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Nonisothermal, nonNewtonian hele-shaw flows, Part II: Asymptotics and existence of weak solutions
NonlinearAndysu,
Pergamon
7’hemy, Methods
&Applicamm,
Vol. 27, No. 5, pp. 539-559, 1996 Copyright 0 1996 Elsevier Saence Ltd Printed in Great Britain. All rights resewed 0362-546X/96 SlS.00 + 0.00
NONISOTHERMAL, NONNEWTONLAN HELE-SHAW FLOWS, PART II: ASYMPTOTICS AND EXISTENCE OF WEAK SOLUTIONS R. P. GILBERTt
and P. SHIS
tDepartment of Mathematical Science, University of Delaware Newark, DE 19716, U.S.A.; and *Department of Mathematical Science, Oakland University Rochester, MI 48309, U.S.A. (Received Key words andphrases:
9 November
1994; received forpublication
10 February
1995)
Hele-Shaw flows, moving boundary problems, asymptotics, weak solutions.
1. INTRODUCTION
In this paper we first give a formal derivation of equations for the injection moulding starting from the basic equations for nonisothermal, nonNewtonian flow in a three-dimensional domain. Let IR, = fl x ( - E, E) be a three-dimensional mold, where R is a bounded domain in R2 with C’ boundary, and 2~ represents the thickness of the mold. Using the method of asymptotic expansions about E we arrive at a two-dimensional model that is independent of E. The second part of the paper is devoted to the study of weak solutions to the resulting model. The present paper is a continuation of our earlier work [l]. We begin with the balance of linear momentum and the energy equations in R,: ,Dv’ =diva’+p’f’ p Dt
and (1.2) where Dv’/Dt is the material derivative of the velocity field vc = Cut, v;, vi>, v ’ = (0;;) is the Cauchy stress tensor, sc = (si;> is the deviatoric part of (T E, p’ is the density of the fluid, f’ is the volume force density, c’ is the specific heat, K’ is the thermal conductivity, and d’=
(d:,)
withd:,=i(z+g)
(1.3)
denotes the strain rate tensor. Summations with repeated indicies are used. Note that the last term on the right hand side of (1.2) represents the heat generated by the deformation of the fluid under the action of the shear forces, the so called dissipation term. The physical 539
540
R. P. GILBERT
and P. SHI
meanings of other terms appearing in (l.l)-(1.2) are self evident. The reader is referred to standard references on fluid mechanics for a more detailed explanation of these equations, for example, see Archeson [2]. We further assume that the fluid is incompressible: div vE = 0. The stress tensor is given by the constitutive Power Law (Eisle [3], Subbiah et al. [4]): with s,; = k’(T’)j,“-‘d17,
q; = -pYijj + s,;
(1.4) (1.5)
where pE is the pressure, +e is the strain rate given by $ = 2J(d:,d,;), n is the Power Law index, and k’ is a given positive function. The product qE = k’(T’&‘‘/2 is known as the viscosity of the fluid. We remark that when n = 1 the fluid is called Newtonian, and when k’ is a constant function the fluid is called isothermal. Moreover, for a Newtonian and isothermal flow, where qE is independent of the temperature T’ and the shear rate +,, the equations (1.1) and (1.3)-(1.5) exactly lead to the standard Navier-Stokes equations. In Section 2 of the paper, we derive the two-dimensional governing equations for the injection molding by use of asymptotic expansions. We scale variables in a manner reminiscent of that used by Ciarlet and Destuynder [5] for the plate model. In Sections 3 and 4 we prove the existence of weak solutions to certain initial boundary value problems associated with the resulting equations. Although the physical models are two dimensional, we shall carry out our proofs in a more general setting. The statements of these problems are given below. For a given time interval (0, T), let fl, = s1 x (0, T). The outward unit normal of aR is denoted by v. We assume that an is decomposed as JR = r;, U r,, where r, and rl are C’ manifolds with r,, 17 r1 = 0. Problem I. Find functions
13and p defined fi such that -Ar3=k(HlVpl’+f
in IR,
-div{k(8)IVpl’-‘Vp}
=g on JR,
(1.8)
P =po
on r,,
(1.9)
=I
r,.
on
8 and p defined in flT such that e,-Ae=k(B)IVpl’+f
in fiT,
-div{k(8)IVp(‘-*VP}
=g,
o= 8,
on JlR x (0, T),
O=p P =po -k(O,lVpl’-‘g
(1.7)
6= 00
-k(O)lVpl’-‘z Problem ZZ. Find functions
in R,
(1.6)
in fiT,
onflx{O], on r. x (OJ), =I
on rI x @,T).
