Nonstationary Navier-Stokes Problem for Incompressible Fluid with Viscosity

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1 Amercan Journal of Mahemacs and Sascs (6): DOI:.59/.ams.6.8 Nonsaonary Naver-Soes Problem for Incompressble Flud wh Vscosy Taalabe D. Omurov Docor of Physcs and Mahemacs professor of Z. Balasagyn Kyrgyz Naonal Unversy Bshe Kyrgyzsan Absrac Exsence and condonal-smooh soluon of he Naver-Soes equaon s one of he mos mporan problems n mahemacs of he cenury whch descrbes he moon of vscous Newonan flud and whch s a basc of hydrodynamc[]. Therefore n hs wor we solve a nonsaonary problem Naver-So es for ncompressble flu d. Keywords Naver-Soes Problem Condonal-smooh Soluon Flud Flow Vscosy Convecve he Acceleraon Dfferenaon Algorh m Newon s Poenal. Inroducon If o desgnae componens of vecors of speed and exernal force as v ) = [ υ ) υ ) υ )] f ) = [ f ) f ) f )] ha for each value = urns ou he correspondng scalar equaon of Naver-Soes υ υ P υ = f µ υ (.) = x x wh condons υ dv ( ) T [ T ]) ν = = (.) υ (x x x ) x x ) (.) = = µ > - nemac vscosy - densy - Laplas s operaor. The addonal equaon s he condon ncompressbly flud (). Unnown are speed ν and pressure P. The wor purpose. The man obec of hs wor - exsence and proofs of sngle and condonal smoohness of he decson of a problem Naver-Soes for an ncompressble flu d wh vscosy. Theorecal and praccal val ue. Our problem does no nclude a dervaon of an equaon n a physcal meanng snce here s a bg amoun of wors reflecng hese quesons[-4 8-]. The eceved decsons on he bass of he developed analycal mehods proves n he general * Correspondng auhor: omurovd@mal.ru (Taalabe D.Omurov) Publshed onlne a hp://ournal.sapub.org/ams Copyrgh Scenfc & Academc Publshng. All ghs eserved applcably of he equaons of Naver-Soes. In a case < µ < he curren s consdered wh very small vscosy. When he curren s consdered wh very small vscosy.e. when eynolds's number s very grea ( e ) [89] here s an border layer n whch vscosy nfluence s concenraed. In many wors n hs area of he decson of he equaons Naver-Soes receved by he numercal analyss also confrm hese conclusons. And n a case < µ = µ = cons < he curren s consdered wh average sze of vscosy. A very slow currens or n currens of s srong-vscous lquds of force of a frcon much more han forces of nera. Hence convecve he acceleraon dong he equaons nonlnear everywhere are supposed dencally equally o a zero[9]. Therefore n a case when convecve acceleraon s no equal o zero problems conneced wh mehods of negraon of he equaons of Naver-Soes n her general vew are arsen. The decson of many problems of heorecal and mahemacal physcs leads o use of varous specal wegh spaces. In wors[5-7] for he frs me have offered a mehod whch gves soluon of problem Naver-Soes n ν υ = { υ L (T )( = ) υ x x ) G (T ): G (T ) x x ) T : C (T ); s connuous and lmed funcons on and x x ) C (T ) C (T )} v = υ G (T ) D (T ) ( υ ; ) = υ D = υ υ = (T ) C (T ) L ( υ ; ) (.4)

