Метод движения граничных точек и мажорантных областей в теории фильтрации
Начало исследований решений краевых задач в зависимости от изменения области было положено в работах М. В. Келдыша и М. А. Лаврентьева. М. А. Лаврентьевым при помощи выделения главных линейных частей относительно малого параметра функций, дающих отображение некоторых простейших областей на област...
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| Дата: | 1953 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7753 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512742145785856 |
|---|---|
| author | Положий, Г. Н. Положий, Г. Н. |
| author_facet | Положий, Г. Н. Положий, Г. Н. |
| author_sort | Положий, Г. Н. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:11:31Z |
| description | Начало исследований решений краевых задач в зависимости от изменения области было положено в работах М. В. Келдыша и М. А. Лаврентьева. М. А. Лаврентьевым при помощи выделения главных линейных частей относительно малого параметра функций, дающих отображение некоторых простейших областей на области, им близкие, были получены качественные выводы о поведении расхода жидкости, выходных скоростей и противодавлений при удлинении шпунтов в случае фильтрации под плоским флютбетом.
В настоящей работе при помощи теоремы о движении граничных точек, установленной нами ранее, исследуется вопрос об изменениях расхода жидкости, выходных скоростей и противодавлений, или, какими будем говорить, интегральных характеристик, в зависимости от всевозможных изменений произвольной области фильтрации. На основе этого дается метод определения указанных интегральных характеристик фильтрационных потоков, основная идея которого заключается в оценке искомых величин, связанных с решением той или иной краевой задачи, сравнением с соответствующими величинами решений той же краевой задачи для таких областей, для которых эти решения известны или находятся просто. Предлагаемый метод применяется к рассмотрению ряда примеров и задач теории фильтрации. |
| first_indexed | 2026-03-24T03:33:37Z |
| format | Article |
| fulltext |
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o6pa3bi TO'Ie K B", C"). ripH oTo6paH<eHHH Q Ha T{' rrpH H e nOJI.BmKHbTX
TO'!K3X b H a TO'IKa b- llp'IIT.HrHBa!Olll.35!, TO'!Ka a- OTT3JIKHB310lli.3H; ,!I.py
rHX HerroJI.BH)I<HbiX TOII,eK HeT, TaK KaK T1" :) Q. CJie;~:oBaTeJibHO, npH nepBoli
HOpMHpOBKe TO'IK3 b 6yJI.eT rrpHTHrHBa!Oili,CH, BCe TO'IKH OTpC3KOB [ab] H [b b"]
npH6Jm3HTC.H K TO'IKe b. To,mo TaK )!<e, oTo6pa)KaH Q
Ha T," npH HCIIOJI.BmKHbiX TOIIKax c H d, y6e)KJI.aeMCH,
'ITO rrpH BTOpOH HOpMHpOBKe TO'IKa C 6y.n.eT npHTHrH-
Ba!Olli.CH, BCC TO'IKH OTpC3KOB [cd] H [cc"] llpH6JIT!35ITCH b b"
C" C
K TO'!Ke c. 3To 3 Ha'!HT, 'ITO JI.aBJierme p Ha JIHHHH BE" -KH;c-.:e-,_- _-,;c-+"--u
yneJIH'IHnaeTCH, Ha JIHIIHII CC" yMeHbiii aeTCH. BeKTop
... dw
CKOpOCTH V = ·dz B6.1H3H TO'IeK B H C 'H paCXO;J: :tKH,!I.-
KOCTH '!epe3 CB5!3Hb!C '13CTll riOTCH!J.H3JlbHblX JIHHHH,
Hl\teiomue CBOHM KOHIJ.OM TOIIKY B H C, yMeHbiiiaJOTC.H.
l.J:T06b! IIO.TIY'!HTb OT06pa:tKCIII-le rrpH TpCTbCH HOp-
b' '
' ,
a' ,
c'
T
a'-----d-,1 -iQ
PHc. 2.
MHpOBKe, OT06pa3HM BH3'13JIC fd Ha npH M oyrOJibHHK T{, 5IBJI5!10lli.Hii:c5! H3JIO
)KeHHCM Q H T{', np'H Heno,n:BH)!<HbiX TC,'IKax a, c, d. IlpH 3TOM TO'IKH a
H d 6y.n.yT rrpHTHrHBaJOlli.HMH, TaK KaK T{ :) Q. Ilpn oTo6pa:tKeHHH T{ Ha
1. , •
I TipH HCllO,!I.BH:tKHOH TO'IKC a 3T3 TO'IKa 6y.n.eT npHTHrHBaJOIJJ;ei{, T3K KaK
K Heir ,LIOJI)!<Hbl 1IpH6JIH3HTbC5I C pa3JIH'!Hb! X CTOpOH TO'!K3 C II o6pa3 TO'IKH a.
CJie.n.onaTeJibHO, npH Tpe_T~eii HopMHpOBKe TO'IKa a 6y.n.eT rrpHT.HrHBaJOmeH:
dw
n BeKTop CKopoCTH v = dz Ha JIHHHH CC" yMeHhiiiaercH. 4To6bi noJiy'IHTb
OT06pa:!KeHHe rrpH t.IeTBepTOH HOpMHpOBKe, OT06pa3HM Q Ha T{ IIpH HC110,U
BH.IKHbiX TOtiKax a, b, a. 3,necb TOliKH a .n a - 11pHTHf"HBmomne. DpH
OTo6pa:~KeHHH Tt' Ha T1" TOliKa a TaJOKe 6y,neT 11pHTHf"HBmome H:, TaK KaK
K ueii: ,noJDKHbi 11pH6JIH3 HTbCH TOtiKa b H o6pa3 TOliKH d. C.le,noBaTCJlhHO, 11pH
'IeTB~e_Toi:i: HOpMHpOBKe TOliKa a 6y,neT 11pHTHrriBaiOII~eti, H Be KTOp CKOpOCTYI
dw
v = dz Ha JIHHHH BE" yMeHbiiiaeTCH. Tarme )Ke paccy)K_nemm 11prr s,naBJIH7
Bamm JIHHHH TOKa AD 11pHBO,nHT K C11paBeJI:.JJHBOCTH YTB ep)J<JLeimi'"I Teope:-.rbr 3.
3 a Me q a H u e. TeopeMa 3 110 CBodi cpoplvrynHpOBJ<e 110 cpam-IeiiHIO
C Teoper,wii: 2 OTJIH<IaeTCH HCKOTOpOH HC110JIHOTOH, a Hi\leriHO: 3,necb HHli e ro
He rOBOpHTCH 0 110BC,nemm JI:aB.lleHH5I Ha HeH3 MeHJ-IOH JIHHHH TOKa H 0 110BC
}I:eiiHH BeKTOpa CKOpOCTH Ha 110Ter!ll,HaJibHhiX JlHHHHX, B351Th!X B II:eJIOt-1. cho
06'h5ICHHeTCH TeM, liTO 3HaK BapHaiJ:HH ,naBJieHHH p Ha IIeH3MeHIIOi:i: JIHHHH
TOKa 3aBHCHT OT xapaKTepa H3MeHeHHH B,naBJIHBaeMOH JIHHHH TOKa, 3HaK
dlV ~
BapHaiJ:HH BeKTOpa CKOpOCTH V = -dz Ha ITOTeHIJ:HaJibHbiX JIHHH5!X BOJJ113H
KOHII:OB HeH3MeHHOH JJHHHH TOKa 3aBHCHT OT MeCTa B,naBJIHBaHHH H3MeHHeMoi-i
JJHHHH TOKa .
2. 0 6 JJ a C T b cp H JJ b T p a II: H H, 0 f" p a H H l! e H H a H ,n By ~I 5I 11 0-
TeHIJ:HaJJbHbll\IH JIHHHHMH H OJI:Hoii JIUHnerr TOKa. Cxc:~ra
Tarmi1 o6JJacTu G coBrra.na~T .co cxeMoi{, H3 o6p<ni<ei-moi-i Ha puc. I , ecJJH cqu
TaTb, liTO TOliKa D coBna,naeT c TOl!KOH A, o6bJliHO .le)KamefJ na 6ecKone1IHO
cnr. KmnypHbie ycJJOBim IIMeJoT BII,n
u [A8=-kh1=const, zz [Ac= -khs=const, v[8c=const, (2)
r.ne h!, hz, H = h! - hz (H > 0)' Q = 00 - TO :IKe Cal\!Oe, 'ITO H B I. ro.no
rpa¢ T KOMI1JJeKcHoro 110TeHII:HaJia w = u + iv HMeeT BII,n 11oJiy11o.nocbr,
yKa3aHHoii: Ha puc. 3. DycTb B ,nanbHeH:rneM, KaK H B I, BeJIIIl!HHhi
H = lz1 - h2 H h2 ocTaroTCH HeH3MeHHbiMH. Tor,na 11pii H3MeneJ-IHH o6JiaCTH
rpHJJbTpaU:HII G 3a ct.IeT CMerii:eHHH Juo6oH: H3 TOt.IeK A, B, C, T. e. 11p11 n eH3-
MermoH: o6nacni G, HO 11p11 11epeMeiUihiX TOl!Kax pa3,ne.na rpmmliHbiX 110TeH
II:HaJihHhiX JIHimi:i: 11 JJHHHH TOKa HMeeT MeCTO CJJe,nyJOLUaH TeopeMa.
T eopeMa 1'. flpu. JJAWHbUteHUU /lOTeHlfUa.llbHOll .llUHlltl 3a C<LeT yae
.llU'teHUJZ .llUHua TOJW: a) oaB.IleHue
y
p = - k U- ?'f/ Ha yse.lltt'teHHO{t .llUHU:ll
TOKa y,\teHbUtaercJZ, ec.ll.u y ,HeHbUWeTCJl JWTeH.t{lWJliJHaJZ JlUHttfl AB, u yaeJZu
'lllBaercR, ec.llu y.ltCHIJUtaercfl JWTCHlf/UUibHaJZ ;1./lHUJl AC; 6) BeKTop cKo-
_. dw .., ., v
pocru v = dz Ha He.tt3AteHHou rwcnt y,\teHblllae,uou JWTeH!{tta.llbHou Armuu
YBCJlU'LUBaeTCJl, a Ha HCU3MCHHOU IWTeHlfUa/lbHOU Jl/.lHU!lt U Ha HC.U3.MeHHO{i
•wcru .llUHUU TOJW yAteHbwaercR. flpu y,HeHbllleJut~t JWTeHqua;1bHOt'i. .llllHUU
3a C'teT CAtCUI1eHUJl TO'lK:U paa8e.lla 110TeHtfUaJlbHblX /lUH/lll." a) OaBJleHUe
P = - r_ u-')'l'J Ha .llllHUU TOKa lj.'rteHblltaeTCR, eCUl y,H.eHbUUWTCJl iiCTeN-k .
l{Ua.llbHaJZ JlUHttfl AB, u yae.llU'utBaercJZ, ecAu yAtelibtuaeroz JWTeHtJ,UaAb-
dw
HaJZ AUHUJZ AC; 6) BeKrop cKopocru v = dz Ha HeU3MeHHott rwcrrt
yMeHbtllaeMOLl /10TCHl{UG/lbHOU .llUHUll U 86Jlll3;U ec KOHt{a Ha Jll/H.UU TOKa
yBe.ll.U'Ul6aeTCJl, a Ha HCU3AtCHHOU 'laCTU yae.lltt'lttBae~toU IWTeHtfUa/tbHOtl
JlUHUU U 66Jltt3U ee KO.Ht{a Ha /lUHUU TOKQ yAteHblllaCTCJl.
B caMOM ,Ll.CJIC, nycTb rl- ro,TJ,orpa(IJ B TIJIOCKOCTH H3MCHCHI!Oro Ko:\1TI
.lJCKCHOrO noTei-II.J,HaJia. ITpH yMeHbWCHHH noTCHIJ,HaJibHO H JrmmH AB 3a cqeT
Cl\ICU.J,CHHH TO'IKH B B rroJimKeiwe TOl!KH B' (p11c. 1) paccl\IOTpHM oTo6pa
)KCHHC T Ha T1 rrpH CJICJ!.YIOlli,HX IIOpMHpormax: 1) HCllOJ!.BH)KHbi.e TO'IKH a H C,
2) HCriO,Ll.BHll(Ha5I TOl!KU a OTpC3Ka [ac] , 3) IIC!lOJJ,IHDKIIall TO'IKa a 0Tp C3 K<I
[ab'], 4) Herro,Ll.BmKH<IH TOl!Ka a oTpe3Ka [be]. ITpH rrepnoH: nopMHpOBKC
TOl!Ka C 6y;J,CT 11pHT5II'HBaiOILI.CH, TO<IKa (I - OTTa.•!KH- v ®
naromei'!. ITprr BTopoi-l nopl\rupoBKC TO'!Ka a 6y,n:cT npH
THrHnaror.u:ei'!, a 11p11 TpCTbCH HOpllmpOBKe - OTTaJIKH
B3IOI.li,CH. 9TO 31Hll!HT, 'ITO lL<IBJICIIHC p ILl _:ll_~HHH TOI<a
_. dw
BC )'i\leHhmaeTcH, neKTop cKopocTH v =-= -dz na AB'
yBeJIH'IHBa eTCH, na A C' yMcnhill<lCTCH. lho6bi rroJiy
l!HTb OT06pmKCHHC npH l!CTBCpTOH 1-!0pl\IHpOBKC, OT0-
T
a
Pnc. 3.
