ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
“„Š 532 + 533.2
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®«ã祭® 04.03.2003  ¥à¥á¬®â७® 15.07.2003
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¨å ¤¢¨¦¥­¨ï.
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஧à å㭪㠮¡'õ¬­®i £ãá⨭¨ ⢥नå ç á⨭®ª i ùå 袨¤ª®áâi.
In this work explores a stream dynamics of the granular continuum in sloping tray on the Ostvald de Uayel's model of
the colliding particles. The boundary problem for the stationary plane motion of granular continuum in sloping tray is
formulated. The boundary condition of slipping is formulated. Two problems are considered: the motion of the granular
continuum with a free surface in sloping tray and the Puazeil 
ow. Algorithm of the numerical solving is developed.
Results of the calculations are presented on the graphs of the volume density distribution of the particles and velocity
eld.
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ç¨ ¤¨­ ¬¨ª¨ £à ­ã«¨à®¢ ­­®© áà¥¤ë ¢ ­ ª«®­­®¬
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à¨á⨪ ¯®â®ª  ¨ ¢ëïá­¥­¨¥ ¢«¨ï­¨ï ­  ­¨å í¬-
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¨ á â®çª¨ §à¥­¨ï ¯à¨«®¦¥­¨© [1,18]. ƒà ­ã«¨à®-
¢ ­­ ï (¨«¨ áë¯ãç ï) á।  ï¥âáï ¤¨á¯¥àá­®©
á¨á⥬®©, ¤¨á¯¥àá­ ï á®áâ ¢«ïîé ï ª®â®à®© á®-
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[13].  §«¨ç îâ âਠ०¨¬  ¤¢¨¦¥­¨ï. ¥à¢ë© {
¬ «®¨­â¥­á¨¢­®¥ ¤¢¨¦¥­¨¥, ª®£¤  ⢥à¤ë¥ ç áâ¨-
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[14]. ‚â®à®© { ç áâ¨æë §  áç¥â ¡®«¥¥ ¨­â¥­á¨¢­ëå
á®ã¤ à¥­¨© 㤠«ïîâáï ¤à㣠®â ¤à㣠, ¯®ï¢«ï¥â-
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ç¥áª®¬ã ⥫ã. ’à¥â¨©, ᮮ⢥âáâ¢ã¥â "¡ëáâà®-
¬ã" ¤¢¨¦¥­¨î á।ë, ª®£¤  ç áâ¨æë ᮢ¥àè îâ
å ®â¨ç¥áª®¥ ¤¢¨¦¥­¨¥ ¯®¤®¡­® ç áâ¨æ ¬ £ §  [9].
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¦¥â ¡ëâì ¢ë§¢ ­ ª ª ¡ëáâàë¬ á¤¢¨£®¢ë¬ ¯¥à¥¬¥-
饭¨¥¬, â ª ¨ ¢®§¤¥©á⢨¥¬ ¢¨¡à æ¨©. ‚ ¯®á«¥¤-
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å à ªâ¥à¨á⨪ íâ¨å ¢¨¡à æ¨© [5,8,13,15]. ¥®¡-
室¨¬®áâì ⥮à¥â¨ç¥áª®£® ®¯¨á ­¨ï â¥å­®«®£¨ç¥-
áª¨å ¯à®æ¥áᮢ ¤«ï £à ­ã«¨à®¢ ­­ëå á। ¯à¨¢¥-
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 ˆ.ˆ ˆ¥¢«¥¢., 2003 41
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
¯®«ì§ãîé¨å í¬¯¨à¨ç¥áª¨¥ § ¢¨á¨¬®á⨠¯à¨ ¯®-
áâ஥­¨¨ ८«®£¨ç¥áª¨å ᮮ⭮襭¨© [5,7,13,15].
‘ â®çª¨ §à¥­¨ï ८«®£¨¨ £à ­ã«¨à®¢ ­­ë¥ áà¥-
¤ë ïîâáï à §­®¢¨¤­®áâìî £¥â¥à®£¥­­ëå á।,
¤«ï ª®â®àëå å®à®è® à §¢¨âë ®¡é¨¥ ¬¥â®¤ë ¯®-
áâ஥­¨ï ¬ â¥¬ â¨ç¥áª®© ¬®¤¥«¨ [16]. ’à㤭®áâ¨
¢®§­¨ª î⠯ਠà¥è¥­¨¨ ª®­ªà¥â­ëå § ¤ ç, ª®£¤ 
âॡã¥âáï ãâ®ç­¨âì ८«®£¨ç¥áª¨¥ ᮮ⭮襭¨ï ¨
¢ë¡à âì ᮮ⢥âáâ¢ãî騥 í¬¯¨à¨ç¥áª¨¥ ª®­áâ ­-
âë [17,18]. ®«ã祭¨¥ ª®­ªà¥â­ëå §­ ç¥­¨© ª®­-
áâ ­â âॡã¥â ¯à®¢¥¤¥­¨ï â®­ª¨å ¨ ¤®à®£®áâ®ï-
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­®©" á®áâ ¢«ïî饩 ⥭§®à  ­ ¯à殮­¨©
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¦¨« á«¥¤ãî饥 ᮮ⭮襭¨¥ [12]:
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 BnZr2r
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£¤¥
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¦¥­¨¥ ï¥âáï  à£ã¬¥­â®¬ ¢ ¯®«ì§ã¢ë¡®à á®®â-
­®è¥­¨© (3),(4) ª ª ®á­®¢®¯®« £ îé¨å ¤«ï ¡ëáâ-
àëå ¤¢¨¦¥­¨© £à ­ã«¨à®¢ ­­®© á।ë. ‡¤¥áì ¥é¥
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0
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£¤¥ k, 
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.
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g ãà ¢­¥­¨ï ­¥à §à뢭®á⨠¨ ¤¢¨¦¥­¨ï ¨¬¥îâ
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dt
+divv = 0; (6)


