 .f¡L¡(C), n'hctc

l¡: l@),ancl the

^functions

Ir(() are the ^inter- polating

Triùgl'Íùn.Çe

i.e. (f21, fB l),

functions constt'uctetl on each ¿ìrc

lespectivel¡',

-\- þj-1 l'or'(e

C;

?j * Ø¡'t

: ^/Jr+r lirr'(eC;*,

aJ eJ+1

0

^othelu'isc

\vo thcn got for the

abor.e cauchy

intcgral ^{up to} tlie

adclitional

constant

^øo

- ^thc

apploxirnation

/*(ø) : i ^1,i,12¡,

^rvhcre

Lr(C).

dC-

Il¡, solying ^{i1 thc}

clzta oll

the

bo¡Ltrrlar¡' p¡obletn,. we gct

tho looliccl ⁿ i(0 ^of

the function.f(Ç-ant1,

implicitly,

^ø-iø

,Cauch¡,'¡s

f

^ioí-,'oÎ

^thc

proposetl bounclar'¡' pl'o¡lcrrn

in all the points

oT

tlie

^dom¿:lin -lJ.

concerning

the

coefficients

1",'' for

/t:

* j thel'could

be

clilcctl¡'

^s¿l-

"crùatctl frorn thc

expression

of i',12¡ using thc' equalit¡' Iim(a-ør,)x lr(a -

^zr)

:0 in the

case

of

t't

- j -1

^{or' /r;}

- j +1'

^{1¡or ¡'}

Ï¡i

"r

^1ve

,have

[2], [3]

it,(r) - *l

^¿¡ _- ^{¿j-l ;-} -.i+t -t-

,

^for .

(

Ê. C _tt+.t. ^,

lrr

1

1-É

ancl

rvher:c

one

choosr¡s

thc ^principal

detertrlin¿-¡tiol'l

fol the

complcx

logalithtu. "

Lel, us

nolv

supllose 1,ha1,

the function /+(z) is

cvaluated

in all

the

norlal points

z¡(h

-i., ^{u,) i.e. /*(zr)} : i

T,n,O,,),lt

-

1, ru.Consictelilg

then

atr approxiniation

of tbe

tr-clualil"y

just

tr'r'ittenr precisel¡'

.f o(tt,,,

* ^ia,) : \, j:r

^.f

iLir,l¡:- ^1¡

^{tt ¡}

It'Ìrere

L¡t :

Ï,,1ør¡

: Mr¡ I

^{ilY,,r, \t'o}

^{alo letl to}

^the

frlllowilg

rea'l sysl,em

af

Ztt, equations

it

2tt, unknorvns

,r,,- \

^M,,¡tt'i

- I

^N,,,r,

+

_?i-?irt J^^l

^* ,, _ln ^.

_Z

^-- _{';,,_}

_2¡

_{| lrr} ++,1,

_?-ìi-t,

tf\

rr-e g^¡L irrrrrrqrliattrl J, L,;

- -!- ^tlr

^f

-h :t- l,

rvlt,'r','ontr Ial<t's f llcsir,lnr:

" 2ni \ z, . :.,-,

⁾

principai rlclelrninatiotr for

logaritltrn.

\\'r-'note that thc

^solvirrg

^{of th} plt'x

r.aliables

(O\rBDlI)

clitl

.or' :urr'

lrurnclic¿l

quaclratruc tl'r¿-¡t conncctctl

l'itlr

tl'tt¡

int

l lre

ìlorrntlrlr'.

¡cìt us Lo\\'suppose

tìlat tho Joldalr loctifiable

^{cur.Ye C has}

in z, e9

:ârì. zrrngulal

point,

1lrôciselv

tho

cout-Ltolclochu'ise

olieutetl

anglt'

of

scrni-

i^r,g"ñt*, in this ^pöint

^bciirg'

n

_¡ræ

rvith _{-1_}

<_

'-t

5.0. It ^is

^{htLowtr tha1}

iìi"ï^.."ir¡'intcgrnl ^{still cxiits in that}

^{case ancl thc} bc'h.avioul of

thc

frLnc-

tiol _/(ø) ^iìr the'neighboulhootl of thc

angular ^{poin'1, ør,}

rl'ill bc given by

./(z)

-

"[(zr)

-

^(ø

-

^2,,)-1-l

tt(z), l-helc¡

^tt(zr)

* ^0r'

^\r'hich

'irnplies for

the

tll

^'¡1

-

tleriyatiye 'v a þehayiorlr of the typo 0 lþ -

^{ør)r-v.1, i,c.}

^{this tleli-}

cIz

v:l,l,ir.e becorncrs unì¡onutletl

in

^e,,

ftlt' -L < ^i, <

^0.

