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Файл:Современные проблемы металлургии и материаловедения гидродинамика и массообмен в многофазных системах металлургии. Учебное пособие
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2
g
F
P P
c ɞ
6.Fru
SP
c0
3
PP
c ɞ
(4.74)
Ɉɩɪɟɞɟɥɢɦ ɫɤɨɪɨɫɬɶ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɤɚɩɥɢ
u
f
, ɞɜɢ-
ɠɭɳɟɣɫɹ ɜ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ. Ʉɚɤ ɢ ɩɪɢ
ɜɵɱɢɫɥɟɧɢɢ ɫɤɨɪɨɫɬɢ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɣ ɱɚɫɬɢɰɵ,
ɩɪɢɪɚɜɧɢɜɚɟɦ ɫɢɥɭ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɢɥɟ ɬɹɠɟɫɬɢ ɫ ɭɱɺɬɨɦ ɜɵɬɚɥɤɢɜɚɸɳɟɣ ɫɢɥɵ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
2
rg
1
u
f
0
3
QU
UPP
§·
c ɫ
ɞɞɫ
1
¨¸
©¹
P P
ɫɞ
. (4.75)
2
3
Ɂɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɤɚɩɥɢ ɜ ɠɢɞɤɨɫɬɢ ɜɩɟɪɜɵɟ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ
Ⱥɞɚɦɚɪɨɦ ɢ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɧɟɝɨ Ɋɵɛɱɢɧɫɤɢɦ. ɉɨɷɬɨɦɭ ɮɨɪɦɭɥɚ
(4.75) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɮɨɪɦɭɥɵ Ɋɵɛɱɢɧɫɤɨɝɨ – Ⱥɞɚɦɚɪɚ. ȼ ɩɪɟɞɟɥɟ
P
ɩɪɢ
ɞ
ofP, ɮɨɪɦɭɥɵ (4.74) ɢ (4.75) ɩɟɪɟɯɨɞɹɬ ɜ ɮɨɪɦɭɥɵ (4.58) ɢ
ɫ
(4.60) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ (ɫɥɭɱɚɣ ɬɜɺɪɞɨɣ ɫɮɟɪɢɱɟɫɤɨɣ ɱɚɫɬɢɰɵ).
ɉɪɢ ɨɩɢɫɚɧɢɢ ɞɜɢɠɟɧɢɹ ɜ ɠɢɞɤɨɫɬɢ ɫɮɟɪɢɱɟɫɤɢɯ ɝɚɡɨɜɵɯ ɩɭ-
ɡɵɪɶɤɨɜ ɩɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ
ɞ
ɫ
0PoP ɢ
ɞ
0UoU.
ɫ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɤɨɪɨɫɬɶ ɜɫɩɥɵɬɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɹ
1
2
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɵɬɚɥɤɢɜɚɸɳɟɣ ɫɢɥɵ ɪɚɜɧɚ
ur
f
3
0
Q
, ɚ ɫɢɥɚ ɫɨ-
c
ɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɝɚɡɨɜɵɣ ɩɭɡɵɪɶ, ɪɚɜɧɚ
ru SP.
04c
4.1.4. ɇɟɫɬɚɰɢɨɧɚɪɧɨɟ ɞɜɢɠɟɧɢɟ ɞɢɫɩɟɪɫɧɵɯ
ɜɤɥɸɱɟɧɢɣ
Ɇɵ ɪɚɫɫɦɨɬɪɟɥɢ ɫɬɚɰɢɨɧɚɪɧɨɟ ɞɜɢɠɟɧɢɟ ɬɜɺɪɞɵɯ ɱɚɫɬɢɰ, ɤɚɩɟɥɶ
ɢ ɩɭɡɵɪɶɤɨɜ ɜ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɟ. ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ ɫɤɨɪɨɫɬɶ
ɱɚɫɬɢɰɵ ɢɡɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ,
ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɩɨɬɨɤɚ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ, ɡɚɜɢɫɢɬ ɧɟ ɬɨɥɶɤɨ ɨɬ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɧɨ ɢ ɨɬ ɭɫɤɨɪɟɧɢɹ
ɱɚɫɬɢɰɵ, ɚ ɬɚɤɠɟ ɨɬ ɟɺ ɫɤɨɪɨɫɬɢ ɜ ɩɪɟɞɵɞɭ-
ɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ, ɬ.ɟ. ɨɬ ɩɪɟɞɵɫɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ. ɗɬɨ
61
ɫɥɟɞɭɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɢɡ ɬɨɝɨ, ɱɬɨ, ɤɚɤ ɭɫɬɚɧɨɜɥɟɧɨ ɪɚɧɟɟ, ɧɚ ɫɮɟɪɢɱɟɫɤɭɸ ɱɚɫɬɢɰɭ ɪɚɞɢɭɫɨɦ
Re 1!!
