Luận án Sử dụng phương pháp số vào một số bài toán cơ học

baitoin duqegiai trendo~n[t"-I,t,,] (III.2.36) ~ ~.~, ~ a~-::;-+U1--t-V1-:::--t-W1-:::-=vxY~+vz~, at ax oy az aT (III.2.37) av Ov Ov &v a2v ---1..+11 ---1.+v ---1..+W ~ =v Llv +v !.a -1 a 1 ~, 1 a xv 1 z.. ~ ,t lX u'y . Z - 02- (III.2.38) Ow Ow Ow Ow a2w --!..+lL --!..+v --!..+w --!..=v .6..w+v ---!.'" " a l~, 1", xy 1 Z a~'at x uy 02 Z~ vai dH~uki~nd'au (III.2.39) ,,-1 ,,-1,,-1 ,,-1 ,,-1 ,,-1 U. =u , VI =V , WI =W -Bucrethuhai,xettrendo~n[t"-1't"+1]'eoh~phuangtrlnh (III.2.40) aU2+.!.Op2+lv2=0, at p ax (III.2.41) Ov2+~Op2-lu2 =0, at p By -59- ~ 3: ~ d-~ uu. ~ 6:p:,~ ~ u Ut~ t1tW.p~ ~ ~ (III.2.42) OP2=pg,az (III.2.43) au2+Ov2+aw2=o. ax 8y az v6i caedieuki~n<fAll (III.2.44) ,,-1 " ,,-1 " ,,-1 "u2 =Ul' V2 =vI, W2 =WI - Bu6ccu6icling,bai loanduqcxettrendo,:\-n[tn,t,,+l]'chungta co h~ phuongtrioovadieuki~nd~utU<1ngt1l(III.2.36)- (III.2.39) (III.2.45) a a a ~ a" ~ +u ~ +v ~ +W ~ =V A 11 +v -~ at 3 ox 3 8y 3 oz xy~ z OZ2 ' (III.2.46) Ov3 Ov3 Ov3 Ov3- 02V3-+u." -+V~-+Wo - - V",,~V3 +v ;;-, at - ox -'oy -,oz -; , 8z- (III.2.47) 0 ~ ~ ~ ~.,W~ ~~ ow3 ow3 u~~~---=...+~---=...+v-+w -= v ~W+v ~at _ 8 3" 3 a X" 3 ' a2'~ oy z ~ z v6i dieuki~ndau (III.2.48) Un=Un+l V"= V,,+1 Wn=W,,+132' 32' 3,,2' H~phuongtdnhd~tiln b9 nghi~m(upvpw1)va (U3,V3,W3)gi6ngooau (cungxuatphattittOaDtirA), d~giaih~naychungtatier t~cphftnraloantir~ thanh3thanhphan: (III.2.49) 1\=All +~2+A13 trongdo, (IIL2.50) j. -"'1;- R 0, 0 l\ 0 0 0 0 0 0 0 0 R 01,, 0 0 '\Ii=1,2,3 -60- e~ 3: ';1l4e<JJJ& ~ 4«9 ~ ~ Ht:iJt4 Jk ~ Ptix4 ;::;~ {i:4t~ D~ tim <P'~,chUng fa dn phiin do~ [tn-p!n]thanh hai do<;mcon [tn-l'~"-:~]'[t,,-;~,tn]'Trendo~n[tn-ptn-~],h~phuongtrinhco d.:mg (IIL2.51) Ba<Pll+ j (f'I = 0- "11'1'11 ,at (UL2.52) 8(P'2;\ - 0B~ +.""2<P12- ,at . (III.2.53) am B~+~CP13 =0,at contrendo~n[tn-;~,tn]chungtaxeth~ (111.2.54) B0<P14+ 4. (f\ =0~ "..3'1'14 ,ot (UL2.55) B O15=0, , at (III,2.56) n a<P16T' 4 <'p =0,b- "4.1 16at B<p;;l =B<p"-1, Bq>;;l = B<p;~;{, B ,,_, B ~_l.\<P;3.= <P;2', B n-)1.- B n-;-'S<P14 - <P13, B<p;;~ =B<p;4'- B<p;~X=B<p~5' (trongd6 <P1i'j =1,...,6kyhi~unghi~mtinhtoanquatUngbuce). Ket quatmhtoan<P;6sauclingchinhlanghi~m<p;. D~tim<p~+1chungtagiai11lnluqtcaephuongtrlnh(III.2.5l) - (UL2.56) cho trenhai do~n[tn,tn+)L.],[tn+)L.';"+1]tuong illlg. Bily gia chungta xetsV xapXl va 6n d~ ctlabai toan(IIL2.51)- (III.2.56).Cacphuongtrinhnaydeucod(~ng (IIL2.57) oU+uoU- v 02U=0 at oJ.. xy aK ' (trongdoA cotheho~cxho~cyho~cz; U, U1acaeleYhi~utuqngtnmg),nen vi~ckhaosatsv xapxi theefj,ttuO1lgtv nhuffil;lCtruac,chungtaco M~nhde sall: -61- ~ 3: 1H#~.~~ ~ ~ ,,~_u Ut~~ ?~~~ M~nhd~1: Bilitoansaiphantheetho;gian ,,-~ ,,-I 11 B<Pll - <p]/+Av<p~-/2=0, 1=1,2,3M " ,,-~ B <fIJ]- <PJ] +1\6-/+1<P~=0, 1=4,5,6 M x[p xi cacbaitoaDtUOllgilp.gc[p haithee~t. H~cacphUOllgtrlnhfanluqttrongnh6m(III.2.51)- (IIL2.53)b6quadiet ki~ncTh.uc6th~vietduai d~g toanri':rIh~Usau: (III.2.58) av - p-+ 1;),V=0at ~~ , Vi =1,2,3. TheokhOnggiancactoaDti'r~,~, R3 duqcxa:pxi bCrlcaetoaDti:rsaipha.n (IIL2.59) ( -" - Y" .,Y 1k i-l)~_ I " l -" I+} R;.v j = P Uijk 2~i 11"1'k- 2f;~ +11"lk J v 1+]' 'F' 1- J xv. (~J2 ' (IIL2.60) ( Tl" IT" Tl" ')TT" TT" \_ I " . " +H - I'" H . " . H - ~r ',L +.'" 1k J 1;)11 = v" I}V< ,)-'" - V I}+'" I}A I}- ~'2 j P ijk 2~Yj xv. (~y)2 ' (III.2.61) _ I " ( V" vn V" - - J p V =P .W -" ijk+1- i}k-l iik+1- 2V'k' +V~ ~"'I " / V . I) I/k-l - k I}: 2&k z (&k)2" Vai cacxapxinhuv~y,chUngtac6M~nhdesau(xem[31][32][33][34][35]): Menhde2:Nghi?mtlmduqctUcacphUOllgtrlnhd':tng(III.2.58)x[p xi phuang trlnhvi phantuOllglIngv6ib~c0(&; ,~y:,lvlax{;].:;;i~}). 1\Thuv~ychungtadlikhio satdugetfnhxa'pxi v~.6ndinhcuacaephuO'Ilg trinhlienqUaildenloantir ~. D~giii duqch~n'ay,chungtal'anluql QuatUngh~ -62- ~ 3: ~ d ~ ~ 4«9 ~ ~ -d &k~ ~ ;D~ {tM~ phuangtrinhphaufa ve d~g phuongtrlnhd~iso tuyentfnhbaduangcheo chfnh,titd6apd\lngphuangphaptroydu6i. IJngvOiroantir1\chungtac6h~ (IU.2.62) V_" V-"-l v_" v_" v_" J V-" +v_"" k - "k '+1'. - ' l 'k '+1,'. - ~ " k ' l 'kIJ IJ +-" I J" I-J 1. " IJ I-J _ 0- U'k -v ~ -, ~ 'J' 2&1 xy (&)~ ho~cviet dumd~ng3duem.gcheo (IIL2.63) ( '-," J l " 2 1'\ ( 1" J v U"k -. V , -~ ,v .l" k -xy L- 'vn" - r'j +_,IV~ + xy +~ vn , = l (L~XJ2 2Axi "'I' (Ax,)' !Jt) 'fr l(Ax,)' 2Ax, ,-ljl- V~-l =-~ 6t Dat GII.2.64) -"v u", A= xy-~ (&)2 2L\X;' (UL2.65) -" v~. Il.J';"B= ~.' +~ (ill;J2 2D.xi' (UL2.66) 2v 1 c= xv+- (&:)2 ~ ' (III.2.67) ~m-1 F=~ fJj' thih~(III.2.63) du<?,cvietl~i (IIL2.68) Av-n rri.l"+JJV-" - 11" +l'k -\...~" ' l 'k --.1.I J 'J" ,- J Tuangtl!truemghqplieUa phanchuthichcuam\lctren,lieUcodieuki~n 2v (IIL2.69) IU;~kl<&~ I thih~troydu6iCUL2.68)dnd~nh(xem[5][40][71][74]). -63- ~ 3: ~. J Ut UM ~ ~ ~ 1M U Ut ~ ~ ?a~ flM.~ BangcachtuangtV,chungtaclingc6 duQ'cdieuki~n6n dtnhchoh? phucmgtrlnhcholoantir~, ~ nhusail: (JILl. 70) 2vxy 1 -" 1 <-, Vijk ~Yj (JILl. 71) 2v I _,. 1 <-2. Wijk &k,. Titd6chungtacoDinhly - -- DinhIv2: N~ucacdi~.uki~n(II!.2.69),(1II.2.70),(111.2.71)thl h$ phuong trlnh(111.2.62)vacach~phucmgtrlnhchotoantv ~, R3c6thegiitiduQ'C. Riengh$ (III. 2.40) - (111.2.43)(ho~cd(;lngt6ngquat(1II.2.32),(111.2.33» chUngtatinhloannhu sail: . . Tich phanphuangtrlnh(IIL2.42)theetrl;lcOz tudayle.nm~thoang,taduQ'c: (111.2.72) P2(x,y,z,t)=-pg(11- z)+Pk<;(x,y,t), saild6thayvaocacphuongtrinh(III.2.40),(1II.2.41),(IIL2.43)vathemdieu ki~nd(>nghQc(IIL2.10),chUngta co h~b6n phuangtrlnh sail dayde (im Up V2'W2'Th: (IIL2.73) ~ 1ap. au2- gd11i+- k<;+lv2=0, at . ax p ax (IIL2.74) ~)2 - g8,,2+~a:q -lu2=0, at By pay (IIL2.75) au2 8v2 aW2- 0-+-+--a ,x oy oz (!ILl.76) On rlr, ~ -.!1.+u l ~+V I 0rJ2=W I . at -1'12 ax -1)2 &v -T)2 - 64- ~ 3: ?1!4ed d..ul'4<M4«9 4«-~ ~ td tit ~ ~ t:'~ {1M.