dv
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㣫®¬  ª £®à¨§®­âã (à¨á. 1). ®«®¦¨¬, çâ®
v = (u(y);0;0);  = (y).
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⮬ â¨ç¥áª¨, «¥¢ ï ç áâìãà ¢­¥­¨ï(7) ®¡à é ¥â-
áï ¢ ­ã«ì,   ᮮ⭮襭¨¥ (2) ¯à¨®¡à¥â ¥â ¢¨¤ [12]
42 ˆ.ˆ ˆ¥¢«¥¢.
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
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¯®«®¦¨âì 
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m = 2
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1
®â  ãáâ ­ -
¢«¨¢ ¥âáï í¬¯¨àç¥áª¨¬ ᮮ⭮襭¨¥¬ [17]

1
= m
0

(
b
BnZr)
: (9)
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b
{ ®¡ê¥¬­ ï ¯«®â­®áâì ⢥म© ä §ë
¯à¨ ¯«®â­®© 㯠ª®¢ª¥; m
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BnZr1

; (10)
	 = 

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:
¥­ã«¥¢ë¥ á®áâ ¢«ïî騥 ⥭§®à  ­ ¯à殮­¨©
¢ ¤ ­­®¬ á«ãç ¥ ¡ã¤ãâ
T
xx
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d
dy

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;
T
xy
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p
2




du
dy




du
dy
; (11)
T
yy
= BnZr	

d
dy

2
;
  ãà ¢­¥­¨ï ¤¢¨¦¥­¨ï ¢ ¯à®¥ªæ¨ïå ­  ®á¨ ª®®à¤¨-
­ â Ox; Oy ¯à¥¤áâ ¢«ïîâáï ¢ ¢¨¤¥
0 =
dT
yx
dy
+
gsin; 0 =
dT
yy
dy
BnZr
gcos: (12)
¥âà㤭® § ¬¥â¨âì, çâ® ãà ¢­¥­¨ï (12) ¨¬¥îâ
¯¥à¢ë© ¨­â¥£à «
T
yx
+ tgT
yy
= const: (13)
  £à ­¨æ¥ , ¯à¥¤áâ ¢«ïî饩 ᮡ®© ⢥à¤ãî
á⥭ªã, ¨¬¥¥â ¬¥áâ® ¯à®áª «ì§ë¢ ­¨¥, ᪮à®áâì
¯à¨«¥£ î饣® ª  á«®ï § ¢¨á¨â ®â ¬­®£¨å ä ªâ®-
஢¨ ¤«ï᢮¥£® ®¯à¥¤¥«¥­¨ï âॡã¥â, ª ª¯à ¢¨«®,
ãáâ ­®¢«¥­¨ï í¬¯¨à¨ç¥áª¨å § ª®­®¬¥à­®á⥩ [18].
‡¤¥áì ¢®á¯®«ì§ã¥¬áï á«¥¤ãî騬¨ á®®¡à ¦¥­¨ï¬¨.
„¢¨¦ãé ïáï ¯® âàã¡¥ ¦¨¤ª®áâì ¨á¯ëâ뢠¥â ᨫã
ᮯà®â¨¢«¥­¨ï ¢ï§ª®£® â७¨ï,   è¥à®å®¢ â®áâì
á⥭®ª ¨ ­ «¨ç¨¥ ­  ­¨å ¢ëáâ㯮¢ ¯à¨¢®¤ïâ ª
¤®¯®«­¨â¥«ì­®¬ã ᮯà®â¨¢«¥­¨î (ᮯà®â¨¢«¥­¨¥
¤ ¢«¥­¨î), ¢¥«¨ç¨­  ª®â®à®£® § ¢¨á¨â ®â ᪮à®-
á⨠¤¢¨¦¥­¨ï ¦¨¤ª®á⨠[19]. ‚ ­ è¥¬ á«ãç ¥ ¡ã-
¤¥¬ áç¨â âì ᮯà®â¨¢«¥­¨¥ ¤ ¢«¥­¨ï ¯à®¯®à樮-
­ «ì­ë¬ ᪮à®áâ¨. ’®£¤  £à ­¨ç­®¥ ãá«®¢¨¥ ­ 
⢥म© á⥭ª¥ ®ª®­ç â¥«ì­® ¬®¦­® § ¯¨á âì ¢
¢¨¤¥
jT
yx
j= f