. In

thcr

cve¡t of

using

thc ^Iltìr\t in thc valiant Cll

onc

must taliq irrto

account

thc

cxisbcrncti

of tìtis

preciscrl;',

thc

"ltiecen,iso has in

V(z:r). of 1,he

furrction

.f( riefl ^lrotlal

^singulal

point

a,

riì

etl

t

rvc shall intcr'-

polriter

thc

^bY

Lt(Q :

L,(z\

-1

: ^I

2r-i

? - 3¡-l ^D

-

ln--'*

D.

È¡ - ?¡-t Z

- ?j-t ?¡ - 7¡+1 ?i - ^7.¡+t

..)

I

' l-t j:l

lt It

: X

^ù[¡,tt,

l Ir]lr,ur

¡Zt i!¡

3 - ^3¡+t

l'l "

^-'+r

-1--

.f,,

l(.f,,-, !1,,)( ( "'-l'=u, ^rìrr'(ec,,

\ 3r,_l - ^27, /

f((\ -

l, I

^(.f

^o*t -

^lo) þ p+l _"1)

752 Ì:IITUS PET\RIILÄ ,t 5 .4N IMPROVED ,CVBEM _{1 53ì}

Conscquontlv,

in

t,hc zlpploximation otl tht¡ s-holo

C,./'(C):

_j:t

\ ^.fil'i, thc

expressjons

of

i,he

pol¡'lornials

^/,i(

() arc iilctrtical to thosc

zill'ca,tt.v

rvritt,t¡ri a,bove ^To'-

j * þ - t,

^1t,

^p

^-1

¹

^rvhìle,

^fol

^{tìLe cases}

j - ^p -

^1,

1J,, 'p

-l I ^t-e

^sh¿r,ll

^not\'

^]ta,\'t',

2.

I.l.tt lls lÌo\\¡ cousitler a, plane, incornpt'cssiblc

ltotontial

^inyiscirl

fllid floÌ'. It is

woll

klrcx'tt

lhal,

it is alll'¿r's

ptxsiìrlo _to

joirr

_{1,o suclt a}

flol'

alr anal5't'ic function _.f(z)

-

^{calletl 1ùo} corr-Lpiox potrxil,iÅt nt

i¡,,

^flou,

-

^ìr-¡o.so

k-non'Ìcrlgo

is entilely equiva,lont n'ilh tlio

cortrplete cletorrnìlratiorr of

thc

flon'.

.

L'ollt'olscl¡',

ant

ltolornolpìri

c function in

a

gìven

durnaitr crt¡lcl bc interpÌcltercì.

ils a comlilcx potoutial

o1 ¿r,

planc

r'ricornltr.cssiìrlc potcrrtia,l

inviscitl

flo_rvltendingadclitiou

of

somc logzr,rithrnic i;orrrÑ

(multif6irn f¡trà- tions) in tho

case oll rnultipl¡'-connector.l clomains.

If

Ne consitlcr onh, a simplc connccterl dornain

likc thc

outside ot :r,lt

obst¿rclo

[C)_ ^thc

cornplcx_potonti:r,l

of a fluitl florv I'it]r t¡e q¡alitiàs

mcntionecl above,

nlounrl

i,tre obst ,cle (C)

-

^1,his

^{lgill bc an}

arialytical

function_in cvery finite point,

ha;r,inu

jn _tho

nciq.ìrbourhood

of

ir,:Éinity

thc

clcvcloprnont

ÍG,Ð

^{= n)æi}

f l. ^{ln e}

i- rrn

¡ ^-4t.¡-42.

a

Zi'i,l J Ð2

\\rcrlenotcd bclt¡

br.

?{)æ:,Iirn t]"f-rn.,

corn¡rlc.x

velocil,y of thc fluitl

¿1,

iz'

,æ

_tlz

gleat

distanccs, ll.y

l'a, r'calfunction

_{of tirne} (rvhich ct¡ukl lte a er)nsl,arrt

ol

evtln.z,cro) calltxl 1,he circiulation of tho

flon'and

n4ricll lepr.trsents tþe

¡rulti-

forrnit-}' ,Ìr.eliorl ^of

-lhc

^lea_Ì.

palt of tl

t', cornplex pol,ontiâl .f, ^arrd

b¡, f,

1,he

1,imo u'hich could crxplicitl¡r

¿lppc?Ìr',

thc Tlilr' Ìreing'ilren a

noñst¿r,-

tiottal'tr

or.lc.

aluos of

thc

frurction ./(e, ^t)

the

contour'

0.

