solutii navier stokes

7/30/2019 Solutii Navier Stokes

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UDHYV BB CHI IEYCV BIB NC]BI\'UVDOIU

Ndvi fi ey|u

Ed|nihby Lcːhcn

Ey{|bnu

3 Bnv|dfyei|i 3

< Udhyv(bb {i|gcninvi {i hbnbb fi ey|inv {c|chihi 5

<.3 Gbu(ec|ic ĳnv|i {hcni {c|chihi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<.< Gbu(ec|ic @cain'[dbuiybhhi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Udhyv(bb {i|gcninvi ĳn gbu

(ecː|b fi |dvcv

(bi =

5.3 Gbu(ec|ic Edyivvi ĳnv|i ebhbnf|b edcpbchb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

5.< Gbu(ec|ic ĳn }cuyh fi |dvcv( bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

5.5 Gbu(ec|ic ĳnv|i fdycː fbuey|b edcpbchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3;

5.6 Gbu(ec|ic ĳnv|i fdycː edny|b edcpbchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Bnv|dfyei|i

Ieycv( bc Nc}bi|'Uvdoiu iuvi finygb|ic cudebcvcː {|bneb{byhyb ednui|}cː|bb bg{yhuyhyb (ieycv( bc fi gbu(ec|i"

c{hbecv ynyb ybf niqvdnbcn ccv ĳn gbu(ec|i hcgbnc|cː3. C}cnf ĳn }ifi|i ecː eihi gcb ĳnvchnbvi ybfi fbn

c{hbecv( bbhi vi`nbei (c{c ci|yh yhiby|bhi ncvy|chi" uynv dgdaini hc nb}ih gce|dued{be u( b |iu{ievcː ey d lyncː

c{|dpbgcv( bi gdfihyh ednuvbvyvb} niqvdnbcn bc| nygi|dcui gbu(ecː|b chi ceiuvd| ybfi cy hde hc }bvizi ey

gyhv gcb gbeb fiecv }bvizc uynivyhyb u( b ĳn ednfbv( bb c{|dc{i bzdvi|gi ued{yh einv|ch ch giecnbebb ybfihd|

}cuedcui iuvi |izdh}c|ic ieycv( bib Nc}bi|'Uvdoiu {inv|y d gbu(ec|i bzded|cː (ybf bnedg{|iublbh" ĳn ec|i

ediffiebinvyh fi }buedzbvcvi iuvi ednubfi|cv ĳnv|'d {|bgcː c{|dpbgcv( bi ednuvcnv α 7 α; bnfi{infinv fi

vig{i|cvy|cː. B{dvizihi ginv (bdncvi ui bcy ĳn ednubfi|c|i ĳn ceiuv ec{bvdh? gcb gyhv( udhyv

(bbhi {|izinvcvi

uynv dlv (bnyvi nygcb {inv|y gbu

(eː c|b {i|gcninvi .

Ieycv( bc Nc}bi|'Uvdoiu {inv|y yn ybf bnedg{|iublbh ui dlv( bni bnv|dfye|cnf |ihcv( bc ednuvbvyvb}cː {inv|yvinud|yh ipv|c'vinubyni

VI 7 < α; F (3"

ĳn ieycv( bc fi gbu(ec|i Ecye`r

σ

∌ }

∌v/ a|cf} }

7 σ l ∜ a|cf { / fb}VI( (<"

3Ui {|iuy{yni ecː nygcː|yh \irndhfu ec|cevi|buvbe iuvi gcb gbe fiecv nygcː|yh \irndhfu e|bvbe \i 8 \ie| ynfi \i 7σ f ];$α; (]; u(b f ffibnf gcːuy|bhi fi |imi|bnv(cː {inv|y }bvizcː |iu{ievb} hynabgi σ finubvcvic ybfyhyb".

3



F ffibnf vinud|yh ubgiv|be }bvizcː fi fimd|gcv( bi.

\ihcv( bc (3" iuvi edg{hivcvcː ey ieycv( bc fi ednvbnybvcvi

fb} } 7 ;( (5"

ynfi } iuvi ecg{yh fi }bvizcː { {|iubynic l md|v(c u{iebffiecː gcubecː u( b σ finubvcvic ybfyhyb.\izyhvcː md|gc }ievd|bchcː c ieycv( bib Nc}bi| ' Uvdoiu

σ

∌ }

∌v/ a|cf} }

7 σ l ∜ a|cf { / α;∍} (6"

ip{|iubi ie`b}chinvcː ey

σ

∌ }

∌v/ a|cf

}<

<

/ |dv} Ñ }

7 σ l ∜ a|cf { ∜ α; |dv(|dv}" ( (9c"

|iu{ievb} {|bn c{hbec|ic ‗|dvd|yhyb‗ |ihcv( bib (6"

σ

∌ ς

∌v/ |dv(ς Ñ }"

7 α;∍ς (9l"

ynfi u'c mdhdubv bfinvbvcvic fb}F ≣ ∍}$< / a|cf (fb}}" ey fb}} 7 ; |iu{ievb} ς 7 3$< |dv} (}ievd|yh

