Димитриенко Ю.И. - Механика сплошной среды (1050318), страница 23
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1.9.5).Ïðîèçâîäíûå îò 1/r è 1/ríèÿÐèñ. 1.9.5. Ê âûâîäó ôîðìó-∂∂nëû Ïóàññîíà³1´∂∂nÇäåñü ââåäåíû îáîçíà÷åíèÿ:r0=−1r02r = x − x0³1´r=−1r20 ïðèíèìàþò çíà÷å-∂rr·n=− 3 ,∂nr∂r0r0 · n= − 03 .∂nrèr0 = x − x00 ,(1.9.57)à òàêæå èñïîëüçîâàíûôîðìóëû ïðîèçâîäíîé ïî íîðìàëè (1.5.19à), (1.5.20à):∂r∂|r|1 ∂rr·n==·r=,∂n∂nr ∂nr∂r0 ∂|r0 |r0 · n= 0 ,∂n ∂nrÏîñêîëüêó íà ñôåðåΣR (0): n = x/R,∂r∂x== n,∂n∂n∂r0∂x== n.∂n∂n(1.9.58)òî ñ ó÷åòîì ôîðìóë (1.9.52) ìîæíîâû÷èñëèòü ñêàëÿðíûå ïðîèçâåäåíèÿr·n=r0 · n =1R1R(x − x0 ) · x =(x − x00 ) · x =|x|2 − x · x0|x|2 − |x0 |2 + r2=,R2R|x|2 − x · x00|x|2 − |x00 |2 + r02=.R2R(1.9.59) 1.9.
Ïîòåíöèàëüíûå äâèæåíèÿ íåñæèìàåìîé èäåàëüíîé æèäêîñòèÓ÷èòûâàÿ, ÷òî íà ñôåðåΣR (0): |x| = R145è âûïîëíÿþòñÿ ñîîòíîøåíèÿ(1.9.53), à òàêæå ïðèíèìàÿ âî âíèìàíèå ôîðìóëó (1.9.51)|x00 | = R2 /|x0 |,ïîëó÷àåìR2 − |x0 |2 + r2,2Rr·n=r0 · n =(|x0 |2 − R2 + r2 )R.22|x0 |(1.9.60)Ïîäñòàâëÿÿ ñîîòíîøåíèÿ (1.9.60) â (1.9.57), à çàòåì â (1.9.50) íàõîäèì∂G1R |x0 |3 0r·n=r ·n− 3 =∂n|x0 | R3 r3rµ¶1RR2 − |x0 |2 + r2|x |2 − R2222= 3(|x|−R+r)−= 0 3 .022Rr2RRr(1.9.61)Ïîñëå ïîäñòàíîâêè ýòîé ôîðìóëû â (1.9.37), äåéñòâèòåëüíî, ïîëó÷àåìôîðìóëó Ïóàññîíà (1.9.55).NÔóíêöèåé Ãðèíà äëÿ çàäà÷è Íåéìàíà (1.9.23), (1.9.25) íàçûâàþò ôóíê-öèþG2 (x, x0 ) =ãäåζ(x, x0 )1r+ ζ(x.x0 )∀x0 ∈ V , ∀x ∈ V ∪ Σ,(1.9.62) ðåøåíèå çàäà÷è Íåéìàíà ñ ãðàíè÷íûì óñëîâèåì âèäà(∆x ζ = 0,x∈V,³1´∂ζ ¯¯∂=−,r = |x − x0 |, x0 ∈ V.Σ∂nÇäåñü, êàê è ðàíåå,Òåîðåìà 1.9.9.