~ D~giii h~phuongtrL.'1hCUL2.73)- CII1.2.76)nay,ngoaicaedieuki~ncho v~ toe iT,chungtac:mco themdieuki~neho112: (IIL2.77) nn-l=","-1,2 .J Caed<;tohameapm9tduqcxapxidumd~g: (IIL2.78) ~""in -"7" -'" oV I - Vi+lift- Vi-lift-I - Ox I 2&;oF' (IIL2.79) ~-I" V-n Vn oVI = ij+1Jc- ij-l~, 8yLik 2,6,Yj (UL2.S0) ~ .In -" -"oVI V, +1- V" 1- 'J~ 'p,- 8zI - 2,6,z liift k D~lllgsaiphancua(III.2.73)- (111.2.76)duqcviet.Mu sau: (III.2.81) ~ - ,,-1 ,,-1- ,,-1 p;' - p ~It , lL. TI2" 1 ' ", 2. 1. 1 ;.-.' 1. ;'r. 1. 1""J" ""jl< "'J ':'-1 + ..~'TJ '."-j +1 ",- 0-(' ' - V = {\+ .:;, ')1\, .'i\' if Zijk ' Ul ._Lj.:.~; p L..:..\-~i (III.2.82) " ",-1 ,,-1 ,,-1 p." - p " V?'ijk- V2ijk- a Tl2ij+!- Tl2ij-1+1 .~qij+lkqij-lI ."=0 A + 0 ,..,A ?~ ;P2;j,!; - , Lli ,,-,tiYj P - Yj (lU.2.83) ,~ " " ." " " U~.+I.' -U~. l ' V~..+lk -V~.. l ' W" +l -W".., 1 w; J.< -,- JI<+ ~'J ~'J-!A + ~;]'< "'<JI<-= 0 2ffii 2,6,Yj 2,1z~ (III.2.S4) ",,-1 ,,-1 .,,-1" ,,-1 ,,-1 112ij-1l2ij + ! llZi+1j -llZi-Ij, 1 llZij+1 -llZij-1 - I ' U! .-1 -r-VI .-1 - W "..1M -1]2;) 2&. -1]'211 2~y . -1]2/), J V6i cachsaiphannhuv~y,chUngtacoM~nhde (xem[74]) Menhde3: H~(III.2.82)- (III.2.84)xapxi h~(III.2.73)- (III.2.76)caphaithee A1:p,6,yj,6z<'CaegiatIi u2,V2'W2'Tl2duqctinhm9teaehtUO'ngminhfitnluqt titcaephuongtrlnh(III.2.81)- (III.2.84). -65- ~ 3: ~ ~6.u ~#<iiK~ ~ ~ --u Ut ~ ~ iD~ ft4t~ III. Mo hlnhd(mgIl;Ich-..ct~ phuangtrinhSaint.Venantl-chieu IILI Mo hlobbattmin 1110ngthuang,dtitfnhdongchaykhong6ndjnhtrenh~thongsongkenh ciiavimg3nhhuangthiiytrieu,nguCri.tasird1;lllgphuangtrinhSaint-Venant1- chieu([32][33]),trongtruangh91>n'ayinh huangcuamasatnhOtbi dKboqua vaxemS1!masatcuachAtlongvathanhranla dangk~.6 dAy,domuonxet hi~uUngnhOltacd\>ngleudongchay,de>ngthaixuAtphattitphuongtrlnhd\>ng 11!chcNavier-Stokes,phuOngtrlnhbaoloankh6iluqng,chungt6iduafa duqc m\>tmohinhtlJaphuangtrinhSaint-Venantl-chieu,trongd6c6s1!thamgiacua thanhph'AnhOtrongphuC1llgtrlnh[20]. ChUngtasirdl:1ngm\>ts6giathi€t saildAy: - ChAtlongde>ngchAt,khOngnenduqc,danghuang, - Ap suAt la t:.lJ.iiytinh. Khi d6tith~phuongtrinhchuy~nd<)ngt6ngquatbaagomcaetensorli'ng suatnhOt't~chUngtaco(xem[38]): (Il1.3.1) au au au au 1ap 1( at~ at~ Ot~ ) -+u-+v-+w-=---+- l-+~+- +Zvat ax By oz pax p ox By 8z ' (III.3.2) 8v 8v 8v 8v 1op 1 ( 81:~ 01:; O't; J -+u-+v-+w-=---+- -+-+- -lu at ax By az pay p ax oy az ' (III.3.3) ap az=-pg vaphuongtrinhbaaloankh6iluqng (III.3.4) OU 8v Ow -+-+-=0. ax 8y Oz (trongd6 I =2msincDd~ctnmgchoIlJc Coriolis) -66- - e,~ 3: ?It#.Ii.~"¥~ 4«9 4a ~ ... d Ut ~ &JWd.fT:1~ {I4t~ MienkhaosatQ cod~g: Q={(x,y,z)/Xd~x~XC'I:t~y~I:, -h(x,Y)~Z~ll(x,y,t)}, trongdo,X,j'Xc' Y.t,I: chicacc~ntrenva dumcuamientheoh1;1cOx,Oy. Dieuki~ntrenm~thoangvadumdayduqcchonhusail: (III.3.5) wi =DT1+ul DT1+vlDT1 1] at 1] ax 1]ay' (I11.3.6) ak ok W I =U I - +v l - .-h -h a. -h '"' X oy Lull Y d day,doch9nt:I¥ct9ad<)cohuangOznguqcvmhaibaitomcuahai ffi1;lCwac, lienu, v, w duqclAyt4i cacgia tti nguqcl~icuaz=T)(x,y t)va z=h(x,y). III.I.! Hephuon~trrnh2-chieu Chungtagiathi€tthemdingd<)sauH =1l+h cuaml;lcnuackh6ngdang kesovaim~tphingflamngang,c6ngmala H « min{Xc- Xd, J: - ~}. Khi dochungtacoth~duavaocaed~iluqngd~etrungchos~phAnb6 -- -- v~ t6etrungblnhtheoehieuthingdUng: (III.3.7) 1] U(x,y,t)=Ju(x,y,z,t)dz, -h " (III.3.8) 1] V(x,y,t) = J v(x,y,z,t)dz. -h Truaehet,chungtaxetphuongtrlnhlient1;le(III.3.4).Tfchphanphuon trlnhn'a.ytheoztU- h(x,y)denl1(x,y,t)taduqc: (III.3.9) 1]au 1J.8v 1]8w f~dz+ J -dz+ f~dz= 0, -h ox -hfJy -h8,. - 67- ~ 3: ~ ~ taL.u.m ~. ~ ~ fIt4- U Ik ~ ~ ~~ ~ ~ X6ttfchphftnthunhifttrong(III.3.9),ta CO 1) au a 1) Br1 BhI-dz= - Iudz-ul1)--uLr.-,-hax ax-h ax ax ho~c qII.3.10) IT) au au Or] ah-dz=--ul ~-ul -, -haX ax 1)ox -hax tll<1ngn;rtfch pha.n'thuhai .. (I11.3.11) ~8v av Or] ahJ -dz=--v\ --vi - -hOy Oy 1)8y -hay va tichphanthllba trong(111.3.9) (111.3.12) raw dz= wiT)-WI-h'-h8z Do di~uki~n(111.3.5),(II1.3.6),suyra (I11.3.13) 1) 8w . Or] Br1 Or] ah BhI-dz=-+ul -+vl -+ul _+vl - -h Bz at T) ax T) ay -h ax I-h 8y Thay(III.3.10),(III.3.11)va(I11.3.13)vao(I11.3.9)taduqcphu<1ngtrinh cho11,U, V: (1II.3.14) Or] au av-+-+-=0, at ax 8y VOi caek~tquatren,chungtacoM~nhd~ Menhde1:PhU<1ngtrlnhbaatow kh6iluqngtrongvungnuacnongtheovipl t6cU,v vad\)sa.uH cod<;lng: (I11.3.15) aH +au +av =O. at ax oy -68- ~ 3: ~ 4J i<u~ ~ ~ ~ --d tUt~ ~ 'P<f-~ pM.~ D6i v6iphuongtrinhchuyendQng,chUngtaco phuongtrinh(III.3.1), (III.3.2)vmdi~uki?n(III.3.4)duqcvii!tl~i: (III.3.16) au+au2+auv+auw=-!op+! ( m~+a't~+m~ J +IV, at ax 8y az pax p ax 8y 8z (III.3.17) av+ouv+av2+avw=_.!ap +.! ( m~+ 81;~+ a't; J -IU. Of ax ay az pay p ax iJy az TichphAnphuongtrinh(III.3.16)theez tU- h(x,y)denll(x,y,t)taduqc: (III.3.18) 1) au 1)au2 1)auv 1)Ouw r-dz+ J-dz+ J-dz+ J-dz=~hat -hax -IIiJy -haz 1 1)a 1 ( 'r8't1 1)81;1 1)O't1 J 11,. = --=-J-.E.dz+-=-J --.ldz+ J~dz+ J -2dz +J lvdz.p -hOX P -hax -hay -haz -II ChUngtallin luqtxetcaes6h<p1gtrang(III.3.18), Soh(;lngthY:nhat 1)au a 1) OT] J -dz=- Judz-ul -.at at . '1)at-t< -t< hay (III.3.19) 1}au au OT] J-dz=--u! --IIat Of 1)at Soh(;lngthY:hai --- .. (III.3.20) 1}au2 a 1) OT] ah J -dZ=- J U2dz-U21 --u2 1 -.ax ax 1} ax -II ax -II -h .. Caethanhphanuvavcothephanthanht6ngcuahaithanhph1ln (III.3.21) 1 u=-U+u', h+ll (III.3.22) 1 v=-V+v' h+ll - 69- ~ 3: ?14ed Gal.~ ~~ ~ ~ - t4 Ut.~ ~ ?~ ~ ~ trongd6, u', V' la-cacd~ilm;mgtrungbinhcuau vav. TichphtUl(III.3.21),(III.3.22)theez tadu<Jc 11 5u'dz=O-h va 11 5v'dz=O. -h Suyra 11 11 [ 1 2J]u' ] 5u2dz; 5 2U2+U,2+- dz -h - (h+T)) h+T) ho~c (III.3.23) 11 U2 11 5u2dz=-+ fu,2dz -h h+T) -h Thay(III.3.23)vito(III.3.20)chUngtanh~ du<Jc 11ou2 a v2 Or] oh f-dz=---u2 1 --u2 1 - -hox axh+T) 11ax -h ox' ho~c (111.3.24) 11ou2 0 U2 Or] 8h 5 oxdz=oxH -u2111ox-uth ox.-h TuCII1gh;T,so hq,ngthrtbatrong(III.3.18)ciingdu<Jcphanthanh . (Il1.3.25) f 1l ouv 0 UV 0 5 11 Or] I ah -h 8y dz=ByH +8y-hu'v'dz-uvlllBy-UVI-hBy. ConSohq,ngthrttutrongvetraicua(III.3.18)du<Jctinhl~inhu sau: J ouwdz=uwjll-uwl-h,-h 8z titdAysuyra (III.3.26) f 1l ouw r Or] Or] 0rJ ) ( oh ah )-dz=u! -+ul -+vl- +ul ul -+vl - .8z 11 at 11ax 11::n} -h I-h ax -h ~}~ , ~ ~ - 70- e.