Ìèíèñòåðñòâî íàóêè è âûñøåãî îáðàçîâàíèÿ Ðîññèéñêîé Ôåäåðàöèè
ÔÃÁÎÓ ÂÎ ¾ÑÃÓ èì. Ïèòèðèìà Ñîðîêèíà¿
Èíñòèòóò òî÷íûõ íàóê è èíôîðìàöèîííûõ òåõíîëîãèé
Êàôåäðà ìàòåìàòè÷åñêîãî ìîäåëèðîâàíèÿ è êèáåðíåòèêè
ÍÀÏÎÐÍÎÅ ÒÅ×ÅÍÈÅ ÆÈÄÊÎÑÒÈ
Ñ ÏÅÐÅÌÅÍÍÎÉ ÂßÇÊÎÑÒÜÞ
Âûïóñêíàÿ êâàëèôèêàöèîííàÿ ðàáîòà ïî íàïðàâëåíèþ ïîäãîòîâêè
02.03.01 Ìàòåìàòèêà è êîìïüþòåðíûå íàóêè
Âûïîëíèë
ñòóäåíò 149 ãðóïïû
Í.Ñ. Ñèçîâ
Íàó÷íûé ðóêîâîäèòåëü
ä.ô.-ì.í., äîöåíò
Í.À. Áåëÿåâà
Çàâåäóþùèé êàôåäðîé ÌÌèÊ
ê.ô.-ì.í., äîöåíò
Þ.Í. Áåëÿåâ
Ñûêòûâêàð 2020
Ñîäåðæàíèå
Ââåäåíèå
Îñíîâíûå ïîíÿòèÿ
1. Ìîäåëü íàïîðíîãî òå÷åíèÿ æèäêîñòè
3
4
4
2. Áåçðàçìåðíàÿ ñèñòåìà ñòàöèîíàðíîãî òå÷åíèÿ æèäêîñòè
8
1.1. Óðàâíåíèå äâèæåíèÿ íàïîðíîãî òå÷åíèÿ. . . . . . . . . . . . . . . . . . . . . .
2.1. Ðåøåíèå â ÿâíîì âèäå. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Èòåðàöèîííûé ìåòîä Íüþòîíà íà îñíîâå ìåòîäà ïðîãîíêè. . . . . . . . . . .
4
8
10
3. Ðåçóëüòàòû ÷èñëåííûõ ýêñïåðèìåíòîâ äëÿ ñòàöèîíàðíîé çàäà÷è
4. Áåçðàçìåðíàÿ ñèñòåìà íåñòàöèîíàðíîãî òå÷åíèÿ æèäêîñòè
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14
5. Ðåçóëüòàòû ÷èñëåííûõ ýêñïåðèìåíòîâ äëÿ íåñòàöèîíàðíîé çàäà÷è
Âûâîäû è ñðàâíåíèå ðåçóëüòàòîâ ÷èñëåííûõ ýêñïåðèìåíòîâ
19
21
4.1. Ìåòîä ïðîãîíêè. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ïðîâåäåíèå ÷èñëåííîãî ýêñïåðèìåíòà ïðè êà÷åñòâåííîì èçìåíåíèè ïàðàìåòðîâ
äëÿ íåñòàöèîíàðíîé çàäà÷è . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cðàâíåíèå ðåçóëüòàòîâ ÷èñëåííûõ ýêñïåðèìåíòîâ . . . . . . . . . . . . . . . . . . .
14
21
21
Çàêëþ÷åíèå
22
Ñïèñîê ëèòåðàòóðû
23
Ïðèëîæåíèå 1. Ïðîâåäåíèå ÷èñëåííîãî ýêñïåðèìåíòà ïðè âàðüèðîâàíèè ïàðàìåòðîâ
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Ïðèëîæåíèå 1. Ñðàâíåíèå ãðàôèêîâ äëÿ íåñòàöèîíàðíîãî è ñòàöèîíàðíîãî
òå÷åíèÿ
28
Ïðèëîæåíèå 1. Ïðîãðàììà ïîñòðîåíèÿ ãðàôèêà ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé è ñêîðîñòè ñòàöèîíàðíîãî íàïîðíîãî òå÷åíèÿ â Visual Studio
íà ÿçûêàõ C++ è C Sharp
30
Ïðèëîæåíèå 1. Ïðîãðàììà ïîñòðîåíèÿ ãðàôèêà ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé è ñêîðîñòè íåñòàöèîíàðíîãî íàïîðíîãî òå÷åíèÿ â Visual Studio
íà ÿçûêàõ C++ è C Sharp
38
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
3
Àííîòàöèÿ
Ïîñòðîåíà ìàòåìàòè÷åñêàÿ ìîäåëü, îïèñûâàþùàÿ íàïîðíîå òå÷åíèå ñòðóêòóðèðîâàííîé æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ ìåæäó äâóìÿ ïëîñêîñòÿìè. Ðàçðàáîòàí
àëãîðèòì è ïî íåìó ñîñòàâëåíà ïðîãðàììà ÷èñëåííîãî ðåøåíèÿ. ×èñëåííîå ðåøåíèå
çàäà÷ ñòàöèîíàðíîãî è íåñòàöèîíàðíîãî òå÷åíèÿ ïîëó÷åíû íà îñíîâå èòåðàöèîííîãî
ìåòîäà Íüþòîíà è ìåòîäà ïðîãîíêè. Ïðèâîäÿòñÿ ãðàôèêè è ðåçóëüòàòû ýêñïåðèìåíòîâ.
Ââåäåíèå
Ðàññìàòðèâàåòñÿ íàïîðíîå òå÷åíèå ñòðóêòóðèðîâàííîé íåñæèìàåìîé æèäêîñòè [1;2].
Ñèñòåìà îïðåäåëÿþùèõ ñîîòíîøåíèé, ñîñòîèò èç óðàâíåíèÿ äâèæåíèÿ Íàâüå-Ñòîêñà,
äèôôóçèîííî-êèíåòè÷åñêîãî è ñîîòâåòñòâóþùèõ íà÷àëüíûõ è ãðàíè÷íûõ óñëîâèé.
Öåëü ðàáîòû ïîñòðîåíèå ìàòåìàòè÷åñêîé ìîäåëè, îïèñûâàþùåé íàïîðíîå òå÷åíèå
ñòðóêòóðèðîâàííîé æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ ìåæäó äâóìÿ ïëîñêîñòÿìè.
Çàäà÷è èññëåäîâàíèÿ:
• ïîëó÷èòü ñèñòåìó äèôôåðåíöèàëüíûõ óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ äëÿ îïèñàíèÿ òå÷åíèÿ;
• îáåçðàçìåðèòü ïîëó÷åííóþ ñèñòåìó;
• ðàçðàáîòàòü àëãîðèòì ÷èñëåííîãî ðåøåíèÿ íà îñíîâå ìåòîäà ïðîãîíêè è èòåðàöèîííîãî ìåòîäà Íüþòîíà íà îñíîâå ìåòîäà ïðîãîíêè;
• ïðîâåñòè ÷èñëåííûé àíàëèç áåçðàçìåðíîé ñèñòåìû ñòàöèîíàðíîãî è íåñòàöèîíàðíîãî
òå÷åíåíèÿ æèäêîñòè ïðè âàðüèðîâàíèè ïàðàìåòðîâ;
• ïðîàíàëèçèðîâàòü ïîëó÷åííûå ðåçóëüòàòû íà îñíîâå ãðàôèêîâ
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
4
Îñíîâíûå ïîíÿòèÿ
Íåíüþòîíîâñêîé æèäêîñòüþ íàçûâàþò æèäêîñòü, ïðè òå÷åíèè êîòîðîé å¼ âÿçêîñòü
~ (ãðàäèåíòà ñêîðîñòè). Îáû÷íî òàêèå æèäêîñòè ñèëüíî íåîäíîðîäíû è
çàâèñèò îò gradV
ñîñòîÿò èç êðóïíûõ ìîëåêóë, îáðàçóþùèõ ñëîæíûå ïðîñòðàíñòâåííûå ñòðóêòóðû.
Âÿçêîñòü µ îïðåäåëÿåòñÿ, êàê ñïîñîáíîñòü îêàçûâàòü ñîïðîòèâëåíèå ïåðåìåùåíèþ îäíîé ÷àñòè æèäêîñòè îòíîñèòåëüíî äðóãîé. ×åì áûñòðåå ïðîèñõîäèò âíåøíåå âîçäåéñòâèå
íà âçâåøåííûå â æèäêîñòè ìàêðîìîëåêóëû(ìîëåêóëà ñ âûñîêîé ìîëåêóëÿðíîé ìàññîé)
ñâÿçóþùåãî âåùåñòâà, òåì âûøå âÿçêîñòü æèäêîñòè.
Íåíüþòîíîâñêèå æèäêîñòè îáðàçóþò øèðîêèé êëàññ ðàçíîîáðàçíûõ ìàòåðèàëîâ, ñâîéñòâàìè êîòîðûõ ÿâëÿþòñÿ òåêó÷åñòü è îòêëîíåíèå îò çàêîíà âÿçêîñòè Íüþòîíà. Ãëèíèñòûå
ðàñòâîðû, ìàñëÿíûå êðàñêè, íåôòåïðîäóêòû, ìíîãèå ïðîìûøëåííûå ñóñïåíçèè, ïëàçìà
êðîâè äàþò ïðèìåðû íåíüþòîíîâñêèõ æèäêîñòåé. Ïðîñòåéøèì íàãëÿäíûì áûòîâûì ïðèìåðîì ìîæåò ÿâëÿòüñÿ ñìåñü êðàõìàëà ñ íåáîëüøèì êîëè÷åñòâîì âîäû. Âÿçêîñòü òàêèõ
æèäêîñòåé íå ÿâëÿåòñÿ âåëè÷èíîé çàâèñÿùåé îò òåìïåðàòóðû, à ñòàíîâèòñÿ çàâèñèìîé
îò äåôîðìàöèè, äâèæåíèÿ, âðåìåíè è äðóãèõ ôàêòîðîâ. Ýòè íåíüþòîíîâñêèå æèäêîñòè
îáðàçóþò ñòðóêòóðèðîâàííûå òåêó÷èå ñèñòåìû.
