!"# !"$%#& ('*)$+$, !&-.0/1234.65798:2$;<.=0.4>?>.

@ACBEDFGH IJJLKMDN$OHPD

Q3RTSUSUV9WYXZRTR R W\[]T^`_aVbXZVdc Vd^TR`e fgX(hgi3^TVd^TRTjk RT^ThlSURT]TVmQ3no[j no[^`i3VdnTpTRTR Sq[XCe

rtsvus6wyxz x|~*osvs*(Cx|

$s&$$ &? ¡£¢¤¡¤$$¥C¦¡$¤&$§¡¨©&¥L¨©&¤$¨$ª$$«¬©& 4­$®<<¯¦2°²± ¦¡$³£¤$$ C¬µ´¡¨©&¥C©¦¡¶¸·¡¡¬¤$¤¡©&¥·´©&¤¡©¹°º·¡©¦C¦¥¡¡Z´¡¶¼»6©&$¡©¯´¡¡½¾<¿«©¦µ´¡¹¸·$$¡¬¤$ $? *¦$´¡ 6©&·$¡½¤$$ À©&¨½¤$$§¡¨$ÁÀ£¬©Â*$¤$¶yÁ6§¡¤$$¥£¡µ¦¡¤$Áà $¡©&¾Ä ©&¡$¨©¯ÆÅ$¢·$·¡©&©&¥L©&§$¨$(®¡¤$$ ($¤$¡Ç·$¡µ¦&&´¡ $Ȩ©&¤¡§$¤¡©¬$¤ Å$¢·¯·­2¦©&·$¢¨& ¢¤¡¤$$ $d¬©¦µ´¡­©¦ ³± d·$ ¹(·$¡©&$®©´¡¹¤$¶yÁdÂ6¢¤$¨$ª$$¥«¡µ¯¬¤$¾É6µ´¡¹°Ê&È©&¶m &´¡ $? ¦¬©&¤¡ª$$ Â6$®$§¡¨©&Ž©*¶µ´2²·$¡©È&®½©& ¤$$¥¡¯¬$$­¬&¨$³£(·¡©¡©¤$¡(©&·$¡<´¡¹¤¡©&¥Y¡¶`·¡©¦2<´¡Å½È&Y<´¡¶yÁ9®½¬$¤¡©¥Z¤¡©&¶yÁC©&§$¤$¶yÁC¡ à ¤$$¥¾Ë¬ÌÍÎ ÏÐÎmÑ&ˬÏÒ¬Ó²¤¡¤$$ ¦¡$¤&$§¡¨©&¥m¨©&¤$¨$ª$$m¬©& 4­Ô©&·$¡<´¡¹¤ ¡·¡©¦2<´¡Å½È&­§&$§$¤¡©¯Æ$¤$ $¤$¤¡©£¡ à ¤$¡¾

Õ2ÖØ×CÙyÚÛÔÜÞÝyÙyßàÞÜgá$ܬâÜãyäåÞæççèæ éêëì$æí<éçîCçëç<éíèæ*ïê4æìðíðëñòÔëð4æè0ëóíçô¡õñô¡õðìíôöëëè0õê$î«÷øFùOúûù&üæòæýÀþCæ îòÔìëÿ

íðë2î9è0íô¡õñìõòoçíêíè0íððõñõéðõç<éýõò9òÔíñç<éìëíèlçë Bõêëõ ëç&æ

ut + uux + vuy + wuz − fv = −ρ−1px,

vt + uvx + vvy + wvz + fu = −ρ−1py,

0 = −ρ−1pz − g,

ρt + uρx + vρy + wρz = 0,

ux + vy + wz = 0.

F

íýgê4æõéºîì 4îí<éçîLëççíòÔõì$æðëíÀçëè0è0í<éêëñçëç<éíè F½ù$æé æô íõç<éêõíðëíÀðíô¡õ ÿéõêðõìdó4æç<éëóðõ ÿ ëðì$æêë4æð2éð9éõóð9êí(íðëñH ëç<éíèæô¡õõêòÔëð4æ éìëê4æí<éçîZôæôõôæ üæðõð4æLêëçHFH íêí<ü"!7ë$#dõõü&ð4æóíð çíìíêÿ

ð6ñZëZý ð6ñ%õ ý£ç&ù θ ')( íõ ( ê4æ*tëóíçôæ î%(ëêõé æéõóôëZð4æ«ç&*tíêíH,+ç& Oxð4æê4æì íð4æ

ð4æLìõç<éõôùOy ' ð4æçíìíêùÔçëÔæCéî íç<éë9òÔíñç<éìïí<é ìLõéêëö4æ éíðõègð4æê4æì íðëëmõçë Oz

H õõéìí<éç<éìïýÀþ(ëè\õê4æ ü&õèZõêíòÔí4îýÀéçîLô¡õèõðíð2éºçô¡õêõç<éë

(u, v, w)H<ACæì íðëíÇë-õé ÿ

ðõç<é«õõü&ð4æó4æýÀéçî.ïôì$æè0ëpëρ / g ' ïçô¡õêíðëíçëTéî íç<éëH@ê4æìðíðë2î9è0õòÔíëì6ìõòî2éçî.êë9çíòïýÀþ(ëêíòõ õ íðë2îÊ÷ Oú0

H/R 1, O

(L/R)2 1, N1

23 4537689:8;<>=?,@A3 BC3EDGFHIFJ8;F?,2 KL7L7MON1P1<>M QF=SRTMOUSFSV N%QW89&C1N1<X8=QY8Z[N1\SMAL]8RM9[8Z^RK=1FL7<XVV8=;M V N1<X8=^8Z5N1\SM-Q_MOF3© acbdbefgAh 3 iG3j?kGlmln be l[oA3 pq3rOsStr13uwvxyzIv|Ey~ _ vyW_&:__:_z:yy_ vw|yW_&:O v:|:y~j$¡ :¢I£ ¤ ¥£¦¥O¤ ¦§ ¨y~ yW|©yzyy© v: jª¬«­®¦W ¯¤ ¥£¦¦¤ ¦§° _ vzIvu±G±² ³¦¦¡£¦¡££&¥ I v § y_ vw|´µ ªª ¶%u1· $¸¹¥O¤ ¦Wº¤ » ¤¼½I¾¿ÀÁ Â:ÃÄÅ&ƽǿÈÉËÊȨÌ:ÍÆIÌÏÎÐ Ñ_Ò

äyÚ0Ö¸Õ2Ö ëç<éíèæLô¡õõêòÔëð4æ é

tgϕ0(L/R) 1, JS

ü½òÔíç&R ' ê4æòÔë2ïç íè]ëù L ù H ' ¡æê4æô2éíêð6í ( õêë2ü&õð2é æð6í«ëYìíê2éëôæð6íLèæçé æÿ òÔìë íðë2îù ϕ0 ' (ëêõé æ / f = 2Ω cosϕ0 = 1 ' 4æê4æè0í<éêoBõêëõ ëç&æ2H"!êë ë íðëí Otõü&ð4æó4æí<éùÞó2éõmê4æççèæ éêëì$æí<éçîéõðôëñ\çõñ ë¡òÔô¡õç<éë\ìêíðí&êí íðëëê4æòÔë4æð6èëçôæ íðë2îè êë íêíè0í<þ(íðëëlõé3õòÔðõ ( õZü&ð4æóíðë2î z