∂n(1.9.63)r∂/∂n = n(x) · ∇x .Ôóíêöèÿ Ãðèíà äëÿ çàäà÷è Íåéìàíà (1.9.23), (1.9.25) îáëà-äàåò ñëåäóþùèìè ñâîéñòâàìè:1) îíà óäîâëåòâîðÿåò óðàâíåíèþ Ëàïëàñà ïî ïåðâîìó àðãóìåíòó ïðèôèêñèðîâàííîì âòîðîì àðãóìåíòåx0 :∆x G2 (x, x0 ) = 0,âñþäó âVêðîìå òî÷êè2) íà ãðàíèöåíóëü:Σîáëàñòèx = x0 ∈ V ;V åå íîðìàëüíàÿ∂G (x, x0 ) =∂n 20,åñëè(1.9.64)ïðîèçâîäíàÿ îáðàùàåòñÿ âx ∈ Σ ∀x0 ∈ V ;(1.9.65)ϕ(x), ÿâëÿþùàÿñÿ ðåøåíèåì çàäà÷è Íåéìàíà (1.9.23),(1.9.25) ñ ïðîèçâîëüíîé ôóíêöèåé ve íà Σ, â îáëàñòè V ìîæåò áûòüâûðàæåíà ÷åðåç çíà÷åíèÿ ôóíêöèè Ãðèíà G2 è çíà÷åíèå ve íà Σ:Z1ϕ(x0 ) =G2 (x, x0 )ve (x) dΣ.(1.9.66)3) âñÿêàÿ ôóíêöèÿ4πΣHÄîêàçàòåëüñòâî óòâåðæäåíèé 1 è 2 î÷åâèäíî.146Ãëàâà 1.
Èäåàëüíûå æèäêîñòè è ãàçûÄëÿ äîêàçàòåëüñòâà óòâåðæäåíèÿ 3 îïÿòü âîñïîëüçóåìñÿ âòîðîé ôîðìóëîéÃðèíà (ò. 1, (3.5.26)) èëè (1.9.37), â êîòîðîé âìåñòî ôóíêöèèψâûáåðåìZ³´∂ϕ∂ζ−ζϕdΣ = 0.òîãäà èìååì∂nζ,(1.9.67)∂nΣÏîñêîëüêó∂ζ/∂n = −∂(1/r)/∂n íà Σ, òî ïîëó÷àåìZZ³ ´∂ϕ∂ 1ϕdΣ = − ζdΣ.∂nr(1.9.68)∂nΣΣÏîäñòàâëÿÿ ýòî ñîîòíîøåíèå â ôîðìóëó (1.9.30), íàõîäèìϕ(x0 ) =1Z³4π1r+ζ´ ∂ϕ∂ndΣ.(1.9.69)ΣÓ÷èòûâàÿ, ÷òî â çàäà÷å Íåéìàíàìóëó (1.9.66).∂ϕ/∂n = ve ,äåéñòâèòåëüíî, ïîëó÷àåì ôîð-N1.9.7. Çàäà÷à îá îáòåêàíèè øàðà ïîòîêîì íåñæèìàåìîé æèäêîñòèÐàññìîòðèì íåñêîëüêî êëàññè÷åñêèõ ïðèìåðîâ çàäà÷ Íåéìàíà â èäåàëüíîéíåñæèìàåìîé æèäêîñòè. Íà÷íåì ñ çàäà÷è îáòåêàíèÿ àáñîëþòíî òâåðäîãî øàðàïîòîêîì íåñæèìàåìîé ïîòåíöèàëüíîé æèäêîñòè.Áóäåì èñêàòü ðåøåíèå çàäà÷è Íåéìàíà:∆ϕ = 0 x ∈ V1 ,¯¯∂ϕ ¯vn ¯ Σ =¯ = 0,R(1.9.70)(1.9.71)∂n ΣRâî âíåøíîñòè øàðàV1 = E3a \ UR (0),ãäåΣR (0) ïîâåðõíîñòü øàðà (ñôåðàR).
