emoi trnern-g lien tuc, d~ thi~t I~p roO-hlnh toan h9C
cho phep gia.i quy~t ca.e van de d~t fa, ngoai cae dinh Juat ea bin
nhu dinh Iu~t bao toan kMi Iuqng (phudng trlnh lien t1,tc), dinh Iy
bien thien de)ng Iuqng (phudng trlnh ehuy~n d9ng), dinh Ii bien thien
moment dQng luqng, din pha.i xay dt;mg phudng trinh trang thai me
ta quan h~ giua Ung suat va. bien d<;tng. Cae quan h~ nay duO'c de
xua:t va. ki~m chUng nha thue nghi~m. Mo hlnh ducrc Slt dung trong
cae bai toan trong luau an 1a.mo hlnh Bingham.
Quy lu~t Ung sua:t - bien dang Bingham co d~ng
de
CT
khi
= (Ts + fJ dt
,..
I 2: 17,;
v0'1117
Irrl< cr, moi twang kh6ng bi bien dang.
~:Jein x~t
.
.EhJ lIng suat khOng doi va vucrt. qua. ung suat gicri han as th\
a- a,
Hong vat the, bitin dang tang theo thai gian ti le veri
=:..
Il
. 5<1do v~t li~u Bingham phan anh mot th\1c tt! 111,
d6i vcri nhieu
vAt li~u su chay dang ke chi xdt bien vt1i tcii tn,mg xac dinh,
dong thm t6c do cMy ph1,lthuQe vao dQ nh&t cua moi twang.
110 hlnh nay do Bingham de xu<1tna.m 1922 cho ta m6i quaIl hE;giua
ttng suat va biEindi;tngthich hqp de mo phong eae qua trlnh va ChID.
Ga.n day, roOhlnh nay con duqc dung d~ tinh toan cae qua trlnh xam
nha.p cna cae va.t tM bi~n dC;tllg.
----
- - - --
(,
CIutl1ng
2
B;\1 TO/\N BI:E~ TT" DO TRaNG
co HQC X..\1\1 'iHAP
Bai toaD cd hQc Vat th~ B veri chien f<;Jng:2H xam nhap VaG vat
the khac dl1&i tac d1,lng cua 1\!c ngoai g(t). Gia. ~n'!rang B la. v~t the
nhci't deo. Hong nen ducic, tuan thee quy Iu~t lIng 5uat - bien dang
cua Bingham:
'111*
- '0 = ~-:-:--(I".t"),
,."Jr"
r'(x",t")
!1i
trong do r" Ia. lIng snat. '0 Ia. l1ng snat gicri han, tJ 111.
he 50 nhal va
(~AIa. van toc thee huang y.
Khi Ung sucH ti~p vuat qua giro han da.n hoi, vat th~ B duqc
chiao thiLnh hai phan (mien chay deo va. mien cUng) ng3.n each bCti
= 8"'(t..)(bien tt,ldo).
:r"
.
Dieu ki~n lIen bien t1,l do c1ia biLi toan duqc thi~t l~p dl,la tIeR cac
gia. thi~t
(HI)
ti~p d~t
(H2)
do.
san:
TIen di~m phan each mien Cling va.mien cleo nhc1t Ung sU.1t
gia tri giai h~
Truang van toc va.truang ting su.1t lien t~c khi qua bien tt,l
Bai tom
tom
hQC. BiLitoan bien bOn hqp cna biLitoin cet h<,>c
(d~ng khong thu nguyen): Cho truc1cTm= > 0, tim u(x, i), s(t) sao
cho
.
..
----
~(t) lien t~c Lipschitz tIen (0, Tm=]j
. u va.
,J211
::
lien tl,1Cvai 0
~x~
8
(t), 0
~ t ~ Tm=;
au
. -&;2va.at lien t~c trong 0 ~ x ~ 8(t) khi 0 < t < Tm=,
. u thoa phuangtrlnh dao ha.mrieng
au - 1 fpu
at
-
f
.
R 8x2 (I, t)
trong 0 < x < 5(t), 0 < t ~ Tmax;
5
+ Rg(t)
(2)
.