ɫɬɢ (
) ɫ ɩɟɪɟɦɟɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɫɨ ɫɬɨɪɨɧɵ ɠɢɞɤɨɫɬɢ ɞɟɣɫɬɜɭ-
ɟɬ ɩɟɪɟɦɟɧɧɚɹ ɫɢɥɚ
r , ɩɟɪɟɦɟɳɚɸɳɭɸɫɹ ɜ ɢɞɟɚɥɶɧɨɣ ɠɢɞɤɨ-
0
Mdt
G
du
. Ɂɞɟɫɶ
U– ɩɪɢɫɨɟɞɢɧɺɧɧɚɹ
3
rMS
0
2
c
3
ɦɚɫɫɚ. ɋɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɫɮɟɪɢɱɟɫɤɭɸ ɬɜɺɪɞɭɸ ɱɚɫɬɢɰɭ, ɩɟɪɟɦɟɳɚɸɳɭɸɫɹ ɜ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɫ ɩɟɪɟɦɟɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɦɨɠɟɬ
ɛɵɬɶ ɜɵɱɢɫɥɟɧɚ, ɟɫɥɢ ɤɪɢɬɟɪɢɣ Ɋɟɣɧɨɥɶɞɫɚ ɹɜɥɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɦ ɢ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɜɢɠɟɧɢɹ ɠɢɞɤɨɫɬɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɭɪɚɜɧɟɧɢɟ ɋɬɨɤɫɚ (4.29). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ, ɪɚɜɧɚ
GG
JG G
SP SP U
66
Fru r
c0 0 c c
SU
2
c0
3
3
r
du du dt
2
dt dt
t
³
0
.
tt
1
(4.76)
Ɂɞɟɫɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɜɞɚɥɢ ɨɬ ɱɚɫɬɢɰɵ ɠɢɞɤɨɫɬɶ ɧɟɩɨɞɜɢɠɧɚ. ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ (4.76) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɢɥɭ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɋɬɨɤɫɚ, ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ – ɢɧɟɪɰɢɨɧɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ – ɬɚɤ ɧɚɡɵɜɚɟɦɭɸ ɫɢɥɭ
Ȼɚɫɫɟ. ɉɪɢ ɜɵɜɨɞɟ ɭɪɚɜɧɟɧɢɹ (4.76) ɧɟ ɭɱɢɬɵɜɚɥɚɫɶ ɫɢɥɚ ɬɹɠɟɫɬɢ.
ɍɱɺɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɜ ɭɪɚɜɧɟɧɢɢ ɞɥɹ ɫɢɥɵ
,
ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɠɢɞɤɨɫɬɢ, ɫɥɚɝɚɟɦɨɝɨ, ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɫɨɛɨɣ ɜɵɬɚɥɤɢɜɚɸɳɭɸ ɫɢɥɭ. ȿɫɥɢ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜ
ɝɚɡɟ, ɚ ɟɺ ɭɫɤɨɪɟɧɢɟ ɧɟ ɨɱɟɧɶ ɜɟɥɢɤɨ, ɬɨ ɜɬɨɪɨɟ ɢ ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɵɟ ɜ
ɩɪɚɜɨɣ ɱɚɫɬɢ (4.76), ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɟ ɩɥɨɬɧɨɫɬɢ ɝɚɡɚ ɢ ɤɨɪɧɸ
ɤɜɚɞɪɚɬɧɨɦɭ ɢɡ ɩɥɨɬɧɨɫɬɢ ɝɚɡɚ, ɦɚɥɵ, ɢ ɢɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɩɟɪɜɵɦ ɫɥɚɝɚɟɦɵɦ.
ȼɵɪɚɠɟɧɢɹ ɞɥɹ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɤɚɩɥɸ ɢɥɢ ɝɚɡɨɜɵɣ ɩɭɡɵɪɟɤ, ɞɜɢɠɭɳɢɟɫɹ ɫ ɩɟɪɟɦɟɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɞɚɠɟ ɜ ɫɥɭɱɚɟ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ ɤɪɢɬɟɪɢɹ Ɋɟɣɧɨɥɶɞɫɚ ɨɤɚɡɵɜɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ ɛɨɥɟɟ ɫɥɨɠɧɵɦɢ, ɱɟɦ (4.76).
ȼ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɡɧɚɱɟɧɢɹ ɤɪɢɬɟɪɢɹ Ɋɟɣɧɨɥɶɞɫɚ ɧɟ ɹɜɥɹɸɬɫɹ
ɦɚɥɵɦɢ, ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɟɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɞɢɫɩɟɪɫɧɵɯ
ɱɚɫɬɢɰ ɜ ɩɨɬɨɤɟ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ ɧɚɬɚɥɤɢɜɚɸɬɫɹ
ɧɚ ɡɧɚɱɢɬɟɥɶɧɵɟ
ɬɪɭɞɧɨɫɬɢ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ ɢɫɩɨɥɶɡɭɸɬɫɹ ɨɛɵɱɧɨ ɩɨɥɭɷɦɩɢɪɢɱɟɫɤɢɟ
ɩɨɞɯɨɞɵ ɢɥɢ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ. ɇɚɩɪɢɦɟɪ, ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨ ɞɜɢɠɟɧɢɢ ɬɜɺɪɞɨɝɨ ɲɚɪɚ, ɜɧɟɡɚɩɧɨ ɩɪɢɜɟɞɺɧɧɨɝɨ ɜ ɞɜɢɠɟɧɢɟ
ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɜɪɟɦɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ ɨɛɬɟɤɚɧɢɹ ɫ ɭɜɟɥɢɱɟɧɢɟɦ Re ɭɦɟɧɶɲɚɟɬɫɹ.
