Decay of Dusty Fluid MHD Turbulence for Four-point Correlation in a Rotating System

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1 Avaabe Onne ubcaon J Sc Re, - JOURNAL OF SCIENTIFIC RESEARCH wwwbangaonfo/ndexh/jsr Decay of Duy Fud HD Tubuence fo Fou-on Coeaon n a Roang Sye A Ba K *, A K Azad, and S A Sae Deaen of Aed aheac, Unvey of Rahah-, Bangadeh Receved ach, acceed n fna eved fo Seebe Abac In h ae we have condeed he duy fud HD ubuence n a oang ye a fou-on coeaon Hee hee and fou-on coeaon beween fucuang uane have been condeed and he unue coeaon ae negeced n coaon o he hd and fouh ode coeaon Fo he convenon of cacuaon, he coeaon euaon ae conveed o he eca fo by ang he Foue anfo Fnay, negang he enegy ecu ove a wave nube, he enegy decay of duy fud HD ubuence n a oang ye fo fou-on coeaon ye obaned Keywod: HD ubuence; Duy fud; Coeaon; Roang ye; Decay aw JSR ubcaon ISSN: - n; - Onne A gh eeved do: h://dxdoog//v J Sc Re, - Inoducon In ecen yea, he oon of duy vcou fud n a oang ye ha deveoed ady The oon of duy fud occu n he oveen of du aden a, n obe of fudzaon, n he ue of du n a ga coong ye and n he edenaon obe of da ve The behavo of du ace n a ubuen fow deend on he concenaon of he ace and he ze of he ace wh eec o he cae of ubuen fud In geohyca fow, he ye uuay oaon wh a conan veocy; uch age cae fow ae geneay ubuen The Coo Effec ongy affec he age-cae Oceanc, Aohec ccuaon, ea, ween bounday cuen, geooc baance, Roby wave, Kevn wave, exena bac fo cacuang he aecoe of vey ong-ange aey he The Coo effec ha gea gnfcance n Aohyc and Sea dynac I ao gnfcan n he Eah cence, eecay eeooogy, hyca geoogy and Oceanogahy Khoe and oefd ] obaned and exeon fo he effec of Coo foce on acceeaon covaance n HD ubuen fow of a duy ncoebe fud Shoua and * Coeondng auho: abubaa_u@yahooco

2 Decay of Duy Fud HD Yohzawa ] deved exeon fo aca anay of an ooc ubuen vcoy n a oang ye Shoua ] ao uded a aca deved woeuaon ode of ubuen caa fux n a oang ye Chandaeha ] obaned he nvaan heoy of ooc ubuence n agneo-hydodynac Khoe and Sngh ] condeed he effec of Coo foce on acceeaon covaance n ubuen fow wh oaona yey Sae and Khoe ] uded he decay of HD ubuence befoe he fna eod Sae and Ia ] obaned he decay of duy fud HD ubuence befoe he fna eod n a oang ye Sae and Ahed ] oned ou ha he fbe oon n duy fud ubuen fow wh wo on coeaon Dx and Uadhyay ] obaned he effec of Coo foce on acceeaon covaance n HD ubuen duy fow wh oaona yey Azad e a ] uded he aca heoy of cean dbuon funcon n HD ubuen fow fo veocy and concenaon undegong a f ode eacon n a oang ye Ia and Sae ] uded he f ode eacan n HD ubuence befoe he fna eod of decay fo he cae of u-on and u-e Azad and Sae ] obaned he decay of duy fud HD ubuence befoe he fna eod n a oang ye fo he cae of uon and u-e onda ] uded he decay of duy fud ubuence befoe he fna eod n a oang ye Azad e a ] uded he f ode eacan n HD ubuence befoe he fna eod of decay n a oang ye Rahan ] obaned he decay of f ode eacan n ncoebe HD ubuence befoe he fna eod fo he cae of u-on and u- e n a oang ye Dee, ] deveoed a heoy on he decay of hoogeneou ubuence befoe he fna eod By anayzng he above heoe we have uded he decay of duy HD fud ubuence n a oang ye fo fou-on coeaon We have condeed wo, hee and fou-on coeaon euaon afe negecng he unue coeaon n coaon o he hd and fouh ode coeaon e Fnay we have obaned he decay of duy fud HD ubuence fo fou-on coeaon n a oang ye Bac Euaon The euaon of oon and connuy fo vcou, ncoebe HD duy fud ubuen fow ae gven by ] a u uu h h h hu uh, h v v v wh u v h w u - u f u v, u, v