6pa3HM T Ha 110JIYI"JOJIOCy T!' , 51BJI5110111,YIOCH HaJIO)KCIHICM T H r,, rrp.H I!C
no,n:BIDKHb!X TO'II<ax a, c, a. ITpn 3TOM TO'!Ka a 6y,J,cT rrpHTHI'HBaiOlli,CH. ITp1r
OT06pa)KCHHH T{ na rl TO'!Ka_ __ (£ TaJOKC 6yJJ.eT npHT5li'I!BaiOill,CH H, CJIC,Ll.Ona-
dw
TC.!IbHO, BCKTop crmpocTH v = dz ua BC yMeHbruaeTCH. TaK )Ke J!.OKa3biBaeTcn
cnpaBeli.JIHBOCTb rrepBoii qacTH T.eopeMbl l' npH yl\rCIIbll.lCHHH noTCIII.I,HaJibiroii
JIHHHH 3a c'!eT cMer.u:eHHH TO'IK'H C. ITycTh Terrepb yMeHbmenHe rroTCHI.I,HaJibiiOH
JIHHHH npoucxo}:I.HT 3a c<reT CMemeHHH TO'IKH A B rroJimKeHHC TOl!KH A'
(pHC. 1). 0To6pa3Hi\I T Ha fiOJJYllOJIOCy r, npn CJICJ!.YIOIUHX HOp!'>mpOBK<LX:
1) I!CriOJ!.BH:iKHbiC TO 'I lUI b H C, 2) HC!lOlJ.Bif)K!IaH TO'!Ka a 0TpC3Ka [ba' ], 3) H C
riO}I,BH)KI!35I TO'!Ka a oTpe3Ka [ac]. ITpH nepBoii HOpMHpOBKe TO'II<a c 6yJJ.CT
rrpHTHrHBaJOUI.CH, TO'!Ka b- OTT3JIKHBiliOLI.I,Ci'!. f1pH BTOpOH HOpMHp0131<C TO'I
Ka a 6y}:I.CT OTTaJIIU!BaJOUI.Cii, a llpH Tpen,eH HOpwmpOBI<C - llpHTHI'HBaiOI.Ilcii.
9To 3Ha'IHT, ~'ITO }:l.aB.Tierme p Ha JJHHHH TOKa BC yMei-IblllaeTCH, BCKTop CKo-
.... dw
pocTH v = dz Ha A' B yBcJm'IHBaeTcH, a na AC yMCHbmaeTCH. TaK)I\ C
YOC)K}:I.aCMC5I B CrrpaBC)(JIHBOCTH BTOpOH l!aCTH TCOpCMbl 1' rrpH Cli!Clli,CHHH
TO'IKH A B llOJ!O)KCHHC TOl!KH C' (pHC. 1) .
ITpn yMeHbmeHHI1 o6JJacTH cpHJ1bTpaHHH G 3a C'!eT rrpoH3BOJibiibiX B;1J113-
JJHBaHHH llOTeHI.I,HaJibHbiX J1H!IHH H JIHHHH TOK3, HO DpH HCH3MCHHbiX TOl!KClX
HX pa:J,n:CJ!Cl fl, B, C HMCIOT 1\!CCTO CJJeJWIOII~HC TCOpCMbl.
T e 0 p e 111 a 2'. flpu aoaa.uwaHUU JlOTCHU,Ua.llbHOii JlUH~IU: a) oaB.IlCHIIC
I' p = - kU-')'!J Ha AUHliU TOKa !JBC.IlU'tuBaeTCR, CC/Ul 60(l6.JlU6aeTCR .llUHUR
AB, u y.MeHbttlaeTcR, ec.llu aoaaAtwanc.n AUHllH AC; 6) aeKrop cKopocru
dw
dz
Ha HCU3AtCHHOll •tacru noreHt{lla.llbHOii /lUHllll
U 66./lliSU ee KOHl{a Ha ./lUHUtl TOKa !.f.\teflbUlaCTCR, a Ha HCrt3~1teHHOU 110TCH
l(lta.ll1JHOU .llUH/Ut U B6.1lU3U ee KOHlfa Ha .1UHllU 1'0Ka JIBC.IlU'tliBaCTCfl.
T e 0 p eM a 3'. flpu 60(l6./lUBGI-Ill1l .llUHlUl TOKa: a) oaaJlC/-1/IC
p = -1 U -)',If Ha HCU3MCHH0tl 1taCTU .U/HUU TOKa C./leBa OT BOa6./lCHHOC:O
!J'UlCTKa (CO CTOpOHIJl Hanopa JIC;li'OKOCTU) !JBC.IlU'tuBaeTCR, a cnpaaa -
dw
!JAtCHbUtaCTCfl,· 6) BCKTO{J CKOfJOCTti V =' - Ha HCU3AtCHHOll ttaCTU .fl,ttHUll
dz
TOKa U B6AU3U ee KOHlfOB Ha nOTCHtfLtaAbHblX A./lHURX lj.MCHbUtaCTCR. PacxoiJ
~ICUOKOCTU Ha BOZKOtl CBf/31-lotl ttaCTU nOTCiil{.ttaAbHOU AUHli,U, 1UhtCIOU{CU CGOU.M
KOHtfO.At OOUH ll3 KOHlf06 A :LtHUU TOKa, !JAteHbUlGCTCR.
B caMOM ,n:eJie, rrycTb r, - ro.n:orpacp II3MeHemroro KOMITJieKCHoro rro
TeHuiiaJia, D- o6pa3 H3MeHemwil: o6JiaCTH cpHJibTpaUHH s rrJIOCKOCTII w.
nycTb B,LI:aBJIHBaeTCH ITOTeHI].HaJibHaH JIHHHH AB, rrpHlJeM qaCTII :noli JIIIHIIII
AA' 'H BB' ocTaiOTCH Heii3MeHI!biMII (pnc. 1). P accMoTpHM OTo6pa)KeHHe D
Ha T1 rrpii CJie.ll,yiOlll,HX 1-!0pMHpOBKax: 1) 1-leiTO.ll,BH)KHble TOlJKH b 11 C, 2) 1-lC
llO.ll,BH:IKHaH TOlJKa a oTpe3Ka [ac], 3) nerro.n:sm1maH TOlJKa a oTpe3Ka [bb' ],
4) Herro,n:BH)KHaH TOlJKa a oTpe3Ka [aa']. DpH rrepsoil: HOpl\mposKe TOlJKa c
OTTaJIKIIBaiOlUaH, TOlJKa b - rrpiiTHrHBaiOlll,aH. Dp11 BTOpoif HOpl\IHpOBKC
TOlJKa a - OTTam<HBaiOlll,aH, a rrpii TpeTbCH 'H qemepTOH HOpl\1HpOBKaX -
rrpHTHrHBa iolll,aH. CJie,n:osaTeJibHO, .n:asJieHne p Ha .TJm-nm TOKa BC yseJIHlJH-
-+ dw
saeTCH, seKTop cKopocTH v = - 1 1-1a BB' H AA' yMeHblllaeTCH, a na AC
rz
yseJII11JHBaeTCH. Te )1\e paccy)K,n:el-1115! rrp11 s,n:aBJII1Bai-IH'H noTeHU11aJibHOH
JTHI-11111 AC rrp11so,n:HT K crrpase.n:JII1130CTII TeopeMbi 2'. DycTb Tenepb 113MeHeH
HaH o6JiaCTb cjJHJibTpau1111 rroJiyqaeTCH s.n:asJmsaHI1CM JI'HHHI1 TOKa BC, np11-
qeM qacn-1 9TOH JII1HI111 BB" H CC" (pnc. 1) oCTaiOTCH Hei13MeiUibiMH. 0To-
6pa311M [] Ha T1 rrpH CJie,n:yiOlll,HX HOpMIIpOBKax: 1) HeiTO.ll,BH)KI-Ibie TOlJK'H b
H C, 2) HefiO.ll,BH)KHaH TOlJKa a 0Tpe3Ka [cc"J, 3) I-IellO}l,I3Il)KHaH TOlJKa a OT
pe3Ka [bb"] (p11c. 3, b", c"- o6pa3bl TOtJeK B", C" s ITJIOCKOCTI1 w). Dp11
rrepBOH I-IOpl\1HpOBKe T01JKI1 b 11 C 6y,n:yT npHTHfHBaiOil1,11MI1. l.JTo6bl fiOJIYlJ'HTb
0T06pa)KeHI1e rrpH BTOpOH HOpMHpOBKe, OT06pa311M Q Ha ITOJIYITOJIOCy T1',
HBJIHIOill,YIOCH HaJIO)KeHHeM [] 11 T1, np11 HefiO.ll,BI1)KI-IbiX TOlJKaX a, C, a",
f / I.C a"- mo6aH TOlJKa mnepsaJia (ac). To1JKI1 a 'H a" np11 3TOM 6y,n:yT
rrpHT5IfliBaiOil1,11MII. Dprr OT06paMeHI1II T{ Ha r, TOlJKa a 6y,n:.eT rrp11T5IfiiBa
IOill,eH, TRK KaK K Heif .ll,OJI)KJ-Ibl rrp110JIH3HTbCH TOlJKa C li o6pa3 T01JKII a.
Tol!HO TaK )Ke y6e)K,n:aeMCH, lJTO nprr TpeTbeli nopM'I1POBKC TOlJKa a 6y,n:eT
rrp11THrrrsmomei1. 0TCio.n:a cJie,n:yeT crrpase.n:mmocTb yTsep)!<.LJ:eHI1H Teo
p.eMbl 3'.
3. 0 6 JI acT h cp 11 JIb T p a u 11 11, or p a 1-1 H q e H 1-1 a H .n: By M H JI rr
II H H M H To K a H rr o T e 1-1 u 11 a JI b H o H: JI u H 11 e H.. CxeMa TaKoil o6Jia
CTI1 cpHJibTpaul1n G cosrra.n:aeT co cxeMoi1, 113o6pa)KeHHoi1 Ha pnc. 1, ecJIH
C1JI1TaTb, 'ITO TOlJKa B cosna,n:aeT c TOlJKOH A, o6bi1JHO Jie)!<ameil: na 6ecKo
He1JHOCTH. I\.OHTypHble ycJIOBHH HMeiOT BH.ll,
u leD=- kh2 = const, u lA = - oo, v Inc= const, 1J lAD = const, (3)
r,n:e h2- TO )Ke CaMoe, lJTO 11 B l. fo,n:orpacp T KOMllJieKCI-!OfO ITOTeHU:11aJia
W = U -+ iv HMeeT BII.ll, fiOJIYfiOJIOCbi, yKa3ai-IHOH Ha pHC. 4.
DycTb B ,n:aJibHeilllleM pacxo.n: )KII.ll.KOCTH Q = v lAc- v lAD H h2 OCTaiOTCH
HeH3MeHHbiMH. Tor.n:a nprr H3MeHei-IrrH o6JiaCTH cjJHJlbTpaui1H G 3a cq.eT cMe
meHHH JIIo6oi1 H3 TOlJeK A, B, C, T. e. rrp11 Hei13MeHHOH o6JiaCTH G, Ho nprr
rrepeMeHHbiX T01JK3X pa3.ll,eJia rpaHI11JHblX JIHI-IHH TOKa H llOTeHU113JlbHOH JIH
RI1H, HMeeT MeCTO CJie.ll,yiOlll,aH TeopeMa.