ô\òÔê2ï ( õèyïH$#¸éõÊçõõéðõ(íðëí$õêõ ÿ(õ9çõ ( Ôæç<ïí<éçî¼çêí&æð6è0ë3õéðõ(íðë2îè0ë ( 4ïëð õô¡í&æð4ædëYê4æòÔë2ïç&æ%íè]ë óëçíðð6íõöíðôë êëìíòÔíð ìº÷ Oú¡½H õ ( Ôæçðõêë ë íðëý N1 ( õêë2ü&õð2é æð6íèæçé æ òÔìë í½ÿðë2î3ç<ï¡þ(íç<éìíððõZè0íð(íLê4æòÔë2ïç&æ%íè]ëH&!õçíòÔðíí êë ë íðëí JS£îì 4îí<éçîÊð4æëõ ííç<éêõ ( ëè6H­+ðõZìõ ð2îí<éçîÊòw4î\çêíòÔðëëë\ðë2ü&ôë(ëêõéùòw4î\ô¡õéõê tgϕ0 6 1H'#¸éõ

ëçôý£ó4æí<é"êëè0íðíðëíè0õòÔíëô«æð4æë2üïéíóíðëñ ìì6çõôë (ëêõé æùéõíç<é(ìYõ 4îêðè0õê$îH éíõêí<éëô¡õ ÿ ( ê2ïõìõñéõóôë(ü&êíðë2îCëð2éíêíçêíòÔç<é æì 4îí<éíçô¡õðíóðõè0íêð4æ î ( ê2ï4æ G ùòÔõ2ïçôæíèæ îlïê4æìðíðë2îè0ë è0õòÔíë ÷øFúûù­õêõ òæíèæ îºçíòïýÀþ(ëè0ëgëð*tëðë2éí<ü&ëèæð6è0ëõíê4æ éõê4æè0ë

X1 = x∂x + y∂y + 2z∂z + u∂u + v∂v + 2w∂w + 2p∂p;X2 = −y∂x + x∂y − v∂u + u∂v;X3 = p∂p + ρ∂ρ;

〈τ〉4 = 2τ∂t + (τ ′x + τy)∂x − (τx− τ ′y)∂y − [(τ ′ + τ ′′′)x2+y2

2+ 2τ ′z]∂z+

+(−τ ′u+ τv + τ ′′x + τ ′y)∂u − (τu+ τ ′v + τ ′x− τ ′′y)∂v−−[(τ ′′′ + τ ′)(xu+ yv) + 4τ ′w + (τ ′′′′ + τ ′′)x

2+y2

2+ 2τ ′′z]∂w − 2τ ′p∂p;

〈α〉5 = α∂x − (α′′x + α′y)∂z + α′∂u − (α′′u+ α′v + α′′′x + α′′y)∂w;〈β〉6 = β∂y + (β ′x− β ′′y)∂z + β ′∂v + (β ′u− β ′′v + β ′′x− β ′′′y)∂w;〈γ〉7 = γ∂z + γ′∂w;〈δ〉8 = δ∂p.

D

+5íê4æ éõê¼æ ( í&ê LçõòÔíê æ é¬2î2é©êõë2ü&ìõ ð ( ÔæòÔôëY*ïðôöëëìêíè0íðë τ(t) ù α(t) ù

β(t)ùγ(t)

ùδ(t)H)( éêë¡æè0ëmõõü&ð4æóíð êõë2ü&ìõòÔð6íõé+*<éë.*ïðôöëñõ

tH

Bõðíóð6íYêíõê4æ ü&õì$æðë2î«òw4î9õíê4æ éõêõì 〈α〉5, 〈β〉6, 〈γ〉7 ëè0íýÀé«ìë¡ò7

〈α〉5 : x = x + α(t), u = u+ α′(t),

z = z + (α′′(t)x− α′(t)y) + 12α(t)α′′(t),

w = w + (α′′(t)u− α′(t)v + α′′′(t)x− α′′(t)y) + 12(α(t)α′′(t))′;

〈β〉6 : y = y + β(t), v = v + β ′(t),

z = z + (β ′(t)x + β ′′(t)y) + 12α(t)α′′(t),

w = w + (β ′(t)u+ β ′′(t)v + β ′(t)x + β ′′′(t)y) + 12(β(t)β ′′(t))′;

〈γ〉7 : z = z + γ(t), w = w + γ ′(t).@ôæ üæðÊéõ ô¡õÀðí<éêëìë4æð6íêíõê4æ ü&õì$æðë2îù&õç<é æð6í¸ìíëóëðêíõê4æ üïýÀéçîéõ ÿòÔíç<éìíððõH"#¸éë êíõê4æ ü&õì$æðë2î êíòÔç<é æì 4îýÀéçõõñºõõþðð6í ( æëííì)íêíðõç& ìð4æê4æì íðëëYõçíñ

OxùOyùOzçõõéìí<éç<éìíððõH !êëè0íðíðëíòæððêíõê4æ ü&õì$æðëñZômêí½ÿ

(íðë2îè6ù¬çõòÔíê æ þ(ëèTõçõíððõç<éë ëç<éõóðëôù¬ç<éõôùìë2êS½ùõü&ìõ 4îí<éAõ 4ïóë2é9êí(íðëíçéíè0ë ítõçõíððõç<éîè0ëù2òÔìë ( æýÀþ(ëè0çîõüæòæððõñ éê4æíô2éõêëëHí&õçêíòÔç<éìíððõíYì6óëçíðëíYô¡õðíóðõ ( õ êíõê4æ ü&õì$æðë2îù¸çõõéìí<éç<éìïýÀþ(í ( õlõíê4æ éõ ÿê2ï 〈τ〉4ùGêëìõòÔë2éºô ðí<îìðõñ *tõêè0íùÇðí<ï&òÔõðõñ òw4î í ( õÊòæðíñ(í ( õ êëè0íðíðë2îH¸AY4îð4æ$õ òÔíðë2î îìðõ ( õ(ìë¡òæ-êíõê4æ ü&õì$æðëíëç&õ ü&õì$æÔæç&(çëè0è0í<éêë2î9ëð*tëðë2éí<ü&ëèæðõíê4æ éõêõìtæ ( í&ê L

õéðõçë2éíðõêíõê4æ ü&õì$æðëñtòÔõ2ïçôæíè0õñ ( ê2ï GH&A(íñç<éìë2éí ÿ

ðõùõçô¡õ ô2ï-êíõê4æ ü&õì$æðë2î ( ê2ï Gçõ:2ê4æð2îýÀéçëç<éíèyï(ïê4æìðíðëñ FùëòÔíñç<éìëí*ð4æ

ý¬õñ(ë2üÇõíê4æ éõêõìæ ( í&ê Lòæ<é¬ëðíñð2ïýlô¡õèëð4æöëýºéí í²õíê4æ éõêõìH !õ üïîç&

*<éëè çõõê4æ íðëíè6ùyæ9é æô íëð*tõêèæöëíñ¼õmìë¡òÔíòÔõ2ïçôæíè0õ ( õ$êíõê4æ ü&õì$æðë2îùçíòï$ÿýÀþ(ëèMë2üdðí<îìðõ ( õ$êíòÔç<é æì íðë2îù¸è0õ ðõ%õ 4ïóë2é3ëçô¡õè0õíAêíõê4æ ü&õì$æðëí çëè0è0í<éêëëîìðõHAY4îCï&òÔõç<éì$ætìõç&êë2î2éë2î+*<éõYêíõê4æ ü&õì$æðëí6üæëç&æðõìtöëëð¡òÔêëóíçôë«ô¡õõêòÔëð4æÿé æ

(r, θ, z)ùx = r cos θ

ùy = r sin θ / U ë V ' ê4æòÔë4æð4æ î ë õôê2ï ð4æ îô¡õèõðíð2éTçô¡õêõç<éëì õçô¡õç<éë

OxyùW ' ô¡õèõðíð2é æLçô¡õêõç<éëìòÔõ Lõçë Oz

êëçH¬F½Ht = ϕ(t),

r = r√

ϕ(t),

θ = θ − 1

2(ϕ′(t)− t) ,

z =1

ϕ′(t)z +

−ϕ′(t)2 + ϕ′(t)4 − 3ϕ′′(t)2 + 2ϕ′(t)ϕ(3)(t)

4ϕ′(t)3r2

2,

u =ur

√

ϕ′(t)+

r

2

(

ϕ′′(t)

2ϕ′(t)3/2

)

,

v = vθ1

√

ϕ′(t)+

r

2

1− ϕ′(t)

ϕ3/2(t),

w =1

ϕ′(t)2w +

1

8(ϕ′(t))5(

2rurϕ′(t)(−ϕ′(t)2 + ϕ′(t)4 − 3ϕ′′(t)2)+

+r2ϕ′′(t)(ϕ′(t)2 + ϕ′(t)4 + 9ϕ′′(t)2)+

+2r(2urϕ′′′(t) + rϕ(IV )(t))− 10r2ϕ′(t)ϕ′′(t)ϕ′′′(t)

)

,

p =p

ϕ′(t).