3 A Ba K e a J Sc Re, - aheaca Fouaon To fnd he fou on coeaon euaon, foowng Dee ] we ae he oenu euaon of duy fud HD ubuence n a oang ye a he on and he nducon euaon of agnec fed fucuaon a, and a u u u h h h h u h u h - u x h h u u h w u - u f u v - u h x h h u u h - u h whee h w he oa HD eue, xˆ, he hydodynac eue, ρ ρ he fud deny, he agnec and nube, he angua veocy λ coonen, he aenang eno, R ρ he a of a nge heca du ace of adu R and ρ conan deny of he aea n du ace KN f, he denon of feuency, K he Soc dug eance, N he ρ conan nube deny of du ace, he neac vcoy, λ he agnec dffuvy, x, h he agnec fed fucuaon, x, v du veocy coonen, he e, u he ubuen veocy, x he ace co-odnae and eeaed ubc ae ued fo o uyng E by h h h by u h h by u h h by u h h and addng he fou euaon, we han ang he angua bace o, we ge uh h h uu h h h - h hhh h uu h h h u - u h h h u u h h h uh hh u uu h h h u uhh h uu hh h x whh h u hh h

4 Decay of Duy Fud HD uh h h uh h h uh h h ] - x u h h h u u h h u u h h ] f u hh h v h h h Ung he anfoaon, u u h h ],, and he nne-denona Foue anfo u h ˆ h ˆ h ˆ u u h u u h u u h u u h ˆ h ˆ h ˆ ˆ h ˆ h ˆ φ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ, φ ˆ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ φ, φ ˆ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ φ, ˆ h ˆ h ˆ ˆ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ φ φ, u u h u u h wh v h wh he fac ˆ h ˆ h ˆ ˆ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ φ dd d, φ ˆ h ˆ h ˆ ˆ h ˆ h ˆ ˆ h ˆ h ˆ ˆ h ˆ h ˆ φ φ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ, φ φ ˆ ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ, ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ δ, η ˆ ˆ ˆ ex ˆˆ ˆ ˆ ˆ ˆ dd ˆ ˆ dˆ, u u h h h u u h h h ; u u hh h u u h h h ; u u h h h u u h h h h ; u u h h h u u h h h h ; and by ang conacon of he ndce and, and, we obaned fou-on coeaon euaon a

5 A Ba K e a J Sc Re, - φ φ K K K KK f φ K K K φ φ - K K K K K K K K K φ φ If we ae he devave wh eec o KK KK ] f η - K K K φ φ δ x of E and uyng by h h h, ung e aveage and wng he euaon n e of he ndeenden vaabe have - δ K K K K K K K K K K K K K K K K K K φ φ K K K K K K K K K K K K - ˆ, ˆ, ˆ, we E and ae he eca euaon coeondng o he fou-on coeaon euaon Thee-on coeaon The eca euaon coeondng o he hee-on coeaon euaon by conacon of he ndce and ae φ K K KK ] K φ K K φ φ K K K φ φ - K K φ φ K K K K K K K φ φ - K K K K Hee he eca eno ae defned by ˆ h ˆ u h φ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ, u u ˆ h ˆ h ˆ φ ˆ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ φ, u u ˆ h ˆ h ˆ φ ˆ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ φ, u h ˆ h ˆ φ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ, u h ˆ h ˆ h ˆ φ ˆ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ,