}c|vij" u( b bfinvbvcvic |dv(a|cfύ" ≣ ;. Bnv|'yn ubuvig ec|vizbcn |ihcv( bbhi (<" |iu{ievb} (6" u( b (5" ec{cːvcː

md|gihi

σ

∌ }b∌v

/ }3∌ }b∌p3

/ }<∌ }b∌p<

/ }5∌ }b∌p5

7 σ lb ∜

∌{

∌pb

/

/ α; ∌ <}b

∌p<3

/∌ <}b

∌p<<

/∌ <}b

∌p<5 ( (b 7 3( <( 5"( (=c"

∌ }3∌p3

/∌ }<∌p<

/∌ }5∌p5

7 ;. (=l"

[inv|y dlv( bni|ic ip{hbebvcː c ieycv( bib Nc}bi|'Uvdoiu ĳn edd|fdncvihi ebhbnf|bei u( b umi|bei ĳn |ihcv( bbhi (3"

― (5" ui bnv|dfye d{i|cvd|bb fbmi|inv( bchb u( b ip{|iubbhi ed|iu{ynzcːvdc|i edd|fdncvihd| |iu{ievb}i. Bn eczyh ĳn

ec|i ybfyh fi hye|y iuvi edg{|iublbh u( b gbu(ec|ic nibzded|cː ui }c mdhdub {i hcnacː ieycv( bc fi ednvbnybvcvi

ĳn md|gc edg{hivcː u( b |ihcv( bc ed|iu{ynzcːvdc|i c vinud|yhyb niqvdnbcn ipv|c'vinubyni d ieycv( bi fi uvc|i

ei ip{|bgcː {|iubynic mynev( bi fi finubvcvi.

Ubuvigyh fi ieycv( bb (=" iuvi yn ubuvig {c|cldhbe ey fi|b}cvi {c|v( bchi nihbnbc| (fcvd|bvcː ceeihi|cv( bib

edn}ievb}i" fi d|fbnyh fdb (d|fbnyh gcpbg fi fi|b}c|i iuvi fdb" ch ecː|yb nieyndueyvi uynv eihi v|ibedg{dninvi chi }bvizib (}3 }< }5" u( b {|iubynic {. [inv|y d |izdh}c|i edg{hivːc ubuvigyh fi ieycv( bb

(=" iuvi ĳnudv( bv fi ednfbv( bb bnbv( bchi u( b hc hbgbvcː ed|iu{ynzcːvdc|i. Iuvi bg{d|vcnv fi |igc|ecv ecː ybfyh

niqvdnbcn cfi|ː c hc uy{|cmcv (c udhbfː c fieb }bvizc ybfyhyb hc ednvcevyh ey uy{|cmcv(c udhbfcː iuvi iachcː ey

}bvizc uy{|cmiv(ib |iu{ievb}i.

Fbn {ynev fi }ifi|i gcvigcvbe ipbuvinv(c u( b ynbebvcvic udhyv( bbhd| {inv|y ieycv( bc Nc}bi|'Uvdoiu uynv

cnchbzcvi ĳn ecvi}c v|cvcvi fi u{iebchbvcvi < . Dlv( bni|ic udhyv( bbhd| cnchbvbei ipcevi u( b ip{yni|ic ĳn fivchby

c {|deify|bhd| {inv|y |izdh}c|ic ynd| ey|ai|b {c|vbeyhc|i Nc}bi|'Uvdoiu mce dlbievyh v|cvcvihd| ehcubei fi

<Hcnfcy H. F. Hbmue`bz I. G. Mhybf Gie`cnbeu }dh. = Dpmd|f [i|acgdn [|iuu 3206Mdbcu E. Gcnhir D. \duc \. Vigcg \. Nc}bi| ' Uvdoiu isycvbdnu cnf vy|lyhinei Ecgl|bfai Ynb}. [|iuu Niq

Rd|o <;;3

<



giecnbec ybfihd|.5.

Bn ceiuv ec{bvdh ui {|izbnvcː ecvi}c udhyv( bb ipcevi Nc}bi|'Uvdoiu {inv|y gbu(ec|ic {i|gcninvcː c ybfihd|

bnedg{|iublbhi (udhyv( bb bnfi{infinvi fi vbg{ {inv|y ybfi ey finubvcvi ednuvcnvcː". \izdh}cː|bhi cnchbvbei

Nc}bi|'Uvdoiu ffii u( b c{|dpbgcvb}i ednuvbvybi ifyecv( bc fi lczcː ĳn fdginbyh giecnbebb ybfihd| ednv|blybnf

ĳn gdf fiffinbvd|by hc md|gc|ic clbhbvcːv( bhd| uvyfinvyhyb fi c gdfihc ey|ai|b fi bnvi|iu {|cevbe chi ybfihd|

|ichi (}cuedcui ucy }buedihcuvbei gbu(ecː|b hcgbnc|i ucy vy|lyhinvi".