4òÔíç&ϕ(t) ' êõë2ü&ìõ ð4æ î*ïðôöë2îH£ACæððõí êíõê4æ ü&õì$æðëí õü&ìõ 4îí<é ( íðíêëêõì$æ éðíç<é æöëõð4æêð6í(êí(íðëëë2üêí(íðëñùÔõëç&6ì$æýÀþ(ë9ç<é æöëõð4æêð6íéíóíðë2î ë¡òÔô¡õç<éëH

ÔÖ ÞÛ¬ä"!Ü$#&%ÝÞÜ('oÚ¬äyÚÛ*)+!Ü, yÙâÜ$#&-)+.0/AY4îmçëç<éíèæ éëóíçô¡õ ( õmõëç&æðë2îmëðì$æêë4æð2éðYëZó4æç<éëóðõ ëðì$æêë4æð2éðYêí(íðëñYçëÿç<éíè FÇðíõ:$õòÔëè0õ^õç<éêõë2é õ2éëèæð2ïýMçëç<éíèyïõòæ ( í&êdòw4îmæ ( í&ê L

H1í<éêëÿìë4æðõç<é(òæððõñdüæòæóëdüæôý£ó4æí<éçîì íçô¡õðíóðõè0íêðõç<éëYæ ( í&ê L

H)!íêì6è Cæ ( õè

õç<éêõíðë2î õ2éëèæðõñçëç<éíè õòæ ( í&ê«îì 4îí<éçîì6óëçíðëí ( ê2ï ìð2ï¡éêíððë æì ÿéõè0õê*të2ü&è0õìAutL

÷ NúûH!êëè0íðíðëí(ç<é æð¡òæê2éðõ ( õdæ ( õêë2éèæZ÷ J ú¬òæ<éçíòïýÀþ(ëñYêí<üïO ÿé æ é

A1 : τ = τ, α = e−t1α, β = e−t1β, γ = e−2t1γ, δ = e−2t1δ;

A2 : α = α cos t2 − β sin t2, β = α sin t2 + β cos t2, τ = τ, γ = γ, δ = δ;

A3 : τ = τ, α = α, β = β, γ = γ, δ = e−t3δ;

A4 : τ = T (t4+χ(t))2χ′(t)

, ( òÔí τ(t) = T (χ(t))2χ′(t)

,

δ = 2χ′(t)D(t4 + χ(t)), ( òÔí δ(t) = 2χ′(t)D(χ(t)),

γ = 2χ′(t)Γ(t4 + χ(t)), γ(t) = 2χ′(t)Γ(χ(t));

α =α0(t4+χ(t)) cos t

2−β0(t4+χ(t)) sin t

2√2χ′(t)

, ( òÔí α(t) = α0(χ(t)) cost

2−β0(χ(t)) sin

t

2√2χ′(t)

α =α0(t4+χ(t)) sin t

2+β0(t4+χ(t)) cos t

2√2χ′(t)

, ( òÔí β(t) = α0(χ(t)) sint

2+β0(χ(t)) cos

t

2√2χ′(t)

A5 : τ = τ, α = α + A5t5, β = β +B5t5,

γ = γ + Γ51t5 + Γ52t52

2, δ = δ,

A5 = [(x1 + τ ′)σ − 2τσ′], B5 = (τ − x2)σ,

Γ51 = σα′′ − ασ′′ − σ′β − σβ ′,Γ52 = σA′′

5 − σ′′A5 − σ′B5 − σB′

5;

A6 : τ = τ, α = α + A6t6, β = β +B6t6,

γ = γ + Γ61t6 + Γ62t62

2, δ = δ,

A6 = (x2 − τ)σ,B6 = [(x1 + τ ′)σ − 2τσ′],

Γ61 = σβ ′′ − σ′′β + ασ′ + σα′,Γ62 = σB′′

6 − σ′′B6 + σ′A6 + σA′

6;

A7 : τ = τ, α = α, β = β, γ = γ + Γ71t7, δ = δ,

Γ71 = 2x1σ − (τσ)′;

A8 : τ = τ, α = α, β = β, γ = γ, δ = δ +∆81t8,

∆81 = 2x1σ + x3σ − 2(τσ′ + στ ′). êëìíò1ðð*tõêèyïOÔæTïôæ üæð êíõê4æ ü&õì$æðð6íô¡õõêòÔëð4æ é ëð*tëðë2éí<ü&ëèæðõ ( õõíê4æ éõê4æ

X = aX1 + bX2 + cX3 + 〈τ〉4 + 〈α〉5 + 〈β〉6 + 〈γ〉7 + 〈δ〉8 G

õò3òÔíñç<éìëíè`ôæ òÔõ ( õdë2ü«æìéõè0õê*të2ü&è0õìH²è0íç<éõðí<îìðõ ( õAêíòÔç<é æì íðë2î\ìð2ï¡éêíððí ( õæìéõè0õê*të2ü&èæA4ï&òÔõðõ9ëç&õ ü&õì$æ éí ( õdîìð2ïý üæëç&ù,õ 4ïó4æíèyïý òÔíñç<éìëíèqêíõ ÿ

ê4æ ü&õì$æðë2î ð4æôæ ò6ñ ë2üõíê4æ éõêõìæ ( í&ê Lùüæëç&æðð6ñdì(ô¡õõêòÔëð4æ éðõèºìë¡òÔí D½H

åÞæççèæ éêëì$æíèyïý æ ( í&ê2ï Lè0õ ðõ ê4æ üõ ë2éTì¹õ 4ïê$îèyïýqç<ïè0èyï

L = L4⊕L∞

ù( òÔí L4 = X1, X2, X3, 〈τ〉4

H¸ï&òÔíèMç<éêõë2éÊõ2éëèæð2ïý çëç<éíèyï õòæ ( í&êºçõè0õþ-ýòÔìï1Cæ ( õìõ ( õlæ ( õêë2éèæT÷ N úûH ð4æó4æÔæ õç<éêõëè õ2éëèæð2ïý çëç<éíèyïlòw4î õòæ ( í&êL4ù²ô¡õéõê4æ îù²ì¼çìõý õóíêíòùòÔõ2ïçôæí<éºê4æ üõ íðëíZìê$îèyïý ç<ïè0èyïlô¡õðíóðõè0íêðõñ ëíçô¡õðíóðõè0íêðõñ ó4æç<éíñH*AY4î

L3 = X1, X2, X3õ2éëèæð4æ î çëç<éíèægç<éêõë2éçîêõ ÿ

ç<éõùÇì¼çë4ïgæííìõç<éë õòæ ( í&êH+ð4æ\çõç<éõë2éºë2üZõòÔðõ ( õ¼éêí2è0íêðõ ( õ êíòÔç<é æìë2éí4î

X1, X2, X3ù¡éêí«òÔìïè0íêð aX1 +X2, bX2 +X3, X1, cX2 +X3, X1, X2

ùéêí õòÔðõ ÿè0íêð aX1 + bX2 +X3, aX1 +X2, X1

ëdð2ïOíìõñ.õòæ ( í&êH+*òÔðõè0íêð6í õòæ ( í&ê ë2ü L4ùëè0íýÀþ(ëíLðíð2ïOíìïý ô¡õõêòÔëð4æ éï 〈τ〉4

ùòÔíñç<éìëíè æìéõ ÿè0õê*të2ü&èæ

A4è0õ ðõ[êëìíç<éëYôZìë¡òïç τ = 1

HA(íñç<éìë2éíðõùEõòZòÔíñç<éìëíè üæè0íð ëð*tëðë2éí<ü&ëèæð6ñZõíê4æ éõê 〈τ〉4

ë2ü&è0íðë2éçîmçíòïýÀþ(ëègõê4æ ü&õè〈τ〉4 = τ(t)∂t + . . . → 〈τ 〉4 = τ(t)ϕ′(t)∂t + . . . .

