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Файл:Современные проблемы металлургии и материаловедения гидродинамика и массообмен в многофазных системах металлургии. Учебное пособие
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ȿɫɥɢ ɞɜɢɠɭɳɚɹɫɹ ɫɪɟɞɚ – ɠɢɞɤɨɫɬɶ, ɬɨ ɡɧɚɱɟɧɢɹ ɤɪɢɬɟɪɢɟɜ
ɉɪɚɧɞɬɥɹ ɢ ɒɦɢɞɬɚ ɢɡɦɟɧɹɸɬɫɹ ɜ ɲɢɪɨɤɢɯ ɩɪɟɞɟɥɚɯ. Ɉɞɧɚɤɨ ɬɨɥɳɢɧɚ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɨɫɬɚɺɬɫɹ ɜɫɟɝɞɚ ɛɨɥɶɲɟ ɬɨɥɳɢɧɵ ɬɟɩɥɨɜɨɝɨ ɢ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɵɯ ɫɥɨɺɜ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɢɡ ɜɵɲɟɢɡɥɨɠɟɧɧɨɝɨ ɫɥɟɞɭɟɬ, ɱɬɨ ɦɟɠɞɭ
G , ɬG ɢ
r
G
ɢɦɟɸɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɹ
ɞ
1
Pr
ɢ
G
1
ɞ
~
G
r
. (4.128)
3
Sc
G
ɬ
~
3
G
r
Ɍɚɤ ɤɚɤ ɞɥɹ ɠɢɞɤɨɫɬɟɣ ɡɧɚɱɟɧɢɹ ɤɪɢɬɟɪɢɟɜ ɉɪɚɧɞɬɥɹ ɢ ɒɦɢɞɬɚ,
ɤɚɤ ɩɪɚɜɢɥɨ, ɛɨɥɶɲɟ ɟɞɢɧɢɰɵ, ɬɨ ɢɡ (4.128) ɫɥɟɞɭɟɬ, ɱɬɨ ɬɨɥɳɢɧɚ
ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɢ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɵɯ ɫɥɨɺɜ ɜ ɠɢɞɤɨɫɬɢ ɦɟɧɶɲɟ
ɬɨɥɳɢɧɵ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ.
ɉɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɩɨɡɜɨɥɹɟɬ ɫɱɢɬɚɬɶ, ɱɬɨ ɝɢɞɪɨɞɢɧɚɦɢɤɚ ɩɨɬɨɤɚ, ɨɦɵɜɚɸɳɟɝɨ ɱɚɫɬɢɰɭ ɜ ɩɪɟɞɟɥɚɯ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɢ ɬɟɩɥɨɜɨɝɨ
ɩɨɝɪɚɧɢɱɧɵɯ ɫɥɨɺɜ, ɩɨɞɱɢɧɹɟɬɫɹ ɡɚɤɨɧɨɦɟɪɧɨɫɬɹɦ
ɬɟɱɟɧɢɹ ɜ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɦ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɢ ɩɪɢ ɟɺ ɨɩɢɫɚɧɢɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɨɥɭɱɟɧɧɵɟ ɪɚɧɟɟ ɡɚɜɢɫɢɦɨɫɬɢ.
4.2.3. Ɇɚɫɫɨɨɛɦɟɧ ɦɟɠɞɭ ɬɜɺɪɞɨɣ ɱɚɫɬɢɰɟɣ ɢ ɩɨɬɨɤɨɦ
ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɯɢɦɢɱɟɫɤɨɣ
ɪɟɚɤɰɢɢ
4.2.3.1. Ɉɫɧɨɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɫɫɨɩɟɪɟɧɨɫɭ
ɫɨɫɪɟɞɨɬɨɱɟɧɨ ɜ ɞɢɮɮɭɡɢɨɧɧɨɦ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ
c
!!
Re 1!! )
c
!!
ɜɵɩɨɥɧɹɟɬɫɹ ɜɨ ɜɫɟɯ ɫɥɭ-
1
t
. Ɂɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ ɒɦɢɞɬɚ ɞɥɹ
c
Ɍɚɤ ɤɚɤ Pe Re Sc
, ɬɨ ɭɫɥɨɜɢɟ Pe 1
ɱɚɹɯ, ɤɨɝɞɚ Re Sc 1!! ɢɥɢ
(Pe 1;
Re
Sc
ɦɟɬɚɥɥɭɪɝɢɱɟɫɤɢɯ ɪɚɫɩɥɚɜɨɜ ɢɦɟɟɬ ɩɨɪɹɞɨɤ
ɭɫɥɨɜɢɟ Pe 1
c
!!
