Vanishing pressure in gas dynamics equations
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(142) ÙºÚÕлÒٺлê
(143) ÒÜ@ÕºÔá4лÞÑÚ
(144) Òj× rts"óû ü ý\þ û ü uvxw õ , Ô{åÑãÑÞÌÔ<ã h 9 * 9 = ÎrÏ1åÑÒ ô 9 ö@"ÑóZh 9 ñ * 9 õ ×{ãÑÞ . óð 9 õrö FÑóOh 9 * 9 õ ×{ÕºÚ , Ô{åÑãÑÞÑÚ
(145) Þ#лã rts"óû ü y1G|{ sçó ü ñüYõõ æ Ô<Õ×{ß»ß ÷ ü = лã¬×+ß»ßsè êÔ<Û Ð»ã@ä ,¬×{êuÖÀÙºÔ ó õ ÓÚYä<ÚHÙ íNï¹ð 9 ñ^ô 9 íNòð 9 ñ^ð 9 íNòLô 9 öø÷ ÓàÏÑлêuÏêL×+ã á<Ú
(146) ß»Ô4êÐsÙgè ô ÷ æ Ô<Õà×{ßsß -43 1 í<òô;9 >QE2R. ,~Ú-лãNÙ`Ú
(147) ä<Õ`×+ÙºÚÞ×{ßsÔ<ãÑäÙ`Ï@ÚOêuÏÑ×{Õ`×{êHÙ`Ú
(148) պлÒEÙ`ÐsêOê
(149) åÑÕá{ÚÒ6×{ÒºÒÔ/ê
(150) Ðo×+ÙºÚÞYÙ`ÔrÙ`Ï@Ú 9 ×+ãÑÞËÜÑÕÔá/ÐsÞÑÚÒàÙ`Ï@ÚÚ
(151) ÒÙ`ÐsÛÀ×:Ù`ÚYÔ<ã ð 9 Ðsã rts"óó|÷ õ þ ó ü ñüYõºõ ü = . g EG7 EG7H7R*? >*?H?'C 28NK I*E V QX8$I I < 8x>QKME > CE;K JTI*U. í<ò4ó . óð:9ºõºõ E2R47H7R*JTI*U ÷ ]QKMQ7'En < [V$W X y . < I*CE2RqI*E < 7'PYI*E4]*?z8|I.
(152) JIuT:U
(153) IuVb~n<JÑKTV{lBF r s óû ü þ û ü u)vxw õ < C8 8 YK ? 7H?'8OK >Q8|7=KY?=EGI*8gCE;K JTI*U ~ U\8 *I U,KMEn?'8 >QEGFH7H. FHI*En?'KOR47'U\KME4];I ~ 7'UP > ~ ~ KM8 . 6 '. y. -. . < PYI ~ CAq>*? i G(I 9 . Ê /'6' -&T/ & +.-% N3 7& ? Ú×+ÕºÚYÕºÚ×{Þ@èÙºÔ Ú
(154) ãÑÞ ÙºÏÑÚYÜÑÕºÔ4Ô æ Ô æ ÎrÏÑÚÔ{ÕºÚÛ = 4лãÑê
(155) Ú ô 9 Ò`×+ٺлÒØÑÚÒ íNï¹;ô 9ñ ;ô 9ºí<ò;ô 9ñ +ð:9ºíNò1ó ó¹ð:9uõõköø÷ ó <õ öøí<ï¹ô 9 ñ^ô 9 íNòLô 9 ñ 0 óð 9 õí<ò/ó . óð 9 õõköø÷ ,Nè ó õ ÓÚÀÞ@ÚÞÑåÑê
(156) Ú æ ÕÔ<Û !kÕºÔ<Ü~Ô<ÒÐsÙ`ÐsÔ<ã Ù`Ϭ×+Ù í<ïô 9 ß»ÐsÚÒлã × , Ô<å@ãÑÞÑÚÞ ÒºÚHÙ Ô æ rtu)s vxw óû ü ýÿþ û üYõ = gãÑÞÑÚ
(157) ÚÞ ð 9 ÐsÒ ,~Ô<åÑãÑÞ@ÚÞâлã rtus vxw ó¹û ü ýÂþ û üYõ ×{ã@Þ ,Nè ó(<õ лÒã@Ô<ãÞÑÚê