õêϕ =

∫

1/τ(t)dtêëìõòÔë2é õíê4æ éõê 〈τ〉4

ômõíê4æ éõê2ïíêíðõç&æ∂tHyæôëèlõê4æ ü&õè6ùì

õ2éëèæð2ïý çëç<éíèyï9ìõñ¡òÔí<é«õòÔðõè0íêð4æ î.õòæ ( í&ê4æ aX1 + bX2 + cX3 + 〈1〉4H

!íêíñ¡òÔíè3ô-õç<éêõíðëýgõòÔðõè0íêð"êíòÔç<é æìë2éííñ«õ2éëèæðõñ(çëç<éíè¼òw4îCæ ( í&êLH õþ(íè ìë¡òÔí3õíê4æ éõêTüæëç&6ì$æí<éçî ìºìë¡òÔí G½H ²ð4æó4æí3ê4æççè0õéêëè ç4ïó4æñ

τ 6=0H0BtæôëìG(íùêë õè0õþ(ëÊòÔíñç<éìë2î æìéõè0õê*të2ü&èæ

A4ìçí ( òæZè0õ ðõ9òÔõë2éçî τ = 1

Hç&õ ü&õì$æðëídæìéõè0õê*të2ü&è0õì

A5ùA6

õü&ìõ 4îí<éYç òÔíÔæ éα = 0

ùβ = 0

H£ìéõè0õê*të2ü&èA7ùA8

õü&ìõ 4îýÀé3üæð2ïOë2é3ô¡õõêòÔëð4æ éγëδHyæôëè õê4æ ü&õè6ùyìYõ2éëèæð2ïý çëç<éíèyï

ìôý£ó4æí<éçîõòæ ( í&ê4æ aX1 + bX2 + cX3 + 〈1〉4H

0çë í τ = 0ù¡ìç4ïó4æí

a 6= 0ùéõ¨õ ÿêí ðíèyïçõè0õþ-ý A5

ùA6òÔíÔæí<éçî

α = 0ùβ = 0

H õè0õþ-ý

A7ùA8üæð2ïO4îýÀéçîdô¡õõêòÔëð4æ é

γ = 0ùδ = 0

ùëì(õ2éëèæð2ïý çëç<éíèyïìõñ¡òÔí<éõòæ ( í&ê4æ X1 + bX2 + cX3

H0çë í a = 0ëb 6= 0

ùéõCç5õè0õþ-ýA7ðíüîüæð2ïOë2é

γH

*<éõè ç4ïó4æíYõ 4ïó4æí<éçîmí<þõòÔëð9õòÔðõè0íêð6ñêíòÔç<é æìë2éí bX2 +X3 + 〈γ〉7H

ç4ïó4æía = 0

ùb = 0

ùæìéõè0õê*të2ü&èA4

õü&ìõ 4îí<émç òÔíÔæ éα = 1

ùβ = 0

HBõõêòÔëð4æ é æγõç<é æ<éçî3õþ(í ( õ ìë¡òæ2HA(íñç<éìïîZõíê4æ éõêõè A8,

õ 4ïó4æíèδ = 0

HÞõ ( òæ«ìõ2éëèæð2ïýçëç<éíèyïdüæëç&6ì$æí<éçî X3 + 〈1〉5 + 〈γ〉7H

+ç<é æí<éçî¼ê4æççè0õéêí<émç4ïó4æñùô¡õ ( òæ c = 0ùéõíç<é

X3ðíì $õòÔë2émì.$æ ü&ëçð6ñÊõíê4æ éõê

õòæ ( í&êH0çë êë *<éõè a = 0ëb = 0

ùéõõ 4ïó4æíèõòæ ( í&ê2ï 〈1〉5+〈γ〉7+〈δ〉8H0çë

í a 6= 0ù2ëè0ííèï íê4æççè0õéêíðð6ñ9ç4ïó4æñH!êë b 6= 0

õ 4ïó4æíèºõòÔðõè0íêð2ïýqõòæ ( í&ê2ïbX2 + 〈γ〉7 + 〈δ〉8

Hyæôëèºõê4æ ü&õè6ùõ2éëèæð4æ î9çëç<éíèæLçõòÔíê ë2éçíòïýÀþ(ëíõòÔðõè0íêð6íYõòæ ( í&êÏ

aX1 + bX2 + cX3 + 〈1〉4, X1 + bX2 + cX3,bX2 +X3 + 〈γ〉7, 〈1〉5 + 〈γ〉7 + 〈δ〉8,

bX2 + 〈γ〉7 + 〈δ〉8, X3 + 〈1〉5 + 〈γ〉7, aX1 − 2aX3 + 〈δ〉8.£ð4æõ ( ëóð6è õê4æ ü&õè ç<éêõë2éçîlõ2éëèæð4æ îçëç<éíèæZòÔìïè0íêð õòæ ( í&êH0@çõìëíõòæ ( í&ê ô¡õè0èyï¡é æ éõêq$æ ü&ëçðTõíê4æ éõêõì õòæ ( í&ê îì 4îí<éçî`ë ëðíñðõñTô¡õèÞÿ

ëð4æöëíñ õü&ìõ 4îí<éYòÔõõ ðë2éíðõÊï¡éõóðë2é¼ìë¡ò êõë2ü&ìõ ð *ïðôöëñù¸ì $õòî2þ(ë ì$æ ü&ëçð6íCõíê4æ éõêHBtæômëdòw4îZõòÔðõè0íêðõòæ ( í&êùð4æëõ ííçõ ð6ñ9òw4îdòæðíñÿ(í ( õëç&õ ü&õì$æðë2îCõíê4æ éõê 〈τ〉4 òÔíñç<éìëíèêíõê4æ ü&õì$æðë2î ìçí ( òæè0õ ðõÏêëìíç<éëLôõíê4æ éõê2ï.íêíðõç&æ[õLìêíè0íðë 〈1〉4 = ∂t

H+ô¡õðó4æ éíðõùÔõ2éëèæð4æ îçëç<éíèæ[õòæ ( í&êì ( 4î¡òÔë2é3çíòïýÀþ(ëèMõê4æ ü&õè *ïðôöëëα1,2(t)

ùβ1,2(t)

ùγ1,2(t)

ùδ1,2(t)

êõë2ü&ìõ ðùÞíçëðíïôæ üæðõ«ëðõíO

X1 + 〈1〉4, X2 + 〈C〉4 + 〈C1et/2 cos( t

2)− C2e

t/2 sin( t2)〉5+

+〈C2et/2 cos( t

2) + C1e

t/2 sin( t2)〉6 + 〈C3e

2t〉7 + 〈C4et〉8,

X1 + 〈1〉4, cX2 +X3 + 〈C〉4 + 〈C1et/2 cos( t

2)− C2e

t/2 sin( t2)〉5+

+〈C2et/2 cos( t

2) + C1e

t/2 sin( t2)〉6 + 〈C3e

2t〉7 + 〈C4et〉8,

dX1 +X2 + 〈1〉4, fX2 +X3 + 〈C〉4 + 〈C1ed

2t〉5 + 〈C2e

d

2t〉6 + 〈C3e

2dt〉7 + 〈C4edt〉8,

aX1 + bX2 + cX3 + 〈1〉4, 〈C〉4 + 〈α2〉5 + 〈β2〉6 + 〈C1e2at〉7 + 〈C2e

(2a+C)t〉8,( òÔíY*ïðôöëë α2, β2îì 4îýÀéçî9êí(íðëíègçíòïýÀþ(íñmçëç<éíèÏ

(

α′

2

β ′

2

)