jT
yy
j+ D

juj; (14)
£¤¥ f

{ ª®íää¨æ¨¥­â â७¨ï ᪮«ì¦¥­¨ï; D

{
ª®íää¨æ¨¥­â ᮯà®â¨¢«¥­¨ï ¤«ï ¯®¢¥àå­®á⨠.
  ᢮¡®¤­®© ¯®¢¥àå­®á⨠BnZr ¯®â®ª  ¢ë¯®«-
­ïîâáï ãá«®¢¨ï ®âáãâáâ¢¨ï ­ ¯à殮­¨©
T
yx
= 0; T
yy
= 0: (15)
3. Ž’ŽŠ ‘Ž ‘‚ŽŽ„Ž‰
Ž‚…•Ž‘’œž
 áᬮâਬ ¯®â®ª ¯®áâ®ï­­®© ⮫騭ë h á®
᢮¡®¤­®© ¯®¢¥àå­®áâìî BnZr. à¨¬¥¬ ¢ ª ç¥á⢥
å à ªâ¥à­ëå ¢¥«¨ç¨­ £«ã¡¨­ã h, ã᪮७¨¥ ᢮-
¡®¤­®£® ¯ ¤¥­¨ï g ¨ ¯«®â­®áâì ⢥म© ä §ë 
.
’®£¤  ᮮ⢥âáâ¢ãîéãî ªà ¥¢ãî § ¤ çã ¢ ¡¥§à §-
¬¥à­ëå ¯¥à¥¬¥­­ëå ¬®¦­® áä®à¬ã«¨à®¢ âì ¢ ¢¨-
¤¥
d
dy
 
	

d
dy

2
!
+
cos
Eu
 = 0; (16)
0 < y < 1;
d
dy

1
M(y)




du
dy




du
dy

+Re sin = 0; (17)
0 < y < 1;
d
dy
= 0; y = 1; (18)
ˆ.ˆ ˆ¥¢«¥¢. 43
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
¨á. 2. ®â®ª ᮠ᢮¡®¤­®© ¯®¢¥àå­®áâìî,
Re

= 1:0; Re = 1:0; Eu = 1:0:
1 - 

= 24

; 2 - 

= 42

; 3 - 

= 56

¨á. 3. ®â®ª ᮠ᢮¡®¤­®© ¯®¢¥àå­®áâìî,


= 42

; Re

= 1:0; Re = 1:0:
1 - Eu = 0:3; 2 - Eu = 1:0; 3 - Eu = 5:0
du
dy
= 0; y = 1; (19)
1
M

du
dy

2
BnZr
Re
Re

u =
= f

EuRe	

d
dy

2
; (20)
y = 0:
‡¤¥áì Eu = 
BnZr1
g
BnZr1
h
BnZr3
{ ç¨á«® ©«¥à ; Re =
= 2
p
2
h
2
m
0
BnZr1
- ç¨á«® ¥©­®«ì¤á ; M(y) = (
b
BnZr
BnZr)=; Re

= D

BnZr1

p
gh { ¡¥§à §¬¥à­ë© ¯ à -
¬¥âà.
  ᢮¡®¤­®© ¯®¢¥àå­®áâ¨, ¢ ᨫ㠮âáãâá⢨ï
¢­ãâ७­¥£® â७¨ï, ¢ë¯®«­ï¥âáï ãá«®¢¨¥
 = 
0
; y = 1: (21)
“à ¢­¥­¨¥ (16) ᮢ¬¥áâ­® á ãá«®¢¨ï¬¨ (18), (21),
¯à¥¤áâ ¢«ï¥â § ¤ ç㠊®è¨, ª®â®à ï ¬®¦¥â ¡ëâì
à¥è¥­  ­¥§ ¢¨á¨¬® ®â u. à¨ í⮬ ­ ¤® ãç¥áâì,
çâ® äã­ªæ¨ï 	 ¨¬¥¥â ®á®¡¥­­®áâì ¯à¨ y !
1. “á«®¢¨¥ ®£à ­¨ç¥­­®á⨠à¥è¥­¨ï § ¤ ç¨ (16),
(18), (21) ­  ¢á¥¬ ®â१ª¥ [0;1] ¯à¨¢®¤¨â ª  á¨¬-
¯â®â¨ç¥áª®¬ã ¢ëà ¦¥­¨î ¤«ï  ¯à¨ y ! 1:
 = 
0