62
ɉɨɥɭɷɦɩɢɪɢɱɟɫɤɢɟ ɩɨɞɯɨɞɵ ɤ ɨɩɢɫɚɧɢɸ ɧɟɫɬɚɰɢɨɧɚɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ ɨɫɧɨɜɵɜɚɸɬɫɹ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɣ,
ɩɨ ɫɬɪɭɤɬɭɪɟ ɚɧɚɥɨɝɢɱɧɵɯ ɭɪɚɜɧɟɧɢɸ (4.76). ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɡɚɦɟɧɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ ɬɚɤɨɝɨ ɠɟ ɜɢɞɚ, ɨɞɧɚɤɨ ɜɹɡɤɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɡɚɜɢɫɹɳɟɣ ɨɬ ɦɝɧɨɜɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ
ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ. ɉɟɪɟɞ ɜɬɨɪɵɦ ɢ ɬɪɟɬɶɢɦ ɫɥɚɝɚɟɦɵɦɢ ɜɜɨɞɹɬ ɩɨɩɪɚɜɨɱɧɵɟ ɦɧɨɠɢɬɟɥɢ, ɜɢɞ ɤɨɬɨɪɵɯ ɧɚɯɨɞɹɬ ɫ ɩɨɦɨɳɶɸ
ɨɛɪɚɛɨɬɤɢ ɨɩɵɬɧɵɯ ɞɚɧɧɵɯ. ɂɧɨɝɞɚ ɫɥɚɝɚɟɦɵɦɢ ɬɢɩɚ ɫɢɥɵ Ȼɚɫɫɟ ɩɪɟɧɟɛɪɟɝɚɸɬ, ɱɬɨ
ɡɧɚɱɢɬɟɥɶɧɨ ɭɩɪɨɳɚɟɬ ɨɩɢɫɚɧɢɟ ɞɜɢɠɟɧɢɹ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ.
Ɉɩɢɫɚɧɢɟ ɞɜɢɠɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɶɤɚ ɩɪɢ ɤɨɧɟɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɤɪɢɬɟɪɢɹ Ɋɟɣɧɨɥɶɞɫɚ ɨɤɚɡɵɜɚɟɬɫɹ ɞɚɠɟ ɩɪɨɳɟ, ɱɟɦ
ɩɪɢ
Re 1 . ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɩɪɢ
Re 1t
ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɭ-
ɡɵɪɶɤɚ ɨɛɪɚɡɭɟɬɫɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɣ ɩɨɝɪɚɧɢɱɧɵɣ ɫɥɨɣ. ɉɪɢ ɷɬɨɦ
ɩɨɥɧɚɹ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɩɭɡɵɪɺɤ ɫɨ ɫɬɨɪɨɧɵ
ɠɢɞɤɨɫɬɢ ɩɪɢ ɧɟɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɞɜɢɠɟɧɢɢ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɩɪɟɞɵɫɬɨ-
ɪɢɢ ɞɜɢɠɟɧɢɹ. ɋ ɬɨɱɧɨɫɬɶɸ ɞɨ ɫɥɚɝɚɟɦɵɯ ɩɨɪɹɞɤɚ
1Re
ɤɨɷɮɮɢɰɢ-
ɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ
ɦɝɧɨɜɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɩɭɡɵɪɹ, ɚ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɩɪɟɞɵɫɬɨ-
3
2
ɪɢɢ ɩɪɨɹɜɥɹɟɬɫɹ ɜ ɦɚɥɵɯ ɱɥɟɧɚɯ ɩɨɪɹɞɤɚ
. Ʉɨɷɮɮɢɰɢɟɧɬ ɫɨɩɪɨ-
Re
ɬɢɜɥɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɹ, ɜɧɟɡɚɩɧɨ ɩɪɢɜɟɞɺɧɧɨɝɨ ɜ
ɞɜɢɠɟɧɢɟ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɢɡɦɟɧɹɟɬɫɹ ɨɬ ɡɧɚɱɟɧɢɹ
ɋ
ɏ
48
Re
ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɨ ɡɧɚɱɟɧɢɹ, ɞɚɜɚɟɦɨɝɨ ɮɨɪɦɭɥɨɣ:
ɩɪɢ
t of
Re 2
48 3
ɋ
ɏ
§·
¨¸
©¹
P
ɞ
1
. (4.77)
P
ɫ
Ɉɛɵɱɧɨ ɩɪɢ ɨɩɢɫɚɧɢɢ ɧɟɫɬɚɰɢɨɧɚɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ
ɩɭɡɵɪɹ ɫ
Re 1!