2 5 Taalabe D. Omurov: Nonsaonary Naver-Soes Problem for Incompressble Flud wh Vscosy T υ = (sup () L (x x x ) d) υ T (): () d = q. To answer he brough aenon o he queson we offer he followng mehod of he decson of a problem Naver-Soes. For ha he phrpose sysem (.) we wll ransform o a nd υ θ = f Px Q x µ υ ( = )(.5) θ = ( υυ x Q x )( = ) (.6) = θ = = θ (x x x ) (x x x ) (.7) Q υ Qx = υυ x ( = ) = = (.8) Qx = [ ] = υ x υυx = = whou breang equvalence of sysem (.) and (.5) (.6). The receved sysems (.5) (.6) conan unnown persons υ θ ( = ) and pressure P. Here θ -nown υ υ. funcons because are nown x The developed mehod of he decson of sysems (.5) and (.6) s conneced wh funcons θ ( = ).e. A) ro θ = θ = ( θ θ θ ); roν or θ ( = ) - any funcons f accordngly as B) necessary condons ae place: а ) ro = = funcons. θ θ ( θ θ θ ) b ) θ - any. A Problem of Naver-Soes wh a Condon (А) In hs paragraph n he subsequen pons a he specfed resrcons on he enrance daa he src subsanaon of compably of sysems (.5) (.6) wll be gven... esearch Wh a Condon (A) Le funcons Then relavely θ ( = ) sasfy o a condon (a ). θ ( = ) we suppose a condon (A) and dvf < µ < (.) where from sysem (.5) and (.6) accordngly we wll receve followng sysems υ θ x Q x = f P x µ υ ( = ) (.) θ = θx (.) θx = ( υυ x Q x )( = ). = Theorem. Le condons (.) (.) (A) and (.) are sasfed. Then sysems (.) and (.) s equvalen wll be ransformed o a nd J= FJ P Q θf f x = υ = f µ υ J x θ = ψ ψ ψ x (x x x ) = (.4) P= Q θ dsdsds F (s s s ) 4 r r = (x s ) (x s ) (x s ). Hence he problem (.) - (.) has he unque decson whch sasfes o a condon (.). Proof. Fro m sysem (.) s vsble f he -equaon (. =) s dfferenaed on x -equaon on x (. =) -equaon on x (. =) and s summarsed we wll receve he equaon of Puasson[] J = F (.5) as ( υx υ x υ x ) ( Q θ P) = = f x µ ( υ x υ x υ x ) = = x dvν = dvf = F. A ha s proved ds ds ds J F (s s s ; ) = 4 r (.6) J = τ F (x τ x τ x τ ; ) x 4 (.7) dτdτdτ ( s x = τ = ). ( τ τ τ ) Algorh m when we wll receve he equaon of Puasson (.5) for brevy we name «algorhm puassonzaon

3 Amercan Journal of Mahemacs and Sascs (6): sysems». In wor of Sobolev[] s specfed ha funcon (.6) sasfes o he equaon (.5) and s called Newons poenal. Therefore f J - he decson of he equaon (.5) hen subsung n (.) we have Jx х Px Q x θ (.8) υ = f µ υ J ( = ; J J ) (.9) x.e. sysem (.) s equvalen by (.9) wll be ransformed o a nd lnear he nonunform equaon of hea conducvy. The equaons (.5) (.9) s here are frs and second equaons of sysem (.4). The sysem (.9) s solved by S.L.Sobolev s mehod: r υ = exp( ) υ (s s s ) 8( µ ) 4µ r dsdsds exp ( ) 8 4 µ ( s) [ f (s s s s ) J (s s s s )] ( µ ( s )) dsdsdsds exp( ( )) τ τ τ υ (x τ µ x τ µ x τ µ ) τ τ τ exp τ τ τ d d d [ ( ( )) f τ µ ( s)x τ µ ( s) x τ µ ( s)s) ; J (x τ µ ( s) x τ µ ( s)x τ µ ( s);s) where All dτ dτ dτ ds H = (.) s x = τ µ or s x = τ µ ( s ). H - s nown funcons and υ x ( = = ) are defned from sysem (.): υx = exp( ( τ τ τ )) υ x τ µ x τ µ x τ µ )dτ dτ dτ ] exp( ( τ τ τ )) f x τ µ ( s)x τ µ ( s)x τ µ ( s)s) ; J (x τ µ ( s)x x τ µ ( s)x τ µ ( s);s) dτ dτ dτ ds H = =. (.) x Then on he bass of (.) (.) and (.) and her prvae dervaves on x we fnd θx = (H Hx H H x ) ψ =. (.) = As ψ - s nown funcons hence from sysem (.) dfferenang equaon on x [(.): =] equaons on х [ (.): =] equaons on х [(.): =] and summarsng we wll receve θ = ψ ψ ψx (.) = a ha ds ds ds θ C ( T): θ = ψ ( s s s ). 4 r The equaon (.) s he hrd equaon of sysem (.4). Therefore from he receved resuls ang no accoun (.6) follows P= θ Q a ds ds ds 4 r F (s s s ) ] (.4).e. funcons υθρ are defned from sysems (.) (.) (.4). Unqueness s obvous as a mehod by conradcon from (.) unqueness of he decson follows υ? T ) =. esuls (.) wh a condon ((A) (.)) are receved where smoohness of funcons υ s requred only on x as he dervave of s order n me has feaure n =. Then ang no accoun (.) (.) (.4) and he sysem (.4) has he unque connuous decson. Furher consderng prvae dervaves of s order υ x = { H } = x (.5) and summarsng (.5) wh ang no accoun (.) we have