ɜɵɩɨɥɧɹɟɬɫɹ ɜ ɲɢɪɨɤɨɦ ɞɢɚɩɚɡɨɧɟ ɫɤɨɪɨɫɬɟɣ ɨɦɵ-
3
10 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
ɜɚɧɢɹ ɩɨɬɨɤɨɦ ɬɜɺɪɞɨɣ ɱɚɫɬɢɰɵ, ɩɪɢ ɤɨɬɨɪɵɯ
Re 10t
3
. (4.129)
ɉɪɢɫɬɭɩɢɦ ɤ ɫɨɫɬɚɜɥɟɧɢɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɪɚɫɫɦɚɬɪɢɜɚɟ-
ɦɨɝɨ ɩɪɨɰɟɫɫɚ.
81
Ɋɚɧɟɟ ɛɵɥɨ ɞɨɤɚɡɚɧɨ, ɱɬɨ ɩɪɢ
Pe 1c!!
ɩɟɪɟɧɨɫ ɦɚɫɫɵ ɩɪɨɢɫɯɨɞɢɬ
ɡɚ ɫɱɺɬ ɤɨɧɜɟɤɬɢɜɧɨɣ ɞɢɮɮɭɡɢɢ ɢ ɥɨɤɚɥɢɡɨɜɚɧ ɜ ɩɪɟɞɟɥɚɯ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɨɛɪɚɡɭɸɳɟɝɨɫɹ ɭ ɩɨɜɟɪɯɧɨɫɬɢ ɬɜɺɪɞɨɣ
ɱɚɫɬɢɰɵ. ȼ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɷɬɨɬ ɩɪɨɰɟɫɫ ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
CC C
ww w
ii i
uvD
xy
ww
2
c
i
.
(4.130)
2
y
w
ɉɪɢɦɟɦ ɨɛɬɟɤɚɟɦɨɟ ɬɜɺɪɞɨɟ ɬɟɥɨ ɡɚ ɲɚɪ ɫ ɪɚɞɢɭɫɨɦ
r . ɉɨɥɨɠɢɦ,
0
ɱɬɨ ɨɛɬɟɤɚɧɢɟ ɲɚɪɚ ɩɨɬɨɤɨɦ ɨɫɟɫɢɦɦɟɬɪɢɱɧɨ. Ɍɨɝɞɚ ɜ ɫɮɟɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɧɚɱɚɥɨ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɚɝɚɟɬɫɹ ɜ ɰɟɧɬɪɟ ɱɚɫɬɢɰɵ,
ɭɪɚɜɧɟɧɢɟ (4.130) ɡɚɩɢɲɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
2
Cw C C C
ww ww
wD
ri
wwT w
T
ii ii
rr r r
00
§·
c
¨¸
w
©¹
2
2
r
(4.131)
.
ɇɚ ɪɢɫ. 4.7 ɩɪɟɞɫɬɚɜɥɟɧɚ ɫɯɟɦɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɢ.
C
∞i
C
δ
д
is
r
0
θ
w
∞
Ɋɢɫ. 4.7. Ʉ ɡɚɞɚɱɟ ɨ ɦɚɫɫɨɨɛɦɟɧɟ ɦɟɠɞɭ ɬɜɺɪɞɨɣ ɱɚɫɬɢɰɟɣ ɢ ɩɨɬɨɤɨɦ
ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ
ȿɫɥɢ
r – ɪɚɞɢɭɫ ɱɚɫɬɢɰɵ, ɬɨ ɢɡɦɟɧɟɧɢɟ ɤɨɨɪɞɢɧɚɬɵ r ɜ ɩɪɟɞɟɥɚɯ
0
ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɪɟɞɟɥɚɯ
ªº
,rr
G
00 ɞ
¬¼
ɧɨɜɭɸ ɩɟɪɟɦɟɧɧɭɸ
82
. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (4.131) ɜɜɟɞɺɦ
y , ɨɛɨɡɧɚɱɚɸɳɭɸ ɤɨɨɪɞɢɧɚɬɭ ɜ ɩɪɟɞɟɥɚɯ ɞɢɮɮɭ-
ɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ
ɫɹ ɜ ɩɪɟɞɟɥɚɯ
0 yddG. ȼɜɟɞɹ y ɜ ɭɪɚɜɧɟɧɢɟ (4.131), ɩɨɥɭɱɢɦ
wD
ri
ɞ
Cw C C C
ww ww
ii ii
wwT w
yr r y
00
yrr , ɤɨɬɨɪɚɹ, ɨɱɟɜɢɞɧɨ ɢɡɦɟɧɹɟɬ-
T
0
2
§·
c
¨¸
w
©¹
2
2
y
(4.132)
.
ȼɬɨɪɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɫɤɨɛɤɚɯ ɜ ɭɪɚɜɧɟɧɢɢ (4.131) ɦɨɠɧɨ ɩɪɟɧɟ-
ɛɪɟɱɶ. Ⱦɥɹ ɨɛɨɫɧɨɜɚɧɢɹ ɷɬɨɝɨ ɭɩɪɨɳɟɧɢɹ ɨɰɟɧɢɦ ɩɨɪɹɞɤɢ ɜɟɥɢɱɢɧ
2
§·
w'
CC
ii
ɨɛɨɢɯ ɫɥɚɝɚɟɦɵɯ ɜ ɫɤɨɛɤɚɯ ɭɪɚɜɧɟɧɢɹ (4.132):
§·
CC
w'
22
o
¨¸
©¹
2
ww
y
w
ii
ry r
wG
00ɞ
CC
2
ii
!!
2
ry
w
0
. Ɍɚɤ ɤɚɤ
ɢ ɩɨɫɥɟɞɧɢɦ ɫɥɚɝɚɟɦɵɦ ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɦɨɠɧɨ ɩɪɟ-
ɞ 0
rG , ɬɨ
2
ɞ 0 ɞ
o
¨¸
22
wG
y
©¹
()rG G , ɨɬɫɸɞɚ
, ɚ
ɞ
ɧɟɛɪɟɱɶ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɪɚɜɧɟɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɣ ɞɢɮɮɭɡɢɢ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ
www
wD
ri
wwT
T
ii i
yr
0
2
c
.
(4.133)
2
y
w
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɜɜɟɞɺɦ ɜ ɭɪɚɜɧɟɧɢɟ
(4.133) ɜɦɟɫɬɨ ɩɪɨɟɤɰɢɣ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ
\
ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɬɨɤɚ
. ɋɜɹɡɶ ɦɟɠɞɭ ɩɪɨɟɤɰɢɹɦɢ ɫɤɨɪɨɫɬɢ ɢ ɮɭɧɤ-
w ɢ w
r
ɢɯ ɜɵɪɚɠɟɧɢɹ
T
ɰɢɟɣ ɬɨɤɚ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɵɪɚɠɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ:
w
r
1
22
r
T
sin
w\
wT
w
;
T
1
rr
Tw
w\
sin
(4.134)
.
ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɷɩɸɪɚ ɫɤɨɪɨɫɬɟɣ ɜ ɩɪɟɞɟɥɚɯ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɜɵɪɚɠɚɟɬɫɹ ɡɚɜɢɫɢɦɨɫɬɹɦɢ
§·
T
ww
r
cos 1
f
31
¨¸
22
©¹
rr
00
r
3
ɢ
3
r
T
ww
Tf
sin 1 .
¨¸
©¹
§·
rr
31
00
44
r
3
3
r
(4.135)
83
ɂɡ (4.134) ɢ (4.135) ɜɵɜɨɞɢɬɫɹ ɮɨɪɦɭɥɚ ɮɭɧɤɰɢɢ ɬɨɤɚ
3
\ T (4.136)
4
22
wy
f
sin .
ɂɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɟ (4.136), ɜɵɪɚɠɚɟɦ ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɬɨɤɚ ɩɟ-
ɪɟɦɟɧɧɭɸ
y :
y
2
3sin
w
T
\
f
.