Êàê áûëî îòìå÷åíî âûøå,ðàäèóñàäëÿ âíåøíåé çàäà÷è Íåéìàíà íåîáõîäèìî ê ñèñòåìå (1.9.70), (1.9.71) ïðèñîåäèíèòü óñëîâèå íà áåñêîíå÷íîñòè (1.9.29),êîòîðîå âûáåðåì ñëåäóþùèì:¯¯v¯∞ = ∇ϕ¯∞ = −v∞ ,ãäåÐèñ. 1.9.6. Ê çàäà÷å îá îáòåêàíèè øàðàïîòîêîì æèäêîñòèãàòü, ÷òîv∞ = v∞ ē3 .v∞äâèæåíèÿçàäàííûéæèäêîñòèâåêòîðíà(1.9.72)ñêîðîñòèáåñêîíå÷íîñòè(ðèñ. 1.9.6). Áóäåì äëÿ ïðîñòîòû ïîëà-147 1.9. Ïîòåíöèàëüíûå äâèæåíèÿ íåñæèìàåìîé èäåàëüíîé æèäêîñòèÄëÿ òîãî, ÷òîáû ñâåñòè çàäà÷ó (1.9.70)(1.9.72) ê ðàññìîòðåííîé âûøåâíåøíåé çàäà÷è Íåéìàíà (1.9.23), (1.9.25), (1.9.29), ðàññìîòðèì ïîòåíöèàëϕâ âèäåϕ = ϕ2 + ϕ1 ,ãäåϕ2 íîâàÿ íåèçâåñòíàÿ ôóíêöèÿ;ϕ1 = −v∞ x3 ïîòåíöèàë ïîñòóïàòåëüíîãî äâèæåíèÿ øàðà.Ïîñêîëüêó= −v∞ ē1 ,ëèíåéíàÿòî äëÿϕ2ôóíêöèÿÿâëÿåòñÿãàðìîíè÷åñêîé,à¯∇ϕ1 ¯∞ =ïîëó÷àåì çàäà÷ó∆ϕ = 0, x ∈ V1 , 2¯∂ϕ2 ¯∂ϕ2== v∞ n̄3 ,¯∂n ¯ ∂r r=R∇ϕ2 ¯∞ = 0,(1.9.73)ïîñêîëüêó∂ϕ1 /∂n = n · ∇ϕ1 = −v∞ ē3 · n = −v∞ n̄3.Çàäà÷à (1.9.73) èìååò ïðîñòîå ðåøåíèå â âèäå ïîòåíöèàëà òî÷å÷íîãîäèïîëÿ (1.9.5):ϕ2 = Cãäår = |x|, x ≡ x3∂∂x³1´r=−C n̄Cx= − 23 ,r3r òðåòüÿ êîîðäèíàòà ðàäèóñ-âåêòîðà(1.9.74)x;òî÷êàx0âûáðàíàñîâïàäàþùåé ñ íà÷àëîì êîîðäèíàò öåíòðîì ñèììåòðèè øàðà (íà ñôåðåΣr (0)âñåãäàn = x/r).Ïîäñòàâëÿÿ (1.9.74) â óñëîâèå íåïðîòåêàíèÿ (1.9.73) íà ñôåðå, ïîëó÷àåì¯∂ϕ2 ¯2C n̄3== v∞ n̄3 .¯∂r r=RR3Îòñþäà íàõîäèì ïîñòîÿííóþϕÒîãäà äëÿC : C = v∞ R3 /2.èìååì îêîí÷àòåëüíîå âûðàæåíèå:v R3 x− v∞ x = −v∞ xϕ=− ∞2r3ÑêîðîñòüvµR31+32r¶.(1.9.76)èìååò âèäv = ∇ϕ = −Íà ñôåðå(1.9.75)ΣR (0),ñêîðîñòåé:ïîëàãàÿv∞ R332rr = R,ē3 −´x− v∞ ē3 .23xrΣR (0): n = x/R2è(1.9.77)ïîëó÷àåì ñëåäóþùåå ðàñïðåäåëåíèå¯3v¯Σ = − v∞ (ē3 − n̄3 n),Ròàê êàê íà ñôåðå³n̄3 = x/R.(1.9.78)148Ãëàâà 1.
