Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 878 ISSN 9-558 Anlil Solions of D Inomrssil Nir- Soks Eqions for Tim Dnn Prssr Grin MAKAz LS Anllh Asr- In his r w rsn nlil solions of wo imnsionl inomrssil Nir-Soks qions D NSEs for im nn onnill rg rssr grin rm g Orlowski n Sozk rnsformion OST n Col-Hof rnsformion CHT To fin h solion of D iffsion qion w l srion of rils mho K-wors- Brgrs qion Col-Hof rnsformion CHT Diffsion qion Orlowski n Sozk rnsformion OST Prssr grin Rnols nmr D NSE INTRODUCTION Th Nir Soks qions r imorn gorning qions in h fli nmis whih sri h moion of fli Ths qions ris from ling Nwon s son lw o fli moion oghr wih h ssmion h h fli fls fors o rssr isosi n rhs n rnl for Th r sfl s h sri h hsis of mn hings of mi n onomi inrs Howr NSEs r nonlinr in nr n i is iffil o sol hs qions nlill In orr o rform his sk som simlifiions r li sh s linrizion or ssmions of wk nonlinri smll flions isrizion Dsi h onnr rsrh on Nir Soks qions hir nirsl solions r no hi Th fll solions of h hrimnsionl NSEs rmin on of h on rolms in mhmil hsis Comionl Fli Dnmis CFD rohs isriiz h qions n sol hm nmrill Alhogh sh nmril mhos r sssfl h r sill nsi n hr ms roimion rrors ssoi wih hm Th lomn of high s Mhmm Al Klm AzPh D sn Profssor Dr Lk Szz Anllh kz34@gmilom nllhls@gmilom Drmn of Mhmis Jhngirngr UnirsiSrDhk Bnglsh omrs nll mks isrizion mhos mor n n i nls h nmril rmn of rln flow 4 h://wwwijsrorg Th solions for h NSEs n oin r of rilr ss E solions on h ohr hn r r imorn for mn rsons Th roi rfrn solions o rif h roim mhos In orr o nrsn h non-linr hnomnon of NSEs on ns o s D NSEs Sin hs qions inl h iffsion ion n rssr grin rms of h fll 3D NSEs hs inoror ll h min mhmil frs of h NSEs Aling OST w h r D NSEs o D isos Brgrs qions n w h sol Brgrs qions nlill g CHT So nmr of nlil n nmril sis on D NSEs n D isos Brgrs qions h n on o sol h gorning qions nlill [][][3][4][5][6][7][8][9] Prof Snj Mil [] si on nlil solions of simlifi form of NSEs Mohmm Mhi Rshii [] rsn nw nlil solions of h 3D NSEs Mri Crml Lomro [3] si on nlil solions of h im nn inomrssil NSEs on n hlf ln Hikm G Hsno [4] fon mho of D nonlinr Ll rnsformion for soling h NSEs Hi H [5] si on h D inomrssil isos flow in whih h lol orii is roorionl o h srm fnion rr niform srm Mhmm R Mohin l[6] rsn nlil solions of D NSEs gorning h ns inomrssil flow Th solions h n oin g honogrh-lgnr rnsformion mho Vin Kmr Sris Ashosh Mohmm Tmsir [7] ] ri gnrl nlil solions of 3D homognos ol ns non-linr gnrliz isos Brgrs qions i CHT CA J Flhr [8] h gnr solions of h D Brgrs qions TX Yn LS Y [9] rsn