Srrpposine 'oLotl'ar¡sltrbiorr

in

f

llc

rlra¡i

lÌ:iìi#,,iåliå'i,, of the

prcfile,

Cis

þir.: ty -,rt,):t -l -l _{tr, 1} yr) -l ar,bitra'y functjonof tirn.l,,.

\Ve'ernâ,Ì,k

2

tLra't

if

ilrstcatl oll

tho

conrplcx potontia,l _,[(2, t) we u'ou]ù constr,rlct i,Jr<¡ corÌi_-

lrltrx

^{vrrìot,i 1,1' ro(;',}

Ð ' ^!'l-,

_tIz ^ühi..

^rvill

^l¡e

^¿

lr<lk¡nror.¡:Ìrirr f urrct,ir¡n

jll

_tht¡-

whole mrtsidt'.

of

tho_¡rlofile (C) s'hicJr also inclurles

lhe poi¡ú at

infirú,b¡,..

$ ^tfe

ncighbour'ìtootl of

this point

the furrctiott to(z1f) has :r, ikxoloprnetì.û

of

tli.e typt:

tu(z;t) -?r,æJ -1- ' +-"' _{Znl ã zz} ^{+-¿"- -l} ,..

þ3

It ìs

_,jrrsú

tltis lcgulaÌity of the

cornplex vcloÚity

that

(leteunines _us tor l.rse

the

abovo dt'-v-cloped

ovllDlvr fol lhis

l'unction

u(ø; t)

and

lrot foi

1,ho <tornlrlex

potential

a,s ìye rvoulcl ¡¿1r1y l¡sslt

tcrlpted tò.'

.

-_-Cgncer'ning tlr,tr bounrlary conrli ions

in thc

points

of ths

cotrtor,u O,

iú'

rvill bc u'l'ittelr fbr thc fulrction

tu(t

; t)

untlel

ille

fornr ,

L, '(() -

t

I-

\ ^¿¿1,-2

?t-t - 2r _z Tot'

( e0n-r,

ll

P \

- -F I

for

^r,

elr,,

othulu'isrt ,

for' (

e01r,

for' (

e

0r*r,

ollìeru'isc

L,(Q -

Lr*r(() :

t Y . \T

^u

t-lz"tt

_{l^, -,,}

I

^j

\ ^P-¡r-1 - þl I

ã.p+t * _à.tt+Z 0

\ ^& n+2

lbr' (

e 0r,.rzr

for' (

^{e C7,r.1r}

othelrviscr

(

lJ,

ù,_,(,)-,|o1,::;,u,:-,-ii-_;..,'-Å,l'/t-u(*)}'

.dl

once

we

also obt¿irr ï¡t¡1

- 7¡

n

_þ ,Il'¡1¡_¡,

i'r,(r): ;;rl

^{2 3 -} _- ^{è¡¡+ttlr-t} ?p-'t - Ìn

)-',,-'(=:i,')l'

},Jr;) : 1 l-Z-:t:t--

_'¿nr

^¡r -i-

^:'o+?'

_-' ¹

^-þ

^/rlr,,-*

13, ^1.1 - E7¡2 à' i,.ptt

I

I

where

^Ì,'u(ù

- j ,:-- ^{¡lt* rvhile firr tlìie otùers}

^'1,,@)

li#

_F,,

_{F -,1,} f + ^{1) tho}

ntr:ttuä.y establìslrctl expl'tìssiolrs al'e ¡i1,i}Ì

valitl.

'¡ l'his inledi;àl conl<l þe nnnlylically pu'fornotl iÊ ç : ¡iillt (a r¿rtirll¡al t.¡¡t¡ç¡- ¡ !r[la

nr < n) [iì].

AN IMPROVED ICVBEM 156.

2a) if

^1,,1

is thc cilculation

of tho basic flon',

this is cqual,o

,Ð,

^{l-,, i.c.}

rvith the

surn

of the

circulatioris

of all thc

given singularities

of

the

flol'.