~3: 11Wd-~ ~ ~ bp:,~ -d Ut ~ &Wt4 p~ ~ ~ Tieptheochungtaphantichvephaicua(III.3.18). Truachet,tit(III.3.3)taco: 1) "" 1) f opdz =- fpgdz , OZ -it-h guyfa (IlL3.27) p(x,y,z,t)=-pg(TJ- z)+~q' Thay (JII.3.27)vaosohqngthz(nhattrongvephaicua(IlL 3.18),taduqc 1)op 1) ( 0Tj ) 0Tj 1) f~dz=- f -pg~ dz=-pg~ fdz, -hffi: -h ox oX-h khido, (IIL3.28) 1 1)op 0Tj - f-dz=-gH-.P -hOx Ox D6iv& cactensorlingsuat,chUngtatfnhtoinwongtt;t (III.3.29) ~ . 1) r 01:~dz=~~f1)1:1dz - ~( t11 0Tj +t11 Oh )"" a 1 1...' ' a 'P '" ox P x P ,r; ox . -fl X-h -h "' (III.3.30) ~f1) m~dz=~~J~1:1dz-~(t~1 0Tj +1:1 1 Oh ] , ;}., "" 2 - 1);}., 2 -h ;}.,P -hv)' P OY-h P v)' . v)' (IlL3.31) 1)~_1 1 1 1 1~f~ dz=t3 1)- 1:3-h'P -h OZ Deiv& thankphanlifcCoriolis,dol=l(x,y)nen .. 1) 1) flvdz=1fvdz, -h -h guyfa (IlI.3.32) 1) J lvdz = lV. -h - 71- ~ 3: '114e444&uu ~ ~ ~ -edUt.~ bWd ?~ Idt ~ Thay (111.3.19),(III.3.24) - (111.3.26),(111.3.28)- (111.3.32)vao(II1.3.18)chung nh~dugcBOdesau B6de1:Phuangtrlnhchuy~nd()ngtheeUtrongvUngnuacn6ngc6d~g (1II.3.33) au au2 aUV Or] a ( f T},2 1f T) 1 ) -+--+--=-gH-+lV+- - u +- tldz + at axH ByH - ax ax -h p -h +~ ( _ru'v'+-.!:.f1}tldZ ) --.!:. ( tl l Or]+t1 1 Or]-tl l ) - By .)-h P -h 2 P 11}ax 2718y 371 1 ( 1 1 ah 1 1 ah 1 1 ) -- t -+t --t p_--l.-hax 2-hBy 3-h . Tllangtt;r,chUngtanh~ dllQ'CphuangtrlnhchoY: B6de2:Phuangtrlnhchuy~nd<)ngtheeV trongvUngnuacn6ngc6d~g: (111.3.34) av 8 UV 8 y2 H Or] lOT 8 ( I 7I , 'd 11J r 2d ) -+--+--=-g -- +- - uv z+- t z + at ax H ByH oy ax -h P:h1 a ( I 1},2 1 J 1) 2 J 1 ( 2 1 Or] 2 1 Or] 2 1 ) ' +- - v dz+- t dz - - t - +t - - t - ~, 2 1 1/ax 2 1/ 3 1}v)' -h P-h P OJ 1 ( 2 1 ah 2 1 8h 2 1 ) -- t -+t --'f' P 1-hax 2-hBy "3,-h ChUngtaduavaocactensor (111.3.35) T} 1} 1} 1} N; =I t~dz-pI u,2dz, N~= I t~dz-PJ v,2dz, -h -h -h-h 1} 1/ NJ2= N~= J t~dz~PJ u'v'dz -h -h vatUB6de2,Bdde3chungtac6M~nhdesau -72- e~ 3: ~ d.I4t U<i.-~ ~ ~ -d Uc.~ bUIt4?~ ~ ~ Menhde2: Giasu (i) M6i trubngla ding huang't~='t;, (ii) H~s6nhOteuaeh~tlongtheecaehu6'ngIanhunhauvz=VX'J= v, (iii)TensorN; duqex!pxidumd~g: 6 1 au z av Nl ( au av ) (IIL3.3 ) Nl =2vp-, Nz=2vp-, z=vp -+- , Ox By By ax (iv)ungsuiftrenb~m~tfnhtheeUngsut(tgi6[26], (III.3.37) { 'tll17=1/p)v/ eoseg, 'tzl =Y zpw2sine17 gag g (v)ung sut(tmasat~ daytfnhtheemasatella'dongOndjnh, r g U(U2+V2)Yz I 'tll-h=2" P Hz ' i C V(U2 +V2)Yz I g I 't2 -h =c2 P Hz (vmPakh6iluqngriengcuakhongkhivac Iah~s6masatday) (III.3.38) Khi do,h~phuO'J1gtrlnhehuy~nd<)ngnu6'enOngcod~g: (III.3.39) au+~u2+i. uv=-gHfJr]+N+V~U+V~ ( au+av J - at axH 8yH ax ax ax iJy 2)h ~ g u(uz+V +1/ ~YV~/coseg, - Hz pCZ (III.3.40) av+~uv+~VZ=-gH8rt-lU-VI1V+v~ ( au+av ) - at OxH ayH fJy fJy Ox 8y v(Uz+vz)h -2- +1z~wzsine CZ Hz g p g g _TJ. - ~ 3: ~d- 4&~ ~ ~ ~_d Ut~~ 'P~fdt~ trongd6, I i l 8rl i l 8rl ' I (°- 12)'t' l =-'t1 --'t2 -+'t3' l-, "TJ 'TJox 'TJf}y 'T) I , j ah , I ah" ( ) 'ti-h ='t~-hax+'t~-h f}y+'t~I-h' i =1,2 Nhuv~y,M~nhde1vaM~nhde2 chochungtah~phu<1I1gtrmhli~nUJ.c chuy~nd<ingd~xacd!nh<;.hieucaoH ciingnhuphfulb6trUCingv~ t6cU, V trongm~tphlingOxy.H~caephuangtrlnhnllykhacsov6i h~phuongtrinh Saint-Venanthaichieubchb,e6thamgiacaethanhphllnd(;tohamb~chai. III.l.2 HephuffilitrinhI-chien Dngd(;tllgm<itehieutVaphuang trlnhSaint-Venant,chUngta gia thi~tday eualong clane6 d(;tllgnila hinh lang 1'1¥,stfthayd6ibem~tvadaytheo1'1¥cOykhongdangk~.Chungtac6d~dang B6de4: GiasirV =o. u lahameuaxvat.lucCorilolis1=0' lue fli6 WE!=0,H 'H " . '-' Khi d6h~phU'<1I1gtrlnhtronglongdAnc6d(;tllg: (III.3.41) aH+au=0,at ax (I11.3.42) au 0 U2 Or] g vial iYU a2u -+--+gH-+--=2v-+v- at oxH ax c2H2 ox2 8y2' (III.3.43) gH Br1- a2uv0 02 . Dinh Iv 1: Cao d<i11euam~tthoangva luu IU'<JllgQ cuadongchaythoa (III.3.44) Br1 aQB-+-=O at ax ' (III.3.45) 1 aQ 1 a Q2 Or] QIQI 2va20--+---+-+ ' =-- gAat gAaxA ax c2A2RgAax2 -74- e~ 3: ?It#!d ~ ~ ~ ~ ~ -cdUt ~ ~ JP~ pk ~ !Tongdo, -A ladi~ntfchm~tcatd13.longda.n, -B litchieuf<?ngm~thoang - R =;labankinhthuyl1;lcualongdiin, Chung minh ChUngtaduavaoham(j cod~gthactri€n tUhamUnhusail: (IIL3.46) U(x,I)'={~ v6i nhftngdi€m bell!Tonglongdiin v6i nhfi'ngdi€m benngoailongdiin Tfchphfulphuongtrlnh(IIL3.4I)'theey tU-ooden+00, chUngtanh~nduqc: 00 aR f 00au J-dy+ J-dy=O-00at -00ax ho~c (IIL3.47) a m a ": -- J Hdy+-:;-J Udy=0at-m ox-m Do Q la hmluqngdi quam~td.t co di~ntfchA vaothCridi€m t,thl (IIL3.48) 00 Q(x,t) =JUdy ~oo va (III.3.49) 00 A(x,t) = J Hdy. .. -m Thayvio (III.3.47),taco (IIL3.50) 8A + 8Q =O. ot ax -7S.. ~ 3: ~ d '" uu 4119bp;~ - u Ut~ ~ ?~ ~ ~ Nell qUaDt.ftmtheo1T1;ICOZva g9i l;=l;(z) la chieur()ngcua long dana caod()z,tIll aA a T} 1/ a~ ilr1 ah - =- J~(z)dz=J -'2dz+~(l1(X,t»)--~(-h(x»)-.at at-I< -I<at at at (0 dAydogiathi€t, n~nc6th~coi 11=l1(x,t)vah=hex»~. Do ~khOngph\! thu\>ct va ; ~i -h bangkhOngn~n aA ilr1 - =~(l1(Xt»)-.at 'at B lachieuf\>ngm~thoeing,nen;(ll(x,t))=B. Khi d6,phuongtrlnh(111.3.50)c6 th~vi€t l~idu6id~g cua(II1.3.<!4). (I11.3.51) D6i v6i phuongtrlnhchuy~nd()ng,chUngta lamtUongtlf. Ti:rphuong trlnh(I11.3.42)suyfa (III.3.52) "'au coa u2 m ilr1 00g vial J-dy+ J--dy+ JgH-dy+ J2~dy=-mat -wOxH -w Ox -",c H w a2u m a2u =]2v~dy+],,~dy -'" ax -m 8y . X6tIanluqttUngs6h~g vetniivaphaicua(III.3.52).Truochetchungta x6tca.c86h~g cuavefrii. Sohqngthu:nhdtdo (I11.3.48) (IIIi3.53) Jau dY=~JUdY=QQ. -OJ at at-'" at Ti:r(III.3.45)sohqngthu:hatc6th~viet '" au2 a '"u2 a '" ~2 ]--dy=- ]-dy=-]U Hdy,-",OxH ax-00H ax-CD do giathietneufrenva do (I11.3.49)nen, '"a u2 a ~,m a ( u ) 2 J --dy= -U~ f Hdy=- - A,-mOxH Ox -00 ax H -76- e~ 3: ?1t4eJ- ~ #u, ~ bp:;~ -d !.k~ tJWd7'~ pMbP1i vadoQ=U%, cu6icimgchUngtanh~ duqc (III.3.54) CD a u2 a Q2 j--dy=--,-CDaxH ax A D6i v6i so hq.ngth((ba,v6i cacgii thietneutren,nen:kh6ngphI thu<)cvaoy, taco (III.3.55) f gH 8rldy=g 8rl jHdy= gAm,. i::CD ax ax-<D ax Sohqngeuoiclingtrongvetriiicua(III.3.52),chungtacoth~viet fCDL UIU!dy=Lfulu!dy=L fU!U!dy2 H2 e2.1 I e2 I-- -a>e -(D 1Ong~ ho~c feD g vial g -1 - 1 f gul l V I I ~-dy=-UU dy=-;;---B. -<De2H2 e2 I IOngd.