(1.10)
Nonisothermal,
nonNewtownian
Hele-Shaw
541
flows
Here f, g, 13,, p,,, 1, cpand k are given functions, while r is a given positive constant related to the power law index n; p is the pressure of the flow and 8 is the temperature. Problem I is a model for a stationary flow and problem II is a model for the time-dependent flow. In both problems, however, moving boundaries are not taken into account. The principle difficulties of the proof lies in overcoming the critical growth in the gradient of p that is present in (1.6) and (1.11). It is interesting to note that when r = 2 equations (l.lM.2) and (1.6)-(1.7) are mathematically identical with the governing equations for thermistor problems which have received extensive study recently [6-111. 2. DERIVATION
OF
THE
EQUATIONS
As is usual, we assume that the channel is narrow with the width 2~. Indicies with Greek letters range from 1 to 2 while indicies with Roman letters range from 1 to 3. For example, we use CC,> := (x,, x2) to designate two coordinates and (xi) := (xi, x2, x3) to designate three coordinates. We assume that the flow is quasistatic; hence, the inertial term is neglected from the balance of linear momentum. This assumption has been widely used in the engineering literature (see [3,4]). Therefore, in the absence of body forces (1.1) and (1.5) become (2.1) the quasistatic form of the balance of the linear momentum. according to the rule
We stretch the x3 coordinate
x3 = Eyj.
(2.2)
At the same time we assume that the physical parameters remain unchanged K’=K,
k’=k,
CE= c,
PE = p,
K’, k’, c’, pE and the time t and
The velocity field U’ :=
;p”+p1+p2c+
(2.3)
T’ transfer accord-
a = 1,2,
... . ..
T’ = To + T’E + T2e2 + . . . We shall not notationally the notation
t=t.
(2.4)
distinguish a function in different variables. For example, we employ
542
R. P. GILBERT and P. SHI
We approximate
the shear rate by computing
4d;dl;
4dimdim _ z,“; s = ’ avi avi (I (I E2dY3JY3
I z?
i: a
p
I ’ av; ‘“,’ E dx, dY3
(2.5)
Hence, from (1.5) we approximate
the stress $I; as (n-*)/2 df, + O(
Using this and the above calculations a a dxs,; = dx,S$ I
l )d,,
.
(2.6)
the left-hand side of (2.1) becomes
1 d + ; dx,G
(2.7)
Nonisothermal, nonNewtownian
Hele-Shaw flows
543
We now distinguish the index i as being 1 or 2 in the following computation.
(2.8) On the other hand when i = 3 we get
Let us now consider the right-hand
side of (2.1). From (2.4), (2.8) and the equation ape -= dx,
d ax,6
Nonisothermal, nonNewtownian
where in the last inequality (4.18) we obtain
557
Hele-Shaw flows
we have used (4.20). Similarly,
using the Cauchy’s inequality
and
(4.26) k(8,N)IVp,Nlru dxdt I cll~ll~~(o,~;~,‘~~(n)). 5 I m=l 11Im fl Therefore, (4.24) follows from (4.231, (4.25) and (4.26). Note that C?[O, T]; H,‘12(0n)) is dense in L2(0, T; IIZ,‘*~(~~)). Hence d0z/dt can be extended uniquely as a bounded linear functional of L2(0, T; H,‘.2(fl)>. Using the notation of the duality pairing between L’(O, T; H,‘s2(fl))* and L2(0, T; H,‘.2(fi>), (4.24) implies a*, I cllvll L”(0.T;H;qIl)) I( dt’v )I We are now ready to apply Simon’s lemma. Let x= H;,2(n),
B =L2(fl)
(4.27)
Vu E L2(0 3T-7H’-2(fl)). 0
Y=H-‘,2(fl).
and
Let F=
{e, - e,;N=
1,2,3...).