=

(

−a b+ 1b− 1 −a

)(

α2

β2

)

,

〈1〉4, 〈C〉4 + 〈α2〉5 + 〈β2〉6 + 〈γ2〉7 + 〈δ2〉8,

X1 + 〈C1〉5 + 〈C2〉6 + 〈C3〉7 + 〈C4〉8, X2 + 〈1〉4,

X1 + 〈C1 cos(

12(c− 1)t

)

+ C2 sin(

c−12t)

〉5++〈C2 cos

(

c−12t)

− C1 sin(

c−12t)

〉6 + 〈C3〉7 + 〈C4et/2〉8, cX2 +X3 + 〈1〉4,

dX1 +X2 + 〈C1 cos(

12(c− 1)t

)

+ C2 sin(

c−12t)

〉5++〈C2 cos

(

c−12t)

− C1 sin(

c−12t)

〉6 + 〈C3〉7 + 〈C4et/2〉8, fX2 +X3 + 〈1〉4,

aX1 + bX2 + cX3 + 〈C1 cos(

t2

)

− C2 sin(

t2

)

〉5 + 〈C2 cos(

t2

)

+ C1 sin(

t2

)

〉6 + 〈C3〉7 + 〈C4〉8, 〈1〉4,

X1, X2,

X1, cX2 +X3,

dX1 +X2, fX2 +X3,

bX2 +X3 + 〈γ1〉7, 〈γ2〉7,

〈1〉5 + 〈γ1〉7 + 〈δ1〉8, 〈α2〉5 + 〈C + α′

2〉6 + 〈γ2〉7 + 〈δ2〉8,

bX2 + 〈γ1〉7 + 〈δ1〉8, 〈γ2〉7 + 〈δ2〉8,

X3 + 〈1〉5 + 〈γ〉7, 〈α2〉5 + 〈C + α′

2〉6 + 〈γ2〉7.

Ö(ÜÚÛ¬äyãyÝyÙgäyÝyßÔÜ /äÞÜÞÝÞÛ¬ÝyÙ") /+) )Ýyä")3Ú¬äyÚÛ*)+!AY4î9ï&òÔõç<éì$æ(òæðíñ(í ( õëççíòÔõì$æðë2îüæë(íèlïê4æìðíðë2î òÔëð4æè0ëóíçô¡õñZô¡õðìíôöëëè0õê$î9ìöëëð¡òÔêëóíçôëmô¡õõêòÔëð4æ é æc

Ut + UUr + r−1V Uθ +WUz − V + ρ−1pr = r−1V 2,Vt + UVr + r−1V Vθ +WVz + U + (rρ)−1pθ = −r−1UV,ρt + Uρr + r−1V ρθ +Wρz = 0,Ur + r−1U + r−1Vθ +Wz = 0,ρ = −pz.

åÞæççè0õéêëègçíòïýÀþïý2î2éëè0íêð2ïýõòæ ( í&ê2ï9æ ( í&êë9çëè0è0í<éêëëmçëç<éíèÏ

L5 = 〈∂z, t∂z + ∂W , ∂θ, p∂p + ρ∂ρ, ∂p〉. I1

Çæ ü&ëçð6í£õíê4æ éõêgæ ( í&ê I1yçõõéìí<éç<éìïýÀéù2ìõê$î¡òÔô¡íÀçíòÔõì$æðë2îù$é æôëèÊõíê4æ éõê4æèæ ( í&ê ë 〈1〉7, 〈t〉7, X2, X3, 〈1〉8

H ( í&ê4æ I1Gõêõ òæí<éLó4æç<éëóðõëðì$æêë4æð2éðõíêí(íðëíê4æð ( æOCëòÔí&*tíô2é æN2ù4ëè0íýÀþ(ííêíòÔç<é æì íðëí

U = U(t, r), V = V (t, r), W = W (t, r, θ, z),ρ = ρ(t, r, θ, z), p = p(t, r, θ, z),

F1êëóíè

ρ > 0ùp > 0

ìLçë4ï ðíõéêëö4æ éíðõç<éë$õéðõç<éëëòæì íðë2îH

!õòÔç<é æìëè êíòÔç<é æì íðëí F1²ìëçW$õòÔð2ïý çëç<éíèyïùõ 4ïóëè çíòïýÀþïýõòÔè0õòÔíUt + UUr − V + ρ−1pr = r−1V 2,Vt + UVr + U + (rρ)−1pθ = −r−1UV,ρt + Uρr + r−1V ρθ +Wρz = 0,Ur + r−1U +Wz = 0,ρ = −pz.

FF

åÞæ ü&êí(ëè õ 4ïóíðð6ítïê4æìðíðë2îõéðõçë2éíðõ[êõë2ü&ìõòÔð*ïðôöëëp

pr = r−1ρ(

rV + V 2 − r (UUr + Ut))

,

pθ = −ρ (U (rVr) + rVt) ,

pz = −ρ.

FO

+ç<é æì(ëíçîïê4æìðíðë2î9çëç<éíèqêëèyï¡éçíòïýÀþ(ëñmìë¡ò7

Wz = −1

r

∂

∂r(rU) ,

FN1ρt + Uρr + r−1V ρθ +Wρz = 0.

F&JS@ê4æìðíðëí FN1²õêíòÔí4îí<é[*ïðôöëý

W (t, r, θ, z)

W (t, r, θ, z) = −z

r

∂

∂r(rU) +W0(t, r, θ).

FDü(ïçõìëñYçõìè0íç<éðõç<éëYïê4æìðíðëñ FOõ 4ïóëèoì6ê4æ íðë2î9òw4î%êõë2ü&ìõòÔð%*ïðôÿöëë.õéðõç<éë

ρr = −r−1(

rV + V 2 − r (UUr + Ut))

ρz, F

ρθ = (U (r + V + rVr) + rVt) ρz. FG

ð2éí ( êëêõì$æðëí«çõõéìí<éç<éìïýÀþ(í ( õmïçõìë2îÊçõìè0íç<éðõç<éë\çëç<éíè FO£çCïóí<éõè õ 4ï$ÿóíððêíòÔç<é æì íðëñ F ½ù FGõü&ìõ 4îí<éLõêíòÔíë2é

Vt

Vt = −rVrU − UV − rU − h(t),

( òÔí h(t) ' êõë2ü&ìõ ð4æ î%*ïðôöë2îYìêíè0íðëùõîìëìCæ îçî$õçíCëð2éí ( êëêõì$æðë2îH!õòÔç<é æÿðõìôæ õ 4ïóíððõ ( õ«ì6ê4æ íðë2î9ìõìéõêõíïê4æìðíðëíçëç<éíè FFyòæí<é«çíòïýÀþ(íípθ = h(t)ρ.

+5õü&ð4æóëèf(t, r) = −Ut − UUr + V +

V 2

r

ùéõ ( òæíêìõíLïê4æìðíðëíçëç<éíè FOÏêëè0í<éìë¡ò7

pr = f(t, r)ρ. ë2éõ ( íùçëç<éíèæ FOíêí&ëç&6ì$æí<éçîZìçíòïýÀþ(íègìë¡òÔí

pθ = h(t)ρ,pr = f(t, r)ρ,pz = −ρ.