1+
k cos
9Eu
(1BnZry)
3

;
ª®â®à®¥ ¬®¦¥â ¡ëâì ¨á¯®«ì§®¢ ­® ¯à¨ ç¨á«¥­­®¬
à¥è¥­¨¨ § ¤ ç¨ Š®è¨.
 å®¦¤¥­¨¥ ᪮à®á⨠u(y) ᢮¤¨âáï ª ç¨á«¥­­®-
¬ã ¨­â¥£à¨à®¢ ­¨î ᮮ⭮襭¨ï (17) ¨ ®¯à¥¤¥-
«¥­¨î ¤¢ãå ¯à®¨§¢®«ì­ëå ¯®áâ®ï­­ëå ¨§ £à ­¨ç-
­ëå ãá«®¢¨© (19), (20). ‚¢¥¤¥¬ ®¡®§­ ç¥­¨¥
W(y) =
y
Z
0
(z)dz:
’®£¤  à á¯à¥¤¥«¥­¨¥ ᪮à®á⨠¬®¦­® § ¯¨á âì ¢
¢¨¤¥
u(y) = C
0
+
y
Z
0
p
Re sin[W(1)BnZrW(z)]dz; (22)
£¤¥ C
0
- ª®­áâ ­â ,
C
0
= Re

cos(tgBnZrf

)W(1): (23)
Žç¥¢¨¤­®, çâ® C
0
¯à¥¤áâ ¢«ï¥â ᮡ®© ᪮à®áâì
᪮«ì¦¥­¨ï ç áâ¨æ ¯® ⢥म© á⥭ª¥ u(0) ¨ ¯®
᢮¥¬ã 䨧¨ç¥áª®¬ã á¬ëá«ã ­¥ ¬®¦¥â ¡ëâì ®âà¨-
æ â¥«ì­®©. â® ¯à¨¢®¤¨â ª á«¥¤ãî饬㠮£à ­¨-
祭¨î ­  㣮« ­ ª«®­  «®âª : tg  f

.
¥§ã«ìâ âë ¢ëç¨á«¥­¨© ¯à¨¢¥¤¥­ë ¢ ¢¨¤¥ £à -
䨪®¢ ­  à¨á. 2, 3, £¤¥ á«¥¢  ¨§®¡à ¦¥­ë à á¯à¥-
¤¥«¥­¨ï ¤«ï (y),   á¯à ¢  { ¤«ï u(y).
44 ˆ.ˆ ˆ¥¢«¥¢.
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
 áç¥âë ¡ë«¨ ¯à®¢¥¤¥­ë ¤«ï á«¥¤ãîé¨å §­ ç¥-
­¨© ¯ à ¬¥â஢: 
0
= 0:56, 
b
= 0:79, f

= 0:45,
k = 1.
Žáâ «ì­ë¥ §­ ç¥­¨ï ¯ à ¬¥â஢ ¯à¨¢¥¤¥­ë ¢
¯®¤¯¨áïå ª à¨áã­ª ¬.
4. Ž’ŽŠ Œ…†„“ „‚“ŒŸ ’‚…„›Œˆ
‘’…Š€Œˆ
 áᬮâਬ ⥯¥àì ¯®â®ª ¬¥¦¤ã ¤¢ã¬ï ⢥à¤ë-
¬¨ ¯ à ««¥«ì­ë¬¨ á⥭ª ¬¨  ¨ BnZr, ­ ª«®­­¥­ë¬¨
ª £®à¨§®­âã ¯®¤ 㣫®¬  (á¬.à¨á.1).
• à ªâ¥à ¯à®áª «ì§ë¢ ­¨ï £à ­ã«¨à®¢ ­­®©
áà¥¤ë ¯® ­¨¦­¥© ¨ ¢¥àå­¥© á⥭ª ¬ ¬®¦¥â
¡ëâì à §«¨ç­ë¬. ®í⮬ã, ¯®¬¨¬® ¯ à ¬¥â஢
f

; Re

, ¢¢¥¤¥¬  ­ «®£¨ç­ë¥ ¯ à ¬¥âàë f
BnZr
; Re
BnZr
,
å à ªâ¥à¨§ãî騥 ¢¥àå­îî ⢥à¤ãî á⥭ªã BnZr.
ã¤¥¬ áç¨â âì, çâ® ¨§¢¥áâ­ë ¤ ¢«¥­¨¥ P
0
­  ¢¥àå-
­¥© á⥭ª¥ ¨ á।­ïï ®¡ê¥¬­ ï ¯«®â­®áâì ¤¨á¯¥àá-
­®© á।ë 
cp
. ‘®®â¢¥âáâ¢ãî騥 ᮮ⭮襭¨ï ¢
¡¥§à §¬¥à­ëå ¯¥à¥¬¥­­ëå ¨¬¥îâ á«¥¤ãî騩 ¢¨¤:
1
Z
0
(y)dy = 
cp
; (24)
	