~ne~H ,HI V~y (III.3.56) fa>L UIU!dy=L QIQIBe2 H2 e2 A2-CD BtlygiGchUngtaxetcac s6h~g cuave phii cua (I11.3.52).SO'h(;mgth nhatduqcbiend6inhu gall: (III.3.57) (D a2u a2 (D yPQ J 2v-d y=2v- J Udy=2v-ax2 ax2 ax2'-(D -a> Sohq.ngthahai duqcguyfa tit(III.3.43)do:=0 nen (III.3.58) riu V fJy2=o. -77- ~ 3: ?/tfe4J ~ U<fUt,~. ~ ~ .-u Ut.~ tIUMI. 'P~ f'4t~ Thay(1II.3.53)- (I1I.3.58)vao(I11.3.52)chungtanh~ duqcphu<1I1gtrinh chuy~nd\)ng(1II.3.45). Nhuv~y,DPili 191chochUngtah~phuangtrlnhtmhloancaode>11va lUll luqngQ cuadongchayme>tchieutlI<1I1gu;rphuangtrlnhSaint- Venant, trongdocotfnhd~nhi~uUngOO6t(xuAthi~ntrongthanhphAnd(!.ohamb~c2). 111.2Phmm!!phap ~iais6 D~giaih~(III.3.44)~(111.3.45)chUngta sirdl;lngphuongphapkhai tri~n ti~mc~ theothams6 be [48][69].D~tf;=2;; , giasircaehamQ va 11duqc khaitri~ntheolily thuacuaf; OOusail: (III.3.59) CD Q(X,t,f;) =LQ",(x,t)~"', =0 (III.3.60) CD 11(X,t,~)= L11",(X,t)f;'" ",=0 v6'if; dunhod~coth~coiSign(Q)OOula Sign(~). Thaycacchu6i(III.3.59),(III.3.60)vaoh~(1II.3.44),(III.3.45)saild6 d6ngnhAtOOftngh~s6ciingb~ccuaf; clingv6'idieuki~nbienva d:iuthich hqp,chungtaOO~-duqccaeh~phuongtrlnhsaildayd~tinhcaeh~s6cua chu6i(III.3.59),(III.3.60). D6i v6'icach~s6Qo,110chungtacoh~.phuangtrlnh (III.3.61) B OrJo+ aQo =0,at ax ~ a~+~~Qo2+OrJo+Qol~1=0 gAat gAax A ax c2A2R (sJ Dieuki~nMU ciingOOudieuki~nbiencuah~naytrilngv6'idieuki~ncua h~phuangtrmh(III.3.44), (1II.3.45). -78- e~ 3: ~ d 6..uUeUt,~ ~ ~ -U 6<k.~ ~ p~ ~ ~ D6ivrncach~sO'Q"" 11""m=1,2, ...chungtacoh~phuongtrlnh (111.3.62) B Or]",+ 8Q", =0 at ax ' ~oQ", + 2 ~(1Q", +811",+21!20IQ",= (Sm) gA at gAox A . Ox c2A2R p",(!20,Qp...,Q",-J vmcaedieuki~nbienvadllud~g thumnhat.Trong(1II.3.62)thanhph1lnve phaieod~g: (II1.3.63) ~(!20)=~02~A ox2' d6ivmnhUngm~2 taco: (111.3.64) pJC20,...,Q",-J=~a2Q~-1 L: [ ~~ OrQs +QrQsSign(~) ]A Ox r+s=", gAox A c2AIRr~l.>'~l ChUngtanh~ thaybaitoan(So)d~g (III.3.63)l?iphi tuyen,trangkhi bailoan(8m)d~g (111.3.64)littuyentfnhvad~giaibitiloan(8m)nftychungta c1lnphai co tatca cae 1mgifu cua cae bill toan (sJ, i =1,2,...,m-1. Trongmts6baitoan,ch!ngh<;tn[43][47],thams6nhi~us xuathi~n nhulitmt?th~s6becuas6h~g phi tuyenvmd<;tohamcapthap.Truemghqp khih~s6nftyxuathi~ntrongs6h<;tngd<;tohamcapcaobon,ketquatinhtoan cho1mgiai6ndinhbon(xem[48]). D~giiiicaebititoan(s(J ), (sm)atrenchUng~asu dl;}ngphuongphapsai phfulvmd()xapxi capm()theothaigian(M) vakh6nggian(l~1ax{!hin. ~(so) D 1 1" '\ !h.ata=- 'Y= va 11..= !... . gA' c2A2R I M -79- ~3: ~44I.4t~ ~ ~ ~ _d Ut~~ ?~~~ Baitom (So)dugcvi~tdumd~g saiphAnnhusau: (I1L3.65) Bn ( n - n-l ) /Y' _/Y' =0i Yi 11oi 11oi +~i ~H , a~A,(nn, - nn,-I)+11n,- T)n, +a~ [ (~J2 - ('4:-1Y ] +Llx,y~nn,lnn, 1 =0I I ~I ~I 01 0,-1 I lin lin I I~' ~I L~ '<~-1 c6 th~tuy~ntfnhh6acac8,(5h~g phi tuy~nnhusau: (II1.3.66) (g;J2 ~ ~i-lg;i' (III.3.67) ~I~il~ ~¥2~i-ll. Khi d6h~phuangtrlnh (III.3.65)duqevietl~ (III.3.68) { nn - F: n +/Y' E n-l~i-l - j Y;T)o; ~; - j Y;T)o;, nn-l n - n + n - 1i~ = n n-l 11oi T)oH XiQ;;i a; '~l ~-l ai AiQ;; trongd6, (111.3.69) 'I/~=a~A.+a~Q;;;-l+~, y~ l nn,-l l . to, II lAin II~I Bi~udi~ntheed~ngtroydu6i,h~(II1.3.68)dugcvietl~i: (III.3.70) { ~i-l =Li11~+1;~i+~i' T)~= P;Q;i + R; trongd6caeh~86troydu6ic6d~g: (111.3.71) Li =BtAi' (111.3.72) 1;=1, (111.3.73) 5;=-B'/AiT)~~t, -80- ~ 3: ?1t1ed 6.uuu, ~ ~ ~ - td Ut ~ PWd jD~ fJ4t ~ (III.3.74) p a~IY'-1H -X. +~-1 p= I A", ' 1 1-AB~P - Ap;~If:nTl-l'. , .-1 . ~.-1 ~"-l (III.3.75) RH - A.If:P ,,71-1+'1 nn-1 A,a~FIY'.-1, . ,-1'10. J\,.~ - . , ,~ 1 71-1 ~= 'K ,,~,-1 - " X.a~If:nTl-11 A.B.P - . . ,~HI . ,-1 ~~1 Nhu v~y,phuangtrlnhtroy du6i (III.3.70) vOi caeh~so (IIL3.71) - (III.3.75)chophepchUngtaxacdjnhduQ'ctiltca,cacgiatq g'ftndungcuabai toan(So)khibietdieuki~nbienvem1!cnUde". III.2.1 B?dtmin(8m) Tuangtl.;lbaitoan(So),phuangtrinhsaiphaneuaphuangtrlnhthunhat trong(III.3.62)c6 d~g hoanlOangiongphuongtrlnhthunhatcua(III.3.65), Conphuongtrinhthuhaidoe6xetthemvephainend~ngsaiphanc6thayd6i: S;Yi("::"-,,';:2)+Q::U-0:-1 =0, (III.3.76) a~Aj(Q: - Q:::;l)+,,::..-"::"-1+2a.~(' ~Q;,.- ~-IQ::U-1J +'..1~ ..1"..~ "~-1 +2.6.x:v"IIY'. IQ" = A" lY'., , I, ~I "" LlA.1 .,, "" trong d6, (III.3.77) Q~Q;: Q~-IQ~-1 l 2 I " ~ - 71 p" = ga~0 Q"'-1 - ~ a~ A; A:-l - '\f~Q " QTlSign(QTI.)MI ':::I 2 L..J, A" I, n $1. 01 ux j r+s=", LlAj r~l; .~1 - 81 - ~ 3: ?!Ue.~~ uu 4419lip:,~ -- r,.lUt~ ~ ;D~,ut ~ D~g troydu6icuah~(III.3.76)nhusau: (III.3,78) {Q n. 1 =L:nn.+Tnn.+S.,J>U- ,""" ,~ , 'nn.=PQn.+R., 'J>U "... , vOicach~s6 (I11.3.79) L. =EA"I I,. , T =1,, S. = -B".A.'nn~l,, I ,""" (I11.3.80) . 1';-1- a~Aj- 2~jY71Q;'il-2a7( Q;;j - Q;;i-l )P.= ..1;' ..11', L'i L'i-1 1- "IBnp 2/..,.ansnnn. ' /\,. . . - ",~,-1, , ,-I ~~1 (I11.3.81) R. - ')".EP 'n1H+A.a~n7l~1-2Aja7B;Q;;i-l'nn~l+ lxxpn. ,-I ",-I""" "~I An ',"" "'" ~= ,-1 1- A.E P - 2A;C1.7B;~-1 I I 1-1 ..11' L'i-1 Nhuv~y,bangcachtUCJngtqbaitoan(So),chungtat1mduQ'C1mgildxAp xi cuabili tom (Sm)'Cu6i cimg1mgiiii cuabai tom (III.3.44),(III.3.45)~i nhfi'ngdi~mrmr~c(Xj,tJ seduQ'cbi~udi~nquat6ng1mgiaicuacacbaitoan (So),(s,J nhu(I11.3.59),(III.3,60), -82- e~ 3: ?J4ed ~ uu. ~ ~ ~ w4U Ut ~ ~ iD~ ~ ~ IV. BaitminIantruyenvakhuechtancUanguongay0nhi~m IV.l M6 blnbbili tmin D~khaosatbailoanveSlJIantruyenvakhwichtancuanguon6 nhi~m, chUngtagiasu<p(x,y,z,t)bi~udi~nIuqngnhi~mbin duqcIantruyenvakhuech tan detheeqiiy d~ocua caeh~tmoi t.n1emgehuy~nd'ing vm.v~ t6e V(x,y,z,t).MienkhaosatQ duqecoiIad6ngchat,kh6ngnenduqcvaduqcbaa bcbbi m~txungquanhL:: ,m~ttrenL::o vam~tdum.day2:H . Khi d6bailoanm6phongslJIantruyenvakhuechtin euanhftngngu6n g~y6nhi~mc6d~gnhusail(xem[22][23][24]): (InA.l) c<{) C([) em O([} 0 C(f) ~ +u-=- +v-~+¥v--'- =ULVD+-V---'-+ i , at ax ev Oz" aT GZ ' Y{V",.~)~n \.-I', In,T; 'i\-~'f"~iC:"""" ;;':<-"~l' trong d6, - Lt,V, 'vi! ha' tb?;-t, Phi.:n "'1'!'1V""'r'tor H:'in ..lU<C lito,' " "- hUlH '-~- '-~... ,_v, ': . , \"""J ,-: - 1-1 h~sokhuechtand9Ctheotll;lcOx, Oy, h~s6khuechtin d9CtheotI1;1cOz, nguongay0 nhit~m. - v -] D~giaibailoannay,ngoaiphuangtrinh(IlIA.!) chungtacanthemdi;.