(4.137)
ɉɪɟɨɛɪɚɡɭɟɦ ɭɪɚɜɧɟɧɢɟ (4.133), ɜɵɪɚɡɢɜ ɜ ɧɺɦ ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ
ɫɤɨɪɨɫɬɢ ɢ ɧɟɡɚɜɢɫɢɦɭɸ ɩɟɪɟɦɟɧɧɭɸ y ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɬɨɤɚ. ɋɧɚɱɚɥɚ ɩɪɟɨɛɪɚɡɭɟɦ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɜɜɟɞɹ ɭ ɧɟɝɨ ɮɭɧɤɰɢɸ ɬɨɤɚ:
ɫɫɫ ɫ
CwCC C
www w
w\ w\ w w\ w\
wD
iii i
ri
yr y y y
w\ w wT w\ w w\ w\ w w
§·§·
T
¨¸¨¸
0
©¹©¹
ɫ
.
ɉɨɞɫɬɚɜɥɹɹ ɜ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɵɪɚɠɟɧɢɟ ɮɭɧɤɰɢɢ ɬɨɤɚ
(4.136), ɩɨɥɭɱɚɟɦ ɩɨɫɥɟ ɨɱɟɜɢɞɧɵɯ ɫɨɤɪɚɳɟɧɢɣ
wC C
T
T T
r
wT w\ w\
0
c
ii
Dwr wr
TT
i
ɫɫ
ww
ɉɪɨɟɤɰɢɸ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ w
sin sin .
00
§·
w
¨¸
©¹
ɡɚɦɟɧɢɦ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ (4.134)
T
ɢ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɣ ɞɢɮɮɭɡɢɢ ɜ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɜ
ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
c ɫ
ww
CC
wT w\ w\
ɫ 22
ii
Dr w
T \
sin 3 .
i
0
§·
w
¨¸
f
©¹
(4.138)
ɗɬɨ ɥɢɧɟɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞ-
ɫ
ɧɵɯ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ, ɜ ɤɨɬɨɪɨɦ ɢɫɤɨɦɚɹ ɜɟɥɢɱɢɧɚ
ɮɭɧɤɰɢɟɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ: ɩɨɥɹɪɧɨɝɨ ɭɝɥɚ
ɢ ɮɭɧɤɰɢɢ ɬɨɤɚ \ .
T
ɋ ɹɜɥɹɟɬɫɹ
i
ɋɮɨɪɦɭɥɢɪɭɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɤ ɭɪɚɜɧɟɧɢɸ (4.138). Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɫɤɨɪɨɫɬɶ ɦɚɫɫɨɨɛɦɟɧɚ ɰɟɥɟɜɵɦ ɤɨɦɩɨɧɟɧɬɨɦ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶɸ ɟɝɨ ɩɨɞɜɨɞɚ ɤ ɦɟɠɮɚɡɧɨɣ ɝɪɚɧɢɰɟ. Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɪɚɧɟɟ, ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɤɨɧɰɟɧɬɪɚɰɢɹ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɧɚ ɦɟɠɮɚɡɧɨɣ ɝɪɚɧɢɰɟ ɪɚɜɧɚ ɧɭɥɸ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɩɨɜɟɪɯɧɨɫɬɢ ɱɚɫɬɢɰɵ
84
ɩɪɢ 0\ 0.
ɋ C (4.139)
iis
Ʉɨɧɰɟɧɬɪɚɰɢɹ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɜɧɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɫɥɨɹ
()\of ɩɨɫɬɨɹɧɧɚ ɢ ɪɚɜɧɚ ɟɝɨ ɤɨɧɰɟɧɬɪɚɰɢɢ ɜ ɧɟɜɨɡɦɭɳɺɧɧɨɦ ɩɨ-
ɬɨɤɟ:
ɩɪɢ
\of const.
ɋ C
(4.140)
ii
f
Ɉɬɦɟɬɢɦ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦɭ ɫɥɭɱɚɸ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɪɟɠɢɦɚ
ɦɚɫɫɨɨɛɦɟɧɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɵɣ ɩɟɪɟɩɚɞ ɤɨɧɰɟɧɬɪɚɰɢɣ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɜ ɨɦɵɜɚɸɳɟɣ ɠɢɞɤɨɫɬɢ ɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɱɚɫɬɢɰɵ, ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɧɚɢɛɨɥɶɲɢɣ ɩɨɬɨɤ ɦɚɫɫɵ, ɜɟɥɢɱɢɧɚ ɤɨɬɨɪɨɝɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɟɪɟɩɚɞɭ ɤɨɧɰɟɧɬɪɚɰɢɣ. ɗɬɨɬ ɩɨɬɨɤ ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥɶɧɵɦ ɞɢɮɮɭɡɢɨɧɧɵɦ ɩɨɬɨɤɨɦ.