F

!¬õéðõç<éρð4æ$õòÔë2éçîmëð2éí ( êëêõì$æðëíèlïê4æìðíðëñ F ½ù FG

ρ(t, r, θ, z) = R (t, λ) , λ = h(t)θ − z + g(t, r), FI1

( òÔí R > 0 ' ðíô¡õéõê4æ îêõë2ü&ìõ ð4æ îA*ïðôöë2îù g(t, r) =∫

f(t, r)drH!õòÔç<é æì 4î2î.õ 4ïóíðÿ

ðõítì F ùð4æ$õòÔëè *ïðôöëýp

p =

∫

R (t, λ) dλ+ p0(t). O 1

çõõéðõ(íðëë FI1£üæìëçëè0õç<éíêíè0íððõñλõéõ 4îêðõ ( õ9ï ( Ôæ θ

ëðíñð4æ2ù7õ *<éõèyïòw4îðí&êíê6ìðõç<éë.õéðõç<éëdìõìçíè êõç<éê4æðç<éìíéíóíðë2îA*ïðôöë2î

RòÔõ ð4æ¨Çé-í½ÿêëõòÔëóíçô¡õñH!õéõñ í"êëóëðíòæì íðëíòÔõ ðõAÇéAíêëõòÔëóíçô¡õñ*ïðôöëíñH(õ ìçë4ïLðíõéêëö4æ éíðõç<éë

ρë2ü5*tõêèyïO O 1yçíòïí<éùó2éõtòæì íðëí*îì 4îí<éçîç<éêõ ( õCìõü&ê4æç½ÿé æýÀþ(íñ *ïðôöëíñ-íêíè0íððõñ

λH íòÔõì$æ éíðõùðí&êíê6ìðõç<éëêí(íðë2îCè0õ ðõÇòÔõë2éçîéõ ô¡õCòw4î

h(t) = 0H

!õòÔç<é æðõìôæ êíòÔç<é æì íðë2î FI1²ìCïê4æìðíðëí F&JS0òæ<éLçíòïýÀþ(íí

Rt +Rλ

(

gt(t, r) + Ugr(t, r) +z

r

∂

∂r(rU) +W0(t, r, θ)

)

= 0.

õè0ðõ ë2éí^êë RλòÔõ íð9üæìëçí<é«éõ ô¡õõé λ H#¸éõ ( õè0õ ðõ(òÔõë2éçî9üæLçó <éì²ÿõê4æ"*ïðôöëñ

W0ëU

1

r

∂

∂r(rU) = f(t), W0(t, r, θ) = gt(t, r) + Ugr(t, r) +

g

r

∂

∂r(rU) ,

U(t, r) = f(t)r

2+

a(t)

r,

( òÔí f(t) ù a(t) ' ðíô¡õéõê6í¨êõë2ü&ìõ ð6íY*ïðôöëëìêíè0íðëH êí<üïOé æ éíëè0ííè ïê4æìðíðëíòw4îA*ïðôöëë

R(t, λ)

Rt − f(t)λRλ = 0. OF

( õLêí(íðëíüæëç&6ì$æí<éçîé æô

R(t, λ) = R

(

λ exp(

∫

f(t)dt)

)

.

Þí&íêLè0õ íèºð4æñ2éë*ïðôöëý Vë2üìéõêõ ( õCïê4æìðíðë2îçëç<éíè FFI

V (t, r) = −r

2+

C(ξ)

r,

OO( òÔí ξ ' Ôæ ( ê4æð íì$æCô¡õõêòÔëð4æ é æ2ùüæòæðð4æ îdçíòïýÀþ(ëègõê4æ ü&õè

dr

dt= U(t, r),

r(0) = ξ.

ON1

Þõ ( òæ"íêìõíïê4æìðíðëíçëç<éíè FF²õêíòÔí4îí<éììíòÔíðð2ïý ð4æè0ëA*ïðôöëýg(t, r)

g(t, r) = −r2

4

(

f ′(t) +1

2f(t)2 +

1

2

)

− a(t)2

2r2− a′(t) ln r +

∫(

C(ξ)2

r3

)

dr. O JS

yæôëèºõê4æ ü&õè6ùõ 4ïó4æíèlçíòïýÀþ(ííéõóðõíêí(íðëí

U =rf(t)

2+

q(t)

2πr,

V = −r

2+

Γ(ξ)

2πr,

W = −zf(t) +W0(t, r),

R = R(

λ exp(

∫

f(t)dt))

,

P =

∫

R(

λ exp(

∫

f(t)dt))

dλ+ p0(t)

g = −r2

4

(

f ′(t) +1

2f(t)2 +

1

2

)

− q(t)2

4π2r2− q′(t)

2πln r +

1

4π2

∫(

Γ(ξ)2

r3

)

dr.

Ñ

²ìíòÔíð`õõü&ð4æóíðë2îa(t) =

q(t)

2π, C(ξ) =

Γ(ξ)

2π.

AY4î.êõë2ü&ìõ ð.*ïðôöëñùì $õòî2þ(ë9ìêí(íðëíùÔëè0íí<éçîîçð4æ î.*të2ü&ëóíçôæ î9éê4æô2éõìôæ2Hïðôöë2î

q(t)üæòæ<éê4æçW$õò3ê4æç&êíòÔí+ððõ ( õ ð4æõçë Oz

ëç<éõóðëôæ2Hïðôöë2îf(t)

õêíòÔí½ÿ4îí<édçô¡õêõç<édê4æòÔë4æðõ ( õ ê4æ ü+<é ædó4æç<éëöHïðôöë2î Γ(ξ)

õü&ìõ 4îí<é üæòæ éêõë2ü&ìõ ðõöëêô2ïO4îöëýlìíô2éõê4æÀçô¡õêõç<éë"õÀõôê2ï ðõç<éë r = const

ìÀðíô¡õéõêõèZöëëð¡òÔêëóíçô¡õèYçõíHæê4æô2éíêð6è0ëdòw4îdêí(íðë2îîì 4îýÀéçî.õìíê2ðõç<éëõç<éõîðç<éì$æ õéðõç<éë

ρ = ρ0

z = g(t, r) + C exp

(∫

f(t)dt

)

. OD

æòæòÔëè *ïðôöëýq(t)

ìLìë¡òÔíq(t) = 2π(1− cos t)

êë0 6 t 6 2π,

q(t) = 0êë

t > 2π, t < 0.

O +ç<é æì(ëíçîêõë2ü&ìõ ð6í*ïðôöëë.õ õ ëèºê4æìð6è0ë9ð2ïOý¨ f(t) = Γ(ξ) = 0

H tæ(êëçHÔOùõôæ üæð õê4æ üïýÀþ(ëíõìíê2ðõç<éíñ õç<éõîððõñ õéðõç<éëgìZòæððõè ç4ïó4æíH ð4æó4æ ÿð6ñdè0õè0íð2éìêíè0íðëAõìíê2ðõç<éë9ê4æìðõñõéðõç<éë îì 4îýÀéçîA4æê4æõ õë¡òæè0ëH!êë

t > 0ù

ô¡õ ( òæ*ïðôöë2î q(t) êëðëèæí<éðíð2ïOíìõí6ü&ð4æóíðëíù1õîì 4îí<éçîLõçõíððõç<éêë r = 0ùçìîÿ

üæðð4æ îdçìôý£óíðëíè ëç<éõóðëôæð4æ(õçëH!õìíê2ðõç<éë9ê4æìðõñõéðõç<éë.êë *<éõèºõé $õòî2éõéõçë

r = 0HÔBõ ( òæ íòÔíñç<éìëíCëç<éõóðëôæ^êíôê4æ þCæí<éçîùõìíê2ðõç<éëYìðõìLç<é æðõìî2éçî4æê4æõ õë¡òæè0ëH