Ñòðóêòóðíûå ïðåâðàùåíèÿ â òàêèõ ñèñòåìàõ îòâå÷àþò çà ïðîöåññû ñàìîîðãàíèçàöèè
- ñàìîïðîèçâîëüíîå ïîÿâëåíèå è ðàçâèòèå íåêîòîðîé ñòðóêòóðû â ïåðâîíà÷àëüíî îäíîðîäíîé ñðåäå. Ê òàêèì ñòðóêòóðàì îòíîñÿòñÿ àâòîêîëåáàíèÿ, àâòîâîëíû, äèññèïàòèâíûå
ñòðóêòóðû è ò.ä. Äèññèïàòèâíàÿ ñòðóêòóðà - ýòî óñòîé÷èâîå ñîñòîÿíèå, âîçíèêàþùåå â
íåðàâíîâåñíîé ñðåäå ïðè óñëîâèè äèññèïàöèè (ðàññåèâàíèÿ) ýíåðãèè, êîòîðàÿ ïîñòóïàåò
èçâíå.
Íåíüþòîíîâñêàÿ æèäêîñòü - ýòî æèäêîñòü ñ µ 6= const.
1. Ìîäåëü íàïîðíîãî òå÷åíèÿ æèäêîñòè
1.1. Óðàâíåíèå äâèæåíèÿ íàïîðíîãî òå÷åíèÿ.
Ïóñòü ñòðóêòóðèðîâàííàÿ
íåñæèìàåìàÿ æèäêîñòü çàïîëíÿåò ïîëîñó ìåæäó ïëîñêîñòÿìè η = h è η = −h. Ïðåäïîëàãàåòñÿ, ÷òî òå÷åíèå ïðîèñõîäèò â íàïðàâëåíèè îñè ξ ïîä äåéñòâèåì äàâëåíèÿ. Îáîçíà÷èì
~ = (Vξ , Vη , Vζ ) âåêòîð ñêîðîñòè æèäêîñòè â ïðîèçâîëüíîé òî÷êå ïîëîñû â ìîìåíò
÷åðåç V
âðåìåíè t, ïðè÷åì
V~ = (V (η, t), 0, 0); Vξ = V (η, t) = V ; Vη = Vζ = 0.
 ýòîì ñëó÷àåì óðàâíåíèå íåðàçðûâíîñòè âûïîëíÿåòñÿ:
∂V
∂V
∂V
η
ζ
ξ
= 0.
+
+
div V~ =
∂ξ
∂η
∂ζ
(1.1)
(1.2)
Óñëîâèå íåñæèìàåìîñòè æèäêîñòè:
ρ = const .
Óðàâíåíèå Íàâüå-Ñòîêñà:
"
#
∂ V~
ρ
+ (V~ , ∇)V~ = − grad p + µ∆V~ + 2(grad µ, ∇)V~ + grad µ × rot V~ + ρF~ .
∂t
Ìàññîâûå ñèëû îòñóòñòâóþò:
F~ = 0.
(1.4)
(1.5)
(1.6)
Âûïèøåì ñëàãàåìûå óðàâíåíèÿ (1.5) â äåêàðòîâîé ñèñòåìå êîîðäèíàò ïðè îòñóòñòâèè
âíåøíèõ ñèë â ðàññìàòðèâàåìîì ñëó÷àå äâèæåíèÿ æèäêîñòè:
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
5
Ïðîåêöèÿ ñêîðîñòè çàïèøåòñÿ êàê
∂ V~
=
∂t
∂V
; 0; 0 ,
∂t
ò.ê. òå÷åíèå æèäêîñòè ïðîñèõîäèò â íàïðàâëåíèè îñè ξ , îñòàëüíûå êîîðäèíàòû áóäóò ðàâíÿòüñÿ íóëþ.
Òîãäà ñêàëÿðíîå óìíîæåíèå ñêîðîñòè íà îïåðàòîð Ãàìèëüòîíà äàñò
∂
∂
∂
∂V (η, t)
~
~
(V , ∇)V = V (η, t) + 0
+0
V (η, t) = V (η, t)
= (0, 0, 0), (V~ , ∇)V~ = 0;
∂ξ
∂η
∂ζ
∂ξ
Ðàñïèøåì ãðàäèåíò âÿçêîñòè
grad µ =
∂µ ∂µ ∂µ
;
;
∂ξ ∂η ∂ζ
= (0;
µ∆V~ = µ (∆Vξ , ∆Vη , ∆Vζ ) = µ
(grad µ, ∆)V (η, t) =
∂µ ∂
∂µ ∂
∂
+
+
∂ξ ∂ξ ∂η ∂η ∂ζ
V (η, t) =
∂µ ∂V (η, t)
=
; 2(grad µ, ∆)V~ =
∂η ∂η
~i
~j
~k
∂
∂ξ
∂
∂η
∂
∂ζ
V (η, t)
0
0
rot V~ =
∂µ
; 0);
∂η
∂ 2V
; 0; 0 ;
∂η 2
∂µ ∂V (η, t) ∂µ ∂V (η, t) ∂V (η, t)
+
+
∂ξ ∂ξ
∂η ∂η
∂ζ
=
∂µ ∂V
2
, 0, 0 ;
∂η ∂η
∂V
(η,
t)
∂V
(η,
t)
= ~i (0 − 0) − ~j 0 −
+ ~k 0 −
=
∂ζ
∂η
∂V (η, t) ∂V (η, t)
= 0;
;−
;
∂ζ
∂η
grad µ×rot V~ =
~i
~j
~k
∂µ
∂ξ
∂µ
∂η
∂V
∂ζ
∂µ
∂ζ
− ∂V
∂η
0
∂µ ∂V
∂µ ∂V
∂µ ∂V
∂µ ∂V
~
~
~
=i −
−
−j −
− 0 +k
−0 =
∂η ∂η
∂ζ ∂ζ
∂ξ ∂η
∂ξ ∂ζ
∂µ
∂V
∂µ
∂V
= ~i −
− 0 − ~j (0 − 0) + ~k (0 − 0) = −
; 0; 0 .
∂η ∂η
∂η ∂η
Ñ ó÷åòîì ïðåîáðàçîâàíèé îïèñàííûõ âûøå, ïîëó÷àåì óðàâíåíèå äâèæåíèÿ â ñëåäóþùåì
âèäå:
!
~
∂ V~
∂
∂
V
ρ
+ ρ(V~ , ∇)V~ =
µ(a)
− grad p.
(1.8)
∂t
∂η
∂η
Çäåñü ρ - ïëîòíîñòü æèäêîñòè, p - äàâëåíèå, µ = µ(a) - âÿçêîñòü æèäêîñòè, îïðåäåëÿåìàÿ
óðàâíåíèåì
−1
µ−1 (a) = µ−1
1 a + µ2 (1 − a)
è çàâèñÿùàÿ îò ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a = a(η, t).
6
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
Ïåðåéäåì â óðàâíåíèè (1.8) ê ïðîåêöèÿì íà îñè êîîðäèíàò:
ξ:
∂V
1 ∂p µ ∂ 2 V
µ ∂V
=−
+
+
2
∂t
ρ ∂ξ
ρ ∂η
η ∂η
(1.9)
η: 0=
1 ∂p
⇒ p 6= p(η)
ρ ∂η
(1.10)
ζ: 0=
1 ∂p
⇒ p 6= p(ζ)
ρ ∂ζ
(1.11)
Ñëåäîâàòåëüíî, èç óðàâíåíèé (1.10),(1.11) ìîæíî ñäåëàòü âûâîä, ÷òî äàâëåíèå çàâèñèò
òîëüêî îò êîîðäèíàòû ξ , òî åñòü p = p(ξ), è â ñèëó óðàâíåíèÿ (1.9)
∂p
≡ b = const.
∂ξ
Ïîëó÷àåòñÿ, ÷òî
− grad p = (b, 0, 0)
Òîãäà ïðîåêöèÿ óðàâíåíèÿ Íàâüå-Ñòîêñà íà îñü ξ ñ ó÷åòîì (1.9) èìååò âèä:
∂V
1
1 ∂
∂V
= b+
µ
.
∂t
ρ
ρ ∂η
∂t
(1.12)
Çàïèøåì ñîîòâåòñòâóþùóþ áåçðàçìåðíóþ ñèñòåìó äëÿ óðàâíåíèÿ äâèæåíèÿ (1.12) è
äèôôóçèîííî-êèíåòè÷åñêîå óðàâíåíèå
∂a ~
da
=
+ V · grad a = D∆a + Φ(a, γ).
dt
∂t
(1.13)
Äèôôóçèîííî-êèíåòè÷åñêîå óðàâíåíèå (1.13) â ðàññìàòðèâàåìîì ñëó÷àå ïðèìåò âèä
∂ 2a
∂a
= D 2 + Φ(a, γ).
∂t
∂ t
(1.14)
ãäå D - êîýôôèöèåíò äèôôóçèè, Φ - ñóììàðíàÿ ñêîðîñòü ïðåâðàùåíèÿ, îïðåäåëÿåìàÿ
ñîîòíîøåíèåì
k0
2
Φ(a, γ) = k2 1 − a − a exp(p0 µ(a) + q0 γ ) .
(1.15)
k2
Çäåñü
γ=
∂V
∂η
ñêîðîñòü äåôîðìàöèè, k0 , k2 , p0 , q0 ïàðàìåòðû æèäêîñòè. Ïîëó÷èì ñèñòåìó äèôôåðåíöèàëüíûõ óðàâíåíèé äëÿ îïðåäåëåíèÿ ñêîðîñòè òå÷åíèÿ è ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé
∂V
∂
∂V
ρ
µ(a)
+ b,
=
∂t
∂η
∂η
(1.16)
∂a
∂ 2a
= D 2 + Φ(a, γ),
∂t
∂η
ñ íà÷àëüíûìè:
V |t=0 = 0; a|t=0 = a0 ;
(1.17)
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
7
è ãðàíè÷íûìè óñëîâèÿìè:
∂a
∂η
∂a
∂η
= 0; V
= 0;
η=h
(1.18)
η=h
= 0;
η=0
∂V
∂η
= 0.
(1.19)
η=0
Ïåðåéäåì îò ñèñòåìû (1.16) − (1.19) ê áåçðàçìåðíîé ñèñòåìå óðàâíåíèé. Äëÿ ýòîãî
ñäåëàåì çàìåíó:
u=
b2 h2
k2 ρh2
k0
V µ2
µ2
η
Dρ
,
p
=
p
bh,
q
=
q
,
κ
=
,
χ
=
.
,
τ
=
t
,
x
=
,
β
=
1
0
1
0
2
bh2
ρh2
h
µ2
µ2
µ2
k2
(1.20)
Çäåñü x áåçðàçìåðíàÿ ïðîñòðàíñòâåííàÿ êîîðäèíàòà, τ áåçðàçìåðíîå âðåìÿ. Îáîçíà÷èì
ν(a) =
µ(a)
1
=
.