Teo p eM a 1". flpu yMeHbuteHu.u noreHtfttaAbHOU AU.Htw: a) iJaBAeHue
/'
p = - k u-yy Ha HCU3.MCHHOU U Ha !JBCAU'teHHOlt AUHURX T()KQ !JBeAU'lll-
dw
6aCTCR; 6) BCKTOp CKOpOCTU V = -dz Ha HCU3.MCHHOii 'taCTU IWTCHlfUGA.bHOtl
A.UHUll U Ha HCIJ3,ltCHHOtl AtiHtlll TOKa yee.1ttttu6aeTCJl, a Ha He.U,"JMeliHOii 'laCTU
yaeAU'leHooii A..tl'Httu roKa yAteHblltaeTcR. IT pu cAteU{eHuu TO'lKU pa3aeA.a A.ttHuii
TOKa: a) iJaBA.CHUC p =- ~ u-yy Ha HL'll3MeHHDii 'taCTll y,\WHblltaeM'oii A.U
HtUl TOKa yeeA.U'l.~tBaeTCJl, a Ha HeU3AteHHOil ttaCTU !JBeA.U'lttBae.Moii A.UHUtl TOKa
-+ dw
!JMeHblltaeTCR; 6) BeKTOp CKOpOCTU 1' = -1 Ha H.et/3,\/.'!HH()Ll ttaCTU yMeHb-
C.Z
1/WCA/Otl A.UHllll TOKa U B6Aa3,U' ee IWHt{a Ha 11DTeHl{IIGA.bfiOtl A.UHUU yeeA.U
<tttBaeTCR, a Ha HCU3AteHHOtl ttaCTil !JBeA.U'llWae.\10tl -?.1/H./1/1 TOKa U B6A.UaU
ee KOHt{a Ha 110TeHt{ltaA.bHOLl A.UHUU ~jAteHbtiWCTCJl.
B caMOM ,n:CJie, rrycTb T1 - ro,n:orpacp l:I3MCIICHHoro KOMITJi eKCHoro rro
TeHu,uaJia. Dprr yMeHbllJeHHH rroreHu,HaJi b HOH mnmH 3a cqer cMem:emm TO'I
Krr c B IlOJ!Q)KCHIIe C' (pHC. 1) OT06pa3 HM T na rl
rrpii CJ!C,.'I,YIOW,HX HOpMIIpOBKax: 1) HeiTO,L\Bll./1\HbJe
TO'IKII a II d, 2) HC!lO,UBll./IUia51 TOl!Ka <t. 0Tpe3Ka [ad],
3) Herro,UBmimaH TOl!Ka a orpe3Ka [ac] , 4) uerro
JlBII)KHaH TOl!Ka a orpe3Ka [de']. Dpii rrepBoii Hop
MIIpoBKe TOl!Ka d - OTTa.JJKIII3aJOI1(35I, TOl!JW a -
rrpiirHriiBaJOIU:aH. Dpii Bropoii HOpMIIpOBKe TOl!Ka a
6y,UeT OTTa,lKliBaJOW,eH, rrpii TpCTbCH HOpMIIpOB
a
a' - c"
-~ -·-
T
PHC. 4.
VG
c'
I
' d'
Ke - rrpiiTHriiBaiOlll,eH. C.JJe,UOBaTe.JJbHO, ,n:aB:re- dw
rme p na .JJIIHIIHX TOKa AC II AD yBeJIHl!IIBaercH, neKrop cKopocrii v = dz
Ha AC yMeHblliaeTCH, Ha AD yBe.rrii'IIIBaercH. 4ro6bi rro.JJyLIIITb oro6pa)J(e
HIIe rrpii LieTBeproii uopMHpaBKe, oro6p a3IIM T na rro.JJyrroJrocy T 1 *, HB.JJH
IOw,yiocH o6w,ei1 qacTbiO T II T 1, rrpii Herro,UBll.IKHbiX TOl!Kax a, d, a. Dpii
3TOM TO'IKII (l II a 6yJlyT OTTaJIKIIBaJOW,II ~Ill . Dpii OT06pa)KCHHII Tl* Ha rl npii
HCfiO,UBH)KHbiX TOl!KaX a 'H a TOl!Ka a 6yzJ,eT OTTa.JJKHBaJOIII.CH, TaK KaK OT Hee
)l,OmKHa OTO)l.BHHYTbCH TOl!Ka d. C.JJe,UoBare.JJbHO, BeKrop CKopoCTII Ha DC'
yBe.JJII'IIIBaercH. TaK)Ke ycraHaB.JJIIBaiOTCH yrnep)!<,UeHHH reopeMbi 1" rrp·II
yMeHbllieHIIII rroreHu,IIa.JJbHOH .JJIIHIIII 3a cqer cMew,eHIIH TO'IKII D. Dycrh
rerreph TO'IKa A CMew,eHa B rro.JJO)!{CHIII! A' (p'IIC. 1). Oro6pa3HM T Ha ro
)l,orpacp T1 Ilpii C.JIC)l.YIOlll,IIX HOpM'IIpOBKax: 1) HCilO)l,BII)I<HbiC TOl!KII C II d,
2) HC!l0)1,13II)J(Ha51 TO'IKa a 0Tp€3Ka [ad], 3) HC!lO)l,Bl:I)!<Ha5I TO'!IW a OTpC3Ka
[ ca']. 0pii rrepBOH HOpMIIpOBKe TOl!Ka C- OTTa.JJKIIBaJOW,a5I, TO'IKa d- rrp.II
T5lfiiBaiOLU:a5I. Dpii BTopoii HOpMHpOBKe T O'IKa a- rrpiiTHriiBaiOnLa51, rrpii rp.e
Tbeii HOpMIIpOBKe- OTTa.JJKIIBaiolll,aH. 3ro 3Ha'IIIT, liTO ,UaB.JJeHIIe p Ha CA'
yBe.TIII'IIIBaercH, Ha AD yMeHhiiiaercH, BeKrop cKopocrii Ha CA' II 136.JJH3II TOl!
KII C na rrorenu,IIa.JJbHOH .JJIIHIIII yBe.JJII'IHBaercH, a Ha AD II B6.JJ'II3II TO'IKII D
ua rror.eHu,IIa.JJbHOH .JJIIHIIH yMeHhmaeTCH. 3TIIM TeopeMa 1" ,n:oKa3aHa.
Dpii II3MeneHIIII o6.JJacrn cpii.JJbrpat(:HII G 3a Cl!eT rrpoii3BO.JJhHhiX B,UaB.JJII
Bmmii .JIIIHIIi{ TOKa II IlOTCHl(Ha.JJbHOH JIHHHII, HO rrpii HCII3MCHHh!X TO'IKaX IIX
pa3)l,e.JJa A, C, D HMeior Mecro c.JJe)l.yiOlll,He TeopeMhi.
T e 0 p eM a 2". Ilpu aaaeA.UBaHUU noreHtfUaA.bHOii A.UH,U.U,' a) aaeA.e
y
HUe p = - k U- yy I-ta A.llHilflX TOKa yMe/-lbutaeTCR; 6) BeKTOp CKOpOCTU
387
dw
V = - IW ftell3AteHHOU ttaCT.U 110TeHl{ltaAbHOit AUHU;ll' U 66AU3tl ee IWH.l{OG
dz
Ha AUHllflX TOKa ljAWHblllGeTCfl.
T e 0 p eM a 3". !Jpu 80a6Alt6aHU;U Al!Hll,ll 1'0/W: a) aaa.~7eH.ue
'Y
p = - k U- ;·y Ha HBU'3.~~teHHOii AUHUU 1'0Ka .U Ha npU.~ltblKaJOU{eii K Heii H e -
ll3.~1teHHOU ttaCTU 60ll6AUBae.~1WU AUH.Utt 1'0Ka yae.~7ll'WBaeTCfl, a Ha opyc'oii
HeuaAteHJWt1 rtacTu aoaa.~utaaehwii AUHUU TOJW, npu .. HbliWJOll{eii K noTeHt.fll -
__. dw .
aAbHOU A UHlHl, y.~lWHblllaeTCfl,' 6) BeKTOfJ CKOfJOCT.'l V = -1- HG H e t.l 3 AWH.HOU
(.Z
'laCTt!l 60a6All6GePtoll AUH.Wl TOKa ll B6AU3U ee KOHlfG Ha IW1'eHlflWAbHO/l
AUHUU y.~lWH.bUWeTCfl, a H.a H.eU3Mefi.HOLi AUHU:l{ TOKa ,(l 66All3.U ee KOH.l{a Ha
tlOTeH.tflWAbH.OU AUHUll yaeAWlUBaeTCfl.
B caMOM ,ll.eJie, rryCTb r, - fO,II.Orpacj:> 113MeHeiiHOf0 KOMilJICKCHOfO !10-
TCHI . .I.HaJla, [J - o6pa3 113Meuemroi1 o6.1acTH cjmJrbTpaul111 n ITJIOCI<OCTH w:
DycTb H3MeHeHHan o6JiacTb cj:>11JibTpau1111 rroJiyllaeTcn B,ll.aBJIHBaHI1eM rroTeH
urraJihHoi1 Jmmm, rrp11lleM llaCTI1 3TOH JII1HHH CC' 11 DD' ocTaroTCH H e l13:\reH
HbiMI1. 0To6pa311M Q Ha T, rrp11 CJie,ll.yiOlll,I1X HOprvmpOBI<aX: 1) HellO,II.BH)I<Hb!C
TOliKI1 C 11 d, 2) H e llO,II.B.H)IU!35I TOliJ<a a 0Tpe3I<a [cc' ], 3) HellO,II.BI1)!U!a5I TOli
Ka a oTpe3I<a [dd'] (p11c. 4). Dp11 nepsoii Hop rvmponi<e TOli KI1 c 11 d 6y,11.yT
I1p'HT5lfHBaiOlll,I1MI1. 4T06bi IlOJIYli11Tb OT06pa:ti<eT-JHe rrp11 BTOpor{ HOpM11pOBI<e;
OT06pa3H~l BH3li3Jie Q !Ia IlOJIYITOJIOCY T,', 5IBJI5IIOlUYIOC5I HaJIO)I(e~meM n
H r,, rrpn HellO)I.BI1)J{[-! b!X TOliKax a, c, a. DpH 3TOM TOCIK11 a 'H a 6y)l.yT np11-
T5lfi1Ba~mi1MI1. 0To6pa:t!<a5I Terrepb T,' Ha r, np11 H e llO)I.BH)!UIOH TOlii<e a,
BH)I.HM, liTO 11pH BTOpOH HOp~·lllpOBI<e TOliJ<a a 6y,11.eT np11THfi1Ba~lll,eH. TaiOJ{e
y6e:tK )I.3eMC5!, liTO TOlii<a a 6y)l.eT rrp11T5If'HB310Tll,e H rrpH TpeTbeH HOpM11pOBKe.
0TCIO)I,a CJie,ll.yeT CrrpaBe)I.JII1BOCTb TCOpCMbl 2".
Dycrh Tenepb B)l.aBJII1BaeTCH JII1HI15I TOI<a AC, rrpHliCM qacrrr 3TOI'i JIH
HHI1 AA', CC' OCTaiOTCH Hei13 M e i-IHbiMH. 0To6pa3HM Q Ha r, rrprr CJi e,ll.y
IOlll,HX HOpMHpOBJ<aX: 1) HellO)I.BH)[(Hbie TOl!K'H a, d, C, 2) I-IellO)I.BH)IUia5! TOl!J<a
a OTpe3Ka [ad], 3) Herro)I.BH:tlmaH TOl!I<a a oTpe3I<a [aa'] , 4) nerro)I.BI-DKHaff
TOl!I<a -a OTpe3Ka [cc"]. Dp11 n epsoii noprvmpoBKe TOl!KH a H c - rrpHTHf'H
BaiOLU11e, TOl!Ka d - OTTaJIKI1BaiO!lla5!. Dp·H BTOpOH HOpM11pOBJ<e TOl!I<a a -
OTTaJirmsaroman, a rrprr TpeThei1 r-rop~mposKe - rrp'HTHrrrsaro u.I,aH. CJie)I.OBa
TCJibHO, .n:asJien11e p Ha CC" yMeHbruaercH, a H a AD H AA' yseJII1l£HBaercn,
neKrop CKopocTrr na AD yseJII1l£11BaeTCH, a Ha AA' yMeHhruaeTCH. 4To6br rro
Jiyl!HTb OT06pa:ti<eHHe I1p11 l!eTBepTOH HOpMHpOBKe, OT06pa3HM Q Ha IlOJiy
IlOJIOCY T{, 5IBJI5IIOlUYIOC5! HaJIO:tKeHI1eM n H r,, npTI HCI10,11.BI1)1Ulb1X TOl£
Kax a, c, a. Dprr 3TOM TOl!Ka a 6y,11..er np11THr.Hsarome i1. Dp11 OTo6pa:tKe i-rHI1 T,'
Ha T, TOliKa a TaiOI<e 6y,11.eT I1pHT5Ifi1Ba!Olll,e H 11, CJie)I.OBaTeJibl-10, Bei<TOp CKO-·
pocnr na CC" yMeHhruaeTCH. TaKI1e :>Ke paccy;.rmeHIIH rrprr s.n:asJmBaHI1H:
JI'HH1111 TOKa AD rrp11BO.JI.HT K cnpase.n:JmBOCTI1 TeopeMbi 3" .