4 5 Taalabe D. Omurov: Nonsaonary Naver-Soes Problem for Incompressble Flud wh Vscosy = exp[ ]{ [ ( τ τ τ ) F x τ µ ( s)x τ µ ( s)x τ µ ( s);s] J[ x τ µ ( s) x τ µ ( s)x τ µ ( s);s]} dτdτdτds = J = F. as Means he sysem (.) sasfes o he equaon (.)... Lmaon of Funcons ( υ υ υ ) n G (T ) The lmng case whch we wll consder concerns resuls of he heorem. Then he decson of sysem (.) s represenng n he form of (.) wh condons (.) (.) (A) (.) and T f : sup D f (x x x s) ds β sup exp( ( τ τ τ ) T τ f (l l l l ;s ) ds β s = T ( sup (s) f (x x x s) ds) β T Jx J sup : D J (x x x s) ds β4 T ( sup (s) J (x x x s) ds) β5 sup exp( ( τ τ τ ) T τ J l (l l l ;s ) ds β6 s = υ : sup D υ β7 ( = ; = ; = )l = x τ µ ( s ) β = max β; β = β ( µ q µ q ) 7 T T ( )d = q ( ) d = q. (.6) eally esmang (.) n G (T ) we have ν [ N β] = M* G (T ) υ = D υ C (T ) N C(T ) = 6 β υ β ( β C(T ) β4 β7 β ) = υ β ( µ q L µ q ) = β = :D υ υ; :D υ = υ = = α( α = ). α α α x x x = (.7) Theorem. In he condons of he heorem and (.6) (.7) he problem (.) - (.) has he unque decs on n. G (T ). The Decson of a Problem of Naver-Soes wh a Condon (B) Here we nvesgae a case (B) when θ ( = ) conanng convecve members of a problem of Naver-Soes are any. esuls of he heorem are no applcable. Therefore for he decson of a problem (.) - (.) we offer followng algorhms... Problem Naver- Soes wh Average Vscosy Le condons (.) (.) are sasfed and: υ(x x x ) = υ(x x x ) V ( ) = γx ; γ : γ = = γ = = = a ha υ V ( )( = ) V ( ) = V ( ); dvf < µ = µ γv m ( ) = ( m = ). = Then on he bass of funcons (.) (.) V ( )( = ) and P x x x ) = γp ( ) υ x x ) = V ( )( = ) υx x x ) = γ V ( ) υ x x ) = γ V ( ); µ υ = µ V x sysem (.) s equvalen wll be ransformed o a nd

5 Amercan Journal of Mahemacs and Sascs (6): LV V ( ) Z( ) V ( ) = = f ( ) γp ( ) µ V ( ) = (.) Z( ) γv ( );Z =. = In he specfed sysems unnown persons conan V P. e mar. Under regular n D = {( ) : < T } he decson we undersand he decson V = he equaon (.) n D whch has a connuous dervave on o he hrd order nclusve and connuous dervave on (>). Fro m sysem (.) consderng condons (.) and havng enered «algorhm puassonzaon sysems».e. γ dfferenang he equaons of sysem (.) accordngly on and hen summarsng we have he equaon: P = F ( ) f ( ) γ = (n) P ( ) = (n = ) (.4) F ( ) = ( F ( ) γ f ( )). = Therefore we wll receve P = ( η ) γ f η ( η )dη = = = γ f ( η )dη = F ( )d η η = P = F ( ); γ F =. = γ ] we have γ ( LV )( ) = γ f ( ) = = P ( ) µ γ V = Z ( ) =. eally on a bass[(.): (.) Then we have he followng (.4). (.5) Furher we have V Z V = Φ µ V = Φ ( ) f ( ) γ F ( ) or for consderaon of unnown funcons (.6) V we have V = exp( τ )V ( τ α )d and exp( τ ) Φ ( τ α( s);s)dτds exp( τ )( γ V ( τ = τ αs; s )) V ( τ αs; s ) αs dτds D [ V V V ]( = ; α = µ ) (.7) as here consder a mehod negraon n pars ( η) ) ( exp( )) 4µ ( s ) µ ( s ) ( γ V ( ηs )) V η ( ηs )dηds' = = exp( τ )( γ V ( τ µ s; s )) = τ V ( τ µ s; s )dτds µ s ( η = τ µ ( s'); s' = s ) Z ( ηs') = γ V ( ηs') = η = η ( η) ) exp( )V ( η)dη µ 4µ ( η) exp( ) µ ( s ) 4µ ( s ) Φ( ηs )dηds = exp( τ )V ( τ µ )d exp( τ ) Φ( τ µ ( s);s)dτds ( η = τ µ or η = τ µ ( s).