µ2
1 + λa
(1.21)
Ïîëó÷èì ñèñòåìó áåçðàçìåðíûõ óðàâíåíèé
∂ 2u
∂a ∂u
∂u
= −λν(a)2
+ ν(a) 2 + 1,
∂τ
∂x
∂x
" ∂x
2 !#
2
∂a
∂u
∂ a
∂u
= β 2 + κ 1 − a − aχexp p1 ν(a)
+ q1
,
∂τ
∂x
∂x
∂x
(1.22)
u|τ =0 = 0; a|τ =0 = a0 ;
(1.23)
ñ íà÷àëüíûìè
è ãðàíè÷íûìè óñëîâèÿìè:
∂u
∂x
∂a
∂x
=u
x=0
=
x=0
= 0;
(1.24)
x=1
∂a
∂x
= 0.
x=1
(1.25)
8
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
2. Áåçðàçìåðíàÿ ñèñòåìà ñòàöèîíàðíîãî òå÷åíèÿ æèäêîñòè
2.1. Ðåøåíèå â ÿâíîì âèäå.
Ðàññìîòðèì ÷àñòíûé ñëó÷àé íåñòàöèîíàðíîãî òå÷åíèÿ. Ñèñòåìà (1.22) è ñîîòâåòñòâóþùèå ãðàíè÷íûå óñëîâèÿ (1.24) − (1.25) äëÿ ñêîðîñòè è
ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé â ñëó÷àå ñòàöèîíàðíîãî ðåæèìà òå÷åíèÿ çàïèøóòñÿ â
âèäå
∂a ∂u
∂ 2u
+ 1 = 0,
ν(a) 2 − λν(a)2
∂x "
∂x ∂x
2 !#
(2.1)
d2 a
du
du
β 2 + κ 1 − a − aχexp p1 ν(a) + q1
= 0,
dx
dx
dx
u(1) = 0;
∂u
∂x
= 0.
(2.2)
x=0
Ðåøèì ïåðâîå óðàâíåíèå ñèñòåìû (2.1). Äëÿ ýòîãî ïåðåíåñåì åäèíèöó â ïðàâóþ ñòîðîíó è
ðàçäåëèì âñå óðàâíåíèå íà ν(a)
da du
1
d2 u
− λν(a)2
=−
2
dx
dx dx
ν(a)
Îáîçíà÷èì
λν(a)2
òîãäà
da
= p(x),
dx
d2 u
du
1
− p(x)
=−
2
dx
dx
ν(a)
Ïðîèçâåäåì çàìåíó
d2 u
= y,
dx2
ïîëó÷àåì
y 0 − p(x)y = −
1
ν(a)
(2.3)
a)Ðåøèì ñîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå
y 0 − p(x)y = 0
Z x
Z x
dy
=
p(x)dx
y
0
0
Z x
ln|y| =
p(x)dx + c
0
Ïîëó÷àåì
y0 = ce
Rx
0
p(x)dx
á)Ïðèìåíèì ìåòîä âàðèàöèè ïîñòîÿííîé c = c(x) äëÿ ðåøåíèÿ óðàâíåíèÿ (2.3), ãäå ν(a) â
ñîîòâåòñòâèè ñ îáîçíà÷åíèåì (1.21)
Rx
y = c(x)e
y 0 = c(x)0 e
Rx
0
p(s)ds
0
p(s)ds
Rx
+ c(x)e
0
(2.4)
p(s)ds
p(x)
(2.5)
Ïîäñòàâèì (2.4), (2.5) â óðàâíåíèå (2.3)
Rx
c(x)0 e
0
p(s)ds
+ c(x)p(x)e
Rx
0
p(s)ds
Rx
− c(x)p(x)e
0
p(s)ds
=−
1
ν(a)
9
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
c(x)0 e
Rx
0
p(s)ds
=−
1
ν(a)
Ðàçäåëÿåì ïåðåìåííûå è íàõîäèì c(x)
dc
1 − R x p(s)ds
=−
e 0
dx
ν(a)
Rx
dc = −(1 + λa)e− 0 p(s)ds dx
Z x
Rx
(1 + λa)e− 0 p(s)ds dx + c,
⇒ c(x) = −
0
ãäå c = const.
Íàéäåì êîíñòàíòó c
Rx
Rx
Rx
p(s)ds = 0 λν(a)2 da
ds = 0
ds
0
x
λ
da
1+λa
= ln|1 + λa|
=
0
= ln|1 + λa(x)| − ln|1 + λa(0)| = ln
1+λa(x)
1+λa(0)
Ïîäñòàâëÿåì âìåñòî èíòåãðàëà ïîëó÷èâøååñÿ âûðàæåíèå
Z x
1 + λa(x)
(1 + λa)
c(x) = −
dx + c
1 + λa(0)
0
⇒ c(x) = −(1 + λa(0))x + c.
Òîãäà
1+λa(x)
y = (c − (1 + λa(0))x)e(ln| 1+λa(0) |)
1 + λa(x)
.
⇒ y = (c − (1 + λa(0))x)
1 + λa(0)
Âåðíåìñÿ ê çàìåíå
du
= y,
dx
⇒
èç óñëîâèÿ
1 + λa(x)
du
= (c − (1 + λa(0))x)
,
dx
1 + λa(0)
∂u
∂x
ñëåäóåò, ÷òî c=0;
=0
x=0
du
= −x(1 + λa(x))
dx
Z x
Z 1
u(x) = −
x(1 + λa(x))dx ⇔ u(x) =
x(1 + λa(x))dx
⇒
1
(2.6)
x
Ïîëó÷àåì óðàâíåíèå äëÿ ñêîðîñòè
Z
u(x) =
1
s(1 + λa(s))ds,
x
ãäå x ∈ [0..1].
Âòîðîå óðàâíåíèå ñèñòåìû (2.1) äëÿ âû÷èñëåíèÿ ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé ñ
ó÷åòîì (2.6) çàïèøåòñÿ êàê
d2 a
2
+
κ
1
−
a
−
aχexp
p
ν(a)(−x(1
+
λa(x)))
+
q
(−x(1
+
λa(x)))
= 0.
(2.7)
1
1
dx2
Åñëè ðàñïðåäåëåíèå íå èçâåñòíî, òî óðàâíåíèå (2.7) ðåøàåì ÷èñëåííî èòåðàöèîííûì ìåòîäîì Íüþòîíà íà îñíîâå ìåòîäà ïðîãîíêè.
β
10
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
2.2. Èòåðàöèîííûé ìåòîä Íüþòîíà íà îñíîâå ìåòîäà ïðîãîíêè.
íàÿ ñèñòåìà ñ ãðàíè÷íûìè óñëîâèÿìè
Z 1
u(x) =
s(1 + λa(s))ds,
Ñòàöèîíàð-
x
(2.8)
d2 a
β 2 + κ 1 − a − aχexp p1 ν(a)(−x(1 + λa(x))) + q1 (−x(1 + λa(x)))2 = 0,
dx
∂u
∂x
∂a
∂x
=u
x=0
x=0
= 0,
x=1
∂a
=
∂x
(2.9)
= 0.
x=1
Çàäàäèì íà÷àëüíîå ïðèáëèæåíèå a0 , óäîâëåòâîðÿþùóþ ãðàíè÷íûì óñëîâèÿì (2.9). Äàëåå, çíàÿ a0 , íàõîäèì u0 . Ñëåäóþùèå ïðèáëèæåíèÿ óòî÷íÿåì ñ ïîìîùüþ ìåòîäà ïðîãîíêè.
Ðàññìîòðèì ìåòîä ïðîãîíêè. Âî âòîðîì óðàâíåíèè ñèñòåìû (2.8) îáîçíà÷èì
exp p1 ν(a)(−x(1 + λa(x))) + q1 (−x(1 + λa(x)))2 = expo
è çàìåíèì ÷àñòíûå ïðîèçâîäíûå ðàçíîñòíûìè ñîîòíîøåíèÿìè:
ai+1,j − 2ai + ai−1,j
d2 a
=
; a = ai,j
2
dx
∆x2
(2.10)
Ïîäñòàâëÿåì (2.10) âî âòîðîå óðàâíåíèå ñèñòåìû (2.8) è ïîñëå íåêîòîðûõ ïðåîáðàçîâàíèé
ïîëó÷èì:
ai+1,j − 2ai,j + ai−1,j
0=β
+ κ − κai,j − κai,j χexpo
(2.11)
∆x2
Äëÿ íàõîæäåíèÿ ñåòî÷íîãî çíà÷åíèÿ ôóíêöèè ai èñïîëüçóåì ïðîãîíî÷íóþ ôîðìóëó:
ai+1,j = Ei+1 ai,j + Fi+1,j ,
(2.12)
ãäå Ei+1 , Fi+1,j ïðîãîíî÷íûå êîýôôèöèåíòû. Ïîäñòàâèì (2.12) â (2.11)
Ei+1 ai,j + Fi+1,j − 2ai,j + ai−1,j
+ κ − κai,j − κai,j χexpo
0=β
∆x2
βEi+1 ai,j βFi+1,j
2βai,j βai−1,j κ∆x2 κai,j ∆x2 κai,j χexpo∆x2
0=
+
−
+
+
−
−
∆x2
∆x2
∆x2
∆x2
∆x2
∆x2
∆x2
2βai,j κai,j ∆x2 κai,j χexpo∆x2 βEi+1 ai,j
βFi+1,j βai−1,j κ∆x2
+
+
−
=
+
+
∆x2
∆x2
∆x2
∆x2
∆x2
∆x2
∆x2
2β + κ∆x2 + κχexpo∆x2 − βEi+1
βFi+1,j + κ∆x2 βai−1,j
ai,j
=
+
∆x2
∆x2
∆x2
βFi+1,j + κ∆x2
ai,j =
∆x2
2β+κ∆x2 +κχexpo∆x2 −βE
∆x2
i+1
+
βai−1,j
∆x2
2β+κ∆x2 +κχexpo∆x2 −βEi+1
∆x2
βFi+1,j + κ∆x2
βai−1,j
+
2
2
2
2β + κ∆x + κχexpo∆x − βEi+1 2β + κ∆x + κχexpo∆x2 − βEi+1
Âîñïîëüçóåìñÿ ñëåäñòâèåì ïðîãîíî÷íîé ôîðìóëû:
ai,j =
ai,j = Ei ai−1,j + Fi,j ,
(2.13)
(2.14)
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
11
è èç (2.13) ïîëó÷èì âûðàæåíèÿ äëÿ ïðîãîíî÷íûõ êîýôôèöèåíòîâ Ei , Fi,j , êîòîðûå ïðåäñòàâëÿþò ðåêóððåíòíûå ñîîòíîøåíèÿ:
β
,
2β +
+ κχexpo∆x2 − βEi+1
βFi+1,j + κ∆x2
.
