Enrg onvrsaon aw n h r moshr or omrssl lows.. alva.. Issakov Pos-ocoral sarchr a h Unvrsé o orran aoraor Enrgéq E chnq nrq EM Nanc ranc. aakh Naonal chncal Unvrs ar.saav lma aakhsan srac hs ar s rsn nrg convrsaon law n h r amoshr or comrssl lows gvn h ach rolm or h hr mnsonal Navr-Soks qaons wh olr connos cocns n h arsan coorna ssm. Usng h aramr mho o hor nonlnar aral rnal qaons wr oan h vloc comonns o h ransn comrssl low whch rov a scron o h consv rlaonshs wn hr hscal qans: h vloc vcor h rnal an nrnal orcs h rssr sron. In rms Navr-Soks qaons comrssl low aramr mho rln moon onal l rssr sron I. INOUION rlnc s low characr rcrclaon an s. Mahmacall rln low s rrsn va a nols comoson n whch h low s rokn own no h sm o an avrag comonn. rln lows can scr wll hrogh h s o h Navr-Soks qaons. h consrvaon laws ar namnal laws o mchancs whch ar arclarl or rln moon. W hav ocs on h gloal snc nqnss an smoohnss o h Navr-Soks rolm or comrssl lows. Eamls o wak solon or ncomrssl lows wr gvn. aarll [] V.Shr []. crcal analss or man analc an nmrcal solons o Navr-Soks qaons was gvn..rman []. W wll n hs nq a o snc o solon gvn n [] sng h nrg consrvaon law or h rnal an nrnal orcs h gran o rssr. hs ar has ocs on h aramr mho wh olr connos cocns o h knmac an namc vscos. hs rsl has shown h n s o mahmacal nrsanng o h lsv comrssl hnomna o rlnc.hr w hav go som aonal namnal normaon ao havor o onal knc an sac nrgs or h scron hr gnral mchancs o h rln comrssl moon. In h rsn work w rv a nmr o rsls concrn wh h m a h m a c a l hor o h Navr-Soks qaons or comrssl ls. W shall al wh h ollowng rolms: on h on han a scron o h known rsls on h snc n h non-lnar an m-nn cass on h ohr han h nqnss a n h rglar o solons or h hr mnsonal Navr-Soks qaons wh olr connos cocns. II. MEMI OMUION O E POBEM rlnc mol s consr n h hr-mnsonal nn sac. no a on n l s consr h vloc vcor n a on M o h hr saal coornas a a gvn m h ormla v v Sos ha nn sacs j k s h vloc vcor s h l rssr l. s consr a ncon n on a on clos s S o. h ncon s sa o olr connos o onn n S hr ss a cons sch ha or all А n S. / W consr comrssl low gvn h gnral Navr -Soks rolm n h ollowng orm с v n hr wh h nal conon on
v с gra h vcor ncon o h Navr-Soks qaons -5 w oan h mahmacal rolm or ha qaon j k nos an rnal an nrnal orcs s a knmac vscos s a l ns h smol nos h gran wh rsc o h ncon h smol nos h hr mnsonal alac oraor s a namc vscos whch s rla o h knmac vscos h ormla hr w assm ha. or h am hr s o nrsan rln ars o h Navr-Soks rolm or comrssl lows. h nal val rolm - whch concrns wh h namnal solon o Posson an ha concon qaons. rln moon s sor h sjc owr rom som rnal orcs an nal vloc. h sha o rln rgon s rmn h ror whch has shown sal or nsal o h vloc moon an h rssr sron. Salng mchansms hav n avoca o lan ars osrv n nmrcal smlaons o rlnc. III. SBE SOUION O E NVIE-SOES POBEM IN E POENI IE In hs ar w consr l whn characr h hr-mnsonal Navr-Soks rolm wh olr connos cocn n h ollowng ssm o qaons: n v n Wh an nal conon on 5 Usng wll-known ormla o vcor analss gra ro 6 w hav go gra ro 7 lng h rsson Wh h nal conon 8 v 9 onsr h ha qaon 5 whr h cocn s n n a clnr s h closr o a on oman cocn sass all olr conons. or / ollowng h classcal rocr [6] w can g solons or h rolm 8- n h ngral sm o h araolc onals namnal solon was oan rom ngral qaon g Usng h qaon 8 was on nknown ncon as solon o h ollowng Volrra-rholm ngral qaon
Z sng sccssv aromaon mho n n Whr n n namnal solon has smaons or hr rvaons 8 Usng rors o h namnal solon an s rvav valaons w hav go a nqnss an sal solon sasng ollowng smaon M onon or h scalar rssr ncon v P Prcs a sa ar whch nrocs a sal rln moon whn ro ro or rssr ncon w hav go h ollowng smaon M onsqnl w s ha a sal o h rln low ns on h conon. IV. VEOIY OMPONENS N UNION O PESSUE O UBUEN SWIIN MOION ssm ha gra hn h Navr-Soks qaon can wrn as : gra ro 5 hr vcor ncon * s a convolon wn vcor an mar 8 onsrng conon ro an sng roor oraor w oan qaon ro ro ro 5 ona ha ro g 6 ro Wh rsc o 6 w hav go vcor qaon g ro 7 Usng rrsnaon s s s s Z w oan ollowng vcor qaon ro Whr vcor 8 has comonns