1 2 3 4

r

- 5

- 4

- 3

- 2

- 1

z

2

1

3

4

5

1 2 3 4

r

- 4

- 2

2

4

z

1

2

4

5

3

äyÚ0Ö ÔÖ !õìíê2ðõç<éëõç<éõîððõñõéðõç<éë OD*òw4î3ê4æìðTêõè0í ï¡éô¡õììêíè0íðët ∈ (0, 2π)

êõð2ïè0íêõì$æð ì5õê$î¡òÔô¡í*ìõü&ê4æç<é æðë2îtêë ' Γ(ξ) = 0

ù ' Γ(ξ) 6= 0

åÞæççè0õéêëè3ç4ïó4æñùô¡õ ( òæ5õè0ëè0õëç<éõóðëôætð4æ£õçëCëè0íí<éçîô¡õðíóð4æ î(üæôê2ï¡éôæéíóíðë2îìðíô¡õéõêõè3öëëð¡òÔêëóíçô¡õè\çõíHAY4î *<éõ ( õüæòæòÔëèT*ïðôöëý Γ(ξ)çíòïýÀþ(ëèÊõê4æ ü&õè

Γ(ξ) = 0êë

ξ < 1, ξ > 2,

Γ(ξ) = 8πêë

1 6 ξ 6 2.ïðôöëý

q(t)õç<é æìëè¼éõñ íùó2éõCëê4æð(íùõêíòÔííððõñ O ùæ f(t) éõ òÔíç<éìíððõê4æì ÿðõñ3ð2ïOýH *<éõèTç4ïó4æí õ 4ïóëè õìíê2ðõç<éëõç<éõîððõñõéðõç<éëùë2ü&õê4æ íðð6íð4æêëçH$Où H Êõ Ôæç<éëùìô¡õéõêõñ

Γ(ξ)êëðëèæí<étðíð2ïOíìõíÇü&ð4æóíðëíùêõëçW$õòÔë2é ì2ïóëì$æÿ

ðëí"õìíê2ðõç<éíñ.õç<éõîððõñõéðõç<éëù4ô¡õéõêõíçõ(ìêíè0íðíèçõ:2ê4æð2îí<éçîù4ðõï&òæ4îí<éçîõéLõçë

OzH

Ö(ßÔÛ¬Ù"!Ùâ ) #&%ÝyÙ") /+) )Ýyä") *<éõñó4æç<éë«ê4æõélê4æççè0õéêëèõç<éêõíðëíëðì$æêë4æð2éðõ ( õêí(íðë2îõéðõçë2éíðõ(çí½ÿòïýÀþ(íñmóí<é6êí2è0íêðõñ%õòæ ( í&ê æ ( í&êë9çëè0è0í<éêëëçëç<éíèÏ

L4 = 〈∂z, t∂z + ∂W , p∂p + ρ∂ρ, r∂r + 2z∂z + U∂U + V ∂V + 2W∂W + p∂p〉, OG

( òÔí5$æ ü&ëçð6íõíê4æ éõêgçõõéìí<éç<éìïýÀéLçíòïýÀþ(ëèºõíê4æ éõê4æèæ ( í&ê 〈1〉7ù 〈t〉7 ù X3

ùX1H

!õòæ ( í&ê4æ OGCëè0íí<é¼ëðì$æêë4æð2é t, θ, U/r, V/rë õêõ òæí<é\ó4æç<éëóðõ ÿ ëðì$æêë4æð2éðõíêí(íðëíê4æð ( æ«O(ëòÔí&*tíô2é æ«N2ù4ëè0íýÀþ(ííYêíòÔç<é æì íðëí

U = ru(t, θ), V = rv(t, θ), W = w(t, r, θ, z),ρ = ρ(t, r, θ, z), p = p(t, r, θ, z).

O !õÇéôæëççíòÔõì$æðë2î%õòÔõðõ ( õêí(íðë2î$êë ρ = raR(t, θ)

ùa = const

Ôæ«ç òÔíÔæð4æì÷ DúûH!õçí õòÔç<é æðõìôëêíòÔç<é æì íðë2î\êí(íðë2î O 5õ 4ïóëè¹*æô2éõêÿ çëç<éíèyï3çíòïýÀþ(í ( õìë¡òæ

rv (uθ − 1) + ru2 − rv2 +1

ρpr + rut = 0,

OI1

rvvθ + u(2rv + r) +1

rρpθ + rvt = 0,

N 1

ρzw + ruρr + vρθ + ρt = 0, N2F

2u+ vθ + wz = 0, N$O

ρ = −pz. NN1

ütïê4æìðíðë2î N$Oõ 4ïóëè êíòÔç<é æì íðëíòw4î.*ïðôöëëw(t, r, θ, z)

w(t, r, θ, z) = −(2u+ v)z + w0(t, r, θ).

NJS@ê4æìðíðë2î OI1ù N 1¬ë NN1ÔòæýÀé£îìðõí¸ì6ê4æ íðëí0òw4î¨êõë2ü&ìõòÔð-*ïðôöëë p(t, r, θ, z) H

6óëçíðëíLçè0íCæððYìéõê$êõë2ü&ìõòÔðYõépòæí<éòÔì$æLïçõìë2îYçõìè0íç<éðõç<éëù¬îì 4îÿ

ýÀþ(ëíçîïê4æìðíðë2îè0ëòw4î.*ïðôöëëρ(t, r, θ, z)

H+ðë9ëð2éí ( êëê2ïýÀéçîZìLìë¡òÔí

ρ(t, r, θ, z) = R(t, λ), λ =r2

2f(t, θ)− z.

N$D²ìíòÔíðõ«õõü&ð4æóíðëí

f(t, θ) = v (uθ − 1) + u2 − v2 + ut. N

!õçí¬õòÔç<é æðõìôë N$D0ìëçW$õòÔð6íïê4æìðíðë2îõ 4ïóëè\ïê4æìðíðëí*òw4î^*ïðôöëëR(t, λ)

Rλλ

(

r2

2(ft + f (4u+ vθ) + vfθ) + λ (−2u− vθ)− w0

)

+Rtλ = 0. N$G

9ðõ ë2éí[êë RλλòÔõ íðdüæìëçí<éLéõ ô¡õLõé t ë λ

ùõéçý¸òæLçíòïí<é

w0 =r2

2(ft + f (4u+ vθ) + vfθ) +

h′(t)

k(t),

u =1

2

(

−vθ +k′(t)

k(t)

)

.

N

4òÔíç&h(t)

ùk(t) ' êõë2ü&ìõ ð6í*ïðôöëëìêíè0íðëH

!õòÔç<é æðõìôæÊçõõéðõ(íðëñ N Cì N$Gòæ<é\õô¡õðó4æ éíðõíì6ê4æ íðëí òw4î õéðõç<éëHAY4îdï&òÔõç<éì$æCüæë(íèlí tììë¡òÔíêõë2ü&ìõòÔðõñ9õéLðíô¡õéõêõñ*ïðôöëëP

ρ = P ′(k(t)λ+ h(t)). NI1

Ò

- 2 - 1 1

x

- 2

- 1

1

y

äyÚ0Ö Ö *ê4æíô2éõêëëmæìéõè0õòÔíðõ ( õLêí(íðë2î

Þõ ( òæòæì íðëítëè0íí<é«çíòïýÀþ(ëñmìë¡ò7

p =1

k(t)P (λk(t) + h(t)).

J 1!õòÔç<é æðõìôæÏêíòÔç<é æì íðëñ NI1ù J 1ìëçW$õòÔð2ïý çëç<éíèyï(ïê4æìðíðëñ OI1 NN1,êëìõòÔë2é

ô9çíòïýÀþ(íñmçëç<éíè0íïê4æìðíðëñ òw4î.*ïðôöëñf(t, θ), u(t, θ), v(t, θ)

fθ = −2vt +

(

−2u− 2vk′(t)

k(t)

)

,

uθ = −1

vut +

(

1− f

v− u2

v+ v

)

,

vθ = −2u+k′(t)

k(t).