3 aM e q a H 11 e . EcJIH B KOHTYPHbiX ycJIOBI1HX (3) ycJIOBI1e u JA = - oo·
3 aMeHHTb ,ll.pyrHM yCJIOBHeM uJA = + 00 (liTO HMe e T M CCTO B CJiyqa e cp11Jib
TpaUHH H3 KaHaJIOB) , TO TeopeMbl 1", 2", 3" IlpH 3TOM OCTaiOTCH CnpaBe,II.JIH
BbiMH rrp·rr ycJIOBHH, .ecJIH se3)1.e B 3THX TeopeMax 3 Har< napnaunn .n:asJienrra-
p = - ~ zz- ;;y Cl!HTaTb rrpOTHBOilOJIO)I(HbiM.
·§ 2. HeKoTophle oap.uau.uoHHbie TeopeMbl Teopuu lj)HJibTpau.uu npu HaJIH'IHH
npoMeJKyTKa BbiCa'IHBaHHJI
YcraHOBHM xapaKrep H3MeHeHH51 ocu o BHbiX muerpaJibHbi X xapaKTepHc
THK T e OpHH cpHJibTpau;HH 11pH 3aMeHe 06JiaCTH cpHJibTpaU.HH, CO,Uep:>J<aiU.eH
B COCTaBe CBOCH rpaHHI~bl 11pOMC)!<yTOK BbJCa'IHBai-IH51, l<aKOH-JIH60 H3 06JiaC
TCH cpHJihTp<n~HH, pacc~IOTpenHbiX B § 1.
1. 0 6 JI a C T b cp .H JI h T p a U. H H, 0 r p a II H lJ C H II a 5I J.L By M 5I JJ H
I! H 51 M H T 0 K a, ,U B y l\1 5I li 0 T e H U. 11 <I J l h H bl M H Jl H II H 5I l\"1 H H 11 p 0 -
:'11 e )f{ y T K oM B hi c a q 11 n a H 11 51. Cxei\1a TaKoi1 o6.!JaCTH cpHJi brpauHH npe.u
cTaBJi cHa Ha p Hc. 5. KOJuypHbie ycJIOB.Hfl IHICIOT BH.U
u /An=-kh1 = const, u /nc• = - kh2 == const, u /cc• = - ky ?> - kh1,
V/nc = const, v AD=const, (4)
r ,n:e /7 1 H /72 - ypOBHH :tKH,UKOCTH B .nep BOM H BO BTOpOM BO,UoeMaX,
H = h , - h2 (H > 0) - Harrop :tKH,UKOCTH. ro.uorp a cp T KOMITJICKCHOro 110-
TCHJ.I;HaJra W == ll + i v HMCCT BH,U, yKa3aiiiibiH Jla pHC. 6.
y
v ®
b .c c" u
h2
~KH 1-- - -~ C'
. I. b I
'7 T I
I
a d' d - iQ
PH c. t) . PH c. G.
f1ycTb BCJIHlJHHbl J-1 H /7.2 OCTaJOTC51 HCH3MCIIIIb!MH, TOr,JJ,a HI\ICIOT MCCTO
CJie,n:y iOLUHC T COpCI\Ibl.
T e o p e M a 4. flpu 3a,lteHe npoAteOtC.tJTKa Bbtca•waaHU.'l CC' n:oreH
LfLta / lbHot'i AUHI/et'i: a) pacxoa 0/CUOIWCT/1 Q yne,w<tuaaeTCR; 6) OaBACHl/C
•• -+ dw
P = -I_ u - ylJ H a A/IH'IHX TOKCl y ,lteHbUlCteTCH; a) BCKTOfJ C!WpOCTU V = dz . ll .
Ha HCil3.\WHHOU IWTCHlfUa/lbHO(i. A ,l/Hrtll, c6AU3 ll ee KOHlfOB U KOHlfa /LfJOJte-
0/C!:/TKa lJbl.CU'ltWaHWl Ha AUH.'L'lX TOIW !JBC.lll 'i!WUCTCH, a H.a HCli3 .1LC!iHOU ttaCTll
!JEWAII'lll6GC.110tl /LOTCHtfUaAI>HOil ,11/HU/l ~~ 66AU3/l ee KO!it{a I-ta AUHUU TOKa
!J.MCHb l/taCTCJl.
B CCIMOM ,UC,;lC, nyCTb T1- ro,n:orpacp B ITJIOCKOCTH H3MCJJCHHOrO KOMTI
JJCKCI!OrO ITOTCHJ.J.}JaJia. PaCCI\IOTplnl OT06p<DKCII'}JC T lia r, npH CJIC.UYIOIUHX
HOpl\IHpOBKaX: 1) HCriOJ.LBIDKI!bJC TOlJKH a H d, 2) H CI10,UBlDKHa51 TOlJI<a b,
3) JJCllO,UBIDKHCI51 T01II<a 0. 0TpC3Ka [ab ] , 4) IICITO)J,BH/JU!aH TO'IKa. a 0TpC3-
Ka [de'], 5) II eno;~BmKIJa51 TOlfKa c. 0ToCipa3IBI T I·Ja npH:-Io_yroJrhiiHK abc"d
(pHC. G) npH IICITO,UBIUKHbiX TOl!KaX b, a, d. f1p11 3TO~I TOl!Ka a 6y,n:eT OTTaJI
I<HBaiOil~CH, TOlJKH b 'H d 6y.uyr rrp HTHnm:tiOill,HMH . CJie,n:oBaTc~IhiiO , npH n ep
noii IIOpl\IHj)OBKC a - OTT(IJIIUI£laiOW:il51 TOlJKil, d - - rij)HT5IrHB <.IJOIJ~a51 , H ,n:pyrHX
ueno;~BH>KHhiX TOlJeK JICT . Oro6pa3IIM T Ha npmviOyroJihiHIK abc"d np11 HC
TIO}J,BH>IUibiX TO'II«tx a, b, c. f1pH 3TOM TOlJKa b 6y,n:cT OTTaJIIUIBaiOlUCH. C Ji e
.uonaTCJibiiO, 11pH BTOpOH HOj)~!Hj)OBKC TO'll<a b - OTTaJIKHB<JIOli.LaH 'H ,n:pyrHX
Hcno;~BmKHbiX TOlJCK n eT. 3ro 3 HC1'1!1T, •no pacxo.u )JUI.UKOCTH Q yneJIHlfH-
389
BaeTCH, ,n:aBJieiUie
I' p=- k zz-yy Ha JIUIHHIX TOKa yMeHblilaeTCH, BeKTOj:}
CKopocni B6JIH3H KOHIJ.OB A 11 B Ha JI'HIHIHX ToKa yBeJIH'IHBaercH. 4ro6bi no
.TIY'IHTb OT06pmKeime rrpH TpeTbeH HOpMHpOBKe, OT06pa3HM BHa'laJie T Ha
rrp.HMoyroJibHHK abc"d rrpH Herro,n:BH)l<HbiX TO'IKax b, a, a. Op11 3TOM TO'IKa a
6y,n:er orraJIKHBaiOill.eii. Dp11 oro6pmKeHHH rrpm\1oyroJlbHHKa abc"d na T1,
TO'IKa a T3K)Ke 6y,n:eT OTTaJIKHBaiOill.eH, Tal< KaK OT Hee ;:J:OJI)KlJbi OTO;:J:BH
I-ryTbCH TO'IKH b 11 a. CJie,n:oBaTeJibHO, rrpn TpeTbeii nopMHpoBKe TO'IKa a 6y.LI.eT
• _., dw .. AB
OTTaJIKHBaiOill.eH H BeKTOp CKOpOCTH V = dz Ha l!OTeHIJ.HaJibHOH JIHHHH
yBeJIH'IHBaeTCH. 4T06bi ITOJry'-IHTb OT06pmKeHHe rrpH 'leTBepTOH HOpMHpOBKe,
oro6pa3HM T I-ra rrp.HMoyroJibHHK T{, HBJI5110ili.HHCH HaJimKermeM T 11 T1, npn
Herro,n:BH)KHbrx TO'IKax a, d, a. Opn 3TOM TO'IKH a n a 6y.LI.yT rrpnnrrHBaiOili.HMH.
Opn OTo6pmKeHHH T{ Ha T1 TO'IKa a 6y,n:eT rrpHTHrHBaiOill.eif, TaK KaK K H e if
.LI.OJDKHbi npn6JIH3HTbCH c pa3JI-H'IHbiX cropoH TO'IKa d 11 o6pa3 TO'IKH b. Eho
- dw
3Ha'IHT, lJTO BeKrop cKopocni v = dz Ha rroreHu.naJrbr-roii JIHI-IHH DC' yMeHb-
maercH. Oro6pa3HM T I-ra rrpmvroyroJibHHK bcd'a (pnc. 6) rrpn Herro,n:BmKHbiX
TO'IKax a, b, c. DpH 3TOM TO'IKH a 11 c 6y,n:yT orraJIKHBaiOlU.HMH. OpH oro6pa
;.KeHHH IIOCJie,n:Hero rrpHMOYfOJ!bHHKa Ha T 1 TO'IKa C 6yJJ:e T OTTaJII<HBaiOULei-i,
TaK K3K OT Hee OTO}:(BHHYTCH TO'IKa b H o6pa3 TOL!KH d. 9TO 3Ha'IHT, 'ITO rrpH
IIHTOH HOpM'HpOBKe TOL!Ka C - OTT3JIKHB3IOULaH, a Bernop CKOpOCTH B6JIB3H
3TOH TOL!KH Ha JIHHHH TOKa yBeJIH'IHBaeTCH. 9THM Teopel\13 ;:J:OKa3ai-!3.
3 a Me 'I a H 11 e. YTBep)K;:J:e~me TeopeMbi 4, B 'laCTHOCTH, 03Hal!aer, 'ITO
;:J:aBJieHHe H paCXO.LJ. )l<H;:J:KOCTH Be,n:yT ce65l rrpH 3aMei-Ie rrpoMe)KYTKa BbiCa'IH
BaHHH IIOTeHU.HaJibHOH JIHI-IHeH TaK )Ke, KaK 'H npH 3al\1eHe ITOTeHU.l13JlbHO~i
.'IIUIHeH JIHHIII-1 TOKa.
Teo p eM a 5. flpu saMeHe npo.uec;JcyrKa BbtcartueaHWI CC' .llUH.ueu roKa:
a) paCXOO CJICUOKOCTU Q y .. ltCHbtuaeTCfl," 6) OaB.IlCH.Ue p=- ~zz-yy HG H.Cll3-
AtCH.IiOU 1taCTU yee.llll'tuBaeJI!Oll ./lUH.Ull TOKa !JBC.Illt'/UBUCTCR, 6) BCKTOp CKOJ:O-
__,. dw • , 6
CTU V = dz H.a H.C.U3AtCH.HOU 'iGCTU YBC.Illi'lll6GCJHOU .Ill/HUll TOKa, 8 .lll/3U ee KOH.-
lfll B H.a rwreH.tfUa.llbHOU .llUH.uu y .. ~teH.buwerc.<z u a6.1lusu TO'tKU C' Ha noTeH.
tfUa.llbH.Oit .llUH.llU yae.lltl'l.U'BaeTCfl. Pacxoa CJIWOKOCTU rtepes BCflKYIO C65l3HYIO
'taCTb nOTCHl{tUl.llbH.OU ./lUH.UU !JA1CH.bUtaCTCfl, CC.IlU KOHlfOAt 3TOit CB513H.OU rtaCTU
RB.IlfleTcfl TO'tKa B, u yee.ll~t'tueaeTcfl, ec.llu KOH.Lfo .. w ee fW.IlfteTCfl TO'iKa C'.
B caMOM .LI.eJre, rrycTb Tt - ro,n:orpa<P B ITJIOCKOCTH I-13MeHeHHoro KOMrr
JieKCI-Ioro rroTeiiu.naJra Wt. PacCI\WTpHM oro6pa)Keime T ua T1 rrpn CJie)ly
IOili.I-IX HOpMHpOBKax: 1) HeiiO}:(BI-I)!<Hbre TO'IKH a H d, 2) HeiiO;:J:BH)J(JI351 TOlJ
Ka b, 3) H eiTO;:J:BH)JUJaH TO'IKa c', 4) HeiiO;:J:B!-I)J{J-IaH TOL!Ka a 0Tpe3Ka [be].