, ɢɫɩɨɥɶɡɭɸɬ ɫɥɟɞɭɸɳɢɣ ɩɨɞɯɨɞ. ɂɡ ɱɢɫɥɚ ɫɢɥ, ɞɟɣ-
ɫɬɜɭɸɳɢɯ ɧɚ ɩɭɡɵɪɶ, ɩɪɢɧɢɦɚɸɬ ɜɨ ɜɧɢɦɚɧɢɟ ɫɥɟɞɭɸɳɢɟ: ɜɹɡɤɭɸ
ɫɨɫɬɚɜɥɹɸɳɭɸ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɡɚɜɢɫɹɳɭɸ ɬɨɥɶɤɨ ɨɬ ɦɝɧɨɜɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ; ɢɧɟɪɰɢɨɧɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɭɱɢɬɵɜɚɸɳɭɸ ɩɪɢɫɨɟɞɢɧɺɧɧɭɸ ɦɚɫɫɭ ɠɢɞɤɨɫɬɢ, ɚ
ɬɚɤɠɟ ɫɢɥɭ ɬɹɠɟɫɬɢ ɫ ɭɱɺɬɨɦ ɜɵɬɚɥɤɢɜɚɸɳɟɣ ɫɢɥɵ.
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɭɡɵɪɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
ɝɞɟ
ɞɞ ɞ c ɞ cc0
dt dt
V
– ɨɛɴɟɦ ɩɭɡɵɪɶɤɚ; u – ɫɤɨɪɨɫɬɶ ɩɭɡɵɪɶɤɚ.
ɞ
du du
VV g ru
U UU U SP
V
ɞ
12
, (4.78)
2
63
Ɍɚɤ ɤɚɤ ɩɥɨɬɧɨɫɬɶ ɝɚɡɚ ɦɧɨɝɨ ɦɟɧɶɲɟ ɩɥɨɬɧɨɫɬɢ ɠɢɞɤɨɫɬɢ, ɭɪɚɜ-
F
Q
Q
ɧɟɧɢɟ (4.78) ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ:
V
du
ɞ
U U SP . (4.79)
ccɞ c0
dt
2
Vg ru
12
Ɋɟɲɚɹ ɭɪɚɜɧɟɧɢɟ (4.79) ɨɛɵɱɧɨ ɢɫɩɨɥɶɡɭɸɬ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɟɪɟ-
2
gr
ɦɟɧɧɵɟ
'u
ɢ
, ɝɞɟ
0
, ɡɞɟɫɶ
'uu
u
f
u
f
ɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɫɮɟɪɢɱɟɫɤɨɝɨ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɶɤɚ;
0
– ɫɤɨɪɨɫɬɶ ɭɫɬɚɧɨ-
9
F
0
ɛɟɡɪɚɡɦɟɪɧɨɟ ɜɪɟɦɹ.
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (4.79) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
'
18 1 'duu
dF
0
. (4.80)
ɍɪɚɜɧɟɧɢɟ (4.80) ɧɟɬɪɭɞɧɨ ɪɟɲɢɬɶ. ȿɫɥɢ, ɧɚɩɪɢɦɟɪ, ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɩɭɡɵɪɺɤ ɩɨɤɨɢɥɫɹ:
t
–
2
r
0
ɩɪɢ
0F
0
, (4.81)
'0u
ɬɨ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4.80) ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ (4.81) ɢɦɟɟɬ ɜɢɞ
18
F
Ɂɚɜɢɫɢɦɨɫɬɶ
Ɋɢɫ. 4.5. ɂɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɝɚɡɨɜɨɝɨ ɩɭɡɵɪɶɤɚ ɩɪɢ ɟɝɨ ɞɜɢɠɟɧɢɢ
64
'1
ue
ɢɡɨɛɪɚɠɟɧɚ ɧɚ ɪɢɫ. 4.5.
'uF
0
1
′
u
0,5
0
0 0,05 0,1 0,15 0,2
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ (Re >> 1)
0
. (4.82)
F
0
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ (4.79), ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
g
0u
ɩɭɡɵɪɺɤ ɛɵɥ ɧɟɩɨɞɜɢɠɟɧ (
ɩɪɢ
ɡɵɪɶɤɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɨ
0t
), ɬɨ ɭɫɤɨɪɟɧɢɟ ɝɚɡɨɜɨɝɨ ɩɭ-
. Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚ-
2g
ɡɨɦ ɦɨɠɧɨ ɨɰɟɧɢɬɶ ɧɚɱɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɤɚɩɥɢ, ɤɨɬɨɪɚɹ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥ ɬɹɠɟɫɬɢ ɢ Ⱥɪɯɢɦɟɞɚ.
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ
ɤɚɩɥɸ, ɦɚɥɚ. ɉɨɷɬɨɦɭ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɤɚɩɥɢ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
U UU U . (4.83)
ɞɞ ɞ c ɞ c
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ
du du
VV g
dt dt
0t
U
§·
ɞ
1
¨¸
du
dt
U
ɫ
©¹
U
1
ɞ
U
2
ɫ
V
ɞ
2
. (4.84)
4.1.5. Ⱦɜɢɠɟɧɢɟ ɫɢɫɬɟɦɵ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ
ɜ ɫɩɥɨɲɧɨɣ ɫɪɟɞɟ (ɫɬɟɫɧɺɧɧɨɟ ɞɜɢɠɟɧɢɟ)
ȼ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɦɟɬɚɥɥɭɪɝɢɢ ɜ ɫɩɥɨɲɧɨɣ ɫɪɟɞɟ
ɨɛɵɱɧɨ ɧɚɯɨɞɢɬɫɹ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɞɢɫɩɟɪɫɧɵɯ ɜɤɥɸɱɟɧɢɣ –
ɬɜɺɪɞɵɯ ɱɚɫɬɢɰ, ɠɢɞɤɢɯ ɤɚɩɟɥɶ ɢɥɢ ɝɚɡɨɜɵɯ ɩɭɡɵɪɟɣ.