ȼ ɬɨɱɤɟ ɧɚɛɟɝɚɧɢɹ ɩɨɬɨɤɚ ɧɚ ɞɢɫɩɟɪɫɧɭɸ ɱɚɫɬɢɰɭ, ɝɞɟ
ɟɳɺ ɧɟ
ɭɫɩɟɥ ɫɮɨɪɦɢɪɨɜɚɬɶɫɹ ɞɢɮɮɭɡɢɨɧɧɵɣ ɩɨɝɪɚɧɢɱɧɵɣ ɫɥɨɣ (ɜ ɩɟɪɜɨɣ
ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ), ɤɨɧɰɟɧɬɪɚɰɢɹ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɪɚɜɧɚ ɟɺ
ɡɧɚɱɟɧɢɸ ɜ ɧɟɜɨɡɦɭɳɺɧɧɨɦ ɩɨɬɨɤɟ:
ɩɪɢ 0\ ɢ 0T
ɋ C
.
(4.141)
ii
f
ɍɪɚɜɧɟɧɢɟ (4.138) ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ (4.139) – (4.141)
ɩɪɟɞɫɬɚɜɥɹɟɬ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɩɪɨɰɟɫɫɚ.
Ⱦɥɹ ɟɺ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ ɩɪɟɨɛɪɚɡɭɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ (4.138) ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɨɛɵɤɧɨɜɟɧɧɵɯ
ɩɪɨɢɡɜɨɞɧɵɯ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɞɜɚɠɞɵ ɜɵɩɨɥɧɢɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɨɣ.
ȼɜɟɞɺɦ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɩɟɪɟɦɟɧɧɭɸ
c2
Dr w
i
K T
3sin2
§ T·
0
f
22
¨¸
©¹
C
.
(4.142)
1
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɥɟɜɭɸ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (4.138) ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɫɫ ɫ ɫ
ɋ CCC
ww w w
ii i i
T T
wK
wT wK wT wK wK
ɫ 2
Dr w
i
3
0
f
2
1cos2 sin .
ɫ 22
Dr
io
ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ, ɩɪɢɜɟɞɺɦ ɭɪɚɜɧɟɧɢɟ (4.138) ɤ
ɜɢɞɭ
ɫɫ
CC
ww
ii
wK w\ w\
§·
w
\
¨¸
©¹
.
85
ȼɧɨɜɶ ɩɪɢɛɟɝɧɟɦ ɤ ɡɚɦɟɧɟ ɩɟɪɟɦɟɧɧɨɣ
ȼɵɪɚɡɢɦ ɩɪɨɢɡɜɨɞɧɵɟ
wK
[
Cw
\
(4.143)
.
1
3
K
ɫ
i
ɢ
ɫ
Cw
i
ɱɟɪɟɡ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɧɨɜɨɣ
w\
ɩɟɪɟɦɟɧɧɨɣ:
ɫɫ ɫ
C ɋ C
ww w
ii i
w[
wK w[ wK w[
ɫɫ ɫ
C ɋ Cww w
ii i
w\ w[ w\ w[
§·
ww w[
¨¸
w\ w\ w\ w[ w[ w\
©¹
ɫɫɫ 2 ɫ
CCC C
www w
iii i
\
§·
11 1
¨¸
11 12
¨¸
¨¸
33 3
22 4
K K \K
©¹
w[
11
2
\
1
1
3
3
K
1
3
K \
(4.144)
,
,
(4.145)
.
w[
(4.146)
ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɹ (4.144) – (4.146) ɜ ɭɪɚɜɧɟɧɢɟ (4.138), ɩɨ-
ɥɭɱɚɟɦ
ɫ 2 ɫ
\
CC
11
3
ww
ii
112
w[
33
K\K
4
ɢɥɢ
w[
2 ɫɫ
CCww
4
ii
2
w[
2
[
3
w[
0.