(158) ÕºÚL×+Һлã@ä ÙºÏ1åÑÒ ó¹:ð 9uõ ÐsÒ , Ô<å@ãÑÞÑÚÞÂлã r s óû ü ý þ û üYõ = ÎrÏÑÚ
(159) ÕºÚ 9 u)vxw æ Ô<ÕºÚ !kÕºÔ{Ü Ô<ÒÐsٺлÔ<ã Ð»Û ÜÑß»ÐsÚÒrÙ`ÏÑ×+ٶٺÏÑÚçßo×+ÒÙ\Ù`ÚÕÛ Ðsã ó A <õ ÐsÒ ,~Ô<åÑãÑÞ@ÚÞ ó ×{ã@Þ ×{ê
(160) Ùºå¬×{ßsßsè Ù`Ú
(161) ãÑÞÑÒÙºÔ ÷ õ = ãÑÚ ÞÑÚ
(162) ÞÑåÑê
(163) ÚÒrÙ`Ϭ×:Ù ô 9 ,~ÚßsÔ<ãÑä<ÒOÙ`Ô × ,~Ô<åÑã@ÞÑÚÞÒÚ
(164) ÙrÐsã y u)G|vx{w s óû ü ýÂþ û üYõ ×{ãÑÞ ,Nè ÒêÔ<ßsÐ
(165) ÒÙ`ÏÑÚ
(166) Ô<ÕºÚ
(167) Û ÓÚêL×{ã ×{ÒºÒåÑÛ Ú ô 9 ôÀùy% ú óºó|÷ õ þ ó ü ñüYõºõ ÷ ü ó A}Nõ. . . . . . . ×+ÙßsÚL×{ÒEÙ æ Ô<Õ× Òå,ÑÒºÚ
(168) ë1åÑÚãÑê
(169) Ú = åÑÕÙºÏÑÚÕÛÝÔ{ÕºÚ ÓÚêL×{ã×{ßsÒºÔ ÒåÑÜÑÜ~Ô<ÒºÚ Ò`×Lè ð 9 f ð Ðsã rts"óó|÷ õ þ ó ü ñYüYõõ= ÎrÏÑÚ
(170) ÒºÚêÔ<ãNá<Ú
(171) Õºä<Ú
(172) ãÑêÚÜÑÕºÔ{Ü Ú
(173) ÕÙ`ÐsÚÒàÜ~ÚÕ 9 ÛÝÐSÙråÑÒàÙ`Ô"ܬ×{ÒÒ\ÙºÔ ÙºÏÑÚß»Ð»Û ÐsÙлãÙºÏÑÚ ÒÚãÑÒÚÍлã óõ ×{ãÑÞÓÚ×{ÕºÚYßsÚÞÙ`Ô ó <õ ×{Ò ä<Ô4ÚÒ¶ÙºÔ D÷ = ? ÚçêuÏ@ÚêuÖâÚL×+ҺлßSèÙ`Ϭ×+Ù¶Ù`ÏÑÚ ÐsãÑÐsÙºÐo×{ßÞ¬×:Ùu× æ Ô<Õ ó¹ð ô õ ×{ÕÚÍÙ`Ï@Ú ß»ÐsÛÝÐSÙ`Ò óð ô 8õ Ô æ óð 9 ô 9 õ = /ÐsãÑêÚ ð Ϭ×+ÒãÑÔ×+Ù`Ô{ÛÝÐsêÍܬ×{ÕEÙ åÑã@лë1åÑÚã@ÚÒºÒ æ Ô<Õ ó <õ æ Ô<ß»ßsÔÓàÒ ÒÚ Ú ' ÓàÏÑÐsêuÏ#ÚãÑÒåÑÕºÚ
(174) Ò Ù`Ϭ×+ÙÍÙ`ÏÑÚÓàÏÑÔ{ß»ÚÒÚë1åÑÚã@êÚêÔ<ãNá{ÚÕºä{ÚÒ ×{ãÑÞÿ×{êuÏÑлÚHá<Ú
(175) ÒÍÙ`ÏÑÚÝÜÑÕºÔ4Ô æ Ô æ ÎrÏÑÚÔ{ÕºÚÛ = 3 Ô+Ù`ÚÝٺϬ×+ÙÓÚÝêL×{ã ×{ß»ÒÔËܬ×{ÒÒÙºÔ Ù`ÏÑÚàßsÐ»Û ÐsÙ6лã ó {õ ÓàÏÑлêuÏÝÒºÏÑÔÓàÒ-Ù`Ϭ×+Ù ô лÒ-×YÕºÚä{åÑßo×{ÕjÒºÔ<ßså@Ù`ÐsÔ<ã"Ô æ ÙºÏÑÚ åÑÕä<ÚÕÒ Úë1å¬×+ٺлÔ<ã =. gE *7'UI2U7H?H?'CU7]>*? RQ[*E2> K FH? лã "! ! $#:- ' 0 % - & %# & +-0/1'& ! #:/23 ( & 0 ) 'D-*Z-*# &A-*! -41$' !ÚÕٺϬ×{Û Ú = Þ = /Ú
(176) Õ = Þ@á =$+ ×+Ù`Ï = ÜÑÜÑß = /êÐ = > ÜÑÜ = ( 9 ÷ó /лã@äN×{Ü~Ô<ÕºÚ ? Ô{Õºß»Þ /êÐsÚãNÙ`ÐSجê : *1õ =.