~u kiendati (III.4.2) <p=<Potrong Q, khit=0 'T>; ,~;;{' rt'l':-'\p ]:1 &-n hi ;5.p\.. -~ ,~ ~'- .~,':'H ~.'-... ,TTT J. "\..11. , ..J J q) ::::(j)s tren L:, t E (O,T]' .. i~,. ,1 i " \"U,J..'-T.-rJ D<p /. .," )'- t -., ( i T'-::;-=O:.<.p iT0il """0' C\-',l I, OZ (('" la ' ha ' m khA no Am ,.ta~c trung' ChosuturVno ta ' c ".AimO~1t""l".~o h,~-,u.., U 0 a , u.' -'.' VHO -< I'j,." -. l'-"'_.'LO '".. ,-,',, (III.4.5) 0«>=0 tren2:", t E(O,T].Oz " - 83- ~ 3: ~ J. tat Uht- ~ tip:.~~.a U Ut ft~, MuM.'PaltaU(k-fd/ m'fbr '---- IV.2 Slfduv"hatB~hiemeRahili tHan BiBbIv 1:Giasu~,v, u:, exIacaegiatr!kh6ng~mva (I11.4.6) w=0 trenLo va Lw rr d'lrong 0, - " {1TI /; '7\\..i.~l ; r11=('7;i) ntfn 11 '>n.. "' " "... ~v... ., ----H - < . " ' ,"" "':':' 1" " " ", ' L o '.. -n';-u l' < r\u~ Or; L- (IlL4.8) - i U" = U;,- Un. Khi do,baitoan(IlIA. I) - (III.4.5)c61aigiJj duynhaL C h11n g min h (xem[80]) . Dodieuki~nkh6ngDenduqcvad6ngchat (IIIA.9) divV=0 cuam6itruemgDenphuongtrlnh(I11.4.1)co th~vietl<;linhusail: 8<p ,- a c<p , , -+dIVV< p=~(D+-V-+ f,at ., ozoz'(IlIA .10) r1h5.nhaive cuaphuongmnh(IlIA.! 0)vill cp,saild6tfchphanthen!tc: oi~TitTf"-l ' ...to 'm (\ 04_~- LCL' '0 " 4,. _4L_4 -- ~:-"-- -,; t ' all"" () "t'roorari"k~ \ l V!.\ ~rl ' Jlh J\'~("""' en ta cc',.'..J..;'h,- ,-" ,'::;:- "~.' "<"" ,',J,-" . " (IIIA,II) (f) 2( ' X "71" )' .(I)2{y,,~nj J .:':,'F'(".'J ',r.-"'... "t""-.I...", . "",1.,'.,.' df:2-1 ""'d~2+ld!'~~(12 .J r) .'.! r;'" " ,,- 02::" -, T { lIf ( o ) 2 ( 0 \2 1 ( a \21 =-I dtI 1;: + -;)tva~)jdO+ +Tr d { U,r <Pacrd') +v( r <Pa<pd) - r<p°cpd')'\ I,l+ r dT! f(i)d~2"' 8 - "' a - ,J' - _ I ' "'," " n \ " - 0'7'" ~0 "" ,.. L. '" L J."o~ -" - /...1'" 0A r!,{«mt7 3: "1'Jt.y-~ii ;:a'(,N,il~ "f.j"t';!;,,-.{~':D'!""i;~i:': ,-'f(~-,'", (bdttyhl16'ngcuavectorphaptuyencuam~t2:0va 2:f{cohu6'ngnguqcnhau, nentrongtfchphiing1lncu6imangd~utim). Khi d6,tfehphantha:batrongvetraicua(IlIA. 11)dl1qcvietl';ti: J dtf (V.ii)(p2d2:=J dtf u: <p2d2:+ J dtf U~<p2 d2:, 0 an 2 0 ao 2 0 an 2 vadodi~uki~nbien(III.4.3)vagii thiet(III.4.6)cuaDinh191,taco ' J dtJ&';'1>' dL> J dtJ U:;' dL;+J dtt~.;; dL;, ' 0 an . 0 ~"': :..",o~"" (IlIA. 12) (IIIA.I3) T£ehphanthtthaitrongvephdicua~.IIA..II),do(III.4.3)c6th~vietl';ti T D( T 8 f d!f.1f<p~d2:=I-lfdtf <ps~d2:. 0 ~ an o. ~ ' On Do dieuki~nbien (III.4.S)nen tfchphan thItba trongvephdi cua . (IlIA. 1I) duqcviet: T o<p (IlIA. 14) fdtvf<p-d2:=o. 0 ~H oZ ' Tfchphtinthututrongvephii cua(III.4.Il) dl1<Jcsuyratir(lII.4A) nhusail: (III.4.1S) T .' a' -' T':, fdtv f<p~d2:=vfdtfa<p2d2:. 0 ~ oz o~- " Thay (IlIA. 12)- (IlIA. IS) vao(lII.4.11)tanh~ dl1qc: , ' (lII.4.I6) ) T + 2 f<p2(x,y,z,TdO+fdtfu,,<pd2:+ " 2 0 ~ 2, (1 .J:) , ,. '-':;'", . -, . - .... T'f { B 000 ] ' a' } T .. +fdtfi (~)2+(~)2 +V(~)2 dO+vfdtfa<p2d2:= 0 ol ax 8y oz 0 Lo 2 T - 2 TOT =f<PodO -5 dt5U"<PsdL: +!-l5dt5<Ps~d2: +5dt5f(pdO 0 2 0 ~ 2 0 ~ an 0 0 - 85- ~ 3: ~ J. &.ituu ~ Up:,~ - e.lUt~ ~ 1D~ ~ au,iM D~chUngminhsq~uynh~tnghi~mcuabaitom,chUngtagiii sitcohai nghi~m<PI'<P2thoaphuongtrlnh(I11.4.IO)cUngvOldieu,ki~,ncTh.u(IIlA.2), cac. dieDki~nbien(IilA3) -(II1.4.5)vacacdieDki~nb6sung(I11.4.6),(IIl:4.9).Khi d6hi~u~=<PI- 'P2thoamanphu<1Ilgtrlnh(xem[2~][24]) . . (IlIA.I?) . . - a ~ 8qi+divV~=~~+ OZv 8z '-at dieDki~n<fAu ~;."!!, . ~=0 trongQ, khi t =0'. vacacdieDki~nbien (I11.4.I9) .n.__,. . '~=O b-en1;,'te(O,T], : . ,~, ~001>tie~'Lo; i e(O,T],az (IIlA.IS) (IIlA.20) ~~0 ~~nLH, te(O,T].az , " D6ivOlbaitomchoham~nhuv~y,thayvito(IlIA. 16),tanh~ duqc: (I11.4.21) J~2(x,y,z,T)dO + J dtJ { ,J(~)2+(~)2 ] +V(~? } dQ 1:- g , 2 0 g L ax oy oz , , . , ., , T U+~2 T +J dtJ-E-dL +vJdtJ a~2dL= O. o',L2 ..0 Lo BCrlvi cacgiatri 1-1,v, u:, a trong(IIlA.21)deukh{)ngAm,Denh~thuc (IIIA.21) chi bang kh{)ngkhi ~=0, conghlala2" IV.3 Phuan~phap~iais6 D~giiiibaitom (IilA.I) - (IIlA.5Y;chUngt6isitdl;UlgphuongphapphAn fatheoquatrinhv~tIy d~duavegiiiihaibaitomsau(xem[79][80]): ~-". Trong(I11.4.1)chungtad~tv=0,1-1=0vab6quanguc'mgAy()nhi~m, d6ngthaichuydendieDki~n(IIIA.~),(IIlA.3), khi d6phuongtrinhc6 d~g phuO'IlgtrinhIantruyench~tgAynhi~mb£n: -86- e~ 3: ~ d t.u~ ~ 4tz~ ..sU &&~ /1ti:Ic4jD~ ~ ~ (111.4.22) o<p OU<p Ov<p i3w<p-+-+-+-=0 at ax By oz ' <P =<Potrong Q, khi t=0, <P=<PstrenL, t E(O,T]. O<P . Oz =a<ptrenLo, t E (O,T], , O<p=0 trenL1n t E(O~T].'"' -..OZ Hill toan(f2) Neu cho u=0,v=0, w=0, phUOTlgtrlnh (III.4.1) co d~g cuam<?t phUOTlgtrinhkhuechtan (IIIA.23) O<p a o<p ~=~<p+-v-+f, ot . OZ,OZ.. <P=<PotrongQ, khi t =0, <p=(PstrenL, t E (O,T], O<p (- =a<ptrenLo, t E O,T],Oz o<p . (- =0 trenLN' t E o,r].Oz D~giii bititoan(III.4.22),(TII.4.23)chungtaphanho~chdo~ [O:T]bai caedi~mt"=nM, '\In=O,[T/MJ.Khi d6i1ghi~mcuacac.'bili:toan.nayIanluqt I duqctlmtrentUngdo~ncon t"~t ~t,,+1. -D6ivOibililoan(Pi) chUngtatim<p~(x,y,z,t"+l)thoaphuangtrlnh .. O<PI OU<PI 8v<PI i3w<PI- 0-+-+-+-- at Ox fJy oz ' (TII.4.24) .<PI=<P;trongQ, khi t =t", <PI= <PStren L, t E (t",t"+1]' - 87- ~ 3: ?14t4i Ut U~ {dt.~ Ol==al!ten 'Lo, te(t"'/"+l]'oz O<pl~0 'tr~nLH:t e(t",t"+1].oz .,. - . .,\ Sail dotlm2(X,y,Z,t"+1)tUbili toan(P2): , O<p 0 o<p1.=I IL).m +- v 1.+f >,at r-T2 oZ 'az' ,,+1 (\ kh ' <P2= <PI !tong ~~, 1t = t", (IIIA.25) 2= <Ps!ten L, t e (t",t"+I]' O<P2= a<p2!ten Lo, t e (t",f"+I]'oz O<P2=0 !tenLH' t e(t",t,,+JOZ . - Dinh Iv 2:Lai gi:ii xApxi <P2titphUc1ngtrli1h(ITIA.25)sethoamanphuang!tlnh xuAtphat(Il1.4.10). C hun g mi nh (xem[23][24]) ChUngtatfchphfultheethaigianphuangtrlnh(ITIA:24)trenkhoang(t",t): (II1.4.26) I ~- J divV <Pldt In Thay<PI dumda:utfchphfulbai chinh(Il1.4.26)taco: (Il1.4.27) <P J x,y, z.l ) = <P;- [ di{ V( <P;- [ divV <p,dl)]dt ho~cvietl~i (IlIA .28) <p,(x,y,z,t); '1'(-'-(t- tJdivV<p; +[di{vIdivV<p,d}t, -88- e~ 3: ?X1e4J d.u~ 4«9 ~ ~ ~ d Ut ~ ~ p~ /1M~ suyfa (1IIA.29) <Pl(x,y,i,t)=<p{- (t- tn)divv<p~'+0(M2). ' Cho t =tM! va d~y & = tn+1- tn,h~thuc (111.4.29)duqc viet l~inhu sau: (II1.4.30) <Pt1=<p{- &divV<p;+0(&2). Tuangn" tichphfulphuangtrlnh (1IIA.25)trendo~ tn5 t 5 tn+!v6idieu ki~nd1lu<p;=<p;+!,tanh~ duqc: (111.4.31) tn+l ( 0 0 )(p~+l=<Pt1+!