6 54 Taalabe D. Omurov: Nonsaonary Naver-Soes Problem for Incompressble Flud wh Vscosy If funcons aes place (.) and ha V = are sysem decsons (.7) hus ( ) D = {( ) : T } () ΦV : sup exp( τ ) Φ ( τ l D α( s);s)dτds β () Φ Φ l ;( ) D : V ( ) r sup exp ( τ ) τ γ D s = rd τds β r ( ) sup V β( = ; = ) l = τ α( s ); β = max β α = µ ; < µ = µ d = d < ( d = βr = ) = α V C (D ) : E = V ( ) : E ( d ) 6 = M. (.8) C = (.9) β Then he soluon of hs sysem (.7) we can fnd on he bass of Pard s mehod V = n D[ V n V n V n] n =...( = ) (.) where V = - n al esmaes and a ha ( ) D V : V V r = cons E = V V En = V V n = = n d< En d E n d< V n V n H C ( D )( = ). (.) Theorem. Under condons () () (.) (.8) problem Naver-Soes has he unque connuous decson. Defnon. The generalsed decson a problems D (.)-(.) (.) n area we name any connuous n < µ = µ. D equaon decson (.7) when.. We wll Consder a Flu d w h Very S mall Vscosy Le condons (.) (.) (.) are sasfed and: υ V ( ) K ( )( = ) K ( ) ( µ ) exp( ) V ( ) = V ( ) dvf ; < µ < ; P ( ) = (m) γv m ( ) = ( m = ) ( ) D = () V V ( m = : Z γ V ) = = V = (V V V )V = (V V V ) (.) a ha P x x x ) = γp ( ) υ(x x x ) = V ( ) ( µ ) exp( ) ( µ ) exp( ) υx (x x x ) = γ V ( ) ( µ ) γ exp( ) υ (x x x ) = γ V ( ) ( µ ) γ x exp( ) γ ( µ ) 4 µ exp( ) µ µ υ = µ V ( µ ) exp( ) ( µ ) exp ( ) υ υ = [ γ (V( ) ( µ ) = x = exp ( ))] [ V ( ) ( µ ) exp( )] = ( = ;Z = ).