Èäåàëüíûå æèäêîñòè è ãàçûÈç ôîðìóëû (1.9.78) äëÿ ìîäóëÿ ñêîðîñòè|v|íà ïîâåðõíîñòè ñôåðûíàõîäèìq¯¯33|v|ΣR = (v¯Σ · v¯Σ )1/2 = v∞ 1 − n̄23 = v∞ | sin ϑ|,Rϑ óãîën̄3 = cos ϑ.ãäåR2(1.9.79)2â ñôåðè÷åñêîé ñèñòåìå êîîðäèíàò (ñì. ðèñ. 1.9.6), òàê êàên = ±ē3¯ (ýòîOx1 ) ñêîðîñòü v¯Σ =Èç ôîðìóëû (1.9.78) ñëåäóåò, ÷òî â òî÷êàõ ñôåðû ñ íîðìàëüþAèEäâå òî÷êè=0(ñì.
ðèñ. 1.9.6), ðàñïîëîæåííûå íà îñèRíà ñôåðå îêàçûâàåòñÿ íóëåâîé, ò. å. ýòî òî÷êè òîðìîæåíèÿ (êðèòè÷åñêèåòî÷êè).EÎáðàòèì âíèìàíèå íà òî, ÷òî òî÷êàðàñïîëîæåíà ñ ïðîòèâîïîëîæíîé(ïîäâåòðåííîé) îò íàáåãàþùåãî ïîòîêà ñòîðîíû ñôåðû.  òî÷êàõ æå, ãäån̄3 = n · ē3 =0 (ýòî âñå òî÷êè, ïðèíàäëåæàùèå îêðóæíîñòèOx1 x2 ,ïåðåñå÷åíèåì ñôåðû è ïëîñêîñòèL,ñì. íàïðèìåð òî÷êèÿâëÿþùåéñÿCèGíàðèñ.
1.9.6), èç (1.9.78) ïîëó÷àåì¯3v¯L = − v∞ ē3 .(1.9.80)2v = |v|Âåëè÷èíà ñêîðîñòèv = (3/2)v∞ ,Ïîëîæèì, ÷òîäàâëåíèåpïðè ýòîì ïðèíèìàåò ìàêñèìàëüíîå çíà÷åíèåêîòîðîå â ïîëòîðà ðàçà áîëüøå ñêîðîñòè íàáåãàþùåãî ïîòîêà.v∞t, òîãäà èç ôîðìóëûµ¶◦|v|2p=ρ f−.íå çàâèñèò îòâ æèäêîñòè:(1.7.44) ëåãêî íàéòè2Êîíñòàíòóf(1.9.81)â ýòîì óðàâíåíèè íàõîäèì, çàäàâàÿ åùå îäíî ãðàíè÷íîå óñëî-âèå äëÿ äàâëåíèÿp∞íàáåãàþùåãî ïîòîêà íà áåñêîíå÷íîñòè:p∞µ¶2v∞=ρ f−.◦2(1.9.82)Èç (1.9.81) è (1.9.82) îêîí÷àòåëüíî ïîëó÷àåì◦ρ2p = p∞ + (v∞− |v|2 ).2Íà ñôåðåΣR(1.9.83)ñ ó÷åòîì (1.9.79) èìååì ñëåäóþùåå ðàñïðåäåëåíèå äàâëåíèÿ:◦pΣR = p∞ +2ρv∞2³1−94´sin2 ϑ .(1.9.84)A è E , â êîòîðûõ v = 0, äàâ◦ 2pΣR = p∞ + (ρv∞/2) ïðèíèìàåò ìàêñèìàëüíîå çíà÷åíèå. Ìèíèìàëüíîå◦ 2çíà÷åíèå: pΣR = p∞ − (5/8)ρv∞ äîñòèãàåòñÿ íà êîíòóðå L (óãîë ϑ = ±π/2,ýòî òî÷êè C è G íà ðèñ.