o hs ac w hav h nq solon o h rolm -5 9 Whr......... Vcor ncon sass ollowng smaon Usng h wll-known rors o rn s ncons w hav go smaon or h vcor vloc M n h nconal sac.ollowng h classcal rocr w g h nqnss an sal o solon or h rolm -5. lso w oan qaon or h rssr ncon v v whr
5 h rolm - whn v s la Ins s Mllnnm rolm whch was ormla rman []. Whn ro s n solon whch has non-connos avs s ncon hror n gnral cas can rmn onl wak solon whch sass h oan alanc qaon. o hs oan alanc qaon or h rssr sron wr n sgncan rors o h ransn ncomrssl low whch rov a scron o h consv rlaonshs wn hr hscal qans: h vloc vcor h rnal an nrnal orcs h scalar rssr sron. h Navr-Socks rolm - n h gnral whn v s alcal o ral rln rocsss whch rrsn an avrag arr rom h rn ons o h sac an w hav rn comnaons o h conons v v v omnng hs conons or h vcor vloc w can lan ransr mchansms o vrgn-convrgn lows. V. E NVIE-SOES POBEM O E IVEEN POENI IE nvsga h havor o h Navr-Socks rolm - n h gnral cas whn v. v n wh h nal conon on or rolm - n h gnral cas w hav go scon orr nonlnar Volrra-rholm ngral qaon n a mar orm sasng 7 Whr Sccssv aromaon mho can sccssll al o solv h nonlnar Volrra-rholm mar ngral qaon. onsrng n h oraor orm w hav qaon r Usng nvrs oraors or h oraors w hav go or
VI. ESUS N ISUSSION rom rvos ss [5]-[6] n cass v an r v or nsal moons w hav h nonlnar v olrra -rholm mar qaon whch gv som sml gra consqncs o h horms gvn n [6]. hs rsl ar shown analogcal conons o snc an nqnss or h hr mnsonal Navr-Soks qaons wh olr connos cocns n h arsan coorna ssms whch n on h nrg consrvaon law an rov havor o moon or comrssl lows. hr Navr-Soks qaons n h gnral cas rrsn h volon o h govrnng sron ncons whch n on h vloc vcor n h oson o arcls as a rsl o hrmal caon a an n rln nrg. Vcor ncon sass h ollowng VII. ONUSION smaon Usng h rors o rn s ncons w hav go smaon or h vcor vloc n h sac Whr c c c c c Usng h Navr-Soks qaon w hav a nq scalar ncon o rssr whch sass v v whr hs ar s rsn convnn rocr n solvng h Navr-Soks rolm or comrssl lows whch allows o rov h nqnss an rglar o h solons n h gnral cas. In h consr cas whn wak solon o h Navr-Soks rolm - sass h nrg convrsaon law w hav scnl rglar solon whch mans an sass classcal ormlaon n hs consr cas. In vw o hs ac h oan alanc rlaon wn comonns o h vloc vcor an h rssr ncon n rms on h nrg convrsaon law rovs qvalnc o h srong an wak solon. In cas whn ro h Navr-Soks rolm - or comrssl lows sass wak ormlaon. onvrsl nsa havor rrsns a arr rom h avrag nrg o h l known as nrg. nvolvng namnal rors o h rln lows whch monsra chnolog an rncal moranc a h orron o classcal aroach whr h rsson v o h rln l nrg s no rglar an os no allow o s classcal ormlaon. In hs wa som cls ars n solvng o h Navr-Soks rolm whch was nconr n sng rln havor or nsal moon sng wak ormlaon. h smlar o h alanc qaons an or h rnal orc an h rssr sron wn h rln vrgn an non vrgn lows monsra sgncan an rncal moranc o h nrg consrvaon law or h comrssl rln lows. 6
VIII. NOWEMEN h ahors grall arca an acknowlg h Plshng Eor or rang arlr ra o hs ar orng commns an ncoragmn. EEENES []. aarllohn an Nrnrg Paral rglar o sal wak solon. Pr an l. Mah.5 98. 77-8. [] V.Shr n n vsc law wh comac sor n sac m.j.om.nalss 99.-. []..rman. Esnc an smoohnss o h N-S qaon. www.clamah.org.-5 [].rman Paral rnal qaons o araolc. lora 987.. [5] К.А. alva Enrg convrsaon law n h r amoshr. Inrnaonal Jornal o Engnrng an Innovav chnolog IJEI Vol. Is. Ma.5-6 lora US I:.895 h://www.j.com/archvscron.h?= [6]. alva. alv Esnc an Unqnss o h Navr-Soks Prolm n Inn Sac. IEN Jornal. ISBN: 978-988-95-7/ Engnrng an omr scnc/ - Jl onon U.. h://www.ang.org/lcaon/we/. UO BIOPY r..alva s workng as a os-ocoral sarchr a h Unvrsé o orran aoraor Enrgéq chnq nrq EM Nanc ranc. Sh has Ph gr n Phscal an Mahmacal scncs: rnal qaons an mahmacal hscs. r..alva has lsh mor han 6 rsarch arcls n h Naonal/Inrnaonal Jornals. r aras o rsarch nrss:. Inal-onar rolms or mlcomonn ha ransr an mass son. Mlhas San rolms or Paraolc Paral rnal Eqaons. Mchancs an namcs o arcls n rln lows..issakov s workng as assoca rossor a aakh Naonal chncal Unvrs nam ar.saav. has ocor gr n chncal scnc. r...issakov has lsh mor han rsarch arcls n h Naonal Jornals. s aras o rsarch nrss:. omr molng o son. Mchancal Eqlrm o as Mr. h hor o mchansms an qmn als 7