Oro6pa3HM BHa'laJie T Ha npHMoyroJrbHIIK abc"d (pHc. 6) npu Heno,n:BmKHbiX
TO'IKaX b, a, d. 0pH 3TOM TO'IKa a 6y.LI.-eT OTTaJIKHBaiOill.eH, a TO'IKH b H d
6yiJ:YT npHTHrHBaiOili.Hl\IH . Op1-1 oTo6pa)Kemm npmvtoyroJibHHKa abc"d Ha T1
TO'IKa a 6y,n:eT npHTHrHB3IOll{eH, a TO'IKa b- OTTaJIKHBalOili.CH, T. e. paCXO;:J:
)I{'H.LJ.KOCTH Q YI\IeHblilaercH. Op11 OTo6pa)l<eHHH npHMoyroJibHHr<a abc"d Ha T1
IIpH HeHO}:(BH)I<HOH TO'IKe b 3Ta HeiiO;:J:BH)!<H35l TO'IKa b 6y;:r:eT rrpHTHrHBaiOIIJ.CH
H ,n:pyrHX HeiiO}:(BH)I<HbiX TO'I-eK l-Ie 6y;:r:eT, TaK KaK K TO'IKC b ;:J:OJJ)!Uibl npH-
390
6Jill3HTbCH C pa3JIHlJHb!X CTOpOH TOlJKH a H d. ::ho 3Hal!HT, liTO ,naBJICHHe p H<'l'
--+ dw
JIHHHH ToKa BC yBeJIHliHBaeTcH, BeKTop cKopocTH v = dz n6Jm 3 H TOl!KH B
H paCXO,ll. )KH,li.KOCTH qepe3 CB513HYIO l!aCTb ITOTCHli.HaJib!IOH JIHHHH, HMCIOW:CH
KOHCll. B TOlJKC B, yMeHbillaJOTCH. 4T06b! IlOJiy<IHTb OT06pa1KCH'HC rrpH TpCThei!
HopMHpOBKe, OTo6pa3HM BHal!aJie T Ha npHMoyroJihHHK adc'b' (p11c. 6) npn
HCITO,li.BH)KHbiX TOlJKaX b', d, C'. JlpH 3TOM TOlJI<a d 6y,ne T rrpHTHfHBaJOlUCH,.
a TOlJKH b', c' 6y,nyT OTTaJIKHBaJOW:HMH. JlpH OT06pa)!<CHHH yKa3aHHOfO rrpH
MOYfOJ!bHHKa Ha T1 TOlJKa c' 6y,neT OTTa.TJKHB310ILLCH, TClK KaK OT Hee ,li.OJI)I<Hbl
OTO,li.BHHYTbCH C pa3JIHlJHbiX CTOpOH TOlJKa d .H o6pa3 TO<IKH b. CJIC,li.OBaTeJibHO,
--+ dw
BeKTop cKopocTH v = · dz n6JIH3H To<IKH C' n pacxo,TJ. iKH,nKocTu qepe3 ncHKYIO
CBH3HYIO lJaCTb ITOTCIIli.HaJibHOH JlliHlili, liMCIOW:YIO KOI!Cll. B TOlJKe C', yBe
JIHl!liBaJOTCH. 4T06bi ITOJIYl!HTb OT06pa)!<CHHe rrpH lJCTBe pTOH HOpM'HpOBKe,
OT06pa3HM T Ha o6.TJaCTb T{, HBJIHIOW:YIOCH HaJIO)KCHHCM T 11 r, np!l H e
ITO,li.B!l)f{HbiX TOl!KaX d, b, rt. JlpH 3TOM TOlJKa a 6y,neT rrpHTHfHBaJOW:CH. Jlpn
OT06pa)KCHHH T{ Ha r, TOl!Ka a TOJKe 6y.IJ.eT rrpUTHrHBaiow:e ii:, TaK KaK K H CH
,noJI)KHbi rrpH6JI'H3HTbCH c pa3JIHl!HbiX CTo poH TOlJKH b 11 d. C Jie,nonaTeJJbHO,
dw
BeKTop cKopocTH v = dz na JIHHHH TOKa BC yMeHhwaeTCH. 3THM TeopeMa
.IJ.OKa3aHa.
3 a M e q a H 11 e. YTBepiK,nenHH TeopeMbl 5, B qacTHOCTH, 03Hal!aJOT, liTO
rrpH 3aMCHC rrpOMC)!<YTJ<a BbiCalJHBaHHH JIHHHCH TOKa paCXO,TJ. )!(!l,li.KOCTH Q'
BC,li.CT ce6H TaK iKe, KaK rrpH 3aMCHC DOTCHUHaJibHOH JIHHHH JIHHHCH TOKa.
2. 0 6 JI a C T b cp H JI b T p a 11. H H, 0 f p a II H <J e H H a H ,ll. B y M H ri 0-
T e II 11. H a Jl b II bl M II JI H H li H i\1 II, JI H H H e H T 0 K a H IT p 0 M e )!( y T K 0 M
n hi c a q 11 n a HuH. Cxei\Ia TaKoii: o6JlaCTli cpHJihTpauHH rrpe,ncTaBJI CHa Ha
pHc. 5, eCJIH ClJHTaTh, lJTO TOLJKa D coBrra.naeT c TOlJKOii: A, o6I>IlJHO JICiK<IW:Cii: .
Ha 6eCKOHClJI!OCTH. l(mnypiihiC YCJIOBHH 111\ICIOT BH,ll.
u/AB= -l<ht =const, u iAc= -kh~. =eonst,
u/cc' = -ky>- kh1, vl 11c=const, (5)
n1.e h 1, h2 , H = h 1 - h2 (H > 0) - TO 11<c caMoe, liTO 11 B cJiyqac I. fo.IJ.o
rpa¢ T KOMITJICKCHOfO ITOTCHll.llaJia W = U + iv HMCCT BH,ll. ITOJIYITOJIOCbl
C Bb!CM!WH, yKa3aiiiiOfl Il<l pl!C. 7; paCXO,J: )KH.UKOCTH Q paBCII 6CCKOIIC11110CTH .
JlyCTb BCJIIllJHHbl fi = lz1 - h2 H h2 OCTaJOTCH HCH3MCHHbiMH, TOf;la
HMCIOT MCCTO CJie,nyiOW:HC TeopeMbl.
Teo p eM a 4'. flpu 3aMeHe npoMeJ~CyTKa Bbtca•utBaHufl CC' noTeH-
l{tta.rtbHOU .flllfiUeu: a) aaB.rteH.ue p = - ~ u- YY Ha .rttmuu TOKa yMefib-
dw
UtaeTCJt; 6) 6eKTOp CKOpOCTll V = dz fill HeU3AtefiHOU IIOTefil{UCl.flbfiOU .fllllillU
U fi(l .flUHl/.11 TOKa 66.rtU3U ee KOfil{08 !J8e.rtU'lit8aeTCfl, a Ha IIOTefil{,UaAbHOil
.rtuHuu A C' yMeHblltaeTCfl.
B caMOM ,neJie, rrycTb r, - ro,norpa¢ B ITJIOCKOCTH H3MeHeHHOfO KOl\!IT
JieKCHOrO IlOTCHli.HaJia w,. 0To6pa3HM T Ha r, rrpH CJie,nyJOW:HX r!OpMHpOBKax:
1) HCITO,li.B!l)!(Hb!C TOlJKH b H a, 2) HCITO,li.BU)f{HaH TOlJKa a OTpC3Ka [ab ], 3) He-
391·
rro)J.BIDKHaH TOlJKa a OTpe3Ka [ac'], 4) Herro)J,BIUKHaH TOlJKa c. Dpn n c pBoi1
HOpMHpOBKe TOlJK3 b 6y)J.eT OTT3JJKHBaiOW.eH, TOlJI<a C - rrpHTHI'HBaJOUI.CH;
LLpyrHx Herro)J,BH)KHbiX TOlJeK He 6y)J.eT. CJJe)J,oBaTeJJbHO, )J,aBJJeime p na JJHHHH
--- dw
TOKa yMeHbUJaeTCH, BeKTOp cKopocTH v = dz B6Jm3H TOlJKH B Ha JJHH'HH
TOKa yBeJJHlJHBaeTC5I. lho6bl llOJJYlJHTb OT06pmKeime rrpH BTOpOH HOpMH
pOBKe, OT06pa3HM T Ha ITOJIYITOJIOCY abc"a (pHC. 7) rrpH HeiTO)J,BH)I<Hb!X TOlJ
KaX b, a, a. DpH 3TOM TOlJKa a 6y)J.eT OTTaJJKHBmomei1, TOlJI<a b - npHTHrH-
v Bmomei1. DpH oTo6pa)Kemm yKa3aHHOi1 noJJynoJJOCbi Ha
t (i) T1 npH HCITO)J,BH)K llb!X TOlJKaX a H a nepBaH !13 HHX
--j-b-----"_-fc_"_..._u 6y.a,eT OTTaJIKHBaJOW.eH, BTOpaH - npHTHI'HBaiOUJeH, T. e .
-KH
c'
T
a
Puc. 7.
ll()H BTOpOH HOp~1HpOBKe TOlJKa a 6y)J.eT OTT3JJI<HBa-
-+ dw
IOI11.ei1 H BeKTOp cKopocTn v = --:;- Ha AB yBeJIHlJHBaeT-
uz
C5I. 4T06b! !lOJJYlJHTb OT06pa)KeHHe npH TpeTbCH HOp
MHpOBKe, OT06pa3HM T Ha ITOJI YITOJIOCY abc" a rrpH
HCITO)J.BmKHbrx TOtiKax b, a, a. DpH 3TOM TOlJI<a a 6yJI.eT
npHTHrHBaiOill.eii. DpH oTo6paR<eHHH yKa3aHHoi1 rroJJynoJJoCbi na T 1 rrpH Herro
)J.BmKHbiX TOlJKaX a H a TOlJI<a a 6y)J.eT np.HT5II'HB3IOW.,eH, T3K K3K TOlJKa b )J,OJI
) 1(1-!3 y)J.aJJHTbCH OT TOlJKH a. 3TO 3H3 lJHT, lJTO llpH TpeTbeH IIOp~mpOBKe TOtii<a a
" -+ riw
6y)J.eT I1pHT5II'HBaiO!lLeH H , CJJe)J,OB3TeJibHO, BeKTOp CKOpOCTH V = dz 113 llOTeH-
l~HaJJbi-!OH .JIHHHH AC' y6hmacT. Dprr lJeTBepToi1 Hop rvmpoBKe TOl!Ka c 6y)J.eT
-+ 1/ W
OTTaJIKHBa!OllJ.eH H, CJJe)J,OBaTCJJbHO, BeKTOp CKOpOC'IiH V = dz Ha .JIHHHH TOKa
B6JJH3H TOtiKH C yBeJJHlJHBaeTCH. 3THM TeopeMa )J.OKa3aHa.
3 a M e q a H H e. YTBep1K,n:emur Te opeMbr 4' 03HalJaJOT, liTO npH 3a~IeHe
y
np0~1C)KYTK3 BbiCal!HBaHH5I llOTeHUH3JibHOH JJHifHCH )J,3BJ1CH'He p=- k u-yy
B e)J,CT ce6H TOl!HO T3K )Ke, KaK H rrpH 3aMCHC .~THHHH TOKa llOTCHUH(lJJbliOii
.TI Ill!HCH .
T e o p eM a 5'. Jlptt s aAteHe npo.JteJIC!JTIW GIJlCWl.l.iBa NWl CC' JPtH.Uet/
UJ !W.' a) daBACHJ.le _!!_ = - ~ U -· Y!J Ha AUH,UU TOIW !JGCAU 1UIBGCTCR ; 6) GCK-
-+ dw " 6 B TOfJ CKOfJO CTII V = dz HG H.CU3AWHH.OU ttaCTU AUHUU TOIW U 6 AU31l TOtlfCU
Ha llOTCHI{Ua.1bHOU At/HUll !J.MCH.blUClCTCR, G6AU3l/ TO'iKU C' H.a IWTCHI(UG.ilbHOU
-1Wtttu yee.~uuwaaercJl . Pacxoo JICUdKOCTU •tepes acRK.tfJO caR3H.IJIO •tacTb no
T CHU,ttaAbH.Ot1 AUHUU !JAtCH.!J lltaCTCR, CCAll KOHLfOA'l 3T0lt ttaCT/1 5l6A5lCTCR T0 1l
l\.!l B, u ,l}ae,ut.•tuaaeTcH, ecAu !(OHt{OAt ee nuAJteTcH TO'll(a C'.