Bn {|bneb{by ui {dv c{hbec fdycː {|deify|b {inv|y acːub|ic udhyv( bbhd| ieycv( bib Nc}bi|'Uvdoiu: (b" fbn

ieycv( bc fi ie`bhbl|y Ecye`r ui dlv( bn ip{|iubbhi edg{dninvihd| ipv|c'vinubynbb fy{cː ec|i ceiuvic ui bn'

v|dfye ĳn (3" u( b ui acːuiue udhyv( bbhi {inv|y }bvizi u( b {|iubyni? (bb" ui |izdh}cː fb|iev ubuvigyh (=" c}cnf ec

nieyndueyvi edg{dninvihi }bvizib u( b {|iubynic. [|bgc {|deify|cː ui c{hbecː fi |iayhcː nygcb gbu(ecː|bhd|

ynbfb|iev( bdnchi ĳn ec|i ceeihi|cv( bc ybfyhyb ui {dcvi niahbjc ucy iuvi bfinvbe nyhcː (}. gbu(ecː|bhi }buedgiv'

|bei ucy ipvinubdnchi".

< Udhyv(bb {i|gcninvi {i hbnbb fi ey|inv {c|chihi

<.3 Gbu(ec|ic ĳnv|i {hcni {c|chihi

Gbu(ec|ic }buedgiv|becː ĳnv|i {hcni {c|chihi iuvi d gbu(ec|i {hcncː (gbu(ec|ic c|i hde nygcb ĳn {hcnyh (p3

p<" ffibnf bnfi{infinvcː fi cpc Dp5" ynbfb|iev( bdnchcː ey }bvizc }(p<" 7 }3(p<". Ieycv( bbhi (=" fi}bn

; 7 ∜∌{

∌p3

/ α;f<}

fp<<(1c"

; 7 ∜ σ a ∜∌{

∌p<

(1l"

; 7 ∜ ∌{∌p5

(1e"

ynfi ∌ }∌p3

7 ; (fbn ieycv( bc fi ednvbnybvcvi" ∌ }∌v

7 ; (gbu(ec|i {i|gcninvcː" bc| imd|vyh vcnainv( bch

ed|iu{ynzcːvd| |ihcv( bib ednuvbvyvb}i (3" iuvi fcv fi ip{|iubc

φ 7 α; ε˒ 7 α;f}

fp<. (0"

Fbn cnchbzc ubuvigyhyb (1" ui dlui|}cː ecː {|iubynic iuvi d mynev( bi fi md|gc

{ (p3( p<" 7 {3 (p3" / {< (p<" (2"

ey

{3 7 o3p3 / o< (3;"

u( b

{< 7 ∜ σ a p<. (33"

Bn (3;" ednuvcnvihi o3(< cy y|gcːvdc|ihi uignbffiecv( bb: o3 7 o |i{|izbnvcː a|cfbinvyh fi {|iubyni ĳn hynayh

5 Li|oi| \. Bnv¶ia|cvbdn fiu isycvbdnu fy gd}iginv f‒yn ybfi }busyiyp bnedg{|iuublhi bn @cnflye` fi| [`rubo }dh.]BBB$< If. Mhyaai Li|hbn U{|bnai| ]i|hca 32=5

Bcedl E. Bnv|dfyevbdn gcv`¶igcvbsyi kc hc g¶iecnbsyi fiu ybfu If. Cecfigbib Lyey|iuvb 3292Ui||bn J. Gcv`igcvbech {|bneb{hiu dm ehcuubech ybf gie`cnbeu bn @cnflye fi| [`rubo }dh. ]BBB$3 If. Mhyaai

Li|hbn U{|bnai| ]i|hca 3292

5



Mbay|c 3: Fbuv|blyv( bc fi }bvizi ĳn gbu(ec|ic {hcn'{c|chihcː. Cbeb o 7 ∜∍ {$h ∍ { ffibnf fbmi|inv(c fi {|iubyni|c{d|vcvcː hc hynabgic h.

fb|iev( bib fi gbu(ec|i {D p3w (o iuvi d ednuvcnvcː niacvb}cː fidc|iei gbu(ec|ic c|i hde fi hc {|iubyni gcb

gc|i hc {|iubyni gcb gbecː" bc| o< |i{|izbnvcː }chdc|ic {|iubynbb hc p3 7 ;. Fbn (33" ynfi u'c ednubfi|cv

p< 7 ;( {< 7 ; |izyhvcː ecː {|iubynic c|i d }c|bcv( bi fi vb{ `bf|duvcvbe {i fb|iev( bc nd|gchcː gbu(ecː|bb ei c|ihde {i hbnbb fi ey|inv {c|chihi.