Tren bien tV do set), u thoa. cae dieu kierl
au
S
5
m(s(t), t)
au
-(set),
t)
ax
=
Rg(t)
=
0,
&u
- R(1-s(t))'
5
(3)
= -I-s(t)'
ox2(s(t),t)
vm 0< t::; Tmax;
. u va s thOa.cae di~u ki~n bien va.dieu ki~n da.usau:
=
seD)
= - tp(x),
u(:c,O)
- ----
-------
b, 0 < b < 1,
=
u(O,t)
(4)
jet),
vm d.c di~u kien tucrng thlch
=
'P(O)
r,o'(b) =
/(0),
0,
= -~1-
'P"(b)
(5)
b'
trong do u la v~ toe, set) la di~m phan each giila mien cleonh&t va
mien tUng, R
pH2 / I-!T la so Reynold (ti so giila h,lc qUail tinh va
11;tcnh&t), 5
ToT/I-! la ti 86 giilal1;tc ngoai va 11;tenh&t.
=
=
:
Dua vao:in ham mm u(x,t) = (x,t), u(x,t) thoa
au
1 &u
S
,
,
=
RO:r2(x,t)+R9(t),
=
R~ft) - H(I - s(t))
=
5
=
tjJ(x),
v(O,t)
=
f(t),
f(O)
=
11'(0),
S
S
-g(O) '- ~R R(1 - b)
8t(x,t)
-,
v(s(1),
S.
t)
av
,,(stiLt)
aT
v(,r,O)
1/,(b) =
,
5
'sfi)
I-s(t)
8
(6)
(7)
(8)
-,'
(9)
(10)
(11 j
( 12)
Ph11crng trinh
tich phAn. Ky hieu k
-T
k
=
~"\.(I,t;~,1")
= R12
exp!
2..)7r(t-1')
i
uung cae ham Green.
k"(x -02
\... -,), it-1'
G(x,t;f;,1')
:= K(x,t;E"r)-K(x,t:-~,i),
.V(x,t;f;,r)
:= K(x,t;E;,1')+I((x,t:-f;,i),
)
(13)
vcii
0 < x < set), 0 < f; < s(t), 0 < T < t.
(14)
Sa.n mot so bi~n d6i xuat phat tu dong nhat thuc Green eila. he
(6)-(12), ta. thu dw;rc phucrng trinh ti'ch pha.n sail:
.)
rei)
= ~(l-s(t»B(r(t»,
(15)
trong do
B(1"(t»
=
1& r//(~)N(s(t),
S
(t
t;f;,O)d{
r( 1")
- R Jo (1 - s(1"»2N(.(t),t; SeT),r)d1"
S
+R t r(r) 8G
0'1-31" ( ) a-,;-($(t),t;s(r}~1")dT'
l
t
-
I
0
S
u
[ fer)
- Rg(1")N(s(t),t;O,1")dT,
]
(16)
va set) duqc Lie dinh nllet
- .
.
--{In
._. s(t)=b+fo~r(T)dr.--trong do fIO(t)=v(s(t),t).
Giai (15) - (17) ta. thn duc;rcset); tit do ta.co th~ unh toan gia.tri
cua. trnetng v~ t6c va.trtldng Ung snat.
511 tOn t~ va duy nhat nghi~m tOM Cl,le-TiI cae bi~u di~n tich
pha.n.eua. nghi~m chung-tOi rut fa. ill<)ts6 kllAngdiu v~ ti'nh chinh
qui c1ianghi~m va.cae d~ ham c1ia.no:
Dinh 1:' 2.2 Dttdi cae dieu ki~n cua Dinh (y 2.1 110cae dieu k~n
trctn: f(t) thuqc (tip C3, get) thutje ldp L- tren (0, +CO), \I'(x) thuijc
,
()3u
OZu .
{(1p C4 tren [0, b]; &tox2'
&t2 hen tf!.C tren 0 ::; I ::; S (t), 0 < t ::; u..
9
D~t
Tmax =
sup{T> 0: (2.17) - (2.20) co nghi~m tIeD [0,T]
va.0 < s(t) < 1 vm m<;>i
t E [0,T]}.
(18)
Tir Dinh Ii 2.1 va.D~n~Iy 2.2 ta co ngay
Bo & 2.1 Dttlii cae dieu ki~n cuo Dinh ly 2.1 (ho~e Dinh ly 2.£),
~t trong cae kef lu~n sou dung
(i) Tmax=+oo.
(ii)
Tmax
< +00
va
= 1, lim sup 18(t)1 < +00.
Jim s(t)
t-+Tm""
t-+Tm""
(ill) 'Tmax<-+00 va Jim 5(t) = 0, Jim sup 18(t)1< +00.
---
--
--
- t"'tT...",,-
.
t-tTm;'",;
(iv) Tmax< +00 va lim sup 18(t)1= +00
t-+T""",
fPu
_-
(i.e. Jim t-+T",ox.
sup I lit&.'r {5(t},t}
1
= +00).