=
2β + κ∆x2 + κχexpo∆x2 − βEi+1
Ei =
Fi,j
κ∆x2
(2.15)
Ôîðìóëû (2.15) ïðåäñòàâëÿþò ñîáîé ðåêóððåíòíûå ñîîòíîøåíèÿ: çíàÿ En , Fn,j è, äâèãàÿñü ñïðàâà íàëåâî (îò i = n−1 ê i = 0) è (îò j = 1 ê j = r, ãäå j - èòåðàöèÿ ïðèáëèæåíèÿ,
r-êîëè÷åñòâî èòåðàöèé), ìîæíî îïðåäåëèòü çíà÷åíèÿ êîýôôèöèåíòîâ (En−1 , Fn−1,j ),
(En−2 , Fn−2,j ), (E1 , F1,j ).
Ïðè ýòîì, äëÿ íàõîæäåíèÿ ïàðû (En , Fn,j ) âîñïîëüçóåìñÿ âòîðûì ãðàíè÷íûì óñëîâèåì
∂a
∂x
= 0,
x=1
èç êîòîðîãî ñëåäóåò, ÷òî
an,j = an−1,j ,
an,j = En an−1,j + Fn,j ,
(2.16)
Ñëåäîâàòåëüíî,
Fn,j = 0,
En = 1,
è, ñëåäóÿ (2.15), îïðåäåëèì ïðîãîíî÷íûå êîýôôèöèåíòû (En−1 , Fn−1,j ), (En−2 , Fn−2,j ), (E1 , F1,j ).
Äëÿ íàõîæäåíèÿ ñåòî÷íûõ çíà÷åíèé èñêîìîé ôóíêöèè ai,j : îïðåäåëèì å¼ çíà÷åíèå â ïåðâîé óçëîâîé òî÷êå, òî åñòü a0,j .
Äëÿ ýòîãî âîñïîëüçóåìñÿ ïåðâûì èç ãðàíè÷íûõ óñëîâèé
∂a
∂x
=0
x=0
è ôîðìóëîé ïðîãîíêè (2.14). Èìååì
a1,j = a0,j ,
a1,j = E1 a0,j + F1,j ,
(2.17)
îòêóäà,
F1,j = 0,
E1 = 1.
Äàëåå, ñëåâà íàïðàâî, ïî ïðîãîíî÷íîé ôîðìóëå (2.14) âû÷èñëèì ñåòî÷íûå çíà÷åíèÿ èñêîìîé ôóíêöèè a2,j , ..., an,j â îñòàëüíûõ óçëàõ ñåòêè.
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
12
3. Ðåçóëüòàòû ÷èñëåííûõ ýêñïåðèìåíòîâ äëÿ ñòàöèîíàðíîé çàäà÷è
Íà ãðàôèêàõ ïðåäñòàâëåííûõ íèæå ìîæíî ïðîñëåäèòü êàê êðèâûå ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé, ñêîðîñòè òå÷åíèÿ è âÿçêîñòè, âûõîäÿò íà ñòàöèîíàð.
 ïåðâîì ãðàôèêå (Ðèñ.1) äëÿ ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a = a(x, r) ìàêñèìàëüíîå çíà÷åíèå r = 100, n = 10, ãäå êîëè÷åñòâî òî÷åê ðàçáèåíèÿ 'r' - ïî èòåðàöèÿì, è 'x'
- ïî êîîðäèíàòàì. Çíà÷åíèÿ êîíñòàíò λ, p1, q1, β , κ, χ áûëè âûáðàíû ìíîé òàêèì îáðàçîì,
÷òîáû ãðàôèê èìåë áîëåå "ãëàäêèé" âèä.
Íà ãðàôèêå ìîæíî óâèäåòü ïðÿìóþ ëèíèþ. Ýòî ëèíèÿ ïåðâîãî ïðèáëèæåíèÿ, ò.å. ìû
ñàìîñòîÿòåëüíî çàäàåì ïåðâîå çíà÷åíèå ñòåïåíè ñòðóêóòðíûõ ïðåâðàùåíèé a0=0.3. Çíà÷åíèÿ íà ãðàôèêàõ ïîêàçûâàþò êóäà ñòðåìÿòñÿ êðèâûå, ÷òîáû âûéòè íà ñòàöèîíàð.
Ðèñ. 1: Ñòàöèîíàðíàÿ ñêîðîñòü u=u(x) è ñòåïåíü ñòðóêòóðíûõ ïðåâðàùåíèé a=a(x): λ =
10, a0=0.3, p1=0.04, q1=0.04, n=10(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè x), β = 0.002, κ =
0.04, χ = 0.4
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
13
Ðèñ. 2: Áåçðàçìåðíàÿ âÿçêîñòü ν =ν(a): λ = 10, a0=0.3, p1=0.04, q1=0.04, n=10(êîëè÷åñòâî
òî÷åê ðàçáèåíèÿ ïî îñè x), β = 0.002, κ = 0.04, χ = 0.4
14
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
4. Áåçðàçìåðíàÿ ñèñòåìà íåñòàöèîíàðíîãî òå÷åíèÿ æèäêîñòè
4.1. Ìåòîä ïðîãîíêè.
(4.4).
Ðàññìîòðèì ìåòîä ïðîãîíêè äëÿ ðåøåíèÿ çàäà÷è (4.1)
∂u
∂a ∂u
∂ 2u
= −λν(a)2
+ ν(a) 2 + 1,
∂τ
∂x
∂x
" ∂x
2 !#
2
∂a
∂ a
∂u
∂u
= β 2 + κ 1 − a − aχexp p1 ν(a)
+ q1
,
∂τ
∂x
∂x
∂x
(4.1)
u|τ =0 = 0; a|τ =0 = a0 ;
(4.2)
ñ íà÷àëüíûìè
è ãðàíè÷íûìè óñëîâèÿìè:
∂u
∂x
∂a
∂x
=u
x=0
=
x=0
= 0;
(4.3)
x=1
∂a
∂x
= 0.
(4.4)
x=1
Äëÿ ñèñòåìû óðàâíåíèé (4.1) çàìåíèì ÷àñòíûå ïðîèçâîäíûå ðàçíîñòíûìè ñîîòíîøåíèÿìè:
∂u
ui,j − ui,j−1 ∂ 2 u
ui+1,j − 2ui,j + ui−1,j ∂u
ui+1,j − ui,j
≈
;
≈
;
≈
,
∂τ
∆τ
∂x2
∆x2
∂x
∆x
∂a
ai+1,j−1 − ai,j−1 ∂a
ai,j − ai,j−1 ∂ 2 a
ai+1,j − 2ai,j + ai−1,j
(4.5)
≈
;
≈
;
≈
,
2
∂x
∆x
∂τ
∆τ
∂x
∆x2
1
ν(ai,j−1 )) =
≈ νi,j−1 .
1 + λai,j−1
Ïîäñòàâèì ðàçíîñòíûå ñîîòíîøåíèÿ äëÿ ïåðâîãî óðàâíåíèÿ ñèñòåìû (4.1)
ui,j − ui,j−1
= −λ(νi,j−1 )2
∆τ
ai+1,j−1 − ai,j−1
∆x
ui+1,j − ui,j
ui+1,j − 2ui,j + ui−1,j
+νi,j−1
+1
∆x
∆x2
Çàìåíèì âûðàæåíèÿ âû÷èñëÿåìûå â ïðåäûäóùåì (j − 1) ñëîå:
ai+1,j−1 − ai,j−1
2
Ai,j−1 = (νi,j−1 )
, αi,j−1 = νi,j−1
∆x
ui,j − ui,j−1
ui+1,j − ui,j
ui+1,j − 2ui,j + ui−1,j
= −λAi,j−1
+ αi,j−1
+1
∆τ
∆x
∆x2
ui,j
λAi,j−1 ui,j 2αi,j−1 ui,j
λAi,j−1 ui+1,j
−
+
=−
+ αi,j−1
2
∆τ
∆x
∆x
∆x
ui+1,j + ui−1,j
∆x2
+1+
ui,j−1
∆τ
ïåðåíåñåì âñå ñëàãàåìûå ñ ui,j â ëåâóþ ñòîðîíó è âûíåñåì ui,j çà ñêîáêó
ui,j
1
λAi,j−1 2αi,j−1
−
+
∆τ
∆x
∆x2
= ui+1,j
Bi,j−1 =
αi,j−1 λAi,j−1
−
∆x2
∆x
αi,j−1 λAi,j−1
−
∆x2
∆x
+ αi,j−1
ui−1,j
ui,j−1
+1+
2
∆x
∆τ
15
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
ui,j
1
λAi,j−1 2αi,j−1
−
+
∆τ
∆x
∆x2
= ui+1,j Bi,j−1 + αi,j−1
ui−1,j
ui,j−1
+1+
2
∆x
∆τ
(4.6)
Äëÿ íàõîæäåíèÿ ñåòî÷íîãî çíà÷åíèÿ ôóíêöèè ui,j èñïîëüçóåì ïðîãîíî÷íóþ ôîðìóëó
ui+1,j = Ei+1 ui,j + Fi+1,j ,
(4.7)
ãäå Ei+1 , Fi+1,j - ïðîãîíî÷íûå êîýôôèöèåíòû. Ïîäñòàâèì (4.7) â (4.6) :
ui,j
1
λAi,j−1 2αi,j−1
−
+
− Ei+1 Bi,j−1
∆τ
∆x
∆x2
= Fi+1,j Bi,j−1 + αi,j−1
ui−1,j
ui,j−1
+1+
2
∆x
∆τ
u
ui,j =
∆x2 Fi+1,j Bi,j−1 + ∆x2 i,j−1
+ ∆x2
∆τ
+
λAi,j−1
2αi,j−1
1
∆x2 ∆τ
− ∆x
+ ∆x
−
E
B
i+1 i,j−1
2
αi,j−1 ui−1,j
+
∆x2
1
∆τ
−
λAi,j−1
∆x
2αi,j−1
∆x2
+
− Ei+1 Bi,j−1
(4.8)
Âîñïîëüçóåìñÿ ñëåäñòâèåì ïðîãîíî÷íîé ôîðìóëû
ui,j = Ei ui−1,j + Fi,j
(4.9)
è èç (4.8) ïîëó÷èì âûðàæåíèÿ äëÿ ïðîãîíî÷íûõ êîýôôèöèåíòîâ Ei , Fi,j , êîòîðûå ïðåäñòàâëÿþò ðåêóððåíòíûå ñîîòíîøåíèÿ:
∆τ αi,j−1
,
− ∆τ ∆xλAi,j−1 + ∆τ 2αi,j−1 − ∆τ ∆x2 Ei+1 Bi,j−1
∆τ ∆x2 Fi+1,j Bi,j−1 + ∆x2 ui,j−1 + ∆τ ∆x2
=
.