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 879 ISSN 9-558 4 h://wwwijsrorg ril srl solions for h imnsionl Brgrs qion In his r w r D NSEs ino D ol Brgrs qions ling OST Thn fr ling CHT D Brgrs qions will r o D iffsion qion B g srion of rils mho w will sol iffsion qion Thn ling CHT n inrs OST w g h nlil solions of D NSEs On ml hs n rri o n hir grhil rrsnion is si MATHEMATICAL FORMULATION Th imnsionlis gorning qions of h fli flow r gin rsil h onini qion -momnm qion ν ρ -momnm qion ν ρ hr n r h loi omonns in h n irions rsil is h rssr ρ is h onsn nsi n υ is h kinmi isosi Ug h imnsionlss finiions in [ ] U U U h h h U ρ Th imnsionlis gorning qions r hn onr ino h non-imnsionl form D NSEs s R R 3 whr R Uh ν - ; g f R is h Rnols nmr As rsl n rwrin s f R 4 n wrin s g R 5 Thn w l OST [] s 6 ih f 7 An g 8 Hr f Agin Now g
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 88 ISSN 9-558 4 h://wwwijsrorg Agin Ths w g 9 3 3 Now Agin Also
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 88 ISSN 9-558 4 h://wwwijsrorg Also Ssiing h rnsform riis in w g R R R 4 Similrl ssiing h rnsform riis in 3 w g R 5 From 4 w oin 6 Agin w know h h non-imnsionl form of D Brgrs qions r R 7 R 8 So 4 5 r h rnsform D Nir-Soks qions fr ling Orlowski n Sozk rnsformion n hs r nlogos o non-imnsionl form of D Brgrs qion 7 8 Now w n o sol 7 wih iniil oniions < < < < whr 4 n 5 n linriz h CHT [7][8][] R 9 An R rform h rnsformion in som ss
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 88 ISSN 9-558 4 h://wwwijsrorg Firs l s ssm ih his rnsformion 7 oms R R R Now ingring h rnsform qion wr hn w h R Thn w mk h rnsformion ln R R R R R R R R Ssiing hs riis in w oin R R R R 4 R 4 R R whih is h wll-known son orr PDE ll h or iffsion qion [] From qion 9 w g ln R R ln C ln R ln R ln C C R C R 3 I is lr from 9 h mliling onsn os no ff so w n wri 3 s z z R 4 For qion 4 gis z z R l z z R Ths w n s h D Brgrs qion n r o D iffsion qion g CHT Agin for 8 w rform h rnsformion in som ss L s ssm ih his rnsformion 8 oms R R R
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 883 ISSN 9-558 Now ingring h rnsform qion wr R z z 8 hn w h 5 R For qion 8 gis R z z Thn w mk h rnsformion ln R R R R z z l whih is h wll-known son orr PDE ll h or iffsion qion n i is l sm s Consir gnrl solion of of h form similr nlog o [7][3] X Y T 9 3 4 R X Y solion R R XY T X YT XY T R R T X Y T X Y 6 Ssiing hs riis in 5 w oin R whih is h wll-known son orr PDE ll h or iffsion qion n i is l sm s From qion 9 w g R ln R ln R ln ln C whih is h sm of h solion 3 4 n h srl T whr 34 r rirr onsns n XYT r fnions of rsil Thn oms Diiing XYT on oh sis w g Sin h lf si is fnion of lon whil h righ si is fnion of n w s h h si ms onsn s Ths λ whih is n for onnss T λ T 3 X Y X Y λ 3 n wrin s X Y X Y λ 3 R ln C R C 7 An h lf si ns onl on whil h righ si ns onl on h si ms onsn s - µ Ths w oin X µ X 3 I is lr from 9 h mliling onsn os no ff so w n wri 7 s 4 h://wwwijsrorg Y λ µ Y 33