6 Decay of Duy Fud HD wh ˆ h ˆ ˆ ˆ ex ˆˆ ˆ ˆ dd ˆ ˆ A eaon beween φ φ and φ can be obaned by eng ˆ n E and coang he eu wh E, we ge φ φ ˆ ˆ ˆ φ ˆ ˆ ˆ dˆ Two-on coeaon The eca euaon coeondng o he wo on coeaon euaon ang conacon of he ndce ϕ ϕ ˆ ϕϕ ˆ α ϕ ϕ ˆ α ϕ ϕ ˆ ] whee, ϕ ϕ and α φφ ae defned by h h ˆ ˆ ex ˆˆ] dˆ ϕ ϕ u h h ˆ α ϕ ϕ ˆ ex ˆˆ] dˆ The eaon beween α ϕ ϕ ˆ and ϕ obaned by eng ˆ n E and coang he eu wh E, hen α ϕ ϕ ˆ φ ˆ ˆ dˆ Souon Negecng unue Coeaon Ung f η R φ, hen negang beween and, we ge, -R, and negecng a he e on he gh de of E φ φ ex f whee φ he vaue of φ a ha aonay vaue fo a vaue of, and Subung of E,, n E and negang wh eec o,, and fahe negang wh eec o e, and n ode o fy cacuaon, we w aue ha a ] ;Tha we aue ha a funcon

7 A Ba K e a J Sc Re, - uffceny genea o eeen he na condon can obaned by condeng ony he e nvovng b] and c] and he negaon efoed, hen ubung he obaned euaon n E and eng H ϕ ϕ,we oban H H whee, φ ˆ, ˆ φ ˆ, ˆ ] ex f ex ] d b ˆ ˆ b ˆ ˆ ] o ex f - ex ex ] ] ex x dx d c ˆ ˆ c ˆ ˆ ] ex - f ex ] ex ] ex x dx d whee H he agnec enegy ecu funcon, whch eeen conbuon fo vaou wave nube o he enegy and he enegy anfe funcon, whch eonbe fo he anfe of enegy beween wave nube In ode o ae fuhe cacuaon, an auon u be ade fo he fo of he baceed uane wh he ubc and n E whch deend on he na condon φ ˆ, ˆ ˆ, ˆ ξ φ ] wheeξ a conan deendng on he na condon fo he ohe baceed uane n E, we ge

8 Decay of Duy Fud HD ] ˆ ˆ ˆ ˆ b b ] ˆ ˆ ˆ ˆ c c ξ Reebeng, d d d co ˆ ˆ θ and θ co,θ he ange beween ˆ and ˆ and cayng ou he negaon wh eec oθ, we ge - ξ ] ex ] ex ξ ex f ] ex - ] ex ] ex - ] ex ex ] - ex ] ex dx x ex ] - / ex ] ex dx x ] d Hee, Inegang E wh eec o, we have ex f whee, ex ξ and Hee,

9 A Ba K e a J Sc Re, - ξ ex ] ξ ex ] ξ ex ex ] dy y Hee, / / / ξ ex ] The nega exeon n E, he uany eeen he anfe funcon ang owng o condeaon of agnec fed a hee on coeaon euaon; ae fo condeaon of he fou-on euaon Inegaon of E ove a wave nube how ha d Indcang ha he exeon fo afe he condon of connuy and

10 Decay of Duy Fud HD hoogeney, hycay, wa o be execed, nce a eaue of anfe of enegy and he nube u be zeo Fo, J d H ex ex ex, whee, N J a conan of negaon and can be obaned a by Con ] Theefoe we oban, H ] ex ] ex N ] ex f ] ex d Fo E, we ge H H H ex f In E H and H agnec enegy ecu ang fo condeaon of he hee and fou on coeaon euaon eecvey E can be negaed ove a wave nube o gve he oa agnec ubuen enegy h h N ξ S R ξ ξ ] ex f whch eeen he euaon of he decay of duy HD ubuence n a oang ye Hee,, S R and, vaue _ /