B caMOM )J,eJJe, paccrviOTpHM oTo6pmKe HHC T Ha ro,n:orpacp H3 M e HeHHoro
KO~!llJieKcuoro noTeHuHaJJa T1 npH CjJC/J:YIOIL~HX 1-ropMnpoBKax: 1) Heno,n:B·mr<
Hble TOlJKH b H a, 2) HCITO)J,BI-DKHb!C TOtJKH C' H a, 3) l!Cl10;J,BIUK!l35I TOl!Ka a
OTpe3Ka [ be]. DpH nepuoi1 HopMHpoBr<e TOl!Ka b 6y,n:e T npHT5IrHBaro m e i1, TOlJ
Ka a - OTTa JIKHBaiOLUCH. DpH BTOpOH HOpMHpOBKC TOl!Ka a 6y)J.eT npHT5II'HBa
IOW.Cif, TOl!I<a C'- OTT3JIKHBaiOllJ.CH H ,n:pyrHX He l10)J,BIUKHblX TO'ICK HC 6yJI.eT.
4T06b! llOJJYlJHTb OT06pa)Kei-me npH TpCThCH HOpMHpOBKC, OT06pa 3HI\I T Ha llO
JIYITO.TIOCy T{, HBJrmomyiOC5I IIaJJO)KCIIHe M T 11 T1 npH Heno,n:BH)l<HhlX TOlJI<ax
a, b, a. Dpn 3TO i\'I TOliKa a 6y,n:e T npHTHrHBaiO!lJ,ei1. Dpn oTo6pmi~eHHH T1'
.892
Ha T1 np.H HCHO,Il,BH)!(Hb!X TO'II<aX a H a TO'II<3 a TO)KC 6y,IJ,eT I1pHTHfHBaiOUJ.CH,
Tal< r<aK K Jieii ,IJ,OJnKHa npH6JIH3HTbCH TOl!Ka b. CJre,IJ,onaTeJibHO, npH TpeTbeH
I!OpMHpOBKe TO'!Ka a - npHTHrHB3IOUJ.aH. 0TCIO,IJ,a CJie,IJ,yiOT BCC YTBep)!(,Il,eHHH
TeopeMbi 5'.
3 a Me 'I a H He. YTBCp)K,IJ,eHHH TeOp(•Mbl 5', B ll3CTHOCTH, 03H3'laiOT, 'ITO
npH 3aMeHe npOMC)KYTKa BbiC3ll'HBaHHH .TIHI-!HCH TOKa ,IJ,aBJIC~IHe B e,IJ,eT ce6H
Tal< )Ke, I\aK H 11pH 3aMCHC I10TCIIUH3JibHOH .TJHIIHH JIHI-IHCH TOKa.
3. 0 6 JI a C T b cp H JI b T p a U H 11, 0 r p a II 11 l[ e II II a 5I ,Il, By M H JI H
H H 5I M H T 0 K a , n 0 T e H U H a Jl b II 0 H JI H H H e j:i H 11 p 0 M e )I( y T K 0 M
n bi c a 'I H n a H 11 H. CxeMa TaKoif o6Jra CTH cp11JibTpaUHH npe,IJ,CTaBJieHa Ha
p11c. 5, ecJIH C'IHTaTb, 'ITO TOlli<a B conna.·~aeT c TOlli<OH A, o6billHO Jie)!(amei-i
Ha 6eCI<OHC1IHOCTH. I\OHTypHb!C YCJIOBHH H MCIOT BH,Il,
u /vc· = - kfl~=const, u lc·c· = -ky, u jA =- :x. ,
1' lAc= const, v !AD= const, (6)
r,IJ,e h2 - ypOBCIIh )J<HJJ.I<OCTH B BO,Il,OCM e . fO,TJ.Orpacp T KOMITJICKCHOfO ITOTCH
Ulla.rra w = u + iv npe,IJ,cTaBJICH IIa pHc. 8.
DycTb B .J:aJi bHci1meM pacxo,IJ, )KH,IJ,KO<"TH Q 11 h2 ocTaJOTCH neH3MCHHbiMH,
TOf.ii:a 'liMCIOT M CCTO CJI C,IJ,YIOUJ.HC TeopeMbl.
T eo p eM a 4". flpu saMeHe npo.MC:JICljTKa BbtcarwaaHUJL CC' noreH-
lfCIG/lbHOU /lUH//Ctt: a) 0(1(]/lCHUC Ha /lUHl/JlX TOKa ljMCftb-
dw
lll£lCTCfl; 6) BCKTO{J CKOfJOCTll V = -- - HCl HC!.I3AteHHOLl •taCTU 110TCHlfU£l/lbft0U
dz
.llttHUtl yMCHbUtaercfl u Ha AllHUflx TOKa a6Auau TO'tKu D yMeHbtuaercfl,
a 8 6AU3 U TO'lKU C ljBC/lll'l ~LBClCTCJl.
B ca!\IOM ,IJ,eJie, rrycTb Tt - ro,IJ,orpacp 113MCHCHHOro KOMDJICI<CHoro no
TCIIUHaJia. 0To6pa3 11M T Ha Tt np11 CJIE·JJ.YIOUJ.HX HOpMHpOBKax: 1) Herro,IJ,
BIUI\Hble TOl!l(!l a H d, 2) HCHO)I.BH)!(Ha51 TOllKa C, 3) HCITO,Il,BH)KHaH TO'!Ka a
mpe3I<a [de']. Dp<11 n e pnofr I-IOpl\mpoBKe T O'!I<a d 6y- v
JI.CT npHTHI'HBaiOLUCH, T04Ka a - OTTa.HKHBa!Olli.CH,
,Il,pymx I-I e no,LIBmi<HbiX T04eK He 6yJJ.eT. Dp11 BTopoi-i
1IOpl\1HpOBKe T04Ka C 6y,IJ,e T OTTa .TJ KHBaiOUJ.CH. 3TO
:ll!atHIT, liTO J(;lil.i!Clli1 C fJ Hc~-~H I!l-15JX TOKa YM CI!bllla CT -
c c" @
c'
C! T
d
--+ dw
CH, I3CKT0p CI<OpOCHI V = , dz Ha JIHHH5IX TOKa B6JIH-
Puc. 8.
3 ll TO'IK'H d y~reHbwaeTCH, BuJI,H3H TOllKH c yneJIH'IH
nae TcH . 4T06bi IHOJIYllHTb OT06pa)!(e !-IHe npH TpCTbCH HOpMHpOBKe, OT06pa3HM
T r-r<l rroJiynoJwcy T{, HBJIHIOLUYIOCH HaJio>KCHHCM T 11 T1 np11 I-Ieno,IJ,BH)!(HbiX
TO'!Kax a, d, a. Dp11 3TOM T04Ka a Gy,IJ,eT npHTHrrrnaiOmeli. DpH oTo6pmKe
HHH Tt' Ha T1 TOllKa a 6y.neT TaroKe npnnnmnaiOmeli, TaK KaK K Heli JJ.OJI)!(Hbi
npH6JIH311TbCH c pa3JIH4HbiX CTopoH TOllKa rl H o6pa3 TO'IKH c. CJie,IJ,oBaTeJibHO,
dw
B CKTOP CKopocTH v = · dz Ha noTCHUH a JibHOH JIHHHH DC' yMei-Ibwa e TCH.
3THM TeopeMa ,Il,OKa3aHa.
3 a M e 'I a H 11 e . YTnep)!(,Il,CHHH TeopeMbi 4", B qacTHOCTH, 03Ha'laJOT, 'ITO
npH 3aMeHe rrpoMe:>~<yTI<a BbiCallHBaHHH CC' noTeHunaJibHOH JIHHHeli ,IJ,aBJie-
::\ YKpaHncKnA MaTeMaT. mypHan, T. V. N<.! 4. 393
u
IUle p = - f zz- yy Be):LeT ce6H TaK :tKe, KaK H npH 3aMeHe JIHHHH TOK3
110TE'HUI-I<iJibi!OH JIHIIHE'H .
.Teo p e ~~a 5". Jlpu sa.'deHe npoMeJIC!JTIW Bbtca•waamm CC' .llUH;•;et'i
r
TOIW: a) oaaACflUe p = - k u-YY Ha HCtl3}.tCHHOli ttaCTU !JBC.IlU'lUBae.~W(i
.1UiiUll TOKu: U IW HCU3J1LC.HHO{i .llllliUU TO/W .tJ6C.Illl'lll6GCTC5l; 6) BCKTOp CIW-
dW
pocru v = dz Ha He,usAteHHOLt .llUHuu TOKa, a6.1lusu ro<tKU D a H a noreH-
t.fUa.llbHott .llUHUU a6.1l u s u TO'tKU C' yae-tu•tuaaeTc<t, a Ha HeU3AteHNOli •tacru
!JBC.IlU'tuBaeJ~!Oli oWHllU TOKa ljo'ttC/ibUtaeTCJl. Pacxoo JIC/IOKOCTll 1tepe3 BCRK!JIO
CD5l3H!JIO •tacTb noreHLflWAbHOti .wHuu, t.t.MCIOilf!JIO a IW•tecTae KoHtfa TO•tK!J D
tLUt TO'LK!J C', !JBCAll'lUB(lCTCJl.
B caMOM .neJie, nycTb T 1 - ro.norpacp H3~IeHeHIIoro I<O:\In.:reKcHoro I!OTC H
uuaJia. 0To6pa3HM T na T1 npn cJie.nyromnx IIOpMHpoBKax : I) neno.nBmKHbll~
TO'IKH a H d, 2) HeTIO):LBH)IUiaH TO'IKa a 0Tpe3K3 [ad], 3) HE'IIO}J,BH)Kl!aH TOC:
Ka a oTpe3Ka [ac] , 4) n e no.nmDKHaH TO'II<a c'. OpH nepBoi1 uop ~mpoBKe TO'IKa d
6y.neT OTTaJIKHBaiOUJ,eH, TOLJKa c' - IIpHT5lfHBaiOlUE'H; JlP)THX !Jel10j ll3l!ji..:JI J,iX
TO'IE'K He 6y.neT. 0pH BTOpo~i HOpMHpOBKe TO'!Ka a - OTT(JJ!KHB(]!O!IlaH, flj)H
TpeTbeH HOpMHpOBKe - 11pliT51fHB(l!OlUaH. 0pH 'Ie TBE'j)TOii HOj)MHpOBKE' TO'I
Ka c' 6y.ne T oTTaJIKHBarometf. 0Tcro.na cJre.nyeT c npaBe;vmnoCTb yTnep)KJJ:emiH
TeopeMbi.
§ 3. MeTO.n Ma:lKopaHTHbiX o6JJacTeif
Ycrar·IOBJi eHHhre Bhrme o6tune Bnp·HaunoHHhie TeopeMhi TeopHH cpnJibTp n
unn B COlJE'TaHHH C Henocpe,'l,CTB eHHbiM HCf!OJ!b30BaHHE'M Teope:IIbl 0 ):LBH:tKeHHI1
rpaHH'IHhiX -roLJer< no3BOJrmoT yCTaJIOBHTh .noBOJihHO ooiUHi'' MeTOJl onpe; Lem>
HHH mnerpaJihHhiX xapmnepHCTHK cpHJihTpal.lHOHHbiX noroKon. CymHOCTb
3TOfO M eTO,[(a, Ha3b!BaeMOfO IIaMH "V!eTO,[J.Ol\I M3)!<0paiiT!Ih!X 06JJaCTE'H, COCTOHT
B CJ1E'l-lYIO!llE'M. )lJIH Toro, cno6bi onpe,[],e.rmTb -ry H.TIH 1-myro mnerpa.:lhHYIO
xapai<TE'p'HCTHI<y CpHJ! bTpaUHOHHOrO f!OTOKa npH 3a,[J.aHHOi'r o6JiaCTH cpHJibTpa
UHH G, HC06XO)J,HMO fiOCTpOHTh cjmJibTpaUHOHHhiC 110TOKH BO BCIIOMOraTCJib
HbiX o6.'laCT5IX cpHJibTpauHH 6 n G TaK, lJT06br, c O/J.HOH CTopoHbi, coOl·Be T
cTnyrmu.an HHTerpaJihii<:lH xapm<Tepucrmw ;vrH oGJiacreH 6 n G 6bi.:1a
H3BE'CTHa, C ,[],pyroH CTOpOHbl - t!'HCJI E'H HOC 3Ha'ICHHC 3TOi'! xapai<TCpHCTHKH
,[J.JIH o6J1aCTH G ,[J.OJDKHO 6hiTh 6oJihme HCKOMOH BeJIH'lHHhi, a ,1J.JI5l o6JiaCTH
9 - MCI!b!lle HCI<OMO~i BE'JIH'IHHbl. )lpyrH~fH CJIOBCIMH, B J<3)K,[J.OH KOHKpeT
IIOH 3a,'l,atiC TCOpHH cjJHJibTpaUH'H, TOtiHOC pemer m e KOTOpOti HCH3BCCTHO,
npen:raraeTCH 3a~IeH5lTb peme tme naHHo~i 3a,[J.a lJH p ellJCHI1CM JI.nyx ,ll.pyrax
3a,[J.a 1l, / l J15! KOTOpbiX 3TH p e m eHHH HaXO):L5lTC5l IIpOCTO !1 }J.a!OT B03MQ)KHOCTh
OUE'IIHTh HCKOMOC 3HiltiCHHC Kai<OI~i -JIH60 mne rp<!JibHOH xaparnep11CTI!KH
cnepxy H cmny *.