O'oncelnin¡¡

the unhlto¡'n. function

to(ø)

- ^the

^cornplcx

^r.elocity

^of-

the lesultant fÌôrv

olltainecl

b)' the

abovc-mentionecl superposition

-

^i1,

s'ill be looketl for in a

class

óf functions

(D)

satisfying the

propertics :

1b) thc¡'

are holorno¡phic functions

in the domain , : Dr\(C)?

except

the

same

points

_{z,l

,:rn ^ri'hich alethe

singulal points

of the

same nal,ule as

for ut;(z); at infinity, their

behaviour

is

irlcnl,ical

with that

^{o1 wn(z)}

i.c. Um tu(z)

:

ro(æ)

:

wn(æ);

izl+æ

2

b) in thc

neighbourhootl

of thc trailirlg

^{eclge ar,}

: ((p.)

e C, rvhcle the' semi-tangents anglc

is

æ

-

^{p?ïr we have}

to(z)

:

(z

- zr)'!''

^{glz), u@r,)}

^{+ o;}

3

b) iri

the points of

thc

curve C, the

furctions

!)(CrcD belons ^1,o the class 11'Fi.e. theyäre Hölclelian functions on C except thc angnlar point ø,

:

((p)'

in

rvhose nc'ighbourhood one has

u(e(Ð) :

^{20*( (( p))} _É

l-((9)

-

^((90)l

^''T

wher,e ^{zo+' e}

IIo in tho

sarnc ncighboulhoocl rvhic]l me¿ìIrs that..eu_*(((9)) is' separ,atel.y Tlöicleliarr on thc upper side and on thc

lot'cr

sitlc of the

plofile irr tho

neigìrboru'hoocl

of z, :

C(9ò) ;

4

b) in the points of the

crit't,e C

they

^{satisT¡', tlxccpt}

thc

angular point.

the follorving ìtounclal]. contlition

^:

there is a l.cal contin[ous fonction

l/(

p)

such

t]rat lor

^e\¡el.y

I

^e

[0, 2")\ìPo] onc

^has

¿ir(((g))

- ^l'(9)

₍₍

^((p) -l Z f ^irn -l icrl((

_þ)

- ^{z.il, t'hclt'}

^{z^}

^e(C)

^{¿¡nrl Z(f),}

fr)

2r,(l), ^co(ü) are the

gir.cn urctions

of

time dotclrnining

^l,lre

rototlanslatio4 of the profilc

(O) _;

5

b) thcy

^Jiulfil

the

ctlualit.v ^w(z)t1ø

: l-s

^rvht¡r'e

tho circulatioll of

the

C

flo-lv

I is

choseu so

that

ono hag

thc

Lxltnthress of the ve.locity

in

ør,,

i.e'i

1 l-,,:f- tt - ) f ru.¿¡(:)d:, ¡ci¡g a sintplc rcctiliablc culvc sttllontttlittg all Lhe sirlgulalitics .zr.

2 _Onc snpposcs LÌrat

À'

cluling thc tìisplacotrcnt of ((J) l'c ltavc (C)cDf, i.c' lhc ^plofile (c) ^tloes not cLoss tìrc points {r,}r=1,orvÌricìr altvn¡'s ìrr:long to thc otrtsitlc o1 ((ì).

3 'fhc singuìalily in:, bcilg rvcak

sa La1

¡¡-1 ^{the intcglal is} conlcl'genL.

754 ^:TITUS PET'R'ILÄ' ⁶

Thcr',.

is

a r.eal

funcl,ion

^{1.( p) so l;ha,t}

for

e\¡cl'-Y p e _10,2æ) s'e ^har'-e

?r'(((p))

: l'(P) :ll] + I +i ^lr f

^ico

^l((p) - erl' ^rvlrerc (: ^((p)

^lho

l(( ^{p) I}

Da, âlnctr.icrr,l ec¡rra1,ion o1

tlre

Jordan

rcctifial¡le

^cul.I,e

9,,:. "

^2æ

-

^per.iotli-

äalfu'rction,

^bou'd.ãä Jnd ,ìLr:i"r¡Ie

in lô, ^Z")

"qo tha1,

erc\ +

⁰

^{a*d ((P)} <

<

^{,4{ u4ren}

^{'l{ is}

^a

^finite

^constant'

l-inall¡r, i,tt"

^possifrle

rnultiformity of the

^funct'ion

-f

^leacls

to

^l'he

ft-Llfilnrent,

ót tne

^cclualit-v

I

_C ^u(2,

L)rlz: f(l)'

.wIìeìI,O

r.(i) is tllo

^Ú,tt, pr.iori'ì giverr

circrrlation. Ir¡ tlre

^casc

^{of tbe}

^protiles

rviLlr

:r,' angrrln' l,,,ii,i'ii,'

^zo'=

^{zr. c,. *;ii't'.'}

I,lre st,rrti-langcnÛs a'ngle ^is equal

to

rr

- ^u,- (Ïì'"*'p 2ìi,t]'"'1.,"1'o*iu..r, ^of the

cotnplex r'elocit)'

I rcuttil't's

-

^to

ttvoitltltc

ttnllottnil-

alioå

such

llLtl,

_[-l

I f

⁼

^L

^',1

⁺

itcl,clminecl t'oefficicnls

f', ^Jl[,

^-ù',

ïlor'.