d
dy

2
= p
0
; y = 1 (25)
£¤¥ Re
BnZr
= D
BnZr
BnZr1

p
gh; D
BnZr
{ ª®íää¨æ¨¥­â ᮯà®â¨-
¢«¥­¨ï ¤«ï ¯®¢¥àå­®á⨠BnZr; p
0
= h
2
P
0

BnZr1
{ ¡¥§à §-
¬¥à­ë© ¯ à ¬¥âà ("¤ ¢«¥­¨¥" ­  BnZr).
’ ª¨¬ ®¡à §®¬, ¯®â®ª ¬¥¦¤ã ¤¢ã¬ï ⢥à-
¤ë¬¨ á⥭ª ¬¨ ®¯¨á뢠¥âáï ªà ¥¢®© § ¤ ç¥©,
¯à¥¤áâ ¢«¥­­®© ãà ¢­¥­¨ï¬¨ (16),(17), ãá«®¢¨ï¬¨
(24),(25),(20) ¨ ãá«®¢¨¥¬
1
M

du
dy

2
BnZr
Re
Re
BnZr
u = (26)
= f
BnZr
EuRep
0
; y = 1
­  BnZr.
‡¤¥áì, ª ª ¨ ¢ ¯à¥¤ë¤ã饬 ¯ã­ªâ¥, § ¤ ç  ­ å®-
¦¤¥­¨ï  ­¥ á¢ï§ ­  á § ¤ ç¥© ­ å®¦¤¥­¨ï u, ¨ ¬®-
¦¥â ¡ëâì à áᬮâ७  ®â¤¥«ì­®. „«ï ¥¥ à¥è¥­¨ï
¢®á¯®«ì§ã¥¬áï ¬¥â®¤®¬ "áâ५ì¡ë" ¯® ¯ à ¬¥âàã

1
= (1). ‚ í⮬ á«ãç ¥ ¯à¨ ¯¥à¥¡¨à ­¨¨ ¯ à -
¬¥âà  
1
¯à¨å®¤¨âáï à¥è âì § ¤ ç㠊®è¨. ®-
á«¥ ®¯à¥¤¥«¥­¨ï  á § ¤ ­­®© â®ç­®áâìî ¯¥à¥å®-
¤¨¬ ª ­ å®¦¤¥­¨î ᪮à®á⨠u(y), 㤮¢«¥â¢®àïî-
饩 ãà ¢­¥­¨î (17) ¨ ãá«®¢¨ï¬ (20),(26). —¨á«¥­-
­®¥ à¥è¥­¨¥ ¯®á«¥¤­¥© § ¤ ç¨ ¯à®¢®¤¨¬ á«¥¤ãî-
騬 ®¡à §®¬.
“ç¨â뢠ï, çâ® ¯à®¨§¢®¤­ ï
du
dy
¬¥­ï¥â §­ ª
¢­ãâਠ¨­â¥à¢ «  [0;1], à §¡ê¥¬ íâ®â ¨­â¥à¢ «
­  ¤¢  ¯®¤¨­â¥à¢ «  [0;y

] [y

;1], ­  ª®â®àëå §­ ª
㪠§ ­­®© ¯à¨§¢®¤­®© ­¥ ¬¥­ï¥âáï, â ª çâ® ¨¬¥¥â
¬¥áâ® à ¢¥­á⢮
du
dy
(y

) = 0: (27)
ˆ­â¥£à¨àãï ãà ¢­¥­¨¥ (17) ­  ¨­â¥à¢ «¥ [0;y

],
¯®«ãç ¥¬ à¥è¥­¨¥ u
I
, ᮤ¥à¦ é¥¥ ¤¢¥ ¯à®¨§¢®«ì-
­ë¥ ¯®áâ®ï­­ë¥ C
0
;C
1
:
u
I
(y) = C
0
+
y
Z
0
p
ResinM(z)(C
1
BnZrW(z))dz; (28)
0 y y

;
£¤¥ W(y) { äã­ªæ¨ï, ¢¢¥¤¥­­ ï ¢ ¯. 4. “à ¢­¥­¨¥¬
¤«ï ­ å®¦¤¥­¨ï y

á«ã¦¨â ᮮ⭮襭¨¥, ¢ë⥪ -
î饥 ¨§ ãà ¢­¥­¨© (17),(27):
W(y) = C
1
(29)
ˆ­â¥£à¨àãï ãà ¢­¥­¨¥ (17) ­  ¨­â¥à¢ «¥ [y

;1],
¯®«ãç ¥¬ u
II
(y)
u
II
(y) = u
I
(y

)BnZr (30)
BnZr
y
Z
y

p
ResinM(z)(W(z)BnZrC
1
)dz; y

< y  1:
ƒà ­¨ç­®¥ ãá«®¢¨¥ (20) ¤ ¥â ¢ëà ¦¥­¨¥ ¤«ï ¯®-
áâ®ï­­®© C
0
:
C
0
= Re

[sinC
1
BnZrf

Eu(p
0
+ Resin
cp
):] (31)
€ £à ­¨ç­®¥ ãá«®¢¨¥ (26) ¯à¨¢®¤¨â ª ãà ¢­¥­¨î
®â­®á¨â¥«ì­® C
1
:
Re
BnZr
[sin(
cp
BnZrC
1
)BnZrf
BnZr
Eup
0
]BnZru
I
(y