Ⱦɥɹ ɪɚɫɱɺɬɚ ɩɨɞɨɛɧɵɯ ɚɩɩɚɪɚɬɨɜ ɢ ɩɪɨɰɟɫɫɨɜ ɩɟɪɟɧɨɫɚ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɧɢɯ, ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɩɨɥɚɝɚɬɶ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɚɩɩɚɪɚɬɟ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ ɫ ɩɨɬɨɤɨɦ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ. Ɍɨɱɧɨɟ ɪɟɲɟɧɢɟ
ɬɚɤɨɣ ɡɚɞɚɱɢ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɵɦ, ɬɚɤ ɤɚɤ ɬɨɱɧɨɟ ɪɚɫɩɨɥɨɠɟɧɢɟ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ ɜ ɩɨɬɨɤɟ
ɢɥɢ ɫɥɨɟ ɧɢɤɨɝɞɚ ɧɟɢɡɜɟɫɬɧɨ. ȿɫɥɢ ɞɢɫɩɟɪɫɧɵɟ ɱɚɫɬɢɰɵ ɦɨɝɭɬ ɩɟɪɟɦɟɳɚɬɶɫɹ, ɬɨ ɧɟɢɡɜɟɫɬɧɵ ɬɚɤɠɟ ɫɤɨɪɨɫɬɶ ɢɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ. ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɜɢɠɟɧɢɹ ɮɚɡ
ɦɧɨɝɨɮɚɡɧɨɣ ɫɢɫɬɟɦɵ ɧɟɨɛɯɨɞɢɦɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜɟɥɢɱɢɧɵ, ɨɫɪɟɞɧɺɧɧɵɟ ɩɨ
ɧɟɤɨɬɨɪɨɦɭ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɨɛɴɺɦɭ, ɫɨɞɟɪɠɚɳɟɦɭ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ, ɧɨ ɦɚɥɨɦɭ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɦɚɫɲɬɚɛɨɦ, ɧɚ ɤɨɬɨɪɨɦ ɨɫɪɟɞɧɟɧɧɵɟ ɜɟɥɢɱɢɧɵ ɫɢɫɬɟɦɵ ɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɹɸɬɫɹ. Ɍɚɤɢɦɢ ɨɫɪɟɞɧɺɧɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ ɹɜɥɹɸɬɫɹ
ɨɛɴɺɦɧɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɮɚɡ, ɨɫɪɟɞɧɺɧɧɵɟ ɩɨ ɷɥɟɦɟɧɬɚɪɧɨɦɭ ɨɛɴɺɦɭ, ɫɤɨɪɨɫɬɢ ɮɚɡ, ɫɢɥɚ ɦɟɠɮɚɡɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɢ ɬ.ɩ. ȿɫɥɢ ɨɛɴɺɦɧɚɹ ɤɨɧɰɟɧɬɪɚɰɢɹ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ
ɨɱɟɧɶ ɦɚɥɚ, ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ
65
ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ ɨɬɫɭɬɫɬɜɭɟɬ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɟɠɮɚɡɧɚɹ ɫɢɥɚ (ɫɢɥɚ,
ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɞɢɫɩɟɪɫɧɵɟ ɱɚɫɬɢɰɵ, ɡɚɤɥɸɱɟɧɧɵɟ ɜ ɟɞɢɧɢɰɟ ɨɛɴɺ-
ɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɨɞɢɧɨɱɧɭɸ ɱɚɫɬɢɰɭ, ɧɚ ɱɢɫɥɨ ɱɚɫɬɢɰ ɜ
ɟɞɢɧɢɰɟ ɨɛɴɺɦɚ. ɉɨɷɬɨɦɭ ɩɪɢ
N
ĺ 0 ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɦɟɠɮɚɡɧɨɣ ɫɢɥɵ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɮɨɪɦɭɥɵ, ɫɩɪɚɜɟɞɥɢɜɵɟ ɞɥɹ ɫɥɭɱɚɹ ɨɦɵɜɚɧɢɹ ɨɞɢɧɨɱɧɨɣ ɱɚɫɬɢɰɵ ɩɨɬɨɤɨɦ, ɤɨɬɨɪɵɟ ɩɨɥɭɱɟɧɵ ɪɚɧɟɟ.