~<P2+ozv ~2+f d~,n khido, ( 0 0 n )(IIIA.32) <p~+l=<Pt!+&~<P~;+oZV:2 +f +0(&2). Thay<p;+ltit(111.4.30)vao(IIL4.32)taco: (1lI.4.33) <p;H='1';- ru( divV '1';- ,.w.<p;- ~v~;-:r)+a(ill' ) Chiahaivecua(IIIA.33)cho& vacho& -*0,tanh~ duqc o<p -+ 0 8<p (III.4A3) ~+divV<P2=~<P2+-v~+ f,at oz oz dieud6conghia<P2thoaphuangtrlnhxwltphiL .U H Degiaibaitoan(Pi), chUngtaduavitotofuti':r (IIL4A4) , . . 0' 0 '0 A=u-+v-+w-, ox ay oz ~ Khi dotaco (IIL4.45) ( ) J o<p o<p o<p ,A<p,<p= (u-+v-+w-)<pdO, n ox ayoz do (1II.4.7),tavietl~i (1II.4.46) 1 &2 8v2 8w2 (A<p,<p)=-J(~+~+~)dO.20 ax 8y oz -89- ~ 3: ~ d I..uU4M,4«9 ~ ~ -u Ut ~ DWI4 1D~ fi4t ~ GiasirmienkhaosatcuachUngtac6d~g hlnhl~pphuang,trencaebiend6i xUngMall,v~ tdcnh~giatrinhunhau,dOngthaidodi~u;ki~n(IIIo4.6),tac6:. (II1.4.47) (A<p,<p)=0, M\)teachhlnhthuc,coi .' i ,...! ' (IIIo4.48) A=~+~+~, trongd6, '. L1 O<p"'1<P=u-+ <Pouox 2ax' (I11.4049) .4z<P= v o<p+ <Pav By '2-By, ~<P=w o<p+<Paw8z 2&' Tuangt1!toaDti:rA, chungtaclingnh~nduQ'c (III.4.50) (.~<p,<p)=0, '\Ii=1,2,3. Tit (III.4.48)va(I11.4.50)chUngtac6M~nhdesan: Menhde1:CaetoantirA va~,~,~ dpmnghlabCri(IIIo4.44)va(I11.4.49)la toanti:rphanHennite. Nhuv~ydoM~nhde 1,chungtac6th~sudl;lngphuangphapphanfa (xem[91]),d~tachtoanti:rA theod~g (I11.4.48).Khi d6baitoaD(PI) duQ'cdua vebabaitoan: . Biii roanthttnhatt1m<P11tit<P2cuabaitoaD(P2) thoa: O<P11+u o<Pn +<P11au =0,at ox 2 ax (I11.4.51) <P11=<P~tfong Q, khi t = tn, <P11= <Pstren 2:, t E (tn,t"t1]. -90- 8~ 3: ?Jt~ d d.:u~ ~ ~ ~ ~ cd tf&~ bWd 'P~ tdt ~ Biii roanthtthai tlm<PntU<PHcilabai loanthunhat: (IIL4.52) o<p 'am m Ov --1l.+v2.!1.+~- =0 O "' t '" ') '" .0' ~0' ,,+1 ~ kl . <P12=<PH trong ~.:, L~ 11t=tn, <P12=<Pstren 2:,t E (t",tn+1]. Biii loanthttba tlmCP13tUCP12cilabaitoanthuhai: oCP13 913CP13Ow- 0-+W-+--- ,ot az 2 oz CP13=<p;;1trong Q, khit = tn, (IIL4.53) Ocp13= aCP13tren2:0, t E(t",t"+l]'az .' oCP13=0 tren2:H,t E(tr"t,,+Jaz D6iv6ibailoan(n), chungta ciingxettuangtv,trongdo loantitit co d~ng: (IIIA.54) A=- ( wl+~V~ ) .. ozoz Khido, (III.4.55) 0 ocp(Acp,cp)=- J (~cp+-v-)<pdQ, \l OZ iJz ti:rdAy suyfa, (A ~,~)=It {( :;)' +(:) ']+ {:;)}o Iad~ilu9'llgkhongfun,nencoth~tachtoantitA thanht6ngcilabaloantu111la xacd~nhduongtheod~ng(IIIAA8), trongdo, (III.4.56) , (III A .57) 02 1\ =-J..!ox2' 02 Al =-J..!fJy2' 02 A...,=-v OZ2' - 91- ~ 3: ~ 41~ ~.~It-4"9 Up:-~ .. e4Ut~ DUd;r>~ ~ ~ Tit (IIlA.56)chungtad~dangchUngminhduqcM~nhdesail: Menh de 2: Cac toanti1A va ~,A.z,~du<JcdPili nghiabffi (111.4.54)va (111.4.57)lani'raxacdiM ducmg. -' - - Nhuv~y,c6thesirdl:1I1gphucmgphanrad~duabaitoan(P2)ve babai toaD: Bai loanthunhatfun <P21~<P13cuabaitoan(IIIA.53) -,. (III.4.58) 'V<p21- a2<p21 at - 1-1-8.2 '- X <P21=<p;;I'trong Q, khit =tn' <P21='Pstren:t,'t E (tn,t~+I]. Bai loanthuhattlm <P22tU<P21cuabaitoan(IIlA.58) a<P22- a2<p22 at- 1-18)72 ' (111.4.59) 'P22='P~;1trongQ, khi t =tn' <P22=<Pstren'L, t e(tn,tn+J Bai loanthubatlm <P23tU<P22cuabaitoan(111.4.59) a<P23- a a~23-Of .-- ---v-+ ,at az az <P23=<p~1trong Q, khi t = tn' (1IIA.60) i a<P23=a<p23tren Lo, t E(t",t"+I]'az 8<p23=0 trenLn, t e(t",t"+I]'az . Tiep h;1c,chUngtasaiphancaebaitoantrend~duave h~-nhUngphucmg trlnhd~ s6tuyentinK Bai toan(IIIA.51) c6d:plgsai.phan:- (I11.4.61) ( ) ,,+1 ( ) " ( ) ,,+1 ( ) ,,+1 ( ) ,-. <PH ijk - <P2ijk ,,<Pu i+ljk- <Pu i-ljk <Pu ijk U~ljk - U~-ljk- 0+U.. + - . M Ilk 2&. 2 2&., , -92- ~ 3: "JJt1e d 64tU4UI.~ ~ ~ ~ td ~ ~ tJWd~~ ~ ~ Vietl~idurnd~ngmatr~ baduangcheo(xem[6]49][59]) (IIIA.62) trongdo, ( ) ,,+1 ( ) ,,+1 ( ) ,,+1 A <PH ;+ljk- C <PHilk +B <PHHjl<:= -.F, (IIIA.63) " A U"k=~ 2&,', " u"k 1B='-~ c=-- 2A~,' ill ', ( '" ( " "P")" k U' +I 'k -u. l 'kF=- -1)+'1 '-I. ill 4~.:"(,, Tir dAysuyranghi~m«PH):kduqctImdurnd~ng (IIIA.64) ( ) ,,+1 ( ) ,,+1 <PHHjk = a. <Pllilk +~;, trongdo,cach~s6truydu6iduqctfnhIDeocOngthuctruyhbinhusan: (III.4.65) , A. ~iB+Fa. - A - i+l - C-a,B' Pi+l- C-a.B'. I Ct2=0, ~2=<Ps' i . TuongtI;CchUngtacod~g saiphanchocacbairoan(IIIA.52) (IIIA.66) ( )"+1 / ) ,,+1 ( . ),,+1 ( ) ,,+1 ' )'" "<P1Z-ilk - \<PH ilk + " <P12 ij+1I<- ,<P12 ij-11: +«P12 ilk V;;+11:- V;j-1< =0V.~ . . 6t IJ 2!1Yj 2 2!1y.: ho~c (III.4.67) ( ) ,,+1 ( ) ,,+1 ( ) ,,+1 A,<p12 Ij+1<- C <P12ij/l:+ B <P12ij-1<=- F, vrn (IIIA.68) " VijkA=-, 2~Yj ( ) ,,+1 V~L 1 <PH'" V~+.c- V~oc B=-~, C=--, F=- 'i" + I)'" 'J-v.. 211Yj M ill 411Yj' Nghi~mdmdurnd~ng " (III.4.69) ( ) 1>+1 ( ) ,,+1 <P12ij-11:= aj <P12ijk +~j' vrncach~s6troydudi (III.4.70) a . = A - ~jB +F )+1 C-a.jB:~j+l-C-a.B' (;(.2=0, P2=<PS.) - 93- ~3: '1/tfed,~~ ~.~ ~_d Ut~t1UM4 fD~~~ D6iv6ibailoan(ITIA.53),phuongtrlnhsaiphfuJ.c6d~g (IlIA.71) ( ) "+1 () II+1 ( ) lItl ( ) lItl ( ' ) " <P13ilk - <P12ilk )1(', <P13iji:+1- <P13ijk-1 <P13ilk W~k+1- W~k-1- 0 ill + ilk 2&k + 2 2&k -, ho~c (IIIA.72) trongd6, , ( C ) ,,+1 ( ) lItl ( ) ,,+1 A <P13ijk+1- C <P13 ilk + B <P12ijk-!=-F, (I11.4.73) "",, ( )"+1" " A= Will: B=- Will:C=-~ F=- <P12ijl:+Wij.';+1-.~ijk-1. 2&k ' 2&k ' M ' ill 4&k Nghi~mclingduQ'Cfun du6id~g (II1A.74) ( ) "+1 ( ) "+1 13Ijk-1=,ak !13 ij{ + f3J:' trongd6cach~s6~z~132duQ'CtimtUdieuki~nbien(IIlA.53)OOu sau: (IIIA.75) A 13,B +F 1a'= 13 =) a= 13 =0k+1 C"- B ' jtl C - B' 2 1 A~' 2 .ak aj + aL1L.k DinhIv3:Neub lapthaigianthlln,chUngtac6 (I11.4.76) (IIlA.77) (I11.4.78) thl cach~phuongtrlnhd:;Li86(I11.4.61),(IIIA.66), (IIIA.7!) c6th~giai duQ'cd~ timnghi~mrOir~c<Pll,<P12,<P13tuongUng. Chung minh " Chungta chi c'AnchUngmiOOchophucmgtrlnh(I11.4.61),cacphucmg mooconl~ihoanloantuongtV. -94- ¥f! {;.I} < {ill:.}M Max{I . G Mill {b} i.j,): viiI: < ,) :Yjill ¥tf {1w;;.1}< {ill.}M e~ 3: 1It<fed d4tf4<Ue~ ~ ~ ~ cdUt ~~ ~ 'PVt~ ,W-rw,I4 Tu giathiet(III.4.76),taCO I "I ffi I U' kl< iIJ -- , &' suyfa I " "I I ,,! 