)+ (32)
+
1
Z
y

p
Re sinM(z)(W(z)BnZrC
1
)dz = 0:
„«ï ç¨á«¥­­®£® ­ å®¦¤¥­¨ï ª®à­ï í⮣® ãà ¢­¥-
­¨ï ¢®á¯®«ì§ã¥¬áï ¬¥â®¤®¬ ¤¥«¥­¨ï ®â१ª  ¯®-
¯®« ¬. „«ï ª ¦¤®£® §­ ç¥­¨ï C
1
­ å®¤¨¬ y

¨§
ãà ¢­¥­¨ï (29), ®¯à¥¤¥«ï¥¬ C
0
¨ u
I
(y) ¯® ä®à¬ã-
« ¬ (31),(28) ¨ ¢ëç¨á«ï¥¬ ¯à ¢ãî ç áâì ãà ¢­¥-
­¨ï (32). ®á«¥ ®¯à¥¤¥«¥­¨ï C
1
á § ¤ ­­®© â®ç­®-
áâìî ­ å®¤¨¬ à á¯à¥¤¥«¥­¨¥ ᪮à®á⥩ ¯® ä®à¬ã-
« ¬ (28),(30).
ˆ.ˆ ˆ¥¢«¥¢. 45
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
¨á. 4. ®â®ª ¬¥¦¤ã ⢥à¤ë¬¨ á⥭ª ¬¨,
Re

= 1:0; Re
BnZr
= 1:0; Re = 1:0; Eu = 2:0; P
0
= 0:005:
1 - 

= 39

; 2 - 

= 67

¨á. 5. ®â®ª ¬¥¦¤ã ⢥à¤ë¬¨ á⥭ª ¬¨,


= 42

; Re

= 1:0; Re
BnZr
= 1:0; Re = 1; P
0
= 0:005:
1 - Eu = 0:5; 2 - Eu = 1:0; 3 - Eu = 3:0
¨á. 6. ®â®ª ¬¥¦¤ã ⢥à¤ë¬¨ á⥭ª ¬¨,


= 42

; Re

= 1:0; Re
BnZr
= 1:0; Re = 1:0; Eu = 2:0:
ªà¨¢ë¥: 1 - P
0
= 0:005; 2 - P
0
= 0:01; 3 - P
0
= 0:1
¥§ã«ìâ âë ¢ëç¨á«¥­¨© ¯à¨¢¥¤¥­ë ­  à¨á. 4-6.
 áç¥âë ¯à®¢®¤¨«¨áì ¤«ï á«¥¤ãîé¨å §­ ç¥­¨©
¯ à ¬¥â஢: 
0
= 0:56, 
b
= 0:79, 
cp
= 0:63, f

=
= 0:45, f
BnZr
= 0:45, k = 1.
‡­ ç¥­¨ï ®áâ «ì­ëå ¯ à ¬¥â஢ ¯à¨¢¥¤¥­ë ¢
¯®¤¯¨áïå ª à¨áã­ª ¬.
‚›‚Ž„›
„«ï ¯®â®ª  ᮠ᢮¡®¤­®© ¯®¢¥àå­®áâìî à á¯à¥¤¥-
«¥­¨¥ ¯® £«ã¡¨­¥ ®¡ê¥¬­®© ¯«®â­®á⨠¤¨á¯¥àá­®©
ä §ë ¨ ᪮à®á⥩ ­®á¨â ¬®­®â®­­ë© å à ªâ¥à. ‘
㢥«¨ç¥­¨¥¬ £«ã¡¨­ë  à áâ¥â,   u ã¡ë¢ ¥â. “¢¥-
«¨ç¥­¨¥ ç¨á«  ©«¥à , á¢ï§ ­­®£® á ¤ ¢«¥­¨¥¬ ¢
á«®¥, ¯à¨¢®¤¨â ª 㬥­ì襭¨î £à ¤¨¥­â  ®¡ê¥¬­®©
¯«®â­®áâ¨.   ᪮à®áâì ¦¥ ç¨á«® ©«¥à  ¯à ªâ¨-
ç¥áª¨ ­¥ ¢«¨ï¥â.
—¨á«  Re; Re