∆x2 − ∆τ ∆xλAi,j−1 + ∆τ 2αi,j−1 − ∆τ ∆x2 Ei+1 Bi,j−1
Ei =
Fi,j
∆x2
(4.10)
Ôîðìóëû (4.10) ïðåäñòàâëÿþò ñîáîé ðåêóððåíòíûå ñîîòíîøåíèÿ: çíàÿ En , Fn,j è, äâèãàÿñü ñïðàâà íàëåâî (îò i = n − 1 ê i = 0) ìîæíî îïðåäåëèòü çíà÷åíèÿ êîýôôèöèåíòîâ
(En−1 , Fn−1,j ), (En−2 , Fn−2,j ), (E1 , F1,j ). Ïðè ýòîì, äëÿ íàõîæäåíèÿ ïàðû (En , Fn,j ) âîñïîëüçóåìñÿ âòîðûì ãðàíè÷íûì óñëîâèåì
u
= 0,
x=1
èç êîòîðîãî ñëåäóåò, ÷òî
un,j = 0,
un,j = En un−1,j + Fn,j ,
(4.11)
Ñëåäîâàòåëüíî,
Fn,j = 0,
En = 0,
è, ñëåäóÿ (4.10), îïðåäåëèì ïðîãîíî÷íûå êîýôôèöèåíòû (En−1 , Fn−1,j ), (En−2 , Fn−2,j ), (E1 , F1,j ).
Äëÿ íàõîæäåíèÿ ñåòî÷íûõ çíà÷åíèé èñêîìîé ôóíêöèè ui,j : îïðåäåëèì å¼ çíà÷åíèå â ïåðâîé óçëîâîé òî÷êå, òî åñòü u0,j . Äëÿ ýòîãî âîñïîëüçóåìñÿ ïåðâûì èç ãðàíè÷íûõ óñëîâèé
∂u
∂x
= 0,
x=0
16
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
è ôîðìóëîé ïðîãîíêè (4.9). Èìååì
u1,j = u0,j ,
u1,j = E1 u0,j + F1,j ,
(4.12)
ñëåäîâàòåëüíî,
F1,j = 0,
E1 = 1.
Îòêóäà:
F1,j
.
1 − E1
Äàëåå, ñëåâà íàïðàâî, ïî ïðîãîíî÷íîé ôîðìóëå (4.9) âû÷èñëèì ñåòî÷íûå çíà÷åíèÿ èñêîìîé
ôóíêöèè u2,j , ..., un,j â îñòàëüíûõ óçëàõ ñåòêè. Èç íà÷àëüíûõ óñëîâèé (4.2) îïðåäåëÿåì
ñåòî÷íûå çíà÷åíèÿ â íóëåâîì ñëîå:
ui,0 = 0.
u0,j =
Òåïåðü ïîäñòàâèì ðàçíîñòíûå ñîîòíîøåíèÿ äëÿ âòîðîãî óðàâíåíèÿ ñèñòåìû (4.1) è
ïðîâåäåì àíàëîãè÷íûå ïðåîáðàçîâàíèÿ. Âî âòîðîì óðàâíåíèè ñèñòåìû îáîçíà÷èì:
2 !
ui+1,j − ui,j
ui+1,j − ui,j
exp p1 νi,j−1
+ q1
= expoi,j
∆x
∆x
Ïîäñòàâèì ðàçíîñòíûå ñîîòíîøåíèÿ (4.5) â íàøå óðàâíåíèå
ai+1,j − 2ai,j + ai−1,j
ai,j − ai,j−1
=β
+ κ [1 − ai,j − ai,j χexpoi,j ]
∆τ
∆x2
ai+1,j − 2ai,j + ai−1,j
ai,j
ai,j−1
=β
+ κ − κai,j − κχai,j expoi,j +
2
∆τ
∆x
∆τ
ïåðåíåñåì âñå ñëàãàåìûå ñ ai,j â ëåâóþ ñòîðîíó è âûíåñåì çà ñêîáêó
1
2β
ai+1,j + ai−1,j
ai,j−1
+ κ + κχexpoi,j +
ai,j
=β
+κ+
2
2
∆τ
∆x
∆x
∆τ
2β
β
β
ai,j−1
1
=
ai,j
+ κ + κχexpoi,j +
ai+1,j +
ai−1,j + κ +
2
2
2
∆τ
∆x
∆x
∆x
∆τ
(4.13)
Äëÿ íàõîæäåíèÿ ñåòî÷íîãî çíà÷åíèÿ ôóíêöèè ai,j èñïîëüçóåì ïðîãîíî÷íóþ ôîðìóëó
ai+1,j = Li+1 ai,j + Di+1,j
(4.14)
ãäå Li+1 , Di+1,j - ïðîãîíî÷íûå êîýôôèöèåíòû. Ïîäñòàâèì (4.14) â (4.13) :
ai,j
1
2β
+ κ + κχexpoi,j +
∆τ
∆x2
ai,j
ai,j
−
β
β
β
ai,j−1
Li+1 ai,j =
Di+1,j +
ai−1,j + κ +
2
2
2
∆x
∆x
∆x
∆τ
1
2β
β
+ κ + κχexpoi,j +
−
Li+1
2
∆τ
∆x
∆x2
1
2β
β
+ κ + κχexpoi,j +
−
Li+1
2
∆τ
∆x
∆x2
=
β
β
ai,j−1
Di+1,j +
ai−1,j + κ +
2
2
∆x
∆x
∆τ
a
βDi+1,j + ∆x2 i,j−1
+ ∆x2 κ βai−1,j
∆τ
=
+
∆x2
∆x2
17
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
ai,j =
+
βDi+1,j + ∆x2
∆x2
∆x2
1
∆τ
1
∆τ
ai,j−1
∆τ
+ ∆x2 κ
+ κ + κχexpoi,j +
βai−1,j
+ κ + κχexpoi,j +
2β
∆x2
2β
∆x2
−
−
β
L
∆x2 i+1
β
L
∆x2 i+1
+
(4.15)
Âîñïîëüçóåìñÿ ñëåäñòâèåì ïðîãîíî÷íîé ôîðìóëû
ai,j = Li ai−1,j + Di,j
(4.16)
è èç (4.15) ïîëó÷èì âûðàæåíèÿ äëÿ ïðîãîíî÷íûõ êîýôôèöèåíòîâ Li , Di,j , êîòîðûå
ïðåäñòàâëÿþò ðåêóððåíòíûå ñîîòíîøåíèÿ:
Di,j
∆τ β
,
+
i,j + ∆τ 2β − ∆τ βLi+1
∆τ βDi+1,j + ∆x2 ai,j−1 + ∆x2 ∆τ 2 κ
=
.
∆x2 + ∆τ ∆x2 κ + ∆τ ∆x2 κχexpoi,j + ∆τ 2β − ∆τ βLi+1
Li =
∆x2
+
∆τ ∆x2 κ
∆τ ∆x2 κχexpo
(4.17)
Ôîðìóëû (4.17) ïðåäñòàâëÿþò ñîáîé ðåêóððåíòíûå ñîîòíîøåíèÿ: çíàÿ Ln , Dn,j è, äâèãàÿñü ñïðàâà íàëåâî (îò i = n − 1 ê i = 0) ìîæíî îïðåäåëèòü çíà÷åíèÿ êîýôôèöèåíòîâ
(Ln−1 , Dn−1,j ), (Ln−2 , Dn−2,j ), (L1 , D1,j ). Ïðè ýòîì, äëÿ íàõîæäåíèÿ ïàðû (Ln , Dn,j ) âîñïîëüçóåìñÿ âòîðûì ãðàíè÷íûì óñëîâèåì
∂a
∂x
= 0,
x=1
èç êîòîðîãî ñëåäóåò, ÷òî
an,j = an−1,j ,
an,j = Ln an−1,j + Dn,j ,
(4.18)
Ñëåäîâàòåëüíî,
Dn,j = 0,
Ln = 1.
Îòêóäà:
an−1,j =
Dn,j
,
1 − Ln
è, ñëåäóÿ (4.17), îïðåäåëèì ïðîãîíî÷íûå êîýôôèöèåíòû (Ln−1 , Dn−1,j ), (Ln−2 , Dn−2,j ), (L1 , D1,j ).
Äëÿ íàõîæäåíèÿ ñåòî÷íûõ çíà÷åíèé èñêîìîé ôóíêöèè ai,j : îïðåäåëèì å¼ çíà÷åíèå â ïåðâîé óçëîâîé òî÷êå, òî åñòü a0,j . Äëÿ ýòîãî âîñïîëüçóåìñÿ ïåðâûì èç ãðàíè÷íûõ óñëîâèé
∂a
∂x
=0
x=0
è ôîðìóëîé ïðîãîíêè (4.16). Èìååì
a1,j = a0,j ,
a1,j = L1 a0,j + D1,j ,
ñëåäîâàòåëüíî,
D1,j = 0,
L1 = 1.
(4.19)
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
Îòêóäà:
a0,j =
18
D1,j
1 − L1
Äàëåå, ñëåâà íàïðàâî, ïî ïðîãîíî÷íîé ôîðìóëå (4.16) âû÷èñëèì ñåòî÷íûå çíà÷åíèÿ èñêîìîé ôóíêöèè a2,j , ..., an,j â îñòàëüíûõ óçëàõ ñåòêè. Èç íà÷àëüíûõ óñëîâèé (4.2) îïðåäåëÿåì
ñåòî÷íûå çíà÷åíèÿ â íóëåâîì ñëîå:
ai,0 = a0 .
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
19
5. Ðåçóëüòàòû ÷èñëåííûõ ýêñïåðèìåíòîâ äëÿ íåñòàöèîíàðíîé çàäà÷è
Íà ãðàôèêàõ ïðåäñòàâëåííûõ íèæå ìîæíî ïðîñëåäèòü êàê êðèâûå âûõîäÿò íà ñòàöèîíàð, ò.å. áîëüøå íå èçìåíÿþòñÿ, íàõîäÿòñÿ â ñîñòîÿíèè ïîêîÿ.