1.9.6).Èç (1.9.84) ñëåäóåò, ÷òî â êðèòè÷åñêèõ òî÷êàõëåíèå149 1.9. Ïîòåíöèàëüíûå äâèæåíèÿ íåñæèìàåìîé èäåàëüíîé æèäêîñòèÈç ôîðìóëû (1.9.84) ñëåäóåò, ÷òî ðàñïðåäåëåíèå äàâëåíèÿ íà ïîâåðõíîñòèñôåðû ñèììåòðè÷íî îòíîñèòåëüíî ïëîñêîñòåéOx1 x3èOx1 x2 .Òîãäà ñëåäóþ-ùèé èíòåãðàë íà ïîâåðõíîñòè ñôåðû:ZF Σs = −Z Zπ2πnpΣR dΣ = −R2ΣRnpΣR sin ϑ dφ dϑ,0(1.9.85)0ïðåäñòàâëÿþùèé ñîáîé ñóììàðíóþ ñèëó, äåéñòâóþùóþ ñî ñòîðîíû æèäêîñòèíà îáòåêàåìûé øàð, áóäåò ðàâåí íóëþ. Äåéñòâèòåëüíî, ïîñêîëüêó âåêòîðíîðìàëèníà ñôåðå ñîâïàäàåò ñ áàçèñíûì âåêòîðîìe2ñôåðè÷åñêîé ñèñòåìûêîîðäèíàò (ñì. ðèñ.
1.9.6) (â ôîðìóëå (1.9.85), êàê è âî âñåì ýòîì ðàçäåëå,íîðìàëüníàΣRÿâëÿåòñÿ âíåøíåé ïî îòíîøåíèþ ê øàðó):n = er = sin ϑ cos φ ē1 + sin ϑ sin φ ē2 + cos ϑ ē3 ,(1.9.86)òî, ïðèíèìàÿ âî âíèìàíèå òî, ÷òî, ñîãëàñíî (1.9.84), äàâëåíèåîòφè ñèììåòðè÷íî íà [0, π] îòíîñèòåëüíî îñèZZπ2π21Z− F Σs = R ē2sin φ dφ pΣR sin2 ϑ dϑ+cos φ dφ pΣR sin ϑ dϑ + R2 ē000Z2π+ R2 ē3íå çàâèñèòZπ2π2pΣRϑ = π/2:0Zπdφ pΣR sin ϑ cos ϑ dϑ = 0.0(1.9.87)0Ýôôåêò, çàêëþ÷àþùèéñÿ â òîì, ÷òî ñóììàðíàÿ ñèëà, äåéñòâóþùàÿ ñîñòîðîíû æèäêîñòè íà îáòåêàåìûé øàð, ðàâíà íóëþ, è, ñëåäîâàòåëüíî, øàð íåèñïûòûâàåò ñîïðîòèâëåíèÿ, íàçûâàþò ïàðàäîêñîì Äàëàìáåðà.Ýòîò ïàðàäîêñ îáúÿñíÿåòñÿ òåì, ÷òî ðåàëüíûå äâèæåíèÿ æèäêîñòè ïðèîáòåêàíèè øàðà íå ÿâëÿåòñÿ ïîòåíöèàëüíûì (÷òî ñóùåñòâåííûì îáðàçîìèñïîëüçîâàíî íàìè ïðè ðåøåíèè äàííîé çàäà÷è), ñ ïîâåðõíîñòè øàðà ñõîäÿòâèõðè è ðàñïðåäåëåíèå äàâëåíèÿ ïî íàâåòðåííîé è ïîäâåòðåííîé ÷àñòÿìèñôåðû óæå íå ÿâëÿåòñÿ ñèììåòðè÷íûì.1.9.8.