; Re
BnZr
­¥ ¢«¨ïîâ ­  à á¯à¥¤¥«¥­¨¥
. ‘ª®à®áâì ¦¥ ¯®â®ª  á 㢥«¨ç¥­¨¥¬ §­ ç¥­¨©
íâ¨å ¯ à ¬¥â஢ ¢®§à áâ ¥â. ‘«¥¤ã¥â ®â¬¥â¨âì,
çâ® ªà¨¢¨§­  ¢á¥å ªà¨¢ëå ¤«ï u(y) ­¥ ¬¥­ï¥â §­ ª
­  ¨­â¥à¢ «¥ [0;1], çâ® ­¥ ­ ¡«î¤ ¥âáï ¯à¨ ¨á-
¯®«ì§®¢ ­¨¨ ¤àã£¨å ¬®¤¥«¥© áë¯ã祩 á।ë [12].
€­ «®£¨ç­ ï á¨âã æ¨ï ¨¬¥¥â ¬¥áâ® ¨ ¢ á«ãç ¥
¤¢¨¦¥­¨ï áà¥¤ë ¬¥¦¤ã ⢥à¤ë¬¨ á⥭ª ¬¨. Žâ-
«¨ç¨â¥«ì­ë¬ §¤¥áì ï¥âáï ­ «¨ç¨¥ ¬ ªá¨¬ã¬ 
᪮à®á⨠¢­ãâਠ¨­â¥à¢ «  0 < y < 1. ‚«¨ï­¨¥
¯ à ¬¥âà  P
0
­  à á¯à¥¤¥«¥­¨¥ (y) ­¥§­ ç¨â¥«ì-
­®¥.
‚ á«ãç ¥ ¯à¨«¨¯ ­¨ï áà¥¤ë ª á⥭ª ¬ (Re