Çíà÷åíèÿ íà ãðàôèêàõ ïîêàçûâàþò êóäà ñòðåìÿòñÿ êðèâûå, ÷òîáû âûéòè íà ñòàöèîíàð.
Ðèñ. 3: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñêîðîñòè u=u(x, τ ) è ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a=a(x, τ ): 4(τ =1.2), 5(1.5), 7(2.1), 8(2.4), 56(16.8); Ïàðàìåòðû: λ=3,
a0=0.4, p1=0.06, q1=0.8, m=100(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè τ ), n=100(êîëè÷åñòâî
òî÷åê ðàçáèåíèÿ ïî îñè x), β =0.2, κ=0.4, χ=0.8, T =30
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
20
Ðèñ. 4: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå âÿçêîñòè ν =ν(a, τ ): 4(τ =1.2), 5(1.5),
7(2.1), 8(2.4), 56(16.8); Ïàðàìåòðû: λ=3, a0=0.4, p1=0.06, q1=0.8, m=100(êîëè÷åñòâî òî÷åê
ðàçáèåíèÿ ïî îñè τ ), n=100(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè x), β =0.2, κ=0.4, χ=0.8,
T =30
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
21
Âûâîäû è ñðàâíåíèå ðåçóëüòàòîâ ÷èñëåííûõ ýêñïåðèìåíòîâ
Ïðîâåäåíèå ÷èñëåííîãî ýêñïåðèìåíòà ïðè êà÷åñòâåííîì èçìåíåíèè ïàðàìåòðîâ äëÿ íåñòàöèîíàðíîé çàäà÷è Äëÿ ïðîâåäåíèÿ ýêñïåðèìåíòà èçìåíÿëèñü ñëå-
äóþùèå ïàðàìåòðû: a0, λ, p1, q1, β , κ, χ, ïðè ýòîì øàã áûë îäèíàêîâ, m=100, n=10. Òàêèå
ïàðàìåòðû êàê λ, p1, q1, κ, χ îêàçûâàëè áîëüøåå âëèÿíèå íà òå÷åíèå æèäêîñòè, ÷åì a0,
β.
Ðåçóëüòàòû óçêàçàíû â Ïðèëîæåíèè 1.
Äëÿ ñðàâíåíèÿ áûëè âûáðàíû ñëåäóþùèå ïàðàìåòðû:
Ñòàöèîíàðíàÿ çàäà÷à: λ=3, a0=0.4, p1=0.06, q1=0.6, β =0.2, κ=0.4, χ=0.8;
Íåñòàöèîíàðíàÿ çàäà÷à: λ=3, a0=0.4, p1=0.06, q1=0.8, β =0.2, κ=0.4, χ=0.8;
Íà ãðàôèêàõ ìîæíî óâèäåòü êàê íåñòàöèîíàðíîå ðåøåíèå ïîêàçûâàåò óñòàíîâëåíèå
ñòàöèîíàðíîãî ðåæèìà, êîòîðûé ñîâïàäàåò ñî ñòàöèîíàðíûì ðåøåíèåì çàäà÷è.
Ãðàôèêè óçêàçàíû â Ïðèëîæåíèè 2.
Cðàâíåíèå ðåçóëüòàòîâ ÷èñëåííûõ ýêñïåðèìåíòîâ
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
22
Çàêëþ÷åíèå
Öåëü àíàëèçà íåñòàöèîíàðíîãî òå÷åíèÿ áûëà äîñòèãíóòà. Ïî ãðàôèêàì ìîæíî ïîíÿòü,
÷òî âÿçêèå æèäêîñòè ñ òå÷åíèåì âðåìåíè âûõîäÿò íà ñòàöèîíàð, ò.å. â ðåçóëüòàòå íåêîòîðûõ ñòðóêòóðíûõ ïðåâðàùåíèé â âÿçêîñòè æèäêîñòè, òå÷åíèå çàìåäëèëîñü è âûøëî â
ñîñòîÿíèå ïîêîÿ.
Äëÿ ïîñòðîåíèÿ ìàòåìàòè÷åñêîé ìîäåëè íàïîðíîãî íåñòàöèîíàðíîãî òå÷åíèÿ ñ ïåðåìåííîé âÿçêîñòüþ áûëà ïîëó÷åíà ñèñòåìà äèôôåðåíöèàëüíûõ óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ ñ íà÷àëüíûìè è ãðàíè÷íûìè óñëîâèÿìè. Áûëà ðàññìîòðåíà íåñòàöèîíàðíàÿ ñèñòåìà è ñòàöèîíàðíàÿ, êàê ÷àñòíûé ñëó÷àé íåñòàöèîíàðíîé, èçìåíåíèÿ ñêîðîñòè è ñòåïåíè
ñòðóêòóðíûõ ïðåâðàùåíèé. Áûëî ïðîèçâåäåíî ðåøåíèå äèôôåðåíöèàëüíîãî óðàâíåíèÿ
ñòàöèîíàðíîãî òå÷åíèÿ â ÷àñòíûõ ïðîèçâîäíûõ äëÿ ñêîðîñòè òå÷åíèÿ â ÿâíîì âèäå. Ñ
ïîìîùüþ ìåòîäà ïðîãîíêè è èòåðàöèîííîãî ìåòîäà Íüþòîíà áûë ïðîâåäåí ÷èñëåííûé
àíàëèç íåñòàöèîíàðíîé è ñòàöèîíàðíîé ñèñòåì è ïîñòàâëåíû ÷èñëåííûå ýêñïåðèìåíòû,
ïîñòðîåíû ãðàôèêè: ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a = a(x), ñòàöèîíàðíîé ñêîðîñòè
òå÷åíèÿ u = u(x), áåçðàçìåðíîé âÿçêîñòè ν = ν(x) ïðè ðàçëè÷íûõ èçìåíåíèÿõ ïàðàìåòðîâ.
Ðåçóëüòàòû ðàáîòû ïðåäñòàâëåíû íà Íàöèîíàëüíîé êîíôåðåíöèè XXVII ãîäè÷íîé ñåññèè Ó÷åíîãî ñîâåòà (Ôåâðàëüñêèå ÷òåíèÿ-2020).
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
23
Ñïèñîê ëèòåðàòóðû
[1] Áåëÿåâà Í.À. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå: ó÷åáíîå ïîñîáèå. Ñûêòûâêàð: Èçä-âî
Ñûêòûâêàðñêîãî ãîñóíèâåðñèòåòà, 2014. 116 ñ.
[2] Àñòàðèòà. Äæ., Ìàððó÷÷è Äæ. Îñíîâû ãèäðîìåõàíèêè íåíüþòîíîâñêèõ æèäêîñòåé,
Ì.: Ìèð, 1978, 312 ñ.
[3] Áåëÿåâà Í.À., Ãîðñò Ä.Ë., Õóäÿåâ Ñ.È. Íåîäíîðîäíîå òå÷åíèå Êóýòòà ñòðóêòóðèðîâàííîé æèäêîñòè, Âåñòíèê Ñûêòûâêàðñêîãî óíèâåðñèòåòà Ñåð.1. Âûï 5. 2003. ñ. 43-48
[4] Áåëÿåâà Í.À., Êóçíåöîâ Ê.Ï. Äèññèïàòèâíàÿ ñòðóêòóðà è îáëàñòü ñâåðõàíîìàëèè êóýòòîâñêîãî òå÷åíèÿ ñòðóêòóðèðîâàííîé æèäêîñòè â ïëîñêîì çàçîðå, Âåñòíèê Ñûêòûâêàðñêîãî óíèâåðñèòåòà Ñåð.1. Âûï 13. 2011. ñ. 61-74
[5] Í. À. Áåëÿåâà, Íåîäíîðîäíîå òå÷åíèå ñòðóêòóðèðîâàííîé æèäêîñòè, Ìàòåì. ìîäåëèðîâàíèå, 2006. òîì.18. íîìåð 6. ñ. 314
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
24
Ïðèëîæåíèå 1. Ïðîâåäåíèå ÷èñëåííîãî ýêñïåðèìåíòà ïðè âàðüèðîâàíèè ïàðàìåòðîâ
Ðèñ. 5: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñêîðîñòè òå÷åíèÿ u=u(x, τ ),
ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé
a=a(x, τ ): 4(τ =1.2), 5(1.5), 8(2.4), 15(4.5); Ïàðàìåòðû: λ=3, a0=0.4, p1=0.06, q1=0.8,
m=50(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè τ ), n=100(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè
x), β =0.2, κ=0.4, χ=0.8, T =30
Ðèñ. 6: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå âÿçêîñòè ν =ν(a, τ ): 4(τ =1.2), 5(1.5),
8(2.4), 15(4.5); Ïàðàìåòðû: λ=3, a0=0.4, p1=0.06, q1=0.8, m=50(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè τ ), n=100(êîëè÷åñòâî òî÷åê ðàçáèåíèÿ ïî îñè x), β =0.2, κ=0.4, χ=0.8, T =30
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
25
Ðèñ. 7: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñêîðîñòè òå÷åíèÿ u=u(x, τ ),
ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a=a(x, τ ):
λ=2, a0=0.3, p1=0.6, q1=0.4, β =0.20, κ=0.9, χ=0.9
Ðèñ. 8: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå âÿçêîñòè ν =ν(a, τ ): λ=2, a0=0.3,
p1=0.6, q1=0.4, β =0.20, κ=0.9, χ=0.9
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
26
Ðèñ. 9: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñêîðîñòè òå÷åíèÿ u=u(x, τ ),
ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a=a(x, τ ):
λ=3, a0=0.4, p1=0.06, q1=0.4, β =0.2, κ=0.3, χ=0.8
Ðèñ. 10: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå âÿçêîñòè ν =ν(a, τ ): λ=3, a0=0.4,
p1=0.06, q1=0.4, β =0.2, κ=0.3, χ=0.8
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
27
Ðèñ. 11: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñêîðîñòè òå÷åíèÿ u=u(x, τ ),
ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé a=a(x, τ ):
λ=0.9, a0=0.5, p1=0.003, q1=0.5, β =0.08, κ=0.9, χ=0.7
Ðèñ. 12: Ïðîñòðàííñòâåííî-âðåìåííîå ðàñïðåäåëåíèå âÿçêîñòè ν =ν(a, τ ): λ=0.9, a0=0.5,
p1=0.003, q1=0.5, β =0.08, κ=0.9, χ=0.7
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
28
Ïðèëîæåíèå 2. Ñðàâíåíèå ãðàôèêîâ äëÿ íåñòàöèîíàðíîãî è ñòàöèîíàðíîãî òå÷åíèÿ
Ðèñ. 13: Èçìåíåíèå ñêîðîñòè òå÷åíèÿ ïîêàçûâàåò âîçðàñòàíèå ñêîðîñòè ñ òå÷åíèåì âðåìåíè, íà ãðàôèêå âèäíî êàê êðèâûå äîñòèãàþò ìàêñèìàëüíîãî çíà÷åíèÿ(ðèñóíîê ñëåâà
ñíèçó, êðèâûå 56-99). Ïðè ýòîì ñòàöèîíàðíàÿ ñêîðîñòü òå÷åíèÿ (ðèñóíîê ñëåâà ñâåðõó,