Fbuv|blyv( bc fi }bvizi ui dlv( bni bnvia|cnf fi fdycː d|b |ihcv( bc (1c":

f}

fp<7

3

α;o p< / E3 (3<c"

} 73

<α;o p<

< / E3p< / E< (3<l"

ynfi ednuvcnvihi fi bnvia|c|i E3 u( b E< ui dlv( bn fbn ednfbv( bbhi hc hbgbvcː? }. Mbay|c 3:

} 7 ∜ ]<( {inv|y p< 7 ; (35c"

} 7 ]3( {inv|y p< 7 ` (35l"

Bn ffinch |izyhvcː ip{|iubc

} (p<" 7o

<α;p< (p< ∜ `" /

]3 / ]<

`p< ∜ ]<. (36"

Ui dlui|}cː ecː fbuv|blyv( bc fi }bvizi (36" ui edg{yni cfbvb} fbnv|'d {c|cldhcː u( b d f|ic{vcː.

Fbuv|blyv( bc {c|cldhbecː fi }bvizcː iuvi aini|cvcː fi gbu(ec|ic [dbuiybhhi (fcvd|bvcː {|izinv(ib a|cfbinvyhyb

fi {|iubyni" bc| fbuv|blyv( bc hbnbc|cː iuvi ec|cevi|buvbecː gbu(ecː|bb fi cnv|inc|i Edyivvi ei c|i hde ĳnv|i {hcni

{c|chihi.

Fbn (0" u( b (36" |izyhvcː fbuv|blyv( bc imd|vyhyb vcnainv( bch |iu{ievb} d fbuv|blyv( bi hbnbc|cː {inv|y gbu(ec|ic

[dbuiybhhi u( b d }chdc|i ednuvcnvcː ĳn eczyh gbu(ecː|bb Edyivvi ĳnv|i {hcni {c|chihi.

Vdv d gbu(ec|i {hcn'{c|chihcː iuvi ey|ai|ic {i|gcninvcː ey uy{|cmcv( cː hbli|cː {i yn {hcn ĳnehbncv ey

yna`byh γ c ynib {ihbeyhi (ffihg‘" fi ybf }cuedu fi a|dubgi ednuvcnvcː ` (}. Mbay|c <". Gbu(ec|ic iuvi

ĳn ceiuv ecz aini|cvcː fi md|v(c gcubecː fi a|iyvcvi? ieycv( bc (1c" ynfi a|cfbinvyh fi {|iubyni iuvi ĳnhdeybv

fi edg{dninvc md|v(ib u{iebffiei fi a|iyvcvi {|dbievcvcː {i fb|iev( bc {hcnyhyb fi}bni

; 7 σ a ubn γ / α;f<}

fp<<(39"

6



Mbay|c <: Gbu(ec|ic {hcn'{c|chihcː {i {hcnyh ĳnehbncv.

ey udhyv( bc

} 7

σ a ubnγ

α; p<

` ∜

p<<

(3="

ynfi u'cy ednubfi|cv ednfbv( bbhi hc hbgbvcː

p< 7 ;( } 7 ;? (31"

p< 7 `(f}

fp<7 ;. (30"

Ednfbv( bc (31" |i{|izbnvcː ednfbv( bc fi cfi|inv( cː c ybfyhyb hc {hcn bc| (30" ednfbv( bc fi imd|v vcnainv( bch

nyh {i uy{|cmcv(c hbli|cː ie`b}chinvcː ey }bvizc gcpbgcː {i uy{|cmcv(c hbli|cː }. |ihcv( bc (0".

<.< Gbu(ec|ic @cain'[dbuiybhhi

Gbu(ec|ic @cain'[dbuiybhhi iuvi ey|ai|ic ei c|i hde ĳnv|'d ednfyevcː eb|eyhc|cː {i hbnbb fi ey|inv {c|chihi

(uy{|cmiv(ihi fi ey|inv ffibnf ebhbnf|b edneinv|beb chyniecːvd|b {i fb|iev( bc cpib edgyni" fcvd|bvcː ynyb a|cfbinv

ednuvcnv fi {|iubyni o. Bn {|izinvyh ipig{hy ednfyevc ui ednubfi|cː d|bzdnvchcː fieb md|v(ihi gcubei ny

ui {|dbieviczcː {i fb|iev( bc ey|ai|bb. C}cnf ĳn }ifi|i ip{|iubc imd|vyhyb vcnainv( bch φ 7 α; ε˒ u( b c }bvizib

fimd|gcv( bib u{iebffiei ε˒ 7 f}$f| fbuv|blyv( bc fi }bvizi c ybfyhyb niqvdnbcn {i fb|iev( bc cpib Dz (bfinvbecː

ey cpc ednfyevib" iuvi fcvcː fi ieycv( bc fbmi|inv( bchcː

f}

f|

73

< α;o | (32"

ynfi }z 7 } (|".