ɉɨɫɤɨɥɶɤɭ ɜ ɩɨɫɥɟɞɧɟɦ ɭɪɚɜɧɟɧɢɢ ɩɪɢɫɭɬɫɬɜɭɟɬ ɬɨɥɶɤɨ ɨɞɧɚ ɧɟ-
ɡɚɜɢɫɢɦɚɹ ɩɟɪɟɦɟɧɧɚɹ [ , ɟɝɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɨɛɵɤɧɨɜɟɧɧɵɣ ɩɪɨɢɡɜɨɞɧɵɯ
2 ɫɫ
dC dC
d
4
ii
2
[
2
[
3
0.
d
[
(4.147)
ɉɨɥɭɱɢɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɤ ɭɪɚɜɧɟɧɢɸ (4.147) ɢɡ ɝɪɚɧɢɱɧɵɯ
ɭɫɥɨɜɢɣ (4.139) – (4.141) ɢɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɩɟɪɟɦɟɧɧɨɣ [
(4.143):
86
[
ɩɪɢ
[of
ɩɪɢ
c
00,
ɋ
i
c
ɋ C
ii
½
°
(4.148)
¾
.
°
f
¿
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4.147) ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ (4.148) ɢɦɟɟɬ
ɜɢɞ [3]:
c3
[ [ [
ɋɋ dC
i
[
exp .
23
³
0
4
§·
¨¸
9
©¹
(4.149)
ɍɱɢɬɵɜɚɹ ɫɨɨɬɧɨɲɟɧɢɹ (4.136), (4.142) ɢ (4.143), ɦɨɠɧɨ ɞɥɹ ɩɟɪɟɦɟɧɧɨɣ ɩɨɥɭɱɢɬɶ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ:
\\
[
11
33
K
3
w
f
4
2
ªº
Dr w
i
«»
«»
¬¼
ªº
Dr w
«»
«»
¬¼
3
0
22
2
0
i
22
§·
f
T
¨¸
©¹
sin
y
T
3
§·
f
T
¨¸
©¹
T
sin(2 )
T
sin(2 )
C
1
(4.150)
.
1
3
C
1
ɉɟɪɟɣɞɺɦ ɞɚɥɟɟ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɤɨɧɫɬɚɧɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
C . ɉɪɢ ɷɬɨɦ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ (4.148),
ɢ
3
C , 2C
(4.141) ɢ ɭɪɚɜɧɟɧɢɹɦɢ (4.149) ɢ (4.150).
ɂɡ ɭɪɚɜɧɟɧɢɹ (4.150) ɫ ɭɱɺɬɨɦ ɩɟɪɜɨɝɨ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ ɫɢɫɬɟɦɵ (4.148) ɫɥɟɞɭɟɬ, ɱɬɨ
ɧɚɯɨɞɢɦ ɡɧɚɱɟɧɢɟ
|
ɋ C
2
C :
2
CC
f
§·
exp
[ [
¨¸
³
©¹
0
ii
4
9
0C . ɋ ɭɱɺɬɨɦ ɜɬɨɪɨɝɨ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ
3
ɫɫ
ff
3
d
1
3
94
§· §·
¨¸ ¨¸
©¹ ©¹
Ƚ
43
0,86 ,
ɫ
i
f
ɝɞɟ Ƚ(ɯ) – ɝɚɦɦɚ-ɮɭɧɤɰɢɹ, ɬɚɛɭɥɢɪɨɜɚɧɧɚɹ ɜɟɥɢɱɢɧɚ.
C
ɉɨɫɬɨɹɧɧɭɸ
ɤ ɭɪɚɜɧɟɧɢɸ (4.138), ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ
ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ (4.141)
1
c ɫ
CC
ɩɪɢ 0\ ɢ 0T .