ȼ ɫɥɭɱɚɹɯ ɤɨɧɟɱɧɨɝɨ ɡɧɚɱɟɧɢɹ ɨɛɴɺɦɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ (
N ĺ 0)
ɞɥɹ ɪɚɫɱɺɬɚ ɦɟɠɮɚɡɧɨɣ ɫɢɥɵ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɣ ɩɨɞɯɨɞ –
ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɹɱɟɟɱɧɚɹ ɦɨɞɟɥɶ (ɪɢɫ. 4.6).
z
r
r
r
c
д
Ɋɢɫ. 4.6. əɱɟɟɱɧɚɹ ɦɨɞɟɥɶ
ȼ ɪɚɦɤɚɯ ɷɬɨɣ ɦɨɞɟɥɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɜɫɹ ɞɢɫɩɟɪɫɧɚɹ ɫɢɫɬɟɦɚ
ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɞɟɥɟɧɚ ɧɚ ɨɞɢɧɚɤɨɜɵɟ ɹɱɟɣɤɢ. ȼ ɰɟɧɬɪɟ ɤɚɠɞɨɣ ɹɱɟɣɤɢ, ɮɨɪɦɭ ɤɨɬɨɪɨɣ ɩɪɢɧɢɦɚɸɬ ɫɮɟɪɢɱɟɫɤɨɣ, ɪɚɫɩɨɥɚɝɚɟɬɫɹ ɨɞɧɚ ɱɚɫɬɢɰɚ, ɪɚɡɦɟɪ ɹɱɟɣɤɢ ɡɚɜɢɫɢɬ ɨɬ ɪɚɡɦɟɪɚ ɱɚɫɬɢɰ ɢ ɢɯ ɤɨɧɰɟɧɬɪɚɰɢɢ.
Ȼɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɜɫɟ ɱɚɫɬɢɰɵ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɣ ɪɚɡɦɟɪ ɢ
ɹɜɥɹɸɬɫɹ ɫɮɟɪɢɱɟɫɤɢɦɢ. ɉɭɫɬɶ ɤɨɧɰɟɧɬɪɚɰɢɹ ɱɚɫɬɢɰ
3
V
ɪɚɜɧɚ
N . ɑɢɫɥɨ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ
N
4
ɞ
. Ʉɚɠɞɚɹ ɱɚɫɬɢɰɚ
3
S
r
0
ɫ ɪɚɞɢɭɫɨɦ
r
0
ɨɤɪɭɠɟɧɚ ɫɩɥɨɲɧɨɣ ɫɪɟɞɨɣ, ɨɛɴɺɦ ɤɨɬɨɪɨɣ ɪɚɜɟɧ
4
Vrr S
ɫ c ɞ
3
33
.
ɗɬɨɬ ɠɟ ɨɛɴɺɦ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɨɛɴɺɦɧɭɸ ɤɨɧɰɟɧɬɪɚɰɢɸ
4
3
Vr S N
ɫ c
3
66
1
, ɝɞɟ
N
V
ɞ
VV
c ɞ
.
ɉɪɢɪɚɜɧɢɜɚɹ ɩɪɚɜɵɟ ɱɚɫɬɢ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɮɨɪɦɭɥ, ɧɚɯɨɞɢɦ ɪɚ-
E
ɞɢɭɫ ɹɱɟɣɤɢ
1
rr
N. (4.85)
c ɞ
3
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɹɱɟɟɱɧɨɣ ɦɨɞɟɥɢ ɩɨɡɜɨɥɹɟɬ ɫɜɟɫɬɢ ɡɚɞɚɱɭ ɨɩɢɫɚɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɢɫɩɟɪɫɧɵɯ ɱɚɫɬɢɰ ɤ ɪɟɲɟɧɢɸ
ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ – ɋɬɨɤɫɚ, ɡɚɩɢɫɚɧɧɵɯ ɞɥɹ ɫɩɥɨɲɧɵɯ ɫɪɟɞ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɨɛɥɚɫɬɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɞɜɭɦɹ ɫɮɟɪɚɦɢ. Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɱɚɫɬɢɰɵ, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɰɟɧɬɪɟ ɹɱɟɣɤɢ, ɡɚɜɢɫɹɬ ɨɬ
ɚɝɪɟɝɚɬɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ ɢ ɫɮɨɪɦɭɥɢɪɨɜɚɧɵ ɪɚɧɟɟ. Ʉɪɨɦɟ ɧɢɯ
ɧɟɨɛɯɨɞɢɦɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɟɳɟ
ɞɜɚ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹ ɧɚ ɜɧɟɲɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɹɱɟɣɤɢ. ȼ ɤɚɱɟɫɬɜɟ ɨɞɧɨɝɨ ɢɡ ɧɢɯ ɡɚɞɚɺɬɫɹ ɨɛɵɱɧɨ ɭɫɥɨɜɢɟ ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ ɧɨɪɦɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɤɨɪɨɫɬɢ ɫɩɥɨɲɧɨɣ
ɫɪɟɞɵ ɧɚ ɝɪɚɧɢɰɟ ɹɱɟɣɤɢ. ɗɬɨ ɭɫɥɨɜɢɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɨɡɦɭɳɟɧɢɟ,
ɜɧɨɫɢɦɨɟ ɜ ɩɨɬɨɤ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɱɚɫɬɢɰɟɣ, ɥɨɤɚɥɢɡɨɜɚɧɨ ɜ ɩɪɟɞɟɥɚɯ ɹɱɟɣɤɢ. ȼɬɨɪɨɟ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɧɟ ɫɬɨɥɶ ɨɱɟɜɢɞɧɨ ɢ ɪɚɡɥɢɱɧɵɦɢ
ɢɫɫɥɟɞɨɜɚɬɟɥɹɦɢ ɡɚɞɚɺɬɫɹ ɩɨ-ɪɚɡɧɨɦɭ. ɇɚɢɛɨɥɶɲɟɟ ɪɚɫɩɪɨ-
ɫɬɪɚɧɟɧɢɟ ɩɨɥɭɱɢɥɢ ɞɜɟ ɦɨɞɟɥɢ – ɏɚɩɩɟɥɹ ɢ Ʉɭɜɚɛɚɪɵ.