Iìetaking,for.l,lresalreofsim¡llicitv,thecaseofotrl¡,oneprofile (C), the

^pltlposed problorn can be

formulatetl

as follorvs'

Lcib

thc

^{funcì,iun 'mn(z\ be}

givtn, tltc ^contplel ¡ctolitf ^{o[ the}

^basio

11o¡r, åì

üunction ."f,i"ft bt'ttlg*'to a

^{class (ø)}

^of functions haviirg

^the propertics ^:

1

a) rìrey alc

rrolornor' _¡rrrT

c

^f

unctt'^ì.,äJi""it*,,

Ï Jl'"llÏ:, ìTffrii'äi

ich ^ulal

^Points

, f"

^{has talren}

be ^the

^trimit

Iìm

lf¡¡da) rvhich obviously exists and

is finite'

^ì

lz'¿æ

AN I]MPROVED CVBEM

ri'hic} s'ill

be completecl

in this

case

by the

complex equation

I

757

\)'r'

^'L

L¡(Od(: I or, equivalently by

ilfC

!

^ø¡Re

pr(0

^<1(

I ^a,Inr _\ ^Jl,(0rl( : I FtJJ

L¡(ÇdC j:1

E

t+¡Trn c

L¡(Qd(:

_j:1

f,

^o¡Re

these

last

trvo real equations allow

to

determine an

ulique

solution of the allot e homogenous system which includes also tbe clata

on

C. This unique

solution

once introclucecl

in the integral

representation

of the

problem (i.c.

in our

case

the

Cauchy formula) Ieads

to the

cornplete clel,ermination of

the

complex

vclocity in

every

point

of the dornain of the florv.

.I.he existence and

the

uniclueness

of the solution of the

^proposecl problern (of

the function

^æ(ø) lookecl

for

uncler

the

above representation) are

not

consiclerecl

here, they

^being sl,udied

eariier

_[1].

Regarrling

the singularities r¡ø,.),_ç.of the fluicl flon'

aclmitting

that they

^are vortices (and so

l*+0)

the absence of external forcesirnplies the

fulfilling of

a so callecl "treeFdo'm conclition"

for them

_[4], i.e.

tf, + I f

^irrr

^.f

^icoe,.

- ^{Iirn I}

^u@\

-l 't'

^.1,

^r :t,

^q.

tlt ^,-,,. I z_2,

^_1,

trlncler these circonstances,

the

displacement of

the profile

C and of

thevorticer:

{zr},,:rnrvith colresponding

circulations are

correlatcct

by

the above adclitional relations.

REFERJ]NCES

[1] ^{P e} t l i I á, "l'., ùI<Llltetnatical ntoclels itt plane hydrodynanics (in lìomanian), Pubtishing llouse oI the Romanian Acacletny, Bucalest, 1981,

[2] ^{H t'o m a d k a} II. ^T., \'., Thc cotnplcr. uatiable ltountlarg clen¡cnt ntcthod, Splilger-Vellag, _- ì3ellin, 1984.

[3] Ilonrentco\¡scIìi, D., Cocola, D., nIãgureânu, Iì., Sotne deuelopnrcnts of the C\¡BEXI. A¡tpliccttiotr lo Ilrc ¡niutl bouttrlat'y-uctlue pLoblem for the Laplcce equaliort,

- INCIìES'I-I3ucarest, pleplint selies in mathcmatics no. 13/1981.

[a] ^{C o n c lr c t.,} G., t.ct co¡tditíon de Joukowsky en mouuenlenls non stelionnair.cs, Faculté clcs Scienccs clc r\Ionl.pelÌicr, Sccrótariat ctes NInthématiques, pubìication No. 74/1969-70.

[5] ^P ctrilãr, 1'., G hcorglìil,C.,F'ittileelunenlnclhods and Àpplicalions (inRomanian), Pnblishiug Ilousc o1 Lhe Romanian Acarlemy of Scienccs, Ilucarest, 1987.