êðèâûå 73-99) íåçíà÷èòåëüíî îòëè÷àåòñÿ îò óñòàíîâèâøåãîñÿ òå÷åíèÿ æèäêîñòè(ðèñóíîê
ñëåâà ñíèçó, êðèâûå 56-99). Óñòàíàâëèâàþùàÿñÿ ñòåïåíü ñòðóêòóðíûõ ïðåâðàùåíèé (ðèñóíîê ñïðàâà ñíèçó, êðèâûå 56-99) ñîâïàäàåò ñî ñòàöèîíàðíîé ñòåïåíüþ ñòðóêòóðíûõ ïðåâðàùåíèé (ðèñóíîê ñïðàâà ñâåðõó, êðèâûå 73-99)
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
29
Ðèñ. 14: Óñòàíàâëèâàþùàÿñÿ âÿçêîñòü æèäêîñòè (ðèñóíîê ñïðàâà, êðèâûå 56-99) íåçíà÷èòåëüíî îòëè÷àåòñÿ îò ñòàöèîíàðíîãî òå÷åíèÿ æèäêîñòè (ðèñóíîê ñëåâà, êðèâûå 73-99)
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
30
Ïðèëîæåíèå 3. Ïðîãðàììà ïîñòðîåíèÿ ãðàôèêà ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé è ñêîðîñòè ñòàöèîíàðíîãî íàïîðíîãî òå÷åíèÿ â
Visual Studio íà ÿçûêàõ C++ è C Sharp
using
using
using
using
using
using
using
using
using
System;
System.Collections.Generic;
System.ComponentModel;
System.Data;
System.Drawing;
System.Linq;
System.Text;
System.Threading.Tasks;
System.Windows.Forms;
namespace Íàïîðíîå_ñòàöèîíàðíîå_òå÷åíèå
{
public partial class Form1 : Form
{
Boolean Ëèíèÿ1;// Ëèíèÿ2, Ëèíèÿ3, Ëèíèÿ4;
int k = 0; // êîëè÷åñòâî ãðàôèêîâ
public Form1()
{
InitializeComponent();
}
//*********************************
//Äåéñâòèÿ ïðè çàãðóçêè ïðèëîæåíèÿ
//*********************************
private void Form1_Load(object sender, EventArgs e)
{
//ïîäïèñü îñè x
ñhart1.ChartAreas[0].AxisX.Title = "x";
//ïîäïèñü îñè v
ñhart1.ChartAreas[0].AxisY.Title = "?";
}
//*******
//Êíîïêè
//*******
private void Graph_Click(object sender, EventArgs e)
{
if (_beta.Text == "" || kolvo_x.Text == "" || q_1.Text == ""
|| P_1.Text == "" || lyambda.Text == "" || this.a_0.Text == "")
{
MessageBox.Show("Åñòü íåçàïîëíåííûå ïîëÿ");
if (_beta.Text == "")
_beta.BackColor = Color.LightCoral;
if (kolvo_x.Text == "")
kolvo_x.BackColor = Color.LightCoral;
if (q_1.Text == "")
q_1.BackColor = Color.LightCoral;
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
if (P_1.Text == "")
P_1.BackColor = Color.LightCoral;
if (lyambda.Text == "")
lyambda.BackColor = Color.LightCoral;
if (this.a_0.Text == "")
this.a_0.BackColor = Color.LightCoral;
return;
}
//ñ÷èòûâàíèå è êîíâåðòàöèÿ ââåäåííûõ çíà÷åíèé
int n = Convert.ToInt32(kolvo_x.Text);
float lya = Convert.ToSingle(lyambda.Text);
float q1 = Convert.ToSingle(q_1.Text);
float hi = Convert.ToSingle(_hi.Text);
float beta = Convert.ToSingle(_beta.Text);
float a0 = Convert.ToSingle(a_0.Text);
float lastt = Convert.ToSingle(q_1.Text);
string moment = Convert.ToString(moment_vremeni.Text);
float p1 = Convert.ToSingle(p_1.Text);
float kappa = Convert.ToSingle(_kappa.Text);
//Êîä Íàïîðíîå òå÷åíèå ìåæäó äâóìÿ ïëîñêîñòÿìè.cpp
double stepx;
int r=100; // êîëè÷åñòâî âû÷èñëåíèé(èòåðàöèé)
double[,] a = new double[n+1,r];
double[,] nu = new double[n+1, r];
double[,] u = new double[n+1,r];
double[] E = new double[n+1];
double[,] F = new double[n+1, r];
double[] x = new double[n+1];
//a[i,j] i - ïî õ, j - ïî èòåðàöèÿì
stepx = 1.0F/n; //Øàã äëèíà îòðåçêà íà êîëè÷åñòâî òî÷åê ðàçáèåíèÿ,
ò.ê. çíà÷åíèå x[0,1]
stepx = 1.0F / n;
//Øàã äëèíà îòðåçêà íà êîëè÷åñòâî òî÷åê ðàçáèåíèÿ,ò.ê. çíà÷åíèå x[0,1]
//Îáíóëåíèå ìàññèâîâ a, u
for (int i = 0; i < n+1; i++)
{
for(int j=0; j<r;j++){
u[i, j] = 0;
a[i, j] = 0;
}
}
for (int j = 1; j < r; j++)
{
//ãðàíè÷íîå óñëîâèå äëÿ u ïðè x=1
u[n, j] = 0.0;
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Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
}
//îïðåäåëåíèå x
for (int i = 0; i < n + 1; i++)
{
x[i] = stepx * i;// x[i=last] = stepx*n = (h/n) * n
}
//ïðîõîä çàäàííîãî çíà÷åíèÿ a0, ïî âñåì òî÷êàì
x íà ïðîìåæóòêå îò 0 äî êîëè÷åñòâà òî÷åê ðàçáèåíèÿ (ïðèáëèæåíèå)
for (int i = 0; i < n + 1; i++)
{
a[i, 0] = a0;
u[i, 0] = x[i] * (1 + lya * a[i, 0]);
nu[i, 0] = 1 / (1 + lya * a[i, 0]);
}
for (int j = 0; j < r; j++)
{
//ãðàíè÷íîå óñëîâèå ïðè x=1
E[n] = 1;
F[n, j] = 0.0;
}
for (int j = 1; j < r; j++)
{
for (int i = n - 1; i > 0; i--) //
äâèæåíèå ïî õ ñ êîíöà, âû÷èñëåíèå èíòåãðàëà è ïðîãîíî÷íûõ êîýôô
{
//çàäàíèå èíòåãðàëà
u[i, j] = u[i + 1, j] + (x[i + 1] - x[i]) *
(x[i + 1] * (1 + lya * a[i + 1, j - 1]) x[i] * (1 + lya * a[i, j - 1])) / 2;
u[0, j] = u[1, j];// èñïîëüçîâàíèå ãðàíè÷íîãî óñëîâèÿ äëÿ u
}
for (int i = n - 1; i > 0; i--)
// äâèæåíèå ïî õ ñ êîíöà, âû÷èñëåíèå èíòåãðàëà è ïðîãîíî÷íûõ êîýôô
{
// ïðîãîíî÷íûå êîýô
double expo = Math.Exp(p1 * nu[i, j-1] *
(-x[i] * (1 + lya * a[i, j - 1]))
+ q1 * (x[i] * (1 + lya * a[i, j - 1])) *
(x[i] * (1 + lya * a[i, j - 1])));
E[i] = beta / (2 * beta + kappa * stepx *
stepx + kappa * hi * stepx *
stepx * expo - beta * E[i + 1]);
F[i, j] = (beta * F[i + 1, j] + kappa * stepx * stepx)
/ (2 * beta + kappa
* stepx * stepx + kappa * hi * stepx * stepx * expo
- beta * E[i + 1]);
32
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
}
for (int i = 1; i < n + 1; i++)
// äâèæåíèå ïî õ ñ íà÷àëà, âû÷èñëåíèå öåëåâîé ôóíêöèè
{
a[0, j] = F[1, j] / (1 - E[1]);
//çíà÷åíèå öåëåâîé à(ïðîèçâîäíîé)
a[i, j] = (E[i] * a[i - 1, j]) + F[i, j];
// ïðîãîíî÷íûå êîýô è ôîðìóëà
}
}
for (int i = 0; i < n+1; i++)
{
nu[i, j] = 1 / (1 + lya * a[i, j]); //çàäàíèå íþ, çàâèñèìîñòü îò à
}
char[] M = moment.ToCharArray();
_ = new string[n + 1];
string[] sravnenie = new string[n + 1];
k = 0;
for (int i = 0; i < M.Length; i++)
{
if (M[i] == ' ')
{
continue;
}
if (M[i] == ',')
{
if(i==M.Length-1)
continue;
k++;
continue;
}
}
sravnenie[k] += M[i];
string[] d = sravnenie.Distinct().ToArray();
k = d.Length - 2;// ñãëàæèâàíèå ìàññèâà
/////////////////////////////////////////////////////////
//Öèêë ïðîâåðêè ñóùåñòâîâàíèÿ âûáðàííûõ òî÷åê t
for (int i=0; i <= k; i++)
{
if (Convert.ToInt32(d[i]) <= n)
{
continue;
}
else
33
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
34
{
MessageBox.Show("Âûáðàííûé ìîìåíò âðåìåíè
âûõîäèò çà êîëè÷åñòâî òî÷åê ðàçáèåíèÿ");
return;
}
}
ñhart1.Series.Clear(); // ÷èñòèò äèàãðàììó
for (int i = 0; i <= k; i++)
{
string s1 = "V[";
string s2 = "][0]";
string s3 = s1 + d[i] + s2;
if (ñhart1.Series.IsUniqueName(s3))
{
ñhart1.Series.Add(s3); // äîáàâëåíèå ñåðèè
ñhart1.Series[i].Name = s3; // Çàäàåò èìÿ i-ãî ãðàôèêà
ñhart1.Series[i].Points.Clear();
}
//chart1.Series[i].Name = s3;
// Çàäàåò èìÿ i-ãî ãðàôèêà
ñhart1.Series[i].ChartType =
System.Windows.Forms.DataVisualization.Charting.