Çàäà÷à î ïîñòóïàòåëüíîì äâèæåíèè øàðàâ ïîòîêå íåñæèìàåìîé æèäêîñòèÐàññìîòðèì ñëó÷àé, êîãäà â ïîêîÿùåéñÿ áåçãðàíè÷íîé íåñæèìàåìîé ïîòåíöèàëüíîé æèäêîñòè ïîñòóïàòåëüíî äâèæåòñÿ øàð, êàê àáñîëþòíî òâåðäîåòåëî ñî ñêîðîñòüþvs ,ïðè ýòîì îòñóòñòâóþò âíåøíèå ìàññîâûå ñèëû.Ïîñêîëüêó øàð ñîâåðøàåò æåñòêèå äâèæåíèÿ (ñì. ò. 2, ï. 3.9.1), òî äëÿ ðåøåíèÿ ýòîé çàäà÷è óäîáíî ââåñòè íåïîäâèæíóþ ïðÿìîëèíåéíóþ äåêàðòîâó ñè-150Ãëàâà 1. Èäåàëüíûå æèäêîñòè è ãàçûñòåìó êîîðäèíàòO0 x0iñ íà÷àëîì â òî÷êå0ðàäèóñ-âåêòîð x âñÿêîé òî÷êèMO0UR (O0 ),ðàäèóñ-âåêòîðîì x öåíòðå ñèììåòðèè øàðàâ êîòîðîé ñâÿçàí ñýòîé æå òî÷êè â íåïîäâèæíîé ñèñòåìå êîîðäèíàòOxiñîîòíîøåíèåì (ðèñ.
1.9.7)x0 = x − a,a(t) = OO0 âåêòîð,âèñÿùèé îò t: da/dt = vs .ãäåÂíåïîäâèæíîé(1.9.88)âîîáùå ãîâîðÿ, çà-ñèñòåìåêîîðäèíàòOxiæèäêîñòü óäîâëåòâîðÿåò ñëåäóþùåé ñèñòåìåóðàâíåíèé:∆ϕ = 0 â V1 ,¯ ¯∂ϕ ¯vn ¯Σ =¯ = vns ,R∂n¯ ΣR¯¯ ¯v ∞ = ∇ϕ¯∞ = 0,ϕ¯∞ = 0,Ðèñ. 1.9.7. Ñõåìà ïîñòóïàòåëüíîãî äâèæåíèÿ øàðà â æèäêîñòèãäå, êàê è ðàíåå,V1 = E3a \ UR (0) âíåøíîñòü øàðà;ΣR (0)(1.9.89) ïîâåðõíîñòüøàðà.Ñêîðîñòü æèäêîñòèvnsíà ïîâåðõíîñòè øàðà îïðåäåëÿåòñÿ òîëüêî ñêîðî-ñòüþ äâèæåíèÿ øàðà:vns = vs · n = v ·3Xx0=v̄sα n̄α = vs cos ϑ,Rvs = |vs |,(1.9.90)α=1ãäåϑ óãîë ìåæäó âåêòîðàìèvsx0è(ðèñ. 1.9.7).Åñëè ïåðåéòè â ïîäâèæíóþ ñèñòåìó êîîðäèíàòïðîèçâîäíûå0∇ =ēi (∂/∂x0i ) è∇=O0 x0i ,òî, î÷åâèäíî, ÷òî âñåēi (∂/∂xi ) áóäóò ñîâïàäàòü, ïîýòîìó è âñÿçàäà÷à (1.9.89) â ïîäâèæíîé ñèñòåìå êîîðäèíàò ñîõðàíÿåò ñâîé âèä.Ñðàâíèâàÿ ïîñòàíîâêó çàäà÷è (1.9.89) ñ (1.9.73), ìîæíî çàìåòèòü, ÷òî ñòî÷íîñòüþ äî çàìåíû îáîçíà÷åíèévns → v∞ n̄1îíè òàêæå ñîâïàäàþò.