[6] Ilrcbbic, C.4.,'l.elles, J. C. F., Wrobel, L. C., Bounclarg elemenllechniqucs.

ll'lrcory and Ap¡tliccrlion in EIIgincet'ing, Splinger Vellag, )3er)itr, I-lciclelbelg, r.\crv York, 'I'ol<yo, _1984.

lìoceived 21.V.1987 Uniuusily of Cht.i-Nctpoctt

F acully ol' fuI atlrcmalics R-3400 CIuj-Napoca

Romûnía

5 - c. ¹⁶²²

15ti TI'I'US P.ÉT.R,ILA,

[4], ^I' : !, 'l + ^Üt

''n1,)- À/

'rt,,

1\41ۓl'c 1;he cocrffioictrts

-t, ^il{, lÍ

^al'c

girnen

u,ith thc

obsl,aclc, (C).

frot us novt¡ considcr,thc funcl,ion +o(zJ

-

^tuq(z\.

ilhis function

l(Ìlo$¡n together

s'ibh

r,o(.ø) being hok-rmorphic

in the

outside

of (C) tho

Cauelry

foirnrrla is valirl in

-D ^ar-rd l\¡c irnlnecli:ltel.v have.

L0@\

o-l

-

tlø

f

8

LU(E\

-

^zun(l\

- t:, I

\ ï':n ^{dp tbr' .e D}

o

c

+

^t

2ni

l'ina,rlyr itr ottlctÍ;o

ustl blrc llcltmdar)r

"utt,ttrtun orr C

^{1\'e pcrfornì}

'ç

- C:

^{((Ê*) e C}

\ iar|

^{nncl ño}

^w0

^get,

2fr

?,,(((p{,)) _-

rpu(((j*)) ;, f il#¡fl*,t,,0-,

,, i rr,,(((p)) I ((p),,uo

+,"i lJ'lp¡ - ^{q1'u*, "r}

lfhis

is l,hc bountla*y

iut,rgr^l

cquai;ion u,hiclt

rvill

lrc uscd for'1;hc cflìccl,ive ,constr,L1ction

of

^¿¡rL

äplr"oiiurrtù¡e

^solution

by

(1VI3ENI. CorLsirleling thorl

;,;i;-fuñ;1

no¿at points ?otzt¡ ..;tÈp-1tø71,Ø1t.r1t

...t2,?30

ont.he culr'e

tt ãiti.Uõt,fro ^{$'ith thc}

systöin-ôf thô pieceu'isé irrtcrpolating' Iragratrge.func-

;ì;ñ;;i

^{¿tch a¡c} Cj (sj;stcrn \\,lìich tãlics

inln

¿r,ccorint

tho

Ì¡ctr¿¡vioul

in tlte

ncighboutltootl clf ^ør,) rve oarl

wlite

'í¡,(((P))

-zuo(((li)) + 5 ^{rr, -

't/1,'¡)[,¡,

tvltole

';(((P)) for j * ? - ^l

^{, þ,}

p -l1 ^{ha'c. llr*}

oxl)t,¿ssiorìs spccil,i_crl irr thc firsl,

part

o1

this

pâpcl'.wllilo 'lor'

i : , * I, ^{þ, i, +} I

^{tr111'¡; could}

^be

olrtainecl

frorn those

_Þreviously

tt'ritttitr by lo¡rlat,irrg

---.,

I s'ilh l;

Usirrg t,horr the [j(r¡(]r,al calt,ulns lL,lrcztt1.1. ptrforrnotl

for irle¡

^{'ù1ut -1,¡^,,}

il

^w(z¡) ---'wn(ør\

:

_,i'r

- ^{it'* anrl}

^-I',,r

^--

^{1t,,,¡ 1-}

iÀrr¡ wc ¿.o lotl

zr'gain to

¡ l¡¡¡ ¡'¡ra,l a,lgcrbraic ltornogeuclo Lls s)rstollr

{ _I,y(q(p)) - ur¡¡(((p)) e 1l# antl (. bcing a scctionally smootlt cur'\'c, tlie intcgral of ,Uauchy tYPc cxisl"s.

0 '¡hs _¡llcmclj fottnrtlas ate still valid'

It ,t

iWtj'tlj

f |

_j:l ^Àrr,ø,

t

æ¡,:

_j.-I

I

,u¿:-E

n

l{r,¡a¡! _\

^N r¡r'¡o