f
ii
1
87
ɉɨɫɤɨɥɶɤɭ ɜ ɬɨɱɤɟ ɧɚɛɟɝɚɧɢɹ ɩɨɬɨɤɚ ɤɨɧɰɟɧɬɪɚɰɢɹ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɧɟ ɪɚɜɧɚ ɧɭɥɸ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ, ɚ ɩɨɫɬɨɹɧɧɚɹ
C ɪɚɜɧɚ ɧɭ-
3
ɥɸ, ɢɡ ɭɪɚɜɧɟɧɢɹ (4.149) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɟɪɟɦɟɧɧɚɹ [ ɧɟ ɞɨɥɠɧɚ ɪɚɜɧɹɬɶɫɹ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɨɛɪɚɬɶ ɬɚɤɨɟ ɡɧɚɱɟɧɢɟ
C
, ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɭɫɥɨɜɢɟ 0[z ɩɪɢ 0T . Ⱦɥɹ ɷɬɨɝɨ ɢɫɫɥɟɞɭ-
1
ɟɦ ɩɨɜɟɞɟɧɢɟ ɮɭɧɤɰɢɢ [ ɩɪɢ 0To . ȼ ɜɵɪɚɠɟɧɢɢ (4.150) ɮɭɧɤɰɢɢ
sin T ɢ sin(2 )T ɩɪɢ 0T ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɵ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɫɥɟ-
ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
35
sin ...;
T T
TT
3! 5!
35
(2 ) (2 )
sin(2 ) 2 ....
T T
T T
3! 5!
Ɉɬɫɸɞɚ ɩɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ɭɝɥɚ
8
sin(2 ) 2
T | T T
3
. ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɪɚɜɟɧɫɬɜɚ ɜ ɜɵɪɚɠɟɧɢɟ (4.150),
ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ
T
sin T|T
6
ɩɨɥɭɱɢɦ
3
w
[
f
4
c2
ªº
Dr w
i
«»
«»
¬¼
ɂɡ (4.151) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɟɪɟɦɟɧɧɚɹ [ ɩɪɢ
ɧɨɥɶ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɩɨɫɬɨɹɧɧɚɹ
y
T
3
03
f
T
2
.
(4.151)
1
3
C
1
0To ɧɟ ɨɛɪɚɳɚɟɬɫɹ ɜ
C
ɪɚɜɧɚ ɧɭɥɸ.
1
Ɉɩɪɟɞɟɥɢɜ ɜɫɟ ɩɨɫɬɨɹɧɧɵɟ, ɜɯɨɞɹɳɢɟ ɜ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4.147),
ɩɨɥɭɱɢɦ ɢɡ (4.149) ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɜɢɞ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ:
ɝɞɟ
c ɫ 3
ɋ Cd
ii
[
0,86 exp ,
[ [ [
3
w
3
c2 1
4
Dr
i
[
f
³
0
f
0
§T·
T
¨¸
©¹
4
§·
¨¸
9
©¹
sin
y
T
sin 2
.
(4.153)
3
2
(4.152)
,
88
Ɉɩɪɟɞɟɥɢɦ ɩɥɨɬɧɨɫɬɶ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɬɨɤɚ ɦɚɫɫɵ ɧɚ ɩɨɜɟɪɯɧɨ-
D
ɫɬɢ ɬɜɺɪɞɨɣ ɱɚɫɬɢɰɵ. ɋɨɝɥɚɫɧɨ ɩɨɫɬɭɥɚɬɭ Ɏɢɤɚ
c
w
C
c
ɩɪɢ 0y
w
mD
ii
i
.
y
ɍɱɢɬɵɜɚɹ ɜɵɪɚɠɟɧɢɹ (4.152) ɢ (4.153), ɩɨɥɭɱɚɟɦ
c ɫ
DC
ii
m
i
94
§· §· § T·
¨¸ ¨¸
43
©¹ ©¹
f
1c2 1
33
Ƚ
3
3
4
w
f
Dr
i
0
sin
T
sin 2
T
¨¸
©¹
2
.
(4.154)
ɂɫɫɥɟɞɭɟɦ, ɤɚɤ ɢɡɦɟɧɹɟɬɫɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧ-
ɬɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɱɚɫɬɢɰɵ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɩɨɥɹɪɧɨɝɨ ɭɝɥɚ ɜ ɢɧɬɟɪɜɚɥɟ 0dTdS. ȼ ɬɨɱɤɟ
ɧɨɫɬɶ ɬɢɩɚ
0
, ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɪɚɫɤɪɵɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ (ɜɨɫ-
T S
i
0
m , ɩɪɢ 0T ɜɨɡɧɢɤɚɟɬ ɧɟɨɩɪɟɞɟɥɺɧ-
0
ɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɞɫɬɚɜɥɟɧɧɵɦɢ ɜɵɲɟ ɮɨɪɦɭɥɚɦɢ ɞɥɹ ɮɭɧɤɰɢɣ
ɢ sin(2 )T ):
sin T
sin 3
lim lim 1,1.