* B cml3l! c 3TH~ ! onJeTu~x. 'ITO 3aMeira pcme mm O;J.HHX cjm:I~>TpauHoH HhlX 3 3/\11'1
pcme mJe M lJ.pyrux, Gonce npocThix 3 a,LJ.a'! H, n '13CTIIOCTH, TaKHx, ;J.JJH KOTOIJI>IX oToClp3j]{a
lOUJ.<lH <}lyllKU,IHl llOJiytzaCTCFI K~lK rJiaDI-IC15I JUIHCi[IraH 1I3CTL OTIIOCHTCJ!hi!O !\ !(1.'10f0 nap<nteTpa,
IICTp C1laCTCa IJ TCOIJHH cjHWbTpaUHii i i,OilOJ!bHO '!aCTO r7, 8, ()]. 0;.(IWKO Tlflll 3TOM, il OTJI!:'IHC
OT rrpe;urarae~!OrO MCTO)J.3, IIC ycTaH31JJIII!l3,10Ch, 13 K3KO~! OT UOlli C'HIIH !JaXO)J.flTCR 'IC B ;m
Hllh!C llCJ!HlJHHbl, COOTIJCTCTllYIOIIl,He HCTIJI!H OMY llOTOKy, C COOTI3CTCTilYIOIIl,fn!H IJCJIIFIH II~D'iH
Bo MHorux 3a.D,at.Iax TeopHH cpHJibTp a u.nn nocrpoeHHe .n.a.tKe caJvrbrx npo
crelirunx Ma.tKopaHTHbiX ooJJaCTeli (; H G npn .n.onoJtbiiO ruupoKOM npoH3BOJie
B HX BbiOOpe npHBO.D,'HT K onpe.D,eJICI-IHIO IICH3BeCHihiX HJITerpaJibHbiX xapaK
TepHCTHK c .n.ocTarotiHoif CTeneiibiO TOtiHOCTH. PaccMoTpHM necKOJibKO npn
Mepon n 3a.D,at.I TeopiiH <jHIJibTpau.un Ha IIpHMe n enue npe;~Jiarael\taro Mero.n.a.
l. 0 11 p e .D. C JI e II H e p a C X 0 .D. a )J< H .D. K 0 C T .H 11 0 ,II, 11 .il 0 C K If M
cp .,1 10 T 6 e T 0 M 11 p H 11 p 0 H 3 B 0 JI b II 0 M K 0 JI If 'I e C T B e ill 11 Y H T 0 B.
Dycrb rpeoyercH o11pe.n.eJIHTb pacxo.n. .IKHII,KOCTH Q 11o.n. 11JIOCKHM cpmoT6eTOM
C OCIIOBai-IHCl\1 [ - r, r] C JIIOObi:vl KOJIHl!eCTBOM y (j)
ru11y1Iron 11pn rJiy6.nHe 3aJieraHH5I no.n.3el\!Horo =--=-,....,-=-=-~-:-~-r-1
no.n.oynopa, paBIIOH R (R > r), npn ycJIOBHH, /' ............. H .... \
'ITO rpaHIIU.bi no.n.oe~ron conna.n.mor c orpe3-
K aMH BCli~CCTBCHI-IOH OCH ( pHC. 9).
I .._
I
l
I
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4ro6b! nocrpoHTb .D,JI5I ::noro C.TJYlla>I oo
JI 3CTL> G, ou.eHHBaroruyro pacxo;1. cmny, 11ponc
;~cM '1Cpe3 TO'mH ( - r, 0), (r, 0) OKpy.lr<I tOCTL>,
oxBaTbiBaromyro Bee mnyrnbr. DycTb Ro -
pa.n.Hyc ::noii oKpynmocTH, (0, - b) - ee
U.CI!Tp. Tor.D,a 13 COOTBCTCTBHH C .D,OK333HHbiMH
I I R
I I
\ G I
',, ,..// I
l?T7T~;:;~:;:~~:-.7r:~~-~~~-
Puc. 9.
nL>rrue napnaU.HOHHhiMH TeopeMaMn B K <tllecTne o6Jiacnr G MOn<CM B3HTb
o6JiacrL>, orpaHnl!er·myro .n.yroli 3Tofr OKpyi!mocnr, npHMbiMH JIHHHHMH, BbiXO
.D.Hl.I.I.HMH H3 TOl!K'H (0, - b), If .n.yroli OKJ>Yn<HOCTH C U.e!ITpO~! B 3TOH TOl!hC
pa.n.·Hyca Rr =' R - b. Dpu 3TOl\I .n.yrn OKp y.IKHOCTCH .D,OJinUibi 6b!Tb JIHHHHMH
TOKa, a npm!bie JIHHHH - 110TeHU.H3JibHbl.\liH JIHHHHM'H (pHC. 9, nyHKTHpHbie
JIHHHH) . B Ka'recTne o6JiacTn G, ou.ennnmou(eH pacxo.n. cnepxy, .D.JIH npocrorbr
MO.IKeM B35ITb 06JiaCTb, 110JIYllaiOlUYIOC5I H3 33.D,3HIIOH 06JI3CTH cpHJibTp3UHH,
nyTCM y.n.aJieiiH5I runyHTOB. [(JI5I paCXO,'I:a )!('H.D,KOCTH Q, COOTBCTCTBYIO!.I.I.ero
o6JracrH G, 1\aK :H3Becnro, MO.IKeM Hamrcarb
- K'
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rcuu.naJr, O'IC!JII,li,IIO, 6y ,li,CT
w = - i l<H(In .!__+ ian )
- an R" 2 '
r.L(C u:r (a > I) yroJr Me.tK.D.Y lllp5IMOJIHHeHHbiMH y tiaCTK<Bill rpaHHllbi
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ll jl i!UJII!)I{CllllOro j}CUJCI!I!H '!aCTO IIO.l\T!lCJl:iK/\<lJlaCb pC3 YJihTaTaMH 3KCIIejlHMCHTO!l U Ha6 JIIO·
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0,84/?H < Q < 1,17/di;
eCJIH cpe.nHee apncj:>MeTWiecKoe sepxueif H Hmlmei-f oue1-101< Q* = 1,005 !?H
ITpHHHTb 33 ITpH6JIH)K€11HOe 3H34€HH€ HCKOMOrO paCXO,Ua, TO ITOrpelllHOCTb
He 6y,neT ITpeBOCXO,UHTb 16,4% .
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IJJITYHTOB C'I'HTa!OTCH paBHbiMH €,'l,HIIH
U €. Tpe6yeTCH oiTpe,rJ,eJIHTb seJIHl!HHY
BeKTOpa CI<OpOCTH cpHJibTp3UHH B TO'l
KC A. B COOTB€TCTBHH c .UOK333HHbl -
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T€pecy!OU-L€H H3C B€JIH4Hllbl CHH3Y ,'J,OCT3T04HO ITO,'l,Cl!HT3Tb B€JIHl!HHY
B€KTOpa -CKOpOCTH V ITpH yCJIOBHH, liTO ,'1,83 Jl€Bb!X lllllYHT3 OXB3Tbi
B3IOTCH K3KOH-JIH60 JrHI-IH€H TOr<a; _.llJI51 011.€ !-IKH CBepxy ,'l,OCT3TOl!HO ITO,UC4H
T3Th B€JIH4HHY B€KTOpa CKOpOCTH V ITpH OTCYTCTBHH .UBYX JI€Bb!X UIITYHTOB.
4T06b! H3HTH KOMITJI€KCHbl€ ITOT€H IJ.H3Jlbl W H 1V, COOTB€TCTBYIO!ll,H€ V H V,
sse.neM ITepeMermylo ~1 = ; 1 + i'71 = Vl + z~. 06pa3 bi .nsyx Jresbrx -rniTyH
TOB B IlJIOCKOCTH sl 6y,'J,yT Jl€)!{3Th BHYTPH rrpHMoyroJibHHKa, orpmml!eHHOro
b y- -
rrpHMhlMH §1 = 2' S1 = l+b 2, '1/1=0, 171 =1 (pHC. 11). )lJIH KOMITJI€KC-
2V1+b2+b :zVi+b2-b
HOrO ITOT€HII.H3JI3 ":_V, BBO,'l,H ITOCTOHHHbl€ {i = 4 , /' = 4 ,
r=Yl+y1 H rrpOM€)!<YT04Hbl€ ITepeM€HHbl€ ~2 =~-1 -{3, (3 = { 2 +r {4 =~!, - ~~- r'
~ 1+{3-r
,. _ ~l+I-2 1 +{3+r
:,- (pHC. 12). IlOJiyl!aeM
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- n
dll} I f:lH V2 (I +:~-r) z I
dz A=- n (1 +~) (1 +{3+r) VI-~- z=O.
Bbrpa)!{aH z H { 4epe3 ~1 , HMeeM
. z -I = V ~~ =- 1 I. = - il/ (1 + {3) (1 + {3 + r) '
ll-?; z=O V1-?; ~,--1 ' 1 +{3-r
dll} I _ ikHV' 2(1 +~-r)
dz z=O- n (1 +{3) (1 +{3+r)
11, CJI C)J.OBaTCJlbHO,
v=kHI 2(1 + (3-r)
- :rr (1 + (3) (1 + (3+ r)"
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11MeeM
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w = n arc Sill c,, ~= --1-<V1+b~-t-2t1 >·
1 + '1 +b~ -
d~ ~ __ l}_!! _ _ V~_z_ ,
dz z= O- :rr 1 + ll + b2 Y1--~' z=o'
-~z=~- 1 =- i V 1 + ll_+_iJ2 , ~-0! I = i kHVJ __ 2 ---
V 1 " z=O dz z= O :rr 1 + V 1 + b~
H, CJH?)J.OB3TCJJbHO,
v= kJl; 1+vT-+ b2
II Jl.'l fl HCI<O'IIOH 13bl:\O.LJ.!IOH CKOpOCTI1 VIA HMCCM OUCHKH
hH 1 /2( I+ (3- r) 1 kH I ( 2
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1ff +b3 =4,1231, (3=3,06155, r= 1,4584,
- kH kH
1) = ---;;- 0,6248, 1J == - 0,4818
.. - :rr
11, 11pHI15IB 3a np116JIIUKCI-!HOC 3H3l!CHHe Bb!XO)l.HOH CKOpOCTH VIA cpe)l.Hee a pHcp
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:rr
i l.flLUYIO 12,92%.
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o6JiaCTH cpHJibrpauHn npe.ncraBJieHa Ha pHc. 13. KomypHbre yc.TJOBH51 .nmr
onpe.neJieHH51 KOMITJieKcHoro rroreHUHaJia w = u + iv ( u = -ll ( ~ + y)) HMeJOT
BH)l, '
uiABc=O, zz lnE=kT, vlcn=const, viAE=const,
Plcn=O, PIAE=O,
rrpH 3TOM ceq.eHHe KaHaJia ABC H rJiy6HHa T 3a,D,arorcH, a BH,D, .nenpeccHoH
HbiX KpHBbrx CD H AE 3apanee He 3a,D,aerc51. Tpe6yerc51 onpe,D,eJIHTb pacxo.n
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,D,aJibHeiiwero rroHHTHe rrapaMerpa KaHaJia
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dx
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dx
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,D,HcpcpepeHUHpyeMaH cpyHKIJ.HH. 4T06bi rrpHMCHHTb K llOCTaBJICHI-!OH 3a,D,aqe Me
TO)], Ma)!(opaHTHbiX 06JiaCTeH, )l,OCTaroqno llOCTpOHTb KOMllJICKCHbie llOTeH
UH3Jibl W H '/J}• COOTBeTCTBYIOW:Jie KOTOpb!M paCXO)l,bl )K'H.LJ:KOCTH Q 'II C] 6biJIH 6bi
TaKHMH, qro Q < Q < Q.