=
Re
BnZr
= 0) â¥ç¥­¨¥ ¯®¤®¡­® ¯«®áª®¬ã â¥ç¥­¨î ã -
§¥©«ï.
46 ˆ.ˆ ˆ¥¢«¥¢.
ISSN 1561-9087 à¨ª« ¤­  £÷¤à®¬¥å ­÷ª . 2003. ’®¬ 5 (77), N 3. ‘. 41{47
1. ƒ®¬¨«ª® €.Œ.,˜« ¯ ª ž.‚. —¨á«¥­­®¥ ¬®¤¥«¨à®-
¢ ­¨¥ â¥ç¥­¨© ­¥«¨­¥©­®-¢ï§ª®© ¦¨¤ª®á⨠¢ ¯«®á-
ª®© ®¡« á⨠// à¨ª« ¤­ ï £i¤à®¬¥å ­¨ª .{ 1999.{
T. 1, N 3.{ ‘. 11-15.
2. ‘¬ià­®¢  ’.Ž., iª®«¥­ª® €.Œ. €­ «i§ ¢ï§ª®i ¯®-
¢¥¤i­ª¨ ¤¨á¯¥àá­¨å á¨á⥬ ¢ â¥à¬i­ å ¬®¤¥«i ¢i«ì-
­®£® ®¡'õ¬ã // „®¯. € “ªà i­¨.{ 1999.{ N 4.{
‘. 91-95.
3. ‘®ª®«®¢ ‚.Œ., Š®¢ «¥­ª® €.€., Š «î¦­ë© ƒ.‘.,
Œ¨­¨­ ‘.€. ˆ­¦¥­¥à­ë¥ § ¤ ç¨ ¤¨ää㧨¨ ¯à¨¬¥-
ᥩ ¢ ¯®â®ª å.{ ‹ã£ ­áª: ‚®áâ®ç­®ãªà. £®á. ã­-â,
2001.{ 168 á.
4.  ª®à祢᪨© €.ˆ.,  á®ª .ˆ. ƒ¨¤à®¤¨­ ¬¨ª 
¨ ⥯«®¬ áᮯ¥à¥­®á ¢ £¥â¥à®£¥­­ëå á¨á⥬ å ¨
¯ã«ìá¨àãîé¨å ¯®â®ª å.{ Š.:  ãª. ¤ã¬ª , 2001.{
346 á.
5. ‡¥­ª®¢ .‹. Œ¥å ­¨ª  ­ á믭ëå £à㧮¢.{ Œ.: Œ -
設®áâ஥­¨¥, 1964.{ 251 á.
6. ƒ¥­¨¥¢ ƒ.€. ‚®¯à®áë ¤¨­ ¬¨ª¨ áë¯ã祩 á।ë.{
Œ.: ƒ®ááâன¨§¤ â, 1958.{ 122 á.
7. ƒ®à⨭᪨© ‚.‚., „¥¬áª¨© €.., ®à¨áª¨­ Œ.€.
à®æ¥ááë ᥯ à¨à®¢ ­¨ï ­  §¥à­®-¯¥à¥à ¡ â뢠-
îé¨å ¯à¥¤¯à¨ïâ¨ïå.{ Œ.: Š®«®á, 1980.{ 304 á.
8. –ë⮢¨ç .€. Œ¥å ­¨ª  £àã­â®¢.{ Œ.: ƒ®á. ¨§¤ â.
«¨â. ¯® áâந⥫ìáâ¢ã,  àå¨â¥ªâ. ¨ áâன¬ â¥à.,
1963.{ 636 á.
9. ƒ®«®¢ ­®¢ ž.‚., ˜¨àª® ˆ.‚. Ž¡§®à ᮢ६¥­­®-
£® á®áâ®ï­¨ï ¬¥å ­¨ª¨ ¡ëáâàëå ¤¢¨¦¥­¨© £à ­ã-
«¨à®¢ ­­ëå ¬ â¥à¨ «®¢.{ ‚ ª­. Œ¥å ­¨ª  £à ­ã-
«¨à.á।. ’¥®à¨ï ¡ëáâàëå ¤¢¨¦¥­¨©.- Œ.: Œ¨à.-
1985.- ‘. 86-146.
10. í£­®«¤ . ªá¯¥à¨¬¥­âë á® ¢§¢¥è¥­­®© áãᯥ­-
§¨¥© ¡®«ìè¨å ⢥à¤ëå áä¥à ¢ ­ìîâ®­®¢áª®© ¦¨¤-
ª®á⨠¯®¤ ¤¥©á⢨¥¬ ᤢ¨£ .- ‚ ª­. Œ¥å ­¨ª  £à -
­ã«¨à.á।. ’¥®à¨ï ¡ëáâàëå ¤¢¨¦¥­¨©.- Œ.: Œ¨à.-
1985.- ‘. 44-63.
11. ƒã¤¬¥­ Œ., Š®ã¨­ ‘. „¢¥ § ¤ ç¨ ® £à ¢¨â -
樮­­®¬ â¥ç¥­¨¨ £à ­ã«¨à®¢ ­­ëå ¬ â¥à¨ «®¢.-
‚ ª­. Œ¥å ­¨ª  £à ­ã«¨à.á।. ’¥®à¨ï ¡ëáâàëå
¤¢¨¦¥­¨©.- Œ.:Œ¨à.- 1985.- ‘. 64-85.
12. ‘í¢¨¤¦ ‘. ƒà ¢¨â æ¨®­­®¥ â¥ç¥­¨¥ ­¥á¢ï§ ­-
­ëå £à ­ã«¨à®¢ ­­ëå ¬ â¥à¨ «®¢.- ‚ ª­. Œ¥å -
­¨ª  £à ­ã«¨à.á।. ’¥®à¨ï ¡ëáâàëå ¤¢¨¦¥­¨©.-
Œ.:Œ¨à.- 1985.- ‘. 86-146.
13. ‡ ¨ª  .Œ. „¨­ ¬¨ª  ¢¨¡à æ¨®­­ëå ®ç¨áâ¨â¥«ì-
­ëå ¬ è¨­.{ Œ.: Œ è¨­®áâ஥­¨¥, 1977.{ 278 á.
14. ‘®ª®«®¢áª¨© ‚.‚. ‘â â¨ª  áë¯ã祩 á।ë.{ Œ.:
 ãª , 1990.{ 272 á.
15. «¥å¬ ­ ˆ.ˆ, • ©­¬ ­ ‚.Ÿ. Ž ⥮ਨ ¢¨¡à æ¨-
®­­®£® à §¤¥«¥­¨ï áë¯ãç¨å ᬥᥩ // ˆ§¢. €
‘‘‘, Œ¥å ­¨ª .{ 1965.{ N 5.{ ‘. 22-30.
16. ¨£¬ â㫨­ .ˆ. Žá­®¢ë ¬¥å ­¨ª¨ £¥â¥à®£¥­­ëå
á।.{ Œ.:  ãª , 1978.{ 336 á.
17. ‘®ã ‘. ƒ¨¤à®¤¨­ ¬¨ª  ¬­®£®ä §­ëå á¨á⥬.{ Œ.:
Œ¨à, 1971.{ 536 á.
18. ‹®§®¢¥æª¨© ‚.‚., Œ®à¤¢¨­æ¥¢ ‚.Œ.  áç¥â ¤¢¨-
¦¥­¨ï è à®¢®© § á믪¨ ª ª ª¢ §¨­ìîâ®­®¢áª®©
¦¨¤ª®á⨠¢ ¡ã­ª¥à¥ ®á¥á¨¬¬¥âà¨ç­®© £¥®¬¥â-
ਨ// à¨ª«. ¯à®¡«¥¬ë ¯à®ç­®á⨠¨ ¯« áâ¨ç-
­®áâ¨.- Œ¥¦¢ã§.á¡. ¨¦¥£®à®¤áª¨© ã­-â.- 1991.-
‘.111-116.
19. Š®­á⠭⨭®¢ ž.Œ. ƒ¨¤à ¢«¨ª .{ Š.: ‚¨é  誮-
« , ƒ®«®¢­®¥ ¨§¤-¢®, 1988.{ 398 á.
ˆ.ˆ ˆ¥¢«¥¢. 47