Fidc|iei }bvizc iuvi nyhcː hc {i|ivihi ednfyevib

} 7 ;( {inv|y | 7 \ (<;"

udhyv( bc ieycv( bib (32" fi}bni

} (|" 7o

6 α;

|< ∜ \<

(<3"

(u'c |iacːubv ceiicu( b fbuv|blyv( bi {c|cldhbecː ec ĳn eczyh gbu(ecː|bb uyl {|iubyni fbnv|i {hcni {c|chihi". Ui

9



Mbay|c 5: Fbuv|blyv( bc fi }bvizi u( b c imd|vy|bhd| vcnainv( bchi ĳn gbu(ec|ic @cain ' [dbuiybhhi (gbu(ec|iynbfb|iev( bdnchcː cpbch ubgiv|becː".

dlv( bni ip{|iubc filbvyhyb v|cnu{d|vcv

S 7ϋ o

< α;

\;

|

|< ∜ \<

f| 7∍ { ϋ \6

0 α; h(<<"

ynfi u'c ednubfi|cv o 7 ∜∍ {$h ∍ { ffibnf ecːfi|ic fi {|iubyni (ĳn hynayh ey|ai|bb" {i hynabgic h fi

ednfyevcː.

Mdhdubnf |izyhvcvyh (<<" ui dlv( bni ip{|iubc }bvizib fi fimd|gcv( bi hc {i|ivihi ednfyevib ε˒ q 7 6]g

\ ey

]g 7 ∍ {\<

0 α; h}bvizc gifbi. [inv|y ceicuvcː gbu(ec|i ui {dcvi echeyhc ipcev ediffiebinvyh fi {bi|fi|b hbnbc|i

fi {|iubyni (Fc|er" {i hynabgic h c ednfyevib? fbn iachbvcvic `|ο :7 ∍ {σa

7 0ϋ< f9

ο h S< |izyhvcnf

}chdc|ic

ο 7

=6

\i (<5"

ynfi \i 7 σ f]g$α; \i 8 \i e| ∸7 <5;; f 7 <\ ffibnf fbcgiv|yh ednfyevib.

Bn Mbay|c 5 uynv |i{|izinvcvi fbuv|blyv( bbhi fi }bvizcː u( b fi imd|vy|b vcnainv( bchi ĳnv|'d ednfyevcː d|b'

zdnvchcː. D aini|chbzc|i c ceiuvib gbu(ecː|b ui {dcvi ednubfi|c gbu(ec|ic [dbuiybhhi ĳnv|i ebhbnf|b edneinv|beb

ccv( b ĳn |i{cyu ucy ĳn gbu(ec|i fi v|cnuhcv( bi.

5 Udhyv(bb {i|gcninvi ĳn gbu

(ecː|b fi |dvcv

(bi

Gbu(ecː|bhi fi |dvcv( bi chi ybfihd| }cuedcui niqvdnbini ĳu( b acːuiue nygi|dcui c{hbecv( bb vi`nbei ĳn u{iebch

ĳn fdginbyh hyl|bffiecv( bib u( b ch |idgiv|bib. Iuvi ncvy|ch ec gbu(ecː|bhi fi |dvcv( bi ucː ui |i{|izbnvi ĳnv|'ynubuvig fi edd|fdncvi ebhbnf|bei ucy umi|bei |dvcv( bc c}cnf hde ĳn ceiuv ecz fy{cː cpc Dύ. Bn eczyh ĳn ec|i

ebnigcvbec gbu(ecː|bb iuvi bnfi{infinvcː fi cpc fi |dvcv( bi gbu(ec|ic {dc|vcː finygb|ic fi cpbch ubgiv|becː

ffibnf ĳnfi{hbnbvcː ednfbv( bc

∌ •

∌ύ≣ ; (<6"

(fi |igc|ecv ecː u( b gbu(ec|ic @cain'[dbuiybhhi uvyfbcvcː ĳn ¡<.< iuvi ĳn edd|fdncvi ebhbnf|bei d gbu(ec|i

cpbch ubgiv|becː mcː|cː ĳnucː c ffi d gbu(ec|i fi |dvcv( bi".

Ieycv( bc Nc}bi|'Uvdoiu (6" {inv|y gbu(ecː|b {i|gcninvi chi ynyb ybf bnedg{|iublbh c|i ĳn edd|fdncvi

ebhbnf|bei {| ( ύ ( zw ip{|iubc:

=



σ

}|

∌ }|∌|

/3

|}ύ

∌ }|∌ύ

/ }z∌ }|∌z

∜}<ύ|

7 σl| ∜

∌{

∌|/

/ α;∌ <}|

∌|< /

3

|

∌ }|

∌| /

3

|<

∌ <}|

∌ύ< /

∌ <}|

∌z< ∜

}|

|< ∜

<

|<

∌ }ύ

∌ύ

(<9c"

σ

}|

∌ }ύ∌|

/3

|}ύ

∌ }ύ∌ύ

/ }z∌ }ύ∌z

/}|}ύ

|

7 σlύ ∜

3

|

∌{

∌ύ/

/ α;

∌ <}ύ∌|<

/3

|

∌ }ύ∌|

/3

|<∌ <}ύ∌ύ<

/∌ <}ύ∌z<

/<

|<∌ }|∌ύ

∜}ύ|<

(<9l"

σ

}|

∌ }z∌|

/3

|}ύ

∌ }z∌ύ

/ }z∌ }z∌z

7 σlz ∜

∌{

∌z/ α;