1' U U u'" IAI +IBI = 1 7 ilk +- ') ilk I =~ ~- =Icl....,~, ...,~, ~, &, , , Tu da.ysuyraphU<1ngphaptroydu6i6ridiM (xem[74]). Baitoan(111.4.58)cod~g saiphAnnhusail ( ),,+1 { ) "+1 ( ),,+1 2( ) ,,+1 ( \,,+1 (IIL4.79) 'P21 <}k-, 'P" ;jk =f! 'P21 H Ifr - ('P21)~+ 'P21)Hik ,& ~,, .Du6id~gbadUOngcheo,chUngtacothevietl~i . . (IlI.4.80) trongdo, ( ) ,,+1 . ( ' ) ,,+1 ' ) ,,+1 . 'A<P21 ' +1'k -C <P~1 :" +B/en?1 ' 1" ='-F,. 'i ~Iji< \'f'-I-Ji< ' (II1.4.81) A=B=~ (L\xiY' 1 '21-Lc=-+- ( ) 2' & ~i F ~ «p13Y,:1 M. Nghi~mt1mcod~g (III.4.64)- (II1.4.66). ',. Baitoan(IIlA.59)cod~ng (III.4.82) ( ) ,,+1 ( ) ,,+1 ( ),,+1 ') ( ) ,,+1 ( ) ,,+1 <P22ilk - <P21 ilk - <P22ij+lk , <P22ilk + <P22 ij-li<. & -~ . ( ) 2 ' ~Yj . ~ ho~c (IIlA.83) , , , ( ) ,,+1 ( ) ,,+1 ( ) ,,+1 A <Pn ij+});- C <Pn ilk +B <Pn ij-lk =- F , v6i (IlI.4.84) A=B=~ (~yJ2' 1 2~C=-+- ( ) 2' & ~Yj ( ) ,,+1 F = P21ijk ' &' -95- ~ 3: ~ d 4.u~ 4«9 ~ ~ Wt4-td Ut ~ ~ iD~ fzM~ Nghi~mtlmco d~g (111.4.70)- (111.4.72). D6i v6i bili loan (111.4.60),chUngtaco ( r+1 ( ) n+l [( ) n+l ( ) n+1 ] [( n+l ] (1IIA.85) 23ift - 22ijk =VHl 23ijk+l - 23ijk - Vk 23)ijk- «f>23)~:~1 n /).t, ' . ,( t.zJ2 + !jk' ho~c (111.4.86) ( ) n+1 ( ) n+1 ( ) n+l A 23i'k+1- C 23i'k +B 'P221*-1=-F,"J ". "J" , J, . trongdo, (111.4.87) . ' ( " ) n+l A= Vk+1 B= Vk C=~ Vk+l+Vk - 'P22 ijk (A- ) 2' ( )2' A++ ( \2' F - ',' D£k ~k Llt &k) ill Nghi~mclingdugcfundumd~g (1IIA.76)- (1IIA.78). Vm caegia1IiA,B,C chob (III.4.81),(1IIA.84),(1IIA.87),chUngtadb dangcodugcD!nhIy sau: DinhIv 3:H~caephuongtrlnhd~is6(1IIA.79),(II1.4.82),(111.4.85)IubnIubn giai duqc d~t1mnghi~mrill r~cZl'22,'P23tuong,ling. V.Baitminvesl1lantruyenvakhuechtan nguonchatb~ntrongnuocdum<tat V.I M6 hlnbbili tmin D~khaosatdi~nbienm~ttI.!do cuanuocdumda:t,clingnhu stJ Ian truyenvakhuechtancuanhiingcha:thoatantrongnuoc,chUngtathietI~pm(>t h~phuC1Ilgtrlnhbaog6mhaiphuC1Ilgtrlnh(xem[77][83][84]): - phuongtrlnhBussinesqmbtam~tt1Jdocuanuocdumda:t,va - phuongtrlnhbieu di~nSlJ Ian truyenva khuechtan cua n6ng d(>ch~t hoa ta.T1trongnuac. -96- ~ 3: ~ d- ~ ~ .l!«9 ~ ~ HI8 U.t&~ ~ .,,~ p4t~ Giasirh~ttvctQadOxyZvmm~t phangxOy songsongv6i phuoogDam ngangcuachuy~nd(>ngphaixet.Khi d6 c6th~coim~tt;rdocuanu6'cdumdatla hams6cuatQadyx,y vathaigiant. z MienklulosatQ trongm~tphing xOy,vilngchuy~nd(>ngcuanuCtcva.chAt luqngnuC1Cnhuhinhve. Zd=Zd(xJ") 0 1'1..Y / ChUngtagiathier, -m~tt;rdocuanuCtcdumdiftZI(x,y,t) Damthifphoo sovdi m~tclift, - chuy~nd(>ngcuanucrcdumcliftganm~tdatlachuy~ndyngkhongap, - lapcliftsetZd(x,y)dummienchuy~nd(>ngcuanu6'cthamthayd6iit, - nuCtcdumdAtIa.chatlongd~ngchat,khOngDenduqc, - dongcharcuanu6'cdumda:tutintheodinhlu~thamDarcy[36][84] (IlL5.I) V(x,y,t)=-KVzJ:r,y,t), (trongd6K la.h~s6thifmcuaclift), - cliftlamoi trUOilgkhongDenduqcvading huang. Khi d6chUngtac6phuoogtrlnhBussinesqsaudAyd~motam~tt;rdo cuanuCtcduqidat: (IIL5.2) (j ~=V.[(ZI- zd)KVz1]+F, trongd6, - (j=a(x,y) : - F =F(x,y,t): dyr6ngcuadat, ... cuOilgdy cuanuC1Cm~thifmxu6ngm~ttt;rdo. D~rutra phuoogtrlnhthllhai (xem[21]),.chUngta gia sirtrongmr6'c chuachathoatanvmn~ngdykh6iluqngduqclaytrungbinhtheechieucaola S(x,y,t).Xet th~tichki~mtrahInhlangt.n;Iday dxx dyvachieucao(Zt- Zd)' kh6iluqngchittboatantrongkh6ithetichna.ysela: (IIL5.3) [a(Zt-zd)dxdy]S. -97- ~3: ~~~uu4"9~~_dUt~~ fDaJt4~pM.~ Suyra 811thayd6i theothOigian (IIL5.4) (;{~[(z,-Z,)S]}dxdY, Theokh6nggian,kh6iluqngn'a.ythayd6idodongthammangdi Ia .(III.5.5) , . V.[(zr-zJVS]dxdY vadokhuechtand6i lUll la" (IIL5.6) -V.[(Zt- z,JDVS]dxdy. Thayd!nhlu~thronDarcy(III.5.1)vito(IIL5.5),13nh~ du'lc (IlL5.7) '"'-V.[(Zt- z<t)KVztS]dxdy. - vathayh~86khuechtand6i luu D =AIVIvito(IlLS.6),13co (IlL5.8) -V.[(Zt-z,,)AKIVztIVS]dxdy. (trongdoA=A(x,y,t)lah~86tanx~). Trongmtdonvi thCrigian,kh6i Iuqngchathoatandongu~nnuacm~tF mangvitothetfchkiemITalit: (IIL5.9) FSdxdy. GiQ =Q(x,y,t) la cuem.gdtraod6i chAtlAytrungblnh theochieucao giuanuocvadAt,khi d6 luqngchAtboatantangthemtrongm\)tdonvi thCrigian dotraod6ichAtvOidAtsela: . (IIL5.10) Q(Zt- z<t)dxdy. Slf thaydoi kh6i 111gllgchaiboatantheothOigianva kh6nggian(do dongthfunmangdi va dokhuechtand6i lull) trongthetfchki~mITa8ebang t6ngcaengu()nbAndonuocm~tmangtOivadotraod6ichAtvOidAt. -98- ~ 3: ~ d-J.u ~ ~ ~ ~ HU.t4 Ut ~ ~ p~ ftk;~ Dodocluingtacophuongtrioo: (IlL5.II) ~~r( 7 - 7' )\S]- '\/ [(- - 7 )' ~ z C'J' - D f(-, - 7")'1 V!D- IID S]-Vatl.\""'t ~d v. ""t '"'<1f'..V tlJ V.L""'t ""d /w'AIV""t v - =FS+O(Zt -Z<1). Tfnhtofutuemgminhcaethanhphllnd~oham,(IlL5.II) coth~vietl<;ti: (IILS.12) (j ~S+(Zt-Zd)(j: - 'V'.[(Zt- zd)K"1z:]S-(Zt- z<t)KVz:VS+ -"1.[(z:- zJAKIVz:I'V'S]=FS+Q(zt - Zd)' ho~e (IlL5.13) [ az ] as 1 I --- '7 -, ("(7--\ --(--'" '" .J.. a Of V.[(L.: ""JKY'.:..:] F UT\..:..: kdJa ot -\"": ""d)J.T(\7""t"1SI r ( ')' I '] , )+'V'.lz: -z<1JAKj'V'z:I'V'S+O(z: -Zd . Dophuongtrlnh(IIL5.2),taco: (III.5.14) ( \8S" r'" I 1 ( ) Zt-Zd}(j &t=(,zt-zd)KVztvS+v.UZt-z,j)AK!'V'Zt!'V'SJ+Q(,Z:-Zd, chiahaivecua(IIL5.I4)cho(z:- Zd)'chUngta nh~ duqc phuongtrlnhcho Dongdch~thoatanStrongt111ooghqpthamphangOOusau: '(III.5.l5) 8S 'V'.[(z:-z<t)AKI'V'z:!'V'S]+KVztVS+Q.(j-= Of Zt- Zd ChUngtanh~ thayv& caeh~so (j> 0, A> 0",K >0, z: >Zd'l'V'z:1>0 thi (III.5.2) va (III.S.15)la phUffilgtrlnh d~g parabolictheo cac an Zt(x,y,t)vaS(x,y,t). V .2PhmYl12phap~jais6 V.2.! Phitan~trinh mat tit do D~giai caephuangtrlnh(III.5.2) a tr~n,chungtaduaved~g quellthu<)c nhu cacm\lc tI11O'c.T111O'Chet,chungta viet l<;liphuangtrinh (III.5.2), trangdo -99- ~ 3: ?14&4J t.i.t~ ~ ~ ~ ..,.d ~ ~ tJUtIk1D~ tW-~ cacth3nhphfu'tv~ t6c dugctinh theo(III.5.l) vOi Zt(x,y,t)ducx!p xi i:Jfan tinh trUc.1c. (III.5.16) OZ o(Zt- zes) o(Zt- zes) ( )( au av ) cr--'-+u +v + Z-z -+- =F. at ax ay t d ax By Kyhi~u (III.5.17) <p(x,y,t)=Zt(x,y,t)~Zd(X,y) thi(III.5.16)dugcvietl~i (IIL5.