(177) z+F8V:Qwf|qLQdVL-p{e|RfBfBTeyR.
(178) gEG76RQK 7'En?'K I*E2>QP 8NU >QEn?|;I*U\87BDC >Q8OK I*En? ~ KM8 RQKY?AFHI*E;8OKME;C;I*C0?"FHIA7 F'K 7'E;8M? 3 Ô<ã@߻лã@ÚL×{Õ ã¬×{ß =J Î + '> &: 9 '' ó Q:& õ )×{ãÑãÑÔ<å@ãÑêÚ
(179) Þ Ð»ã BDCn>Q8NK I*En? R47"8NU >QEn?|;I*U\8LCE;KORQK 7'En?'K I*E;EG7'PMPY7H?> FHIA7 F'K 7'E;8Z? RQKY?AFHI*E;8OKME;C? # ' {÷> ÷ ( 9 ÷ ó : <õ = '
(180) Cn>QPKM8N[%?AI*PC8OK I*En? JTI*UL2U7H?H?'CU7'PY7H?H?]>*?A7H?H. I*EGI*8$I*EG7 ?AF>QP >QU FHI*En?A7'U\*>Q8OK I*E P > ~ ?L>QE2R@C4E;KOBDC7'EG7H?H? k! ÕºÚÜ@պлãNÙ ó ::& õ ×{ãÑãÑÔ{åÑãÑêÚ
(181) Þлã I*PC48NK I*En?L7'E%RQCn>QPKM8 7 ;I*C4USPY7H? ]'> S?D>QEn? 2U7H?H?'K I*E
(182) # ' QD} ÷ ( '"9 ÷ (& ó Q:& õ= * ! #" $% ! '& 8NK )F (*[gn>QU\8NK F'PY7H? >QE2R?AF>QP >QU FHI*En?A7'8NU\*>Q8OK I*E P > ~ ? Ù`Ô ×{Ü@Ü Ú×{Õàлã> + D = 3 å@Û = ã¬×{ß = = * $% ! +& :KY?'8|7'EGFH7 ]QPYI < >QPY7;I*C4U PY7t?'[A?',8 7 7gR47H? ]'> g?D>QEn?.2U7H?H?'K I*E # ' >> ( 9 ( *ó : < õ= } . -/ 021 43 65 [\;7'U < I*PK FS?'[ ?'8|7 ,? I$JzFHI*En?A7'U\*>Q8NK I*E P > ~ ?>QE2R 8 7 >Q8 7 . >Q8NK F>QP8 7HI*U\[LI$J ?AF IA)F ( ~ >QQ7H? T# Úä<ÐsÔ<ã¬×{ß: Ô<ã æ ÚÕÚãÑê
(183) Ú 4ÚÕºÐsÚÒÐsã ÜÑÜÑßsлÚÞ + ×+ÙºÏÑÚÛÝ×+Ù`ÐsêÒ á<Ô<ß = ó k! ÏÑл߻×{ÞÑÚßsÜÑÏÑÐ»× 5 + :( 'Nõ= )(*. 1 78 9 6: PYI < >QPQ? IAI*8 ?AI*PC48NK I*En? I$J ?A$I 7BDC >*?' K |PKMEG7>QU [H;7'U < I*PK F ?'[ ?'8|7 ,? ~ KM8 P >QU ]47zR>Q8x> k! ÕÚÜÑÕлãNÙ Q:& = &. 1 78 9 ;<
(184) >=? A@ 1 7>*?'C4U7 ?AI*PC48OK I*En?8|I 8 7"PKMEG7>QU CP8O K RQ K 7'En?'K I*E2>QP;8NU >QEn?|;I*U\87BDCn>Q8OK I*E ~ KM8 EGI*E ? IAI*8 FHIA7 F'K 7'E;8M? Ô<ÛÝÛ = ! > 9>%'' ( 9' & ó :( õ= * BC =?9$ D4 E" F: U>QAKM8 >Q8OK I*E2>QP KMEn?'8x> < KMPKM8OH[ G >QE >'2UI : K >Q8|7L8 7HI*U'[ JTI*ULP >QU ]47zR47'En?'KM8N[;7'U\8NC < >Q8NK I*En? ÒÙºÕºÔ<Û = ÒEÙ`ÕÔ<ÜÑÏ =J2> & * 9 &: ó ( ÷Nõ =.
(185)
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