11 A Ba K e a J Sc Re, - / / / ] / / / ] ] / /

12 Decay of Duy Fud HD E can be wen a ] h A B C D ex f whee f nϕ he Coo aaee, he du ace aaee, h denoe he oa enegy ha, ean uae of he agnec fed fucuaon n fou-on coeaon, he e, and A, B, C, D, and ae abay conan deened by he na condon The f e of gh hand de of E coeond o he enegy of agnec fed fucuaon of wo-on coeaon; he econd e eeen agnec enegy fo he hee-on coeaon; he hd and fouh e eeen agnec enegy fo fou-on coeaon If Coo foce and du ace ae aben hen E of he fo h A B C D whch he enegy decay of HD ubuence fo fou-on coeaon n non oang ye If ξ hen he E becoe h A B Th wa obaned eae by Sae and Khoe ] Fo age e, he econd e n he euaon becoe neggbe, eavng he -/ owe decay aw fo he fna eod Reu and Dcuon Th udy how ha he e aocaed wh he hghe-ode coeaon de ou fae han hoe aocaed wh he owe ode one Hee hee and fou-on coeaon beween fucuang uane have been condeed and he unue coeaon ae negeced n coaon o he hd and fouh ode coeaon If he uadue and unue coeaon wee no negeced, E conan oe e n negave hghe owe of and woud be added o E In h cae, he enegy decay of HD ubuence geae han he fou-on coeaon n a oang o non-oang ye Hee E and ae eeened by y,y,y,y,y and y,y,y,y,y eecvey coeondng o he vaue,,,, Fo dffeen vaue of f and, he enegy decay cuve obaned fo E and ae oed n Fg o, eecvey

13 A Ba K e a J Sc Re, - <h > y y y y y <h > y y y y y e e Fg Enegy decay cuve fo f, Fg Enegy decay cuve fo f, <h > y y y y y <h > y y y y y e e Fg Enegy decay cuve fo f, Fg Enegy decay cuve fo f, <h > y y y y y <h > y y y y y e e Fg Enegy decay cuve fo f, Fg Enegy decay cuve fo f,

14 Decay of Duy Fud HD Concuon If f and ae eua hen he ouon of educe o E ha he decay of fouon coeaon of HD ubuence We ee ha n a oang ye he enegy decay owe han he non oang ye fo fou-on coeaon n HD ubuence due o f> Fo Fg o, we obeve ha n a oang ye f f< he enegy de ou vey fae han he non oang ye Theefoe, he decay of enegy of duy fud HD ubuence n a oang ye fo fou-on coeaon geae han he decay of HD ubuence fo non oang ye f f- negave gn and ae f f- ove gn n he E Acnowedgeen Th wo uoed by he eeach gan unde he Naona Scence & Infoaon and Councaon Technoogy NSICT, Bangadeh and he auho Ba, Azad acnowedge he uo The auho ao exe he gaude o he Deaen of Aed aheac, Unvey of Rahah fo ovdng u a face egadng h ace Refeence N Khoe and Y T oefd, A Sace Sc, h://dxdoog//bf Y Shoua and A Yohzawa, hy Soc Jaan, h://dxdoog//jsj Y Shoua, J hy Soc Jaan, h://dxdoog//jsj S Chandaeha, oc RoySoc London A, h://dxdoog//a N Khoe and SR Sngh, A Sace Sc, h://dxdoog//bf S A Sae and N Khoe, In J Eng Sc, h://dxdoog//-- S A Sae and A Ia, J ah ah Sc, S A Sae and S F Ahed, J Sc Re, do:/v T Dx and B N Uadhyay, ASace Sc,, A K Azad, A Azz, and S A Sae, Bangadeh J Sc Ind Re, A Ia and S A Sae, Indan J ue a ah, AK Azad and SA Sae, Bha ah Soc, N I onda, J Eng A Sc, A Azz, A K Azad, and S A Sae, Re J ah Sa, L Rahan, J ech Con ah Sc, R Dee, hy Fud, h://dxdoog// R Dee, hy Fud, h://dxdoog// S Con, J A hy, h://dxdoog//