∌ <}z∌|<

/3

|<∌ <}z∌ύ<

/∌ <}z∌z<

/3

|

∌ }z∌|

(<9e"

|ihcv( bc fi ednvbnybvcvi ffibnf fcvcː fi

∌ }|∌|

/}||

/3

|

∌ }ύ∌ύ

/∌ }z∌z

7 ; (<="

Ubuvigyh fi ieycv( bb (<9"― (<=" c|i ecvi}c udhyv( bb cnchbvbei c{|dpbgcvb}i {inv|y gbu(ecː|b cpbch'ubgiv|bei

fi bnvi|iu {|cevbe fbnv|i ec|i ginv( bdncːg: gbu(ec|ic ĳn }iebncːvcvic ynyb fbue ĳn |dvcv( bi6 gbu(ec|ic fi

|dvcv( bi {i ei|ey|b edneinv|bei ĳnv|i {hcni {c|chihi9 gbu(ec|ic ĳn }iebncːvcvic ynyb {ynev fi bg{cev= u( b

yn nygcː| fi udhyv( bb cnchbvbei ipcevi fbnv|i ec|i ui }d| v|cvc ĳn ednvbnyc|i: gbu(ecː|bhi }buedgiv|bei fi vb{

Edyivvi ĳnv|i ebhbnf|bb edneinv|beb u( b fbuey|b edneinv|bei |iu{ievb} gbu(ec|ic fi |dvcv( bi |babfcː‘ c ynyb

ybf }cuedu ĳnv|'yn }cu ĳn |dvcv( bi.

5.3 Gbu(ec|ic Edyivvi ĳnv|i ebhbnf|b edcpbchb

Bn eczyh gbu(ecː|bb fi |dvcv( bi ynbfb|iev( bdnchi ĳnv|i ebhbnf|b edcpbchb ednubfi|cv( b fi hynabgi bnffinbvcː u( b

ccv( b ĳn gbu(ec|i fi |dvcv( bi |ihcvb}cː |ihcv( bbhi (<9" uynv ie`b}chinvi ey ubuvigyh

∜σ}<

|7 ∜

∌{

∌|(<1c"

; 7∌φ

∌|/

< φ

|(<1l"

; 7 ∜ σ a ∜ ∌{∌z

(<1e"

\ihcv( bbhi (<1" ui dlv( bn fbn (<9" ey }ύ 7 } (|" }| 7 }z 7 ; ∌ }$∌z 7 ; u( b |iuv|bev( bc (<6"? (<9l"

ffibnf ie`b}chinvcː ey (<1l" ynfi imd|vyh vcnainv( bch c|i ip{|iubc

φ 7 α; ε˒ 7 α;

f}

f|∜

}

|

( (<0"

6Ue`hbe`vbna @. Ldynfc|r hcri| vìd|r =v` Ifbvbdn Ge'A|cq @bhh Niq Rd|o {. <35 32=09Li|oi| \. Cn ipcev udhyvbdn dm vì Nc}bi 'Uvdoiu isycvbdn Bnv. J. Ina.Ueb. <;(<" <31 ― <5; 320<=E`y|e`bhh U. Q. ]buedyu dqu? vì {|cevbech yui dm vìd|r Lyvvi|qd|vù {. <9; 3209

1



(<1l" c}cnf udhyv( bc

φ 7E

|<. (<2"

Gbu(ec|ic fiue|bucː fi ubuvigyh (<1" iuvi d gbu(ec|i cpbch ubgiv|becː ĳn ec|i fbuv|blyv( bc fi }bvizi iuvi

bnfi{infinvcː fi cpc Dz.

Fbn (<0" u( b (<2" ui dlv( bni fbuv|blyv( bc fi }bvizi ĳnv|i eib fdb ebhbnf|b

f}

f|∜

}

|7

E3

|<. (5;"

Udhyv( bc aini|chcː c ieycv( bib (5;" ui dlv( bni ec uygc fbnv|i udhyv( bc ieycv( bib dgdaini }; 7 E<| u( b d

udhyv( bi {c|vbeyhc|cː } { 7 ∜ E3

<| |iu{ievb}

} 7 }; / } { 7 E< | ∜E3

< |(53"

Ednuvcnvihi fbn (53" ui fivi|gbncː fbn ednfbv( bbhi hc hbgbvcː fi ipig{hy

| 7 \3 ( } 7 ς \3 (5<"

| 7 \< ( } 7 ; (55"

ynfi ς iuvi vy|cv( bc ednuvcnvcː c ebhbnf|yhyb bnvi|bd| ebhbnf|yh ipvi|bd| ffibnf ednubfi|cv ĳn |i{cyu (}. Mbay|c

6". [|bnv|'yn echeyh ubg{hy ui dlv( bni fbuv|blyv( bc fi }bvizi ĳnv|i eib fdb ebhbnf|b ĳn ednfbv( bbhi inynv(cvi

} 7 ς |

\<<

|<∜ 3

\<<

\<3

∜ 3(56"

ynfi u'cy mdhdubv }chd|bhi ednuvcnvihd| fbn (53"

E3 7 < ς\<3\<

<

\<3 ∜ \<

<

( E< 7 ς\<3

\<3 ∜ \<

<

. (59"

Bn eczyh ynib fbmi|inv(i |ihcvb} gbeb fbnv|i eib fdb ebhbnf|b ζ $\ 88 3 (ζ 7 \< ∜ \3( \3 7 \"

fbuv|blyv( bc fi }bvizi iuvi mdc|vi c{|dc{i fi eic hbnbc|cː bc| }bvizc fi fimd|gcv( bi u{iebffiecː iuvi ednuvcnvcː

ε˒ ∸7 ς\ζ

(}. Mbay|c 6".