JF

¸ï&òÔíè ëçôæ é«õçíçëè0è0í<éêëóðõí(êí(íðëíçëç<éíè JFIu = u(t), v = v(t), f = f(t).

!õ 4ïóëèlçíòïýÀþ(ííêí(íðëí

u =k′(t)

k(t), v =

A

k(t)− 1

2,

f =1

4

(

k′(t)

k(t)

)2

− k′′(t)

k(t)− 1

4+

(

A

k(t)

)2

.

4òÔíç&k(t) ' êõë2ü&ìõ ð4æ î*ïðôöë2îù

A ' êõë2ü&ìõ ð4æ î ô¡õðç<é æð2é æ2H)!õç<éêõëèéê4æíô2éõêëëòÔìë íðë2î9ó4æç<éëöì õçô¡õç<éë Oxy

ù$òw4î*<éõ ( õ«ìíêíèk(t) = 2 + sin t, A = 1.

ACæððõíêí(íðëíîì 4îí<éçîíêëõòÔëóíçôëè ìõLìêíè0íðëH1tæêëçHÔNCë2ü&õê4æ íð4æ-¡æê4æô2éíêð4æ îéê4æíô2éõêë2îmó4æç<éëö`ìéíóíðëë ë¡òÔô¡õç<éëùõêíòÔí+ððõè õ 4ïóíðð6ègêí(íðëíè6H

Ò

ÜÞà(#3ã")Ýyä")AY4î íçô¡õðíóðõè0íêðõñ ( ê2ïùòÔõ2ïçôæíè0õñLïê4æìðíðë2îè0ë(òÔëð4æè0ëóíçô¡õñ«ô¡õðìíôöëë«è0õ ÿê$îùõç<éêõíð4æYõ2éëèæð4æ îçëç<éíèæ3õòÔðõè0íêðºëÊòÔìïè0íêð õòæ ( í&êH $õòÔídëççí½ÿ

òÔõì$æðë2îlì6óëçíðõ\ìÊîìðõè ìë¡òÔíô¡õðíóðõíêíõê4æ ü&õì$æðëídòw4î õíê4æ éõê4æZòÔõ2ïçôæíè0õñ( ê2ïùêíõê4æ üïýÀþ(í ( õ ìêíèyî t

ì[êõë2ü&ìõ ð2ïý*ïðôöëýMõétH2ACæðð6ñmìë¡ò$êíõê4æ ü&õ ÿ

ì$æðë2î9ëè0ímôý£óíìõíü&ð4æóíðëíòw4îAõç<éêõíðë2îõ2éëèæðõñçëç<éíèHyæô íÏõ 4ïóíð ðõì6íéõóð6íêí(íðë2îïê4æìðíðëñ òÔëð4æè0ëóíçô¡õñmô¡õðìíôöëëè0õê$îHåí½ÿ(íðë2îYîì 4îýÀéçîYó4æç<éëóðõdëðì$æêë4æð2éð6è0ë3ê4æð ( ædO«ë9òÔí&*tíô2é æ9N2H!íêìõí(õëç&6ì$æí<édéêí¡ÿè0íêðõí«ìë2êíìõíLéíóíðëíù]õêõ òÔíððõíLìüæëè0õòÔíñç<éìëíè7ê4æç&êíòÔí+ððõ ( õZð4æ9ìíê2éëôæ ÿðõñ^ê$îè0õñ ëç<éõóðëôæ"êõë2ü&ìõ ðõñè0õþ(ðõç<éë ë^êõë2ü&ìõ ðõ ( õ(ìê4æ þ(íðë2î ì(öëëð¡òÔêëóí½ÿçô¡õè`çõíùõôê2ï æýÀþ(íèTëç<éõóðëôH ¸éõêõíLêí(íðëí«çõõéìí<éç<éìïí<éYìë2êíìõèyï$íêëõòÔëóí½ÿçô¡õèyï.õLìêíè0íðëmõçíçëè0è0í<éêëóðõèyïéíóíðëýMìõôê2ï ( ð4æó4æÔæô¡õõêòÔëð4æ éH!êëè0íð2î2î òÔõ2ï$ÿç<éëè6íÏêíõê4æ ü&õì$æðë2îçëè0è0í<éêëëù4è0õ ðõ ( íðíêëêõì$æ é *ôìëì$æíð2éð6ítêí(íðë2îù2ì(ô¡õéõ ÿêìë2ê²òÔìë í<éçîYõêõë2ü&ìõ ðõñtéê4æíô2éõêëëì ( õêë2ü&õð2é æðõñYõçô¡õç<éëëìê4æ þCæí<éçî íç<éô¡õLç5êõë2ü&ìõ ðõñ ï ( õìõñ çô¡õêõç<éýHAY4îíêìõ ( õêí(íðë2îAêëìõòî2éçî.*të2ü&ëóíçôëí¡æê4æô2éíêëç<éëôëõìíê2ðõç<éíñ[õç<éõîððõñ õéðõç<éë[õéõôæ2ùòw4îìéõêõ ( õtêí(íðë2î[õç<éêõ ÿíðTéê4æíô2éõêëëmó4æç<éëöH

r~«rxtUz«~Þ ¾¸? ¤$¤$$¨©&t¾ Ǿ !#"$&%"##(')"#*+ !,.-0/12-0/3 !-04#"#"5')/67$98+:Ç¡·$$$¤$­;ÇÅ7t­ Ä =<>­ (?@0A ¾B ¾C*D´©&¤2:¾v­$¿¥¡¨t¾=EF/HG##IJK/-L M3 N8¡¿¾PO$¿L$­ (?Q+ ¾R ¾¸? ¤$¤$$¨©&St¾ ǾTVUW/X#YZ"')(G#[0#I]\^,."#,.Y_ M')N\JX#/%(G`H !U6baa&c£©&¨´Ô¾y=<>¾ (??R ¾d¾ RRR ­e @ ¾Ä ¾ A3fB(g7A3fh ¾

h ¾ji©´©&$¤ Ä ¾ Ǿv­9k6¤¡©&¨©&<£¾ <£¾9l=m0X#X#/3/1n3#(G#"+o5%"pqpj r9 !#4#"#(G#[0#I]\nm! !#"#19stm0*+ !U#/3 ZX#/,./uU" N8#;Ç®¦$¯Æ©K>6©&©¡È&$¡¨©&Ž©Å½©¢¤$$¡$&­ BffQ ¾

v ¾jDy©¶yÁ&>¾ w ¾v­=kÇ¢·<Á$¤&<£¾ :¾TVU/%#/H'x*3,.YZ"#*#/ "#y="#3#YZ#/H'z9 !_ !#"#"&m! !#"#1|`y"#%6=/%"#u#(')"#-0"3Y')/,6pj r=I­#y¤$$¨&>)i4Ó¾ Ä $$ ~O¿$¨­2¬µÁ¤$$¨­$¤¡Â*©&¡$¨­ Bf+(f ­d¾ (f ­e h ­ Ä ¾ B@(g+Rv ¾ íê ( íñ Çæíêíìëóõ õìëðùðç<éë2éï¡é ( ë¡òÔêõòÔëð4æè0ëôëëè6H 3H H æìêíð2éíì$æ +Må Cùêÿ:é«æôH æìêíð2éíì$æ2ù¬FDù N I 2ù ( H+õìõçëëêçôùåõççë2îÞÿ&09##L39=##9Fr#VM+æêë2î êíìð4æBtæ üæô¡õì$æ2ùõìõçëëêçôëñ ( õç<ï&òæêç<éìíðð6ñmïðëìíêçë2éí<éùïOH)!ëêõ ( õì$æ2ùOù N I 2ù ( H+õìõçëëêçôùåõççë2îÞÿ&0+r)9+=~L977q9+FH7