ȼ ɦɨɞɟɥɢ ɏɚɩɩɟɥɹ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ ɹɱɟɣɤɢ ɜɵɫɬɚɜɥɹɟɬɫɹ
ɭɫɥɨɜɢɟ ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ. ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɜɧɟɲɧɹɹ ɝɪɚɧɢɰɚ ɢɝɪɚɟɬ ɪɨɥɶ ɫɜɨɛɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ȼ ɦɨɞɟɥɢ Ʉɭɜɚɛɚɪɵ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ ɜɵɫɬɚɜɥɹɟɬɫɹ ɭɫɥɨɜɢɟ
ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ
ɡɚɜɢɯɪɺɧɧɨɫɬɢ ɩɨɬɨɤɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɠɢɞɤɨɫɬɢ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɹɱɟɣɤɟ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɡɧɚɱɟɧɢɟ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ ɹɜɥɹɟɬɫɹ ɦɚɥɵɦ. ɀɢɞɤɨɫɬɶ ɧɟ-
constU
ɫɠɢɦɚɟɦɚ (
). ɉɨɦɟɫɬɢɦ ɜ ɰɟɧɬɪ ɱɚɫɬɢɰɵ ɧɚɱɚɥɨ ɫɮɟɪɢɱɟ-
ɫɤɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɧɚɩɪɚɜɥɟɧɢɟ ɩɨɥɹɪɧɨɣ ɨɫɢ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɫɤɨɪɨɫɬɢ ɠɢɞɤɨɫɬɢ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. ɉɪɢ
Re 0o
ɢɧɟɪɰɢɨɧɧɵɟ ɱɥɟɧɵ ɜ ɭɪɚɜɧɟɧɢɢ ɇɚɜɶɟ – ɋɬɨɤɫɚ ɦɚɥɵ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɜɹɡɤɢɦɢ, ɢ ɞɜɢɠɟɧɢɟ ɠɢɞɤɨɫɬɢ ɜ ɹɱɟɣɤɟ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɪɚɧɟɟ, ɨɩɢɫɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ
422
0EEE\ \ , (4.86)
2
ɝɞɟ
– ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ:
2
wTw w
2
E
sin 1
22
rr
w
§·
¨¸
wT T wT
©¹
sin
; (4.87)
67
f
– ɮɭɧɤɰɢɹ ɬɨɤɚ, ɫɜɹɡɚɧɧɚɹ ɫ ɫɨɫɬɚɜɥɹɸɳɢɦɢ ɫɤɨɪɨɫɬɢ
\
w
ɠɢɞɤɨɫɬɢ ɫ ɩɨɦɨɳɶɸ ɫɨɨɬɧɨɲɟɧɢɣ
W
1
sin
w\
. (4.88)
w\
w
r
1
2
r
T
sin
wT
;
w
T
rr
Tw
w
ɢ
r
Ɋɚɫɫɦɨɬɪɢɦ ɨɛɬɟɤɚɧɢɟ ɫɮɟɪɢɱɟɫɤɨɣ ɬɜɺɪɞɨɣ ɱɚɫɬɢɰɵ. ȼ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɫ ɱɚɫɬɢɰɟɣ, ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɟɺ ɩɨɜɟɪɯɧɨɫɬɢ ɛɭɞɭɬ ɢɦɟɬɶ ɜɢɞ
ɩɪɢ
rr
ww
0
r
0
. (4.89)
T
ɋɮɨɪɦɭɥɢɪɭɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ ɹɱɟɣɤɢ. ȼ
ɧɟɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɤɚɤ ɭɤɚɡɵɜɚɥɨɫɶ ɜɵɲɟ, ɜ ɧɭɥɶ ɨɛɪɚɳɚɟɬɫɹ ɪɚɞɢɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɤɨɪɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɱɚɫɬɢɰɟɣ (ɬ.ɟ. ɩɨɞɜɢɠɧɨɣ), ɧɚ ɜɧɟɲɧɟɣ
ɝɪɚɧɢɰɟ ɹɱɟɣɤɢ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ
ɩɪɢ
rr
c
wu T
cos
r
. (4.90)
ȼ ɤɚɱɟɫɬɜɟ ɜɬɨɪɨɝɨ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ ɹɱɟɣɤɢ ɢɫɩɨɥɶɡɭɟɦ ɭɫɥɨɜɢɟ ɏɚɩɩɟɥɹ, ɬ.ɟ. ɭɫɥɨɜɢɟ ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ ɤɚɫɚ-
W
ɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ
.