Fbn cnchbzc ieycv( bbhd| (<1c" u( b (<1e" ui dlv( bni md|gc fbuv|blyv( bib fi {|iubyni

{ (| ( z" 7 {; / a (z" / m (|" (5="

ynfi {; iuvi d ednuvcnvcː (}chdc|ic {|iubynbb hc z 7 ` 7 ednuvcnv " bc| mynev( bbhi a(z" u( b m (|" ui dlv( bn {|bn

bnvia|c|ic ieycv( bbhd| ginv( bdncvi ynfi ui v( bni ednv fi (56". \izyhvcː

{ (| ( z" 7 {; / σ a (` ∜ z" / σ

E<< |<

<∜ <E3E< hn | ∜

E<3

<|<

(51"

ednuvcvihi fi bnvia|c|i ffibnf fcvi fi (59".

0



Mbay|c 6: Fbuv|blyv( bc fi }bvizi ĳn gbu(ec|ic Edyivvi ĳnv|i ebhbnf|b edcpbchb ({|iuy{yu( b fi hynabgi bnffinbvcː".

Mbay|c 9: Uy{|cmcv(c hbli|cː c ybfyhyb ccv ĳnv|'yn }cu ĳn |dvcv( bi.

5.< Gbu(ec|ic ĳn }cuyh fi |dvcv

(bi

Gbu(ec|ic ynyb ybf }cuedu ĳnv|'yn }cu ccv ĳn |dvcv( bi (}. Mbay|c 9" iuvi fiue|bucː fi ceihicu( b ieycv( bb

(<1" ednfbv( bbhi hc hbgbvcː ffibnf fi ceicuvcː fcvcː

| 7 ; ( } 7 ; (50"

| 7 \ ( } 7 ς \ (52"

Fbn (53" |izyhvcː

} 7 ς | (6;"

ynfi E3 7 ; u( b E< 7 ς.

2



Mbay|c =: Gbu(ec|ic fi |dvcv( bi ĳnv|i fdycː fbuey|b edneinv|bei ĳn hb{uc ceeihi|cv( bib.

Bn ceiuv ecz fbuv|blyv( bc fi }bvizcː fbn ybf iuvi bfinvbecː ey eic c ynyb udhbf |babf ccv ĳn gbu(ec|i fi

|dvcv( bi. Fbuv|blyv( bc fi {|iubynb (51" ĳn ybf fi}bni

{ (| ( z" 7 {; / σ a (@gbn ∜ z" / σς< |<

<(63"

Ieycv( bc uy{|cmiv(ib hbli|i ui dlv( bni {yncnf ednfbv( bc ec { (| ( z" 7 {; ĳn (63" |iu{ievb}

z 7 @gbn /ς< |<

<a. (6<"

Fbn ednfbv( bc fi ednui|}c|i c }dhygyhyb fi ybf fbn }cu ui {dv dlv( bni ip{|iubbhi ĳncːhv( bgbhd| @gbn u( b

@gcp ĳn mynev( bi fi ĳncːhv( bgic ybfyhyb ĳn uvc|i fi |i{cyu @; (}. Mbay|c 9"

] dh 7 < ϋ

\;

z | f| 7 ϋ \< @; (65"

Fbn (6<" u( b (65" |izyhvcː

@gbn 7 @; ∜ς<\<

6a(66"

|iu{ievb}

@gcp 7 @; / ς<

\<

6a. (69"

5.5 Gbu(ec|ic ĳnv|i fdycː fbuey|b edcpbchi

Fbuv|blyv( bc fi }bvizi ĳn gbu(ec|ic cpbch'ubgiv|becː fi |dvcv( bi fbnv|i fdycː fbuey|b edneinv|bei (}. Mbay|c

=" iuvi udhyv( bc ieycv( bib

; 7∌ <}

∌|</

3

|

∌ }

∌|/

∌ <}

∌z<∜

}

|<(6="

dlv( bnyvcː fbn (<9l" ĳn b{dvizc }| 7 }z 7 ; u( b }ύ 7 } (| ( z".

3;



Ednfbv( bbhi hc hbgbvcː {inv|y (6=" uynv

z 7 ; ( } 7 ; (61"

z 7 `( } 7 | ς (60"

Fbuv|blyv( bc fi }bvizi ec|i }i|bffiecː (6=" ― (60" iuvi

} 7| ς

`z. (62"

\izyhvcː ε˒ 7 |ς`

(}bvizc fi fimd|gcv( bi u{iebffiecː" u( b φ 7 α;|ς`

(imd|v vcnainv( bch".