7 Amercan Journal of Mahemacs and Sascs (6): Then on he bass of funcons V( )( = ) sysem (.) s equvalen wll be ransformed o a nd: LV V ( µ ) exp( ) = = f ( ) γp ( ) µ V ( = ) (.) n he specfed sysems unnown persons conan V P. Therefore P ( ) = F γ f ( ) = (.4) P = F ( η )d η ( ) D as ( γv ) γ ( µ ) exp( ) = = = = γ f ( ) P ( ) µ γv ( = ). = = Theorem 4. Le funcons V are sysem decsons V ( ) = Φ( ) µ V ( = ) (.5) Φ f γ F ( µ ) exp( ). Then V ( ) = exp( τ )V ( τ α )dτ exp( τ ) Φ( τ α( s);s)dτds H ( ) = ; α = µ (.6) where nown funcons and H G (D ): V = G (D) ( V V C ) M L = (m) V = C H m M = C T V = (sup (s)h L ( s) ds) M ( sup (s) Φ( s) ds) M D () H H ( = ; = ) T T q = ( ) dq = ( )d. Theorem 5. If funcons V P = are sysem decsons (.4) (.5) (.) ha (.) s he decson of sysem (.) n G (D ): υ = H ( ) ( µ ) exp( ). (.7) emar. In parcular f condons (.) (.) (.) and are sasfed. Then aes place K ( ) : υ = V ( ) = (.8) V = f ( ) γ P µ V =. (.9) Fro m here consderng (.4) (.5) we have P = F ( η )d η V = Φ µ V Φ( ) f ( ) γf ( ) = γ F =. = (.) Hence all condons of he heorem 4 and 5 are sasfed. Example. The specfed mehod of he heorem 5 can be used n par cular and for he decson on a problem (.)-(.) as a es example when aes place: ha υ V ( ) = υ = V( ) K ( ) Ω( ) K ( ) ( µ ) exp( ) = γx < γ = γ = = = = γ = γ = γ = = = = = γ d = ( = 6d = = 6 ) = = = υ = [ γ V γ K] = x = = γω = [ γ V( ) γk ( )] = = =

8 56 Taalabe D. Omurov: Nonsaonary Naver-Soes Problem for Incompressble Flud wh Vscosy or V ( µ ) exp( ) = f ( ) γ P µ V =. V = Φ ( ) µ V d P = F ( η )d η P = F γ f = γ F = = F = ( f f f ) 6 Φ 6 ( µ ) exp( ) f. = (.) Then V = exp( τ )V ( τ µ )dτ exp( τ ) Φ( τ µ ( s);s) d dτds H( ) υ = H( ) K ( ) H ( ) =. emar. As V ( ) C () ha lmedly he decson of a problem of Naver-Soes (.) - (.) wh a condon (B) s possble o prove lmaon and n W (D )- wegh space of ype of Sobolev: ν = W υ W ( υ ) = W ( υ { } ) = ( ) D : υ υ L ( D ) = T υ { sup ()[ ( )] d W = υ ( υ ) = T sup ( ) υ } ( ) dd = as V ( ) W ( D )V = (V V V ). 4. Conclusons I. From he receved resuls follows ha sysem Naver-Soes (.) n he condons of (.) (.) (A )-(A ) can have he analycal unque s condonal-smooh decson. A leas such decson answers o a mahemacal queson and possbly o consruc he decson of a problem of Naver-Soes (.)-(.) for an ncompressble lqud wh vscosy. II. esuls of he heorem and 5 can be appled o a problem of Nav er-soes of an ncompressble flu d wh vscosy when n n x = [ ). EFEENCES ν [] Naver-Soes Exsence and Smoohness Problem//The Mllennum Problems saed n by Clay Mahemacs Insue. [] G. Brhoff (98) Numercal flud dynamcs. SIAM ev. Vol.5 No pp.-4. [] B.J. Canwell (98) Organzed moon n urbulen flow. Ann. ev. Flud Mech. Vol. pp [4] Ladyzhensy Island And (97) Mahemacal quesons of dynamcs of a vscous ncompressble lqud. - Moscow: Scence 88 p. [5] T.D. Omurov () Nonsaonary Naver-Soes Problem for Incompressble Flud (n ussan) //Mahemacal morphology. Elecronc mahema. and he medo-bologs magazne. X. elease. - Idenfcaon 44\.hp:// -hml/titl-9.hm [6] T.D. Omurov () Nonsaonary Naver-Soes Problem for Incompressble Flud (n ussan). - J.Balasagyn KNU Bshe p. [7] T.D.Omurov () Naver-Soes problem for Incompressble flud wh vscosy//vara Informaca. Ed. M.Mlosz PIPS Polsh Lubln pp [8] L. Prandl (96) Gesammele Abhandlungen zur angewanden Mechan Hudro- und Aerodynamc. Sprnger Berln. [9] H. Schlchng s (974) Boundary-Layer Theory. Moscow: Scence 7 p. [] L.S. Sobolev (966) Equaons of Mahemacal Physcs. - Publ. 4. Moscow: Scence 44 p.