Th,ta tIeD 811tOn t~ va. duyuhat nghi~m dia phu<1Ilg (duqc chi ra
trongbai Mo cua D.D. Ang et al.), ba.ng ca.ch ap d~ng nguyen If C\,IC
d~ cho phu<1Ilgtrinh parabolic chUng toi chUng minh duqc ca.c k~t
qua. sau:
Dinh
sou:
If 2.3 Dttlii cae dieu ki~n cua Dinh ly £.2 va cae dieu ki~n
1. i'(t)-Jdl(t)
+oc5
~ 0 v~i mQit? 0 va M =RsUPt~O Il(t)-ftg(t)1
<
5
.
2, t~~oo [ Rg(t) - J(t) ] = g*< +00 (g* ? R(1- b»'
3. !pfl/(x) ? 0 vdi mQi 0 =s;x =s;b,
To co
"
Tmax =
- -'.
-
q ..::.y(x,t)
+00 (Tmaxnhu trong B6 d~ 2.1), s(t)?
0- .: ;
0 ("It ~ 0),
1
..::.:;Mp
" = ~ x) + Fe Or;}:~bI 2). Them va.o diem t",< vm gia. tri lap
ban dkll duae di<;ln Ia.
r(tn+1) = 2r(tn) - r(tn-LJ
f19)
Dungphepco X.1£dinh Mi (15)-(16),trongd6 a 7e pha.ir(t), s(t)
lay gia.t.r:ixap xi eRa bu6c l~p trucrc (hod.cgia :q l~p ban clan),
xac dinh r(t"'+1),roi s(tn+1) (nher(17))
Sf! hi?i tt,l, sat s6 Nh.Ungk~t qua. v?!s11Mi tll cua tJ,lli,LttoaD va. sai
s6 thll~t toem pAn thu9c cae aanh gia trong chUng :ninh s1,tton tai
nghi~m dia phudng cua phuong trlnh tich pha.n. Mi:Ltkhae, trong tinh
toan chUng toi da sir dT,Ulgn9i BUYLagrange d~ tinh gkn dung r(t) va.
s (t). Cae tich pha.n xdt hi~n trong (16), (17), baa gom cae tieh phan
co kY di (thu~ l~ (t - 1')-1/'l)
, d~n ton t~ TIl1i roe tieD phan s6
vdi sai s6 tin.h t.oa.ntut1ng Ung:
1
~2
1.£'1 F(.7:)dI:=h[-Fl
2
1
+ -F1] +O(h3P").
(20)
2
Truirng hqp tich pha.a ky di
($2
F(.7:)tb:
Jr:l (.7:2_z)1/2
= -
= h1/'l[Fl+F'l] +O(h3/2P).
d day h 3:2 3:1 (tmng truang
h L1r, bi~n khO1lg-gi;urh=~&)-:--
=
(21)
hqp cua. chung ta., 'vdi bi~n th€ri. gia.n
,-
-- -_d..
-
Ky hi~u T Ia. ci.uh~ co tic B".(O,M) vao chinn no xac dinh bdisup q(t),
v~ phai deL (16) viii M s6 co a (0 < a < 1); IIqU",
9:°99"
(q E C[O,in]); T Ia.taU tit .dp xi da. T. Tit co dinh If san:
Dinh Iy 2.6 Dtrtii aie dieu ki~n cua D;nh Iy 2.1, J.~ttde l~p thti
n ta eo cae daM gid lien quan de'n sa; si{ thfj~t loan va sa; sit t{nh
toan sau:
(i)
IIF
- rlln ~ tr'!vf.
Do do thuq.t toan fa hql t~. Hun niia, ta co
111- sUn ~ anMu,
,
c.;
".,
.'. ""-'.
12
..!
-
(ii) IIT'70
(iii)
- Tnroll"
,
,"
< (CjnL1{!/-' + C:d17l- IjL1xj)-
1
:
l -Q
~1 -
IIT"ro - rUn < o:n}vi+ (C1nL1t3/Z+ '-:"-'z(m
- 1).::1x3)
0
do r la nghi~m cMnh xcic; f(= Tr'ro) IiI nghi~m gan dung d
bttdc I(lp thtf n; ro la gici try igp ban dati; s(t), s(t) dttqc :uic dinh
irony
tti(17)vdiFl.,t),
r(t) tttetngting;C1,CzlG
ccichdngst{cMphtj
thuQc
vao dil "i~n cho trtJdc cua bai loan va gici tri I(lp ban dttu; m Ia s6'
diem nut !.hong gian ph4n ho(,lch [0,bJ.