ɬ
ȼ ɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɫ ɱɚɫɬɢɰɟɣ, ɷɬɨ ɭɫɥɨɜɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
ɩɪɢ
rr
W P
ɬ c
c
§·
¨¸
rrr
wT w
©¹
T
rr
0
. (4.91)
w
w
ww
w
1
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4.86) ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ (4.89) – (4.91)
ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɮɨɪɦɟ
2
sinfr\ T. (4.92)
ɉɨɞɫɬɚɜɥɹɹ (4.92) ɜ (4.86), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ
ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
r
, ɤɨɬɨɪɚɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
42
df df df f
dr r dr r r
488
42 23 4
dr
.
0
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɟɧɨ ɪɚɧɶɲɟ:
68
f
D
D
rCrCrCrCr
. (4.93)
42 1
4211
ɉɨɫɬɨɹɧɧɵɟ ɭɪɚɜɧɟɧɢɹ (4.93) ɞɨɥɠɧɵ ɛɵɬɶ ɧɚɣɞɟɧɵ ɫ ɩɨɦɨɳɶɸ
ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ (4.89) – (4.91).
ɋ ɭɱɺɬɨɦ (4.88) ɢ (4.92) ɩɟɪɟɩɢɲɟɦ ɷɬɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ:
ɩɪɢ
ɩɪɢ
rr
ɩɪɢ
22 0
c
rr
rr
fr r
df
fdr
0
f ; (4.95)
c
df d f
cc
dr
; (4.94)
0
0
2
ur
c
2
2
2
c
dr
. (4.96)
2
c
ɉɟɪɟɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ (4.93) ɜ ɜɢɞɟ
42
ɝɞɟ
§· §· §·
rrr
fr C C C C
c
440
cccc
¨¸ ¨¸ ¨¸
4211
rrrr
000
©¹ ©¹ ©¹
4
c
;CCr
220
2
c
;CCr
110
;CCr
c
C
1
C
r
0
1
.
r
§·
0
. (4.97)
¨¸
©¹
ɉɨɞɫɬɚɜɥɹɹ (4.97) ɜ ɭɪɚɜɧɟɧɢɹ (4.94) – (4.96), ɩɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ
ɱɟɬɵɪɺɯ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɱɟɬɵɪɺɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
,Cc 2,Cc 1,Cc
4
c
C
. ɗɬɚ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɟɟ
1
ɪɟɲɟɧɢɟ:
*5
\[
c
C
, (4.98)
4
D
*4
32
\ [ [
c
C
2
C
1
*5
32
\ [
c
, (4.100)
\
c
C
1
, (4.99)
D
*
, (4.101)
69
ɝɞɟ
F
F
*
\
ur
2
0
2
r
0
;
; (4.102)
[
r
c
23 3 2 .D [ [ [ (4.103)
56
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɤɰɢɹ ɬɨɤɚ ɞɥɹ ɬɟɱɟɧɢɹ ɠɢɞɤɨɫɬɢ ɜ ɹɱɟɣɤɟ
ɧɚɣɞɟɧɚ. Ɋɚɫɩɨɥɚɝɚɹ ɫɨɨɬɧɨɲɟɧɢɟɦ (4.100), ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫɢɥɭ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
F, ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɬɜɺɪɞɭɸ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ
ɠɢɞɤɨɫɬɢ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ
8.
C SP (4.104)
c1
ɋ ɭɱɺɬɨɦ (4.100) ɢ (4.104) ɩɨɥɭɱɢɦ
432
SP [
F
23 3 2
ur
c0
[[[
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ur SP (4.106)
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Ⱥɧɚɥɨɝɢɱɧɵɟ ɦɨɞɟɥɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɨɩɢɫɚɧɢɹ ɫɬɟɫɧɟɧɧɨɝɨ
ɞɜɢɠɟɧɢɹ ɝɚɡɨɜɵɯ ɩɭɡɵɪɟɣ ɢ ɤɚɩɟɥɶ. ɋɭɳɟɫɬɜɭɸɬ ɛɨɥɟɟ ɫɥɨɠɧɵɟ
ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɬɚɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɉɪɢ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɨɛ ɨɛɬɟɤɚɧɢɢ ɫɮɟɪɢɱɟɫɤɨɣ ɱɚɫɬɢɰɵ ɢɞɟɚɥɶɧɨɣ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɶɸ ɩɪɢɧɢɦɚɟɬɫɹ, ɜ ɱɚɫɬɧɨɫɬɢ, ɱɬɨ ɨɛɬɟɤɚɧɢɟ ɱɚɫɬɢɰɵ ɩɨɬɨɤɨɦ ɨɫɟɫɢɦɦɟɬɪɢɱɧɨ. ȼ ɤɚɤɨɦ ɫɥɭɱɚɟ ɷɬɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ?
2. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɨɛɬɟɤɚɧɢɹ ɫɮɟɪɵ ɢɞɟɚɥɶɧɨɣ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɶɸ, ɟɫɥɢ ɟɺ ɞɜɢɠɟɧɢɟ ɛɟɡɜɢɯɪɟɜɨɟ?
70
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