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 884 ISSN 9-558 hr λ µ α Solions o 3 33 n 3 r gin X os µ µ Y osα α T λ 3 I follows h solion o is gin λ os µ µ osα α 3 35 Ths h gnrl solion of D iffsion qion is gin 3 4 λ os µ µ osα α 3 36 B g CHT w oin h gnrl solion of D isos Brgrs qions s λ 4 µ µ µ os µ osα α 3 λ Bonr oniions: R 3 4 os µ µ osα α 3 37 R π λ 3 4 os µ µ α osα 3α π λ ; R 3 4 os µ µ osα α 3 R 38 R π π ; R 39 ; ; Now ling inrs OST w g h nlil solions of D NSEs s 4 whr will rmin from qions 9-3 4π R 34 Iniil oniions: π R R os R R R π π π π π os π π π 4π os π π π π Ω π π os π π π Ω R R ; ; π R R π ; 4 DISCUSSION 4 Prolm π R R π ; In his rolm h omionl omin is Ω { : } Th solions for n r l from D Brgrs qions n hn g IOST solions of NSEs r om n lo in fig n fig Th solions of Brgrs qions 4 n 5 n gnr g h CHT [7] [8] [3] whih r: 4 h://wwwijsrorg
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 885 ISSN 9-558 [] Prof Snj Mil Anlil solions of simlifi form of Nir-Soks qion 8 h Ino-Grmn inr Am Mhim Misr Inin insi of hnolog 73 Knr Ini [] Mohmm Mhi Rshi Nw nlil solion 7 of h Thr-Dimnsionl Nir-Soks Eqion 7 Morn Phsis lrs B worlom9 Vloi 7 69 5 Sil o-orin 4 6 8 Sil o-orin [3] Mri Crml Lomro Anli solions of h Nir-Soks Eqions Rnioni l Cirol Mhmis i Plrmo M Vol 5 Iss 99-3 [4] Hikm G Hsno Mho of wo-imnsionl nonlinr Ll Trnsformion for soling h Nir- Soks Eqion DOI: 576/ 5-343_347 Fig Th l of gris for R 5 n im ll wih 5 im ss [5] Hi H E solions of h Uns Two imnsionl Nir-Soks Eqions Arizon Unirsi TsonAZssHrr/s/9877n 38 689H Vloi 65 6 65 6 65 6 5 Sil o-orin Fig Th l of gris for R 5 n im ll wih 5 im ss [6] Mhmm R Mohin l E solions of h im nn Nir-Soks Eqions Hoogrh-Lgnr rnsformion mhos Tmsi Ofor Jornl of Mhmil Sins 43 7 4 6 8 Sil o-orin [7] Vin Kmr Sris Ashosh Mohmm Tmsir Gnring solion of hr imnsionl ol ns nonlinr gnrliz isos Brgrs qion Inrnionl Jornl of Mhmil Sins 35-3 [8] C A J Flhr Gnring solion of h woimnsionl Brgrs qion In J Nmr Mh Flis 3983 3-6 [9] T X Yn LS Y Vril srl solions for h -imnsionl Brgrs qion Chin Phs L 33 335-337 6 CONCLUSION In his s w h shown how o sol nlil solions of D NSEs g OST n CHT wih h hl of srion of rils mho B g sm mho on n sil fin o h nlil solions 3D n D NSE Th mho is siml n n s o fin gnrl solion of h gorning qions whn w s iniil n onr oniions REFERENCES [] Nrslwi Rsli l Nmril omion of wo imnsionl Nir- Soks qions g n imro fini iffrn mho Mhmik Vol 7Nmr -9 [] A Orlowski n K Sozk R Mh Phs 7989 59 [] E Hof Commn Pr Al Mh 3-95 JD Col Q Al Mh 9 5-3 95 [3] MC Kw A Mnong A Koross V Ssmgn Nmril solions of h Brgrs ssm in wo imnsions nr ri iniil n onr 4 h://wwwijsrorg
Inrnionl Jornl of Sinifi & Enginring Rsrh Volm 5 Iss 6 Jn-4 886 ISSN 9-558 oniions Ali Mhmil Sins Vol 6 no 3 563-565 [4] Mohmm Tmsir Vin Kmr Sris A smi-imlii fini-iffrn roh for woimnsionl ol Brgrs qions Inrnionl Jornl of Sinifi & Enginring Rsrh Vol Iss 6 Jn- [5] Vin Kmr Sris Mohmm Tmsir Ukrsh Bhrwj YVSS Snsirj Crnk-Niolson shm for nmril solions of wo imnsionl ol Brgrs qions Inrnionl Jornl of Sinifi & Enginring Rsrh Vol Iss 5 M- [6] Li Zhng Lish ng Xiohn Ding E finiiffrn shm n nonsnr fini-iffrn shm for ol Brgrs qions Ans in Diffrn Eqions Sringr on Jornl 4// Aril Hisor: Ri M 4 A : 7 M 4 Ri in ris form: 9 Jn 4 4 h://wwwijsrorg