Ïîýòîìóãîâîðÿò, ÷òî çàäà÷è (1.9.89) è (1.9.73) îïðåäåëÿþò àáñîëþòíîå äâèæåíèåøàðà, à ðåøåíèå çàäà÷è (1.9.89) ñòðîèòñÿ àíàëîãè÷íî ðåøåíèþ (1.9.74):ϕ=3Xα=1ãäå∂Cα 0α∂x³1´r0=−3Xα=1Xn̄x0αCα 0α2 ,C∞ 03 = −rr3rα = |x0 |; Cα = const.Ïîäñòàâëÿÿ (1.9.91) â óñëîâèå íåïðîíèöàåìîñòè íà ñôåðå÷òî(1.9.91)α=1Cα = v̄sα R3 /2,ΣR ,ïîëó÷àåì,è, ñëåäîâàòåëüíî,3R3R3 Xv̄sα x0α = − 03 vs · x0 .ϕ = − 032r2rα=1(1.9.92)151 1.9. Ïîòåíöèàëüíûå äâèæåíèÿ íåñæèìàåìîé èäåàëüíîé æèäêîñòè ñëó÷àå, êîãäà âåêòîððåøåíèå (1.9.92) ñîâïàäàåò ñÄëÿ âåêòîðà ñêîðîñòèvvs −Ox1 ,êîëëèíåàðåí îñèòîv̄s2 = v̄s3 =0, è(1.9.74).æèäêîñòè èìååì ñëåäóþùåå âûðàæåíèå:ÃRv = ∇ ϕ = − 3032r0vsϕ23X3xr 02!0αx̄ vsαR3032r=α=1Ïîñêîëüêó íà ïîâåðõíîñòè ñôåðû³E−ΣR : r 0 = Rè´00x⊗x· vs .
(1.9.93)023rx0 = nR,òî èç (1.9.93)ïîëó÷àåì ñêîðîñòü íà ñôåðå:¯11v¯Σ = − (E − 3n ⊗ n)vs = − (vs − 3vns n),R22qp1|v |2|v|ΣR =vs2 + 3vns= s 1 + 3 cos2 ϑ .2(1.9.94)2Äëÿ òîãî, ÷òîáû íàéòè äàâëåíèåpâ æèäêîñòè, âîñïîëüçóåìñÿ ôîðìóëîéOxi :µ¶◦∂ϕ|v|2p=ρ f−−.(1.7.44) â íåïîäâèæíîé ñèñòåìå êîîðäèíàò∂t(1.9.95)2Ïðè ïåðåõîäå â ïîäâèæíóþ ñèñòåìó êîîðäèíàòïðîèçâîäíàÿ∂/∂tOx0iñëåäóåò ó÷åñòü, ÷òîáóäåò ïðåîáðàçîâûâàòüñÿ ñëåäóþùèì îáðàçîì:∂ϕ∂ϕ ∂x0i∂ϕ∂ 0iϕ , t) =+ 0i=− vs · ∇0 ϕ,∂t∂t∂t∂t∂xòàê êàêvs = da/dt = −∂x0 /∂t.(1.9.96)Òîãäà èíòåãðàë Êîøè Ëàãðàíæà (1.9.95) âïîäâèæíîé ñèñòåìå êîîðäèíàò ïðèíèìàåò âè䵶∂ϕ|∇0 ϕ|20p=ρ f−+ vs · ∇ ϕ −.◦∂t(1.9.97)2¯f (t) ñ ó÷åòîì óñëîâèé íà áåñêîíå÷íîñòè ∇0 ϕ¯∞ = 0,¯¯(∂ϕ/∂t)¯∞ = 0, p¯∞ = p∞ , îêîí÷àòåëüíî ïîëó÷àå춵◦ ∂ϕ|v|2p = p∞ − ρ− vs · v +.(1.9.98)Èñêëþ÷àÿ ôóíêöèþ∂tÏîñêîëüêó íà ñôåðåΣR2èç (1.9.90) è (1.9.94) ñëåäóþò ñîîòíîøåíèÿ¯∂ϕ ¯R dvs· n,¯ =−∂t ΣR2dtvs · v −|v|22111288(1.9.99)222= − (vs2 − 3vns) − (vs2 + 3vns) = (9vns− 5vs2 ) ==vs28(9 cos2 ϑ − 5) =vs22(1 −94sin2 ϑ),152Ãëàâà 1.
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