SeriesChartType.Spline; //ïëàâíûå
ñhart1.Series[i].Points.Clear();
}
//*******
//Ïîñòðîåíèå ãðàôèêîâ(âûáîð a, u, nu)
//*******
for (int i = 0; i < n+1; i++)
{
for (int j = 0; j <= k; j++)
{
ñhart1.Series[j].Points.AddXY(x[i], a[i, Convert.ToInt32(d[j])]);
ñhart1.Series[j].Points.AddXY(x[i], u[i, Convert.ToInt32(d[j])]);
ñhart1.Series[j].Points.AddXY(x[i], nu[i, Convert.ToInt32(d[j])]);
}
}
}
//*******
//Ãðàôèê
//*******
private void Ñhart1_Click(object sender, EventArgs e)
{
Ëèíèÿ1 = !Ëèíèÿ1;
if (Ëèíèÿ1 == true)
{
for (int i = 0; i <= k; i++)
{
ñhart1.Series[i]["DrawingStyle"] = "Line";
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
35
}
}
else
{
ñhart1.Series[0]["DrawingStyle"] = "Default";
}
//////////////////////////////////////////////////////////////
//Óâåëè÷åíèå ðèñóíêà
//////////////////////////////////////////////////////////////
//ïî îñè X
ñhart1.ChartAreas[0].CursorX.IsUserEnabled = true;
ñhart1.ChartAreas[0].CursorX.IsUserSelectionEnabled = true;
ñhart1.ChartAreas[0].AxisX.ScaleView.Zoomable = true;
ñhart1.ChartAreas[0].AxisX.ScrollBar.IsPositionedInside = true;
//ïî îñè Y
ñhart1.ChartAreas[0].CursorY.IsUserEnabled = true;
ñhart1.ChartAreas[0].CursorY.IsUserSelectionEnabled = true;
ñhart1.ChartAreas[0].AxisY.ScaleView.Zoomable = true;
ñhart1.ChartAreas[0].AxisY.ScrollBar.IsPositionedInside = true;
}
//*****************
//Òåêñòîâûå ôîðìû
//*****************
private void Text_vyazkost_Click(object sender, EventArgs e)
{
}
private void Text_max_y_Click(object sender, EventArgs e)
{
}
private void Text_max_t_Click(object sender, EventArgs e)
{
}
private void Text_kolvo_y_Click(object sender, EventArgs e)
{
}
private void Text_kolvo_t_Click(object sender, EventArgs e)
{
}
//****************************
//Äåéñòâèÿ ïî íàæàíèþ êëàâèøè
//****************************
private void Vvod_drobnogo_chisla(object sender, KeyPressEventArgs e)
{
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
36
char number = e.KeyChar;
if ((e.KeyChar <= 47 || e.KeyChar >= 58) &&
number != 8 && number != 44 && number != 127)
{
e.Handled = true;
}
if (e.KeyChar == ',' && (sender as TextBox).Text.IndexOf(',') > -1)
{
e.Handled = true;
}
}
private void Perechisleniya(object sender, KeyPressEventArgs e)
{
char number = e.KeyChar;
if ((e.KeyChar <= 47 || e.KeyChar >= 58) &&
number != 8 && number != 44 && number != 127)
{
e.Handled = true;
}
}
private void Vvod_naturalnogo_chisla(object sender, KeyPressEventArgs e)
{
char number = e.KeyChar;
if ((e.KeyChar <= 47 || e.KeyChar >= 58) &&
number != 8 && number != 127)
{
e.Handled = true;
}
}
//***********************************************
//Äåéñòâèÿ ïî íàæàíèþ íà TextBox (TextBox_Enter)
//***********************************************
private void Podsvetka_text_polya(object sender, EventArgs e)
{
if (_beta.Text != "")
_beta.BackColor = Color.White;
if (kolvo_x.Text != "")
kolvo_x.BackColor = Color.White;
if (q_1.Text != "")
q_1.BackColor = Color.White;
if (P_1.Text != "")
P_1.BackColor = Color.White;
if (lyambda.Text != "")
lyambda.BackColor = Color.White;
if (a_0.Text != "")
a_0.BackColor = Color.White;
if (_hi.Text != "")
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
}
}
}
_hi.BackColor = Color.White;
if (_kappa.Text != "")
_kappa.BackColor = Color.White;
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Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
38
Ïðèëîæåíèå 4. Ïðîãðàììà ïîñòðîåíèÿ ãðàôèêà ñòåïåíè ñòðóêòóðíûõ ïðåâðàùåíèé è ñêîðîñòè íåñòàöèîíàðíîãî íàïîðíîãî òå÷åíèÿ
â Visual Studio íà ÿçûêàõ C++ è C Sharp
//ñ÷èòûâàíèå è êîíâåðòàöèÿ ââåäåííûõ çíà÷åíèé
int n = Convert.ToInt32(kolvo_x.Text);
int m = Convert.ToInt32(m_tau.Text);
float lya = Convert.ToSingle(lyambda.Text);
float q1 = Convert.ToSingle(q_1.Text);
float hi = Convert.ToSingle(_hi.Text);
double Tau = Convert.ToDouble(max_Tau.Text);
float beta = Convert.ToSingle(_beta.Text);
float a0 = Convert.ToSingle(a_0.Text);
string moment = Convert.ToString(moment_vremeni.Text);
float p1 = Convert.ToSingle(p_1.Text);
float kappa = Convert.ToSingle(_kappa.Text);
//Êîä Íàïîðíîå òå÷åíèå ìåæäó äâóìÿ ïëîñêîñòÿìè.cpp
double stepx;
double steptau;
double[,] a = new double[n + 1, m + 1];
double[,] nu = new double[n + 1, m + 1];
double[,] u = new double[n + 1, m + 1];
double[] E = new double[n + 1];
double[,] F = new double[n + 1, m + 1];
double[] L = new double[n + 1];
double[,] D = new double[n + 1, m + 1];
double[] x = new double[n + 1];
//a[i,j] i - ïî õ, j - ïî âðåìåíè
stepx = 1.0F/n; //Øàã ïî îñè 'x', äëèíà îòðåçêà íà êîëè÷åñòâî òî÷åê ðàçáèåíèÿ,
ò.ê. çíà÷åíèå x[0,1]
steptau = Tau / m; //Øàã ïî îñè '?' ìàêñèìàëüíîå çíà÷åíèå T
íà êîëè÷åñòâî òî÷åê ðàçáèåíèÿ
//Îáíóëåíèå ìàññèâîâ a, u
for (int i = 0; i < n+1; i++)
{
for(int j=0; j<r;j++)
{
u[i, j] = 0;
a[i, j] = 0;
}
}
//ïðîõîä çàäàííîãî çíà÷åíèÿ a0, ïî âñåì òî÷êàì x íà ïðîìåæóòêå îò 0
äî êîëè÷åñòâà òî÷åê ðàçáèåíèÿ (ïðèáëèæåíèå)
//íà÷àëüíûå óñëîâèÿ
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
for (int i = 0; i < n + 1; i++)
{
u[i, 0] = 0;
a[i, 0] = a0;
nu[i, 0] = 1 / (1 + lya * a[i, 0]);
}
for (int j = 1; j < m + 1; j++)
{
//ãðàíè÷íîå óñëîâèå äëÿ u ïðè x=1
u[n, j] = 0.0;
}
//îïðåäåëåíèå x
for (int i = 0; i < n + 1; i++)
{
x[i] = stepx * i;// x[i=last] = stepx*n = (h/n) * n
}
//èç ãðàíè÷íûõ óñëîâèé ñëåäóåò
for (int j = 0; j < m + 1; j++) {
D[n, j] = 0.0;
L[n] = 1;
}
for (int j = 1; j < m + 1; j++)
{
// äâèæåíèå ïî õ ñ êîíöà, âû÷èñëåíèå èíòåãðàëà è ïðîãîíî÷íûõ êîýôô
for (int i = n - 1; i > 0; i--)
{
double znam_u = (stepx * stepx) / steptau + lya *
Math.Pow(nu[i, j - 1], 2) *
(a[i, j - 1] - a[i+1, j - 1]) + 2 * nu[i, j - 1] - E[i + 1] *
(lya * Math.Pow(nu[i, j - 1], 2)*
(a[i, j - 1] - a[i + 1, j - 1]) + nu[i, j - 1]);
E[i] = nu[i, j - 1] / znam_u;
F[i, j] = (F[i + 1, j] * (lya * Math.Pow(nu[i, j - 1], 2) *
(a[i, j - 1] - a[i + 1, j - 1]) + nu[i, j - 1]) + Math.Pow(stepx, 2) *
u[i, j - 1] / (steptau) + Math.Pow(stepx,2)) / znam_u;
}
for (int i = 1; i < n + 1; i++)
{
u[0, j] = F[1, j] / (1 - E[1]);
u[1, j] = u[0, j];
u[i, j] = (E[i] * u[i - 1, j]) + F[i, j];
}
// äâèæåíèå ïî õ ñ êîíöà, âû÷èñëåíèå èíòåãðàëà è ïðîãîíî÷íûõ êîýôô
for (int i = n - 1; i >= 1; i--)
{
double expo = Math.Exp(p1 * nu[i, j - 1] *
39
Ñèçîâ Í.Ñ. Íàïîðíîå òå÷åíèå æèäêîñòè ñ ïåðåìåííîé âÿçêîñòüþ
((u[i + 1, j] - u[i, j]) / stepx) +
q1 * Math.Pow((u[i + 1, j] - u[i, j]) /
double znam_a = Math.Pow(stepx,
Math.Pow(stepx, 2) *
kappa + Math.Pow(stepx, 2) * kappa * hi
L[i] = beta / znam_a;
D[i, j] = (beta * D[i + 1, j] +
kappa + Math.Pow(stepx, 2) *
(a[i, j - 1]) / steptau) / znam_a;
}
stepx, 2));
2) / steptau + 2 * beta +
* expo - beta * L[i + 1];
Math.Pow(stepx, 2) *
//ïðîãîíî÷íûå ôîðìóëû äâèæåíèå ñëåâà íàïðàâî
for (int i = 1; i < n + 1; i++)
{
a[0, j] = D[1, j] / (1 - L[1]);
a[1, j] = a[0, j];
a[i, j] = (L[i] * a[i - 1, j]) + D[i, j];
a[n, j] = a[n - 1, j];
}
for (int i = 0; i < n + 1; i++)
{
nu[i, j] = 1 / (1 + lya * a[i, j]);
}
}
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