Udhyv( bc (62" iuvi ĳnucː }chclbhcː nygcb ĳn eczyh ĳn ec|i edg{dninvc ceeihi|cv( bib einv|bmyai fbn (<1c"

ui niahbjiczcː edg{hiv (ĳn hb{uc ceiuvib b{dvizi ny iuvi {dublbhcː fivi|gbnc|ic fbuv|blyv( bib fi {|iubynb fbn

(<1c" u( b (<1e" ec ĳn eczyh gbu(ecː|bhd| fi |dvcv( bi {|ieifinvi". Bg{dublbhbvcvic dlv( bni|bb ynib fbuv|blyv( bb fi

{|iubynb { 7 { (| ( z" ĳn ednmd|gbvcvi ey fbuv|blyv( bc fi }bvizi (62" figdnuv|iczcː mc{vyh ecː {inv|y \i

> ; ebnigcvbec gbu(ecː|bb fi |dvcv( bi ĳnv|i fbuey|b edneinv|bei v|ilybi ucː ednv( bncː eih {yv( bn u( b edg{dninvc

}bvizib {i fb|iev( bi |cfbchcː.

5.6 Gbu(ec|ic ĳnv|i fdycː edny|b edcpbchi

Ubgbhc| ey gbu(ec|ic ĳnv|i fdycː fbuey|b edcpbchi gbu(ec|ic ĳnv|i fdycː edny|b edcpbchi c|i udhyv( bi ipcevcː

ĳn hb{uc md|v(ihd| fi bni|v( bi (gbu(ec|ic ffibnf uvyfbcvcː ĳn edd|fdncvi umi|bei". Bn ceicuvcː b{dvizcː imdvyh

vcnainv( bch (φ 7 vύΰ" ui dlv( bni ec udhyv( bi c ieycv( bib fi gbuec|i ; 7 ∌φ ∌ΰ

/ < edv ΰφ |iu{ievb}

φ 7E

ubn<ΰ(9;"

fbuv|blyv( bc fi }bvizi {i fb|iev( bc fi |dvcv( bi c edny|bhd| }ύ 7 } (| ( ΰ" (}| 7 }ΰ 7 ;" u( b }bvizc ed|i'

u{ynzcːvdc|i fi fimd|gcv( bi c}cnf ip{|iubbhi:

} 7 | ubn ΰ m (ΰ" (93"

ε˒ 7 ubn ΰfm

fΰ(9<"

Cnchda ey fbuv|blyv( bc (62" }bvizc ybfyhyb fbnv|i eihi fdycː edny|b (93" ui ednubfi|cː c}cnf d fbuv|blyv( bi

hbnbc|cː ĳn |c{d|v ey fbuvcnv(c mcv( cː fi cpyh fi |dvcv( bi ĳn ceiuv ecz (| ubn ΰ". Bn ceiuv ecz {inv|y μ 7 γ<∜γ3

gbe u( b γ< 7 ϋ$< }bvizc fi fimd|gcv( bi u{iebffieːc fi}bni ε˒ ςμ

.

Ednfbv( bbhi hc hbgbvcː {inv|y gbu(ec|ic uvyfbcvcː uynv:

ΰ 7 γ3 ( } 7 | ubn γ3 ς (95"

ΰ 7 γ< ( } 7 ; (96"

ynfi u'c ednubfi|cv ecː yna`by|bhi hc }c|m chi edny|bhd| uynv < γ3 |iu{ievb} < γ< (}. Mbay|c 1".

Fidc|iei φ 7 α; ε˒ |izyhvcːfm

fΰ7

E3

ubn5ΰ(99"

33



Mbay|c 1: Aidgiv|bc gbu(ecː|bb fi |dvcv( bi ĳnv|i fdycː edny|b edneinv|bei.

{|bn bnvia|c|i dlv( bncnfy'ui

m (ΰ" 7 E< / E5

hnva

ΰ

<∜

edu ΰ

ubn<ΰ

(9="

Mdhdubnf (95" u( b (96" ui fivi|gbncː ednuvcnvihi E< u( b E5 fbn (9=" fbuv|blyv( bc fi }bvizi (93" ecː{cːvcnf

ip{|iubc

} 7 | ς ubn ΰ

hn

vaγ<<

va ΰ<

∜eduγ<ubn<γ<

/ edu ΰubn<ΰ

hnva

γ<<

vaγ3<

∜eduγ

<

ubn<γ</ eduγ3

ubn<γ3

. (91"

Bnv|'yn gdf ubgbhc| ui {dcvi uvyfbc u( b gbu(ec|ic fi |dvcv( bi c ynyb ybf niqvdnbcn ĳnv|i fdycː umi|i

edneinv|bei (}. ec{bvdhyh 6 fbn Hcnahdbu1".

1Hcnahdbu Q. I. Uhdq }buedyu dq Gcegbhhcn Edg{. Niq Rd|o 32=6

3<

solutii navier stokes

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