B~ qua' Co\d;nh tn, dieu kil]n oond,nh cua set do tinh xiip xi:
.::1tlj2
"
&2
>
t1/Z
n
-
bZ----
(22)
ThuM toau duQc ap d1,lugcho ba. VI dl,l 86. K~t qua phil hqp vai
cae daub gia If thuy~t, d~c bi~t phU hqp vai k~t qua ciia cae tic gici
trucrc day.
JC
Va Ch9ffi cua thanh
Chuo'ng 3
deo nhdt v~w v<).tcan dan hili tuyifn tinh
Bai toan co' hQc. Thanh ehi~u diti L dv6i tae dung clia lHe ngoiti
ehuy~n dong doc true. va cham vito vat can d~1llh6i tuyen t{nh ti1-idau
thanh theo hucfng phap tuyen. Thanh du<1egii thiet thanh la. v~t th~
khong Hen du<1Cva. co th~ mo ta quail he giua Ung su<1tva bien di1-ng
nher qui lu~t Bingham:
au"
a"(x",t")+ao=J1.ax..(x"t"),
neula"l>ao,
(23)
trong do a* la. Ung suat phap, <70lit Ung su<1t tdi hi;tn, J1.la. M so nMt
va u* v~n toe theo huang x.
Bai toan toan hQc. D<,tng khong thu nguyen ciia bai toan bien hon
h<;1p:
au
-at(X, t)
du
=
1 fPu
R ax2 (x, t) khi 0 < x < set), 0 < t < T', (24)
5
(25)
= R(I - s(t»'
dt(s(t),t)
au
ox (s(t),t) = 0,
:(O,t)
(26)
= 5{I+Q[~tu(O,r)dT+a)}.
(27)
Di~u ki~n da.u:
~-
_.~-
=
~(x), 0 < x < b,
s(0) = b, 0 < b< 1.
u(x,O)
(28)
(29)
trong do u la. van tOe, s (t) la. di~m phan ~a.eh giiIa mi~n cleo Rhett va.
mi~n CUng, R va. 5 nhu chu<1ng2.
Dih ki~ntu<1ngthich: ham
0 sao cho ne'u -E <
2). Them vaa di~m nut in+! vc1i
gia tri l~p ban da.u duqc eh(;mla
=
rein+!) = 2r(tn) - r(tn-l),
Vl(tn+l) = 2Vl(tn)- Vl(tn-l)'
(45)
(46)
Dung phep co xac dinh bm (36) va. (37), tinh dp xi r(tn+!),
vl(in+!), roi s(tn+!) (nher(41)).
Dinh ly 2.6 va.n dung vai sa do t{nh gall dung (; day. N6i khac di
thu~t toaD h(>itv va. On dinh vai sai s6 duqc danh gia theo Dinh ly
2.6.
Thu~t tOaD duac ap dung cho mot vi du s6.
17
K~t lu~n
Tom l<;ti,Juan an d<;ttdUC1Cmot s6 k~t qua mdi:
(1) ChUng minh duqc 51,!ton t<;tinghi~m toiw CI;J.C
clia. bai toa.n
bien tu do trong C(JhQc xam nM-po Cac ket qua nay cho phep mo ta
dang dieu cua ham s(t) khi t t~ng (51,1phat trieR cila mien deo nhat
theo thCri gian) Han mIa, chi ra ch~n tren cila then diem het pha
(deo) dum cae truerng hap d~t tai khac nhau.
(2) Mo hinh hoa bai toan va ch~m cua. thanh cleO:nhdt vito vat
can dim hOi tuye'n tIn}). Gia thiet vat can dan h6i phil hap vm thuG
t,~ han gia thiet v~t can rein tuY$t d6i igia thiet cua Barenblatt va.
15hliskii). Han mia. khi dQ cUng c1ia.v~t can t~ng leu .(tien ra +00 j
thi bili toan a day trer thanh bai toan vm dieu ki~n vat d.n dng tuY$t
doi.
(3) ChUng minh duqc St,ltOn t~ va duy nha:t nghi~m (dia. phuang,
toan C"l;J.c)
cua bili toem va. ch~ ctia. thanh deo nhdt va.o:v~t can dan
hoi.
(4) Thiet l~p dl1qc cae thn~t toan cho nghi~m gitn dung c1ia.hai
bai toem cia:nen. ChUng minh St,lhQi t"I;J.,
tinh On dinh cua. sa dOtinh
ga.n dung. Chi fa. sai so cua. thu~t teaR ding nhl1 sai so xa:p xi.
0
18