-18)- cro<p+u acp+v acp+ <p ( au+av ) =F. at ax By ox By Dogillthiet,chAtlongkh6ngnendugc (IIL5.l9) au av -+-=0 .ax By , nen (I1L5.20) craep+uocp+vaep=F. at ax By D~t a a (III.5.2l) A=uax+vBy' d6i vOi tOM ti':rA chUng ta c6 (A q>,ep)=J ( uBcp +v Bcp ) <fkiQ Q ox By ho~c (III.5.22) 1 ( au2 av2 J(Acp,cp)=21 0: + ;; dO. -100- e~ 3: ~ d.6atf4ht,4"9 ~ ~ ... ~..lUt~ ~ p~ {t4t~ Gia S11mienkhaosatQ eod~nghlnhchunhc).t,v~ toetrenbienbang khOng,khid6chUngtac6, (II1.5.23) (Acp,cp)=O. Til (III.5.23)chUngtac6M~nhdesau M~nhde1:Vaimi~nkhaosatQ=[X<1'Xc]x[~,YJthlloantUA xacd!nhbill (III.5.21)latoant11phanHermite. Do dieuki~n(111.5.19)toantUA duQ'Cvietthanht6ng (III.5.24) A= Ay+~, trongd6, (IlL5.25) <\<P= u O(P+ cpOUox 2ox' va (II1.5.26) .1 8cp cp '"' "-'2cp-=v-+- ov Oy 2ay' N dangthilyding (111.5.27) (~<p,<p)=f( u ocp+ cpou J <PdQ=f( u O<p2+cp2au ) dQ = n ox-- 2 Ox n 2 ox . 2 ox =f( .!.OU<p2 - cp2 OU +<p2ou ) dQ.=r .!.OUcp2dQ. n 2 ax 2 Ox 2 ox 02 ox... Tu dAysuyfa (IlL5.28) (r\p,cp)=O. Tucmgtl!chUngtaclingnh~duqc (III.5.29) (~<p,cp)=O. TiI (III.5.28),(III.5.29)chUngtac6M~nhdesau -101- ~ 3: "/It&.4J t4ct4U ~ 4a~ -d Ut ~ /JtiMk ?~ ~ ~ Menhde2: VOi mienkhilosat0= [Xd,Xc]x[y",J:]thl caetofu ti'r~,~ xae djnhben(II1.5.25),(II1.5.26)latofuti'rphfu1Hennite. Nhuv~y,tUcacM~nhde1vaM~nhde2chUngill co th~sirdl..lIlgphuong phapphanratheet9ad(?d~giiliphuongtrlnh'(III.5.20)(xem[79][80]). Caetofu ti'rl\, 1\ du<JesRiphanbencaetofu ti'rAt' A2nhusau: (1II.5.30) 11 11 u~+u~, ~ Uj+lj +Ujj 11 - 'J ,-IJ ,<p1l , 11-1 11 - 11<Pi+lj 2 dJ <Pi Uj+lj UHj( )11 2 + J, A1<P ij = 2~ 2 2~j, (111.5.31) 11 11 v~ +v~V"+l+ Vij 11 'J 'J-l <p1l 11-1 11 11IJ <Pij+l - 2 jj-l <Pij Vij+l- Vij-l( )11 2 + - , A2<P jj = 2!J.Yj 2 2!J.Yj . trongdoeacthanhphfinv~ t6edu<Jcxapxi dumd~g (1II.5.32) _11 _11 U n K Zti+l ' - Z ij = - J ti-lj 2!J.x ', va (III.5.33) _11 11 Zt' -z1l Vij= -K 'J+1 tij-l 2!J.Yj (vm Zt~la gia tti CURZti:J11mtinh l~ptru6c CUR<). VOiketquab cacml;lctru6c,chUngill coDinhly vesl;!xapxi vadndinh CUR1mgiilixapxi tuongUng. Quatrlnhphfulraduqegiilitufultl;!quacaebu6cnhusaudcitm nghi~m <p1l,<p1l+1tU <p1l-1CUR (II1.5.20) tren do~ [t1l-l't1l+1]: - TIm <p1l-1/2tUh~phuong trlnh (1II.5.34) 11-1f2- 11,-1 11-1/21 11-1<Pij <p'J +(A<p) =-F"cr 1" 2 I)M IJ - 102- ~ ~ 3: ~ d 64tuu. ~ ~ ~ f4 d Ut ~ ~ ;:;~ {141.hP!k - Tun <p"tith~phuongtrinh (IIL5.35) " ,,-1/2 (j <Pij- <Pij +(A <p)"=.!.F7-1/2, M 2 'f 2 " - Tun <p,,+1/2tUh~phuangtrlnh (IIL5.36) (p~,+J/2- (O~, ( ' ),,+1/2- 1F"If 't'lf + A {O - - '",.". 2't' " 2 IfV M If - Cu6i clingt1m<p,,+ltith~phuangtrlnh (IIL5.3?) ,,+1 "+1/2 <Pij - <Pij (A ),,+1;::;.!.F n+1/2(j + 1(0 ".M 't' ij 2 If Thay(IIL5.30),(II!.5.3l)vaocach~phuongtrlnh(IIL5.34)- (III.5.3?)chUngta nh~ duQ'ch~phuangtrlnhd<;\i86tuyentinhbaduem.gche<Yd<;\llg[6][49]: (IIL5.38) A<p a +l, - C{f'l~+Bma1 =-F,I J 't"J 't'1-. J d~tinh <p,,-J/2va <p,,+1,trong do " A - Ui+lf ' +u~ - IJ 4~ 'j B - u;+u~- - J I-lj 4~, 'I (IIL5.39) c =- .!2..M' - (j(P~ 1F =---Y...- -Fa vOia.=n-l hayn+1/2. M 2 If ' "' Bangcachtuongt1!chungtadingnh~ duQ'ch~ (IlLS.40) A<p~+1- C<p~+B<p~-l=-F, d~t1m<p"va <pn+1/2. Nghi~mtroydu6iduQ'cdmhoanloangi6ngcacm1;lctruacvatuongtV, chungtacodjnhly vesl;lgiaiduQ'cuah~phuangtrlnh(IIL5.38)nhusail: -103- ~s: 7JtfeJ.~~~~~_e.iUt~~ ?~~~ l!inIL.lY.l: Neucacdieuki~nsandftythoaman ~~aX&i S ~~{IU;+1jl,lu~I,IU;-ljl}, (IIL5.42) ~MF~YjS ~~{jV~+ll,lv~I,IV~-ll}. Thlh~(IIL5.38),(III.5.40)e6th~giii duQ'c (I1I.5.41) Sankhicaegiat:rirqir~e<p~t1mdU9ca lapthaigianthun,chUngt2.suy ra nghi~mcua phuongtrlnhm~tt1!do (III.5.2)a lap thCri.gianthu n theo (IIL5.17)nhusan: (I11.5.43) " "ZI " =H\..+Z..'J -r'J U'J V.2.2Phllan~trlnhIantrovenvakhuechtan D6ivmphuongtrlnhIantruyenvakhueehtin (IIL5.IS), chUngt2.c6th~ vietl~idumd~g (I1I.5.44) as as ascr-+U-+V-= fJt ox ay a [ ~ ] a [ . ~ ] =- (Zt-zd)AK/Vztl- +- (z,-Zd)AKIVztl- +Q. ax ax ay By Bangeachphftnratheoquatrlnhv~tly nhuM¥cIV, chUngt2.duavegiai hIDphuangtrlnhriengbi~t: (III.5.45) as as as cr-+u-+v-=O at ax 8y va (IlLS.46) ~ 0 [ as ] a [ as ] (j-=- (Zt-zJ)-K!Vztl- +~ (zt-zJAKIVztl- +Qat ax ox oy 8y Pbuangtrlnh(IIL5.45)c6 d~g gi6ng(III.5.20),Denphuangphapgiai hoantoantuangtt;r.Nghi~mdu<?etlmtrendo~ [t,,-pt,,+l]ranlu<?ttitcaeh~san: (IIL5.47) S"-1f2- S,,-l ) ,,-1/2 A., ij I (A S =, (j' 'I T 1 ij& -104- e~ 3: 1It&,d d4t~ ~ ~ tt~ ..;;.d Ut ~ ~ ?~ ~ ~ (IIL5.48) S" S"-1/2 (j ii - ii +(A 8)n =0 /:)J 2 ij , (IIL5.49) ,,+1/2- 8~ "'+1/"2.- 0Sjj IJ +(A28)" - ,(j IJ/:)J (IlLS .50) 8"'+1 8,,+1/2 (j ij - ij +(A 8)"+1=0 /:)J 1 ij , trongd6cactoanti':rAI' i\ duQ'cxacdinhthee(I1I.5.3D),(111.5.31). CilngoouPhanV.2.1, chUngtac6dieuki~nve s~on dlnhcuaphuong phaptroydu6i. . D~giaiphuongtrlnh(III.5.46),tru6'chetchUngtaxApxi h~s6khuechtan d6ilUlldumd~g: . . (IlLS.51) ( " ) 2 ( )2Z -" "" D ;::::D" =='),K i li+1j zli-lj + Zlij+1- Zlij-1 V 4(&,)2 4(lWY'j' Khi d6,phuongmoo(III.5.46)duQ'cx~pxi nhusau (III.5.52) (j as_ ( 828+828 ) = Q.. (ZI~ - Zdij)D" 8t 8x2 8y2 (Zt~- Zdij)D" Phuongmoo(IIL5.52)duQ'CQuav~d()l1gphanratr~ntUngdo~ [t,,-pt,,+J nhusail: (IIL5.53) (j: =(Zt~- zd,JD"~~+~Q, (111.5.54) cr: =(z,;-z"JD, ~~+~Q. X6t caetoanti':rsaiphantheekh6nggian (IIL5.55) (A,S):=- (~y(8;'.'1-2S;j +S;:,), -105- ~3: ~d ~~~~~~uUt~bUId 1D~~~ (IlL5.56) (A,SJ:= - ( 1)' (S.;.,- 2S""+.s:;J. ,b.Yj Sadc>saiphAncua(III.5.53),(III.5.54)c6d~g (IlLS.57) 0' (Z/~,- z ..)D A+ (S~-1/2- S~-l )+(A S) "-1/2 Q,,-l/2" I) "IS') h~ 'J 1" = ij'J (2 z"- 'lij ZlSij)Dh (IIL5.58) 0' (" ) (S~- S"-112) Z"i- zd;i D,JI.t 'i u' +(A,St= Q;'J '4Zt~~-zd.)D 'I) h (III.5.59) 0' (Zt~,- Z , )D A+(S~+1/2- S~ )+(A S)"+1/2 rY'+1/2 I) dlj h~ 'J 2" = ~ij'J (2 z" - 'Iii Zd;j)D;, (III.5.60) 0' (Zt~,- Z ..)D A+ (S~+l- S;"+1/2 )+(A S) n+l Q~+l ) d'J h~ J 2 Ij = ( 'j2 z,.- .lij Zd;j)D" Bangcachxa:pxi nhuv~y,chUngtading nh~ du<Jcacxapxi ca:phai.. Cach~phucrngtrlnh(IIL5.57)- (Il1.5.60)du<Jcduave d(~lllgba duemg . ch60,phucrngphapgiai hoantoantUcrng11;t. Trongcach~phucrngtrlnhnAy,do dieuki~n (IIL5.61) 0" (Zt~,-z)D A+>0, 'J dij ;,~ Denphucrngphaptroydu6ilu6nlu6n6nd!nh. ~ --- _n- . .. .. ... ':::'",.,..:.".. - 106-