JlJIH rnocrpoeHHH KOMITJieKcHoro rrorenuHaJia w npe,K,n:e ncero H3 paccMo
rpeHHH fO)l,Orpacpa BCKTOpa CKOpOCTH 3aJI1eqaeM,- 'ITO He3aBHCHMO OT BH,D,a
ceqeHHH KaHaJi a ABC .nerrpeccHOHHbre KpHBbie CD 'H AE rrpe.ncraBJimor co-
6oii KpHBhie, y.naJimow;necH or ocH y c y6biBaHHeM y, no ropa3,n:o Me,D,JieJme'e,
* .[{JUI HCKOTOpbiX CClJCHHH KaHaJIOB, }lJill TpaneU.OH,l\aJibHOro, TpeyrOJibHOrO H HCKO·
TOpbiX ,npyrHX '!aCTHbiX CJiy'laCB H3BCCTHO TOlJHOe pCliiCHHC :HOl't 3a,nalJH (7, 10, II). .[{ml
11pOH3BOJibHO 3a,l\aHHOrO CClJCHH!I KaHaJia TOlJHOC peliiCime 3TOif 3a,l\a'IH 11pe,ncTaBJI!!CT 11pHH
l.\1!11HaJ!bHbiC rpy ,nHOCTH, nocKOJibKY MeTo.n H. E. )KyKoucKoro 3.1\CCb oKa3biBaeTC!I He
npnMeHHMbiM.
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n ::r
f:J'-- ':) ll -(/- 0 H npoxo,n:HmHe coOTBeTCTBeHHO l!epe3 TOl!KH A 11 C - ~
(pHC. 13) . 0TCIO)I:a B CHJIY TCOpeM § 1 B !<al!CCTBe HH)KHCH OU.eHKH JI.JIH Q
MO)!(Cl\1 B351Tb p3CXO)I: )!UI) (KOCTH 9 llflH cpHJibTp3U.HH B npHMOyrOJibHHKC
A CD' E' (pHc. 13) npH CJICJI.YIOW.HX KOIITYPHhiX ycJIOBHHx
u iAc=O, ub'E'= kT, vlcJJ·=const, viAE'=const.
C.Tie,J.OBaTeJihiiO, HMee~I
Q ;:> Q = 2kR.
DycTb G H w = u + iv o6o3Hal!<HOT TO )!(C cal\10e, liTO H B § 1, rrpHl!eM
u iAF1=0, v l11c=const, n lcn=f(z) :;> O, viDA=const,
f }l.C r (z)- 3a,naHH35I cpyHKU.HH TOl!KH .li.YrH CD. Tor,na TOl!HO T3K )!(C, K3K
H 13 § 1, H3 TCOpC~Ihl 0 JI.BH:tKCIIHH rpaHHlfHbiX TOl!CK [6] Bb!TCK3CT CllpaBC}(JIH
BOCTb CJIC,'l.Y!Oil(CH TeOpCMbl.
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c paH/lftHou noreHtflULlbHDt'i AllHllll AB u 2paHWlHbtX AttHUflX roKa BC u DA
parxocJ :J/CtlcJKOCT'l lj.lteHbl/taeTCfl, eCA/l llOAO:JICUTb
u lcD=/1 (z) < min I (z),
•C C/J
r,ne f, (z) - cj)yHKUHH TOl!KH z ,nym CD.
B CllJlY :HOH TCOp CMbl H TCOpeM § 1 B K3l!CCTBC BepXHCH OUCHKH )I:JI H Q
1\10)1\CM 13351Tb paCXOJI: )!{Hj(KOCTH Q npH cjmJibTpaU.HH B llpHMOYfOJibHHKC abed
(puc. 13) npH cJrc,nyiomHx KOHTYPHhiX yCJIOBHHX
u lab =0,
J1enw no.Tiyl!aeM
'V I =const
be ' '''I =const da
Q = i·~k~~H(2R+T0 tgr/+T0 tg(J').
06o3Ha'laH Tenepb qepe3 Q0 M'HHIB13JibHOe 3Ha'leHH e Q npH T0 .:;;; T, HMeeM
Go=fdl y--,--,--c =2kR 1 ~Yl+o-2 [l+o2 (1+Vl+o2)]
T0o:H(l+ Ho- ') • jl +o-2
If. c:IC/ J;OBaTCJibHO, rrpu T, y,noBJICTBopmm.u.eM ycJIOB'HIO
T ::> T* = H(l + Vt + cJ-!),
2kR < Q< 2kR(l +2o j/1 +al+2c.l'2)
H 11p11 npH6JIH)!(CHHOM 3H3l!CHHH Q, paBHOM cpe,n:HeMy apHcpMCTHl!CCKOMY
Q*=2kR(l +o Vl +o'+o),
0
norpemuoCTh B npou.eHTax onpe,neJIHeTCH BCJIHl!HHOH V 100. Dp.H Tex )!(C
l+ol!
)'C.TIOBHHX, B35IB 33 ,n:JITIHY npOMC)!(YTKa llbiCa l!HB3HH51 BCJIHl!HHY
2L*=R(l +<tVl. +o2+o2),
d
.11.."151 IIOrpeiiiHOCTH B npOUCHTaX OY.ll.CM HMCTb BCJIHlJHHY v 100.
1 +a~
399
5ICHO, liTO JlJ1 5I rTJ,OBO.TibHO Il1Hp01<01'0 KJI<1CCa K<1H3.!IOB npOH3BOJ1bHbiX CC
l!-CHHH napaMeTp K<1H3Jia <5 6y,D.eT )J.OCTaTOl!HO M3JIOH BCJIIFIHIIOH. 3TO, HanpH
H
\Iep , 6y.n:eT npH l\13 Jib!X yrJJaX a' H [J' IWH np:H :\Ia:IOi:I BC:I'HtJHHC -1-,. n.'IH ..
3T0f0 KJJaCCa KaHCJJIOB OlJCBH).l,HO, 11TO fJOJIYl!CIIHbiC OIJ,CHKH ,.'!JI5! pacxo,TJ,a ;.KH)J.
KOCTH H )J.JIHHbl ·npOMC.II<yTI<a Bb!CC14HBC11UI5I onpc;~e.15IIOT HX ~ JLOCT3TOl!IIO B bi
COKOH -cTeneHbiO TOlJHOCTH. TaK, HanpHMc p, ,TJ,JIH Tpeyro.TibHOro cetJCIIHH KaHaJ1a ,
CIHfMCTp'Hl!HOfO OTHOCHTC.JibHO OCH !J, HMCCi\1
d'= tg u., ! Q-Q* I } _ < sin cc
I Q''' '
H
H, CJI C/LOBaTCJlbHO, Il[HI 1' > H + -,-- cpe,D.Hee apmjnieTHlJCCI<Oe B e.ilHl!HH '?
Sill u.
H Q )J.3CT 3H34CHHC pacxo.n:a C norpCilliiOCTbiO, HC npeBblillCliOI.I.I;CH 5,3 °~,
rrpu a < 3" 11 He rrpeBbiillaiOmeii I 0,5 % -- npH a < 6 1 .
JUHEPATYPA
I. M. B. K e JJ A him 11 M . A. JI a B p e H The n, 06 yCToii<IHBOCTII pemcmu1 JaJJ.;Jtm
JlHpiiXJJe, l1AH, cepHH MaT. ( 1937), 551 - 596.
2. M. B. K e JJ JT. bllll, 0 pa3pem.IIMOCTH H yCToii<mnocTH :JaJT.a 'IH JlnpiiXJie, liAH
CCCP, 18 (1938), 315-318.
3. M. B. KcJJ;<hiill, 0 3a;<a 'le llnpnxJJc, JlAH CCCP (1941), 308-309.
4'. M. B. K e JJ A him, 0 p<l3pemHMOCTH 11 yCToii<muoCTH 3<1Aa'IH JlnpnxJJe, Ycnexn
'I3TCM. HayK, 8 (J941J), )71-2311.
B. M. A. JI all p c H T h e B, KompopMHbiC OT06paiKCIIIHl, rocTCXH3JT.<IT, MocKIJa ( 1946 ).
6. f. H . f1 0 Jl 0 iK H ii, 0 ABHiKCHHII rpaHH'IHhiX TO'ICK OT06paiKaCMbiX 06J!aCTeii,
Ycnex n MaTeM. nayK, T. 7, o. ii (52), (1952).
7. n. 5I. n 0 J[ y 6 a p II II 0 ll a- K 0 'I II H a, Teopml i\BHiKCHHH rpyHTOllhiX BOA, roc
TCXH3;<aT, MocKBa ( 1932).
8. A. M. C e H K 0 B, f1. <!>. ¢II Jl h 'I a K 0 ll, f1pn6Jl!liKCHHI~ e MCTO;<hl paC4CTa CTal\HO
HapHOrO ;<IJ!l)KCHH51 rpyHTOBhiX BOA ITO;< fH;<pOTCXHH4CCKHMH COOpy)KCH!lH~III, Y!J;<-00
AH YCCP, Kneo (1952).
9. ct>. 5. H e •1 h co 11 - C K o p n .R K on, c!>HJJhTpal\HH o o;<Impo;umfl cpe,il,e, Moc
Kna (1949).
10. B. B. B c A e p II II K 0 B, <l>nJ!hTPai.\HH 113 KaHaJJOB, rocCTpoiinJ;<aT, MocKBa (1934).
11. IO. Jl. Co K o JI o o, 0 pacqeTe QJIIJ!hTpaunH w1 KaHa.~a TpaneuonaaJJL>Horo ceqe
HHH, JlAH CCCP, T. 79 (1951), 759-762.
f1o.~y4CHa 3 IUOHH 1953 r.
Knen.
0380
0381
0382
0383
0384
0385
0386
0387
0388
0389
0390
0391
0392
0393
0394
0395
0396
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|
| id | umjimathkievua-article-7753 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:37Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bc/e9d4cbeccc39b8b0a0d8c6150b9a90bc.pdf |
| spelling | umjimathkievua-article-77532023-08-15T10:11:31Z Метод движения граничных точек и мажорантных областей в теории фильтрации Метод движения граничных точек и мажорантных областей в теории фильтрации Положий, Г. Н. Положий, Г. Н. Начало исследований решений краевых задач в зависимости от изменения области было положено в работах М. В. Келдыша и М. А. Лаврентьева. М. А. Лаврентьевым при помощи выделения главных линейных частей относительно малого параметра функций, дающих отображение некоторых простейших областей на области, им близкие, были получены качественные выводы о поведении расхода жидкости, выходных скоростей и противодавлений при удлинении шпунтов в случае фильтрации под плоским флютбетом. В настоящей работе при помощи теоремы о движении граничных точек, установленной нами ранее, исследуется вопрос об изменениях расхода жидкости, выходных скоростей и противодавлений, или, какими будем говорить, интегральных характеристик, в зависимости от всевозможных изменений произвольной области фильтрации. На основе этого дается метод определения указанных интегральных характеристик фильтрационных потоков, основная идея которого заключается в оценке искомых величин, связанных с решением той или иной краевой задачи, сравнением с соответствующими величинами решений той же краевой задачи для таких областей, для которых эти решения известны или находятся просто. Предлагаемый метод применяется к рассмотрению ряда примеров и задач теории фильтрации. Начало исследований решений краевых задач в зависимости от изменения области было положено в работах М. В. Келдыша и М. А. Лаврентьева. М. А. Лаврентьевым при помощи выделения главных линейных частей относительно малого параметра функций, дающих отображение некоторых простейших областей на области, им близкие, были получены качественные выводы о поведении расхода жидкости, выходных скоростей и противодавлений при удлинении шпунтов в случае фильтрации под плоским флютбетом. В настоящей работе при помощи теоремы о движении граничных точек, установленной нами ранее, исследуется вопрос об изменениях расхода жидкости, выходных скоростей и противодавлений, или, какими будем говорить, интегральных характеристик, в зависимости от всевозможных изменений произвольной области фильтрации. На основе этого дается метод определения указанных интегральных характеристик фильтрационных потоков, основная идея которого заключается в оценке искомых величин, связанных с решением той или иной краевой задачи, сравнением с соответствующими величинами решений той же краевой задачи для таких областей, для которых эти решения известны или находятся просто. Предлагаемый метод применяется к рассмотрению ряда примеров и задач теории фильтрации. Institute of Mathematics, NAS of Ukraine 1953-11-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7753 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 4 (1953); 380-400 Український математичний журнал; Том 5 № 4 (1953); 380-400 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7753/9441 Copyright (c) 1953 Г. Н. Положий |
| spellingShingle | Положий, Г. Н. Положий, Г. Н. Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_alt | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_full | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_fullStr | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_full_unstemmed | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_short | Метод движения граничных точек и мажорантных областей в теории фильтрации |
| title_sort | метод движения граничных точек и мажорантных областей в теории фильтрации |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7753 |
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