To To
00
§·§·
T
¨¸
©¹
TT
sin(2 )
2
| |
11
33
T
TT
¨¸
©¹
T
8
12
3
3
2
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ 0
ɇɚ ɪɢɫ. 4.8 ɩɪɟɞɫɬɚɜɥɟɧ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ
T ɢɦɟɟɦ
f
3
4
3
0,95 .
mDC
iii
c ɫ
w
f
c2
r
i
(4.155)
0
fTT
sin
sin 2
§T·
T
¨¸
©¹
, ɜɯɨ-
1
3
2
ɞɹɳɟɣ ɜ ɤɚɱɟɫɬɜɟ ɫɨɦɧɨɠɢɬɟɥɹ ɜ ɜɵɪɚɠɟɧɢɟ (4.154).
89
f T
D
1,0
0,5
0
S
4
S
2
34S
S
T
Ɋɢɫ. 4.8. Ƚɪɚɮɢɤ ɮɭɧɤɰɢɢ
f T
ɉɨɫɤɨɥɶɤɭ ɞɪɭɝɢɟ ɫɨɦɧɨɠɢɬɟɥɢ, ɜɯɨɞɹɳɢɟ ɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ, ɧɟ
ɡɚɜɢɫɹɬ ɨɬ ɭɝɥɚ
, ɧɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɩɥɨɬɧɨɫɬɶ ɞɢɮɮɭɡɢɨɧɧɨɝɨ
T
ɩɨɬɨɤɚ ɦɚɫɫɵ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɩɪɢɧɢɦɚɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɜ ɬɨɱɤɟ ɧɚɛɟɝɚɧɢɹ ɩɨɬɨɤɚ ɧɚ ɱɚɫɬɢɰɭ (ɩɪɢ 0T ). ɗɬɚ ɜɟɥɢɱɢɧɚ
ɭɛɵɜɚɟɬ ɩɪɢ ɜɨɡɪɚɫɬɚɧɢɢ ɭɝɥɚ T ɢ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɩɪɢ T S ɧɚ
ɡɚɞɧɟɣ ɤɪɨɦɤɟ ɱɚɫɬɢɰɵ.
Ɉɰɟɧɢɦ ɞɚɥɟɟ ɡɚɜɢɫɢɦɨɫɬɶ ɬɨɥɳɢɧɵ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨ-
ɝɨ ɫɥɨɹ ɨɬ ɭɝɥɚ
T. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɤɨɧɰɟɧɬɪɚɰɢɹ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨ-
ɧɟɧɬɚ ɩɨ ɬɨɥɳɢɧɟ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɢɡɦɟɧɹɟɬɫɹ ɥɢɧɟɣɧɨ. Ɍɨɝɞɚ ɩɨ ɩɨɪɹɞɤɭ ɜɟɥɢɱɢɧ ɝɪɚɞɢɟɧɬ ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɚɜɟɧ
ɫ
C
c
i
grad .
C
f
i
(4.156)
G
ɞ
ɉɥɨɬɧɨɫɬɶ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɩɨɬɨɤɚ ɦɚɫɫɵ ɰɟɥɟɜɨɝɨ
ɤɨɦɩɨɧɟɧɬɚ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɝɪɚɞɢɟɧɬɭ ɤɨɧɰɟɧɬɪɚɰɢɢ (ɫ ɤɨɷɮɮɢɰɢ-
ɟɧɬɨɦ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ
c
). Ɉɬɫɸɞɚ ɩɪɢɛɥɢɠɺɧɧɨ
i
ɫ
C
c
i
mD
G (4.157)
ii
f
.
ɞ
ɉɨɞɫɬɚɜɢɜ ɜ ɥɟɜɭɸ ɱɚɫɬɶ (4.154) ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɞɢɮɮɭɡɢɨɧ-
ɧɨɝɨ ɩɨɬɨɤɚ ɦɚɫɫɵ ɰɟɥɟɜɨɝɨ ɤɨɦɩɨɧɟɧɬɚ, ɜɵɪɚɠɟɧɧɨɟ ɱɟɪɟɡ (4.157),
90
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