Then from (4.20) to (4.21), it follows that F is a bounded subset of L*(O, T; X). Moreover, from (4.271, it follows that dF/dt is a bounded subset of L2(0, T; Y). Therefore, lemma 4.4 is proved. n LEMMA
4.6. Let 3, be the function defined by
(x,t) E R,.
if(m--1)6It
i&xx, t> = 0,“(x)
Let tI~L~(flr) and let (Nj} be a sequence of natural numbers. Then {e,) converges to 8 strongly in L2(n),) if and only if {ON,} converges to 8 strongly in L2(Ln,). Proof Setting v = 0,” - O,“_, in (4.16), we obtain
(4.28) = /ktt’;)lV~:l’te,h. - B,N_Jdx + f(O,“- B;mJdx. /n R We now multiply both sides of (4.28) by a2 and sum over m = 1, 2.. . N. Using the Cauchy’s inequality and the estimates (4.18), (4.20), it is straightforward to show that
N 6 (e,“- e/~J2dx=o(v%). =/ m=l n Using (4.11), it is easy to establish the following relations:
/ fh
(8,,-
O)‘dxdt=
f
f
6 /(f?;-
m=l
n
(4.29)
‘-’ (3, - 0)‘dxdt / 0 / bl
8,N)2dx+
[t
- (m - l)S](e,N_,
- r3)dxdt.
(4.30)
Nonisothermal, nonNewtownian
Hele-Shaw flows
545
where (2.13) and we have replaced To by its average To over the interval (0, E). It is interesting to note that (2.12) and (2.13) constitute a nonlinear Darcy’s law, which follows from our much more basic assumptions (2.1)-(2.4X Using the second part of the assumption (2.111, the incompressibility condition (1.4) averages to asjo L=() dx,
’
which with (2.12), (2.13) yields a pressure equation (2.14) This can be improved upon using (2.10*). First we sum the square of both sides of (2.10) to obtain 2 c?pO
ape
au;
0
4y3dx,c)x,=kz(T
)
i
dvp"
dy,2jq
n-1 I
av,O
au;
dY3 dY3
or (2.15) Hence
(!&y”-;
= ( &,vpo,)“li(,
from which follows that s = I’[,‘(
&)“n,Vpo,l’n-l
dJ’dy,
= ~vpo~‘/“-‘m(~o)
(2.16)
where m(T”> Substituting
= [JG’(
&)-“‘didy3=
[k~~“~l-l’”
[JliC21)1”‘diilg,.
this s(x~) into the equation (2.14) yields (2.17)
which is the standard pressure equation such as used by Guceri et al.
546
R. P. GILBERT
and P. SHI
We now turn to the energy equation. From (2.6) and a further estimate on the term dz it is clear that (2.18)
+ O(E)
so that (1.2) becomes dT”
PC
x+ex+...
dT’
+E(u,O+EU;+...)
+~(u:t2+u:E3+
dT’ +
. ..) +
-
E dy,
*-.
+ O(E).
In light of (2.15), it is straightforward
(2.19)
to show that
(n+1)/2 =
C?y,)
(n+l)/n[k(T(I)]-(‘/n)IVpol(n+l)/~,
(2.20)
We now assume that the temperature profile is symmetric over the mold thickness and that on the surface of the mold it is insulated. This amounts to saying that JT’ -=
at
0
x3 =o,
+E.
3x3
Taking the average on both sides of (2.19) and making desired two-dimensional model for the energy equation:
use of (2.20)-(2.21)
aT0 a aT0 pcat=ay, ( K- dy, 1 + Am(?;“>lvpop+
1)/n,
we obtain the
(2.22)
where A is the constant that equates 1
(2[)(‘+1”nd(=
[k(T”>]m’l’“’
Am@‘).
(2.23)
/0 Equations molding. The r = (n + 1)/n, (2.221, with f
(2.17) and (2.22) are the form of the equations are it is easy to see that (1.11) and g added as appropriate
desired two-dimensional model for the injection independent of the mold thickness 2~. By setting and (1.12) are nondimensional forms of (2.17) and source terms.
Nonisothermal, 3. WEAK
nonNewtownian SOLUTIONS
Hele-Shaw OF
547
flows
PROBLEM
I
In this section, we assume that R is a bounded C’ domain of R”, where n is a natural number that is greater or equal to 2. For s > 1, let Hi*”
= {v; v E Zf’~‘(fi>,
v = 0 on I,}
denote the usual Sobolev space equipped with the standard norm. Let r* = r/(r ifln. i We assume that the boundary values 0, and p,, for problem defined on fi such that
- 11,
n/n-l
Cl=
8, E H’,“(Q)
and
I can be extended to functions
po E H1xT(fln),
(3.1)
where T is a fixed number that is greater than r. We further assume that gELqR)
fe L.“1(s1),
and
(3.2a)
I E L~~(r,>,
where a,, i = 1, 2, 3 satisfy * u, > !!2’
t-1.Y,
*2>
and
3
if 1
otherwise, we assume that and if r>n. I dug, f, ga51(RL Finally, we assume that there exist positive numbers k, > k, > 0 such that k,
V8ER1.
(3.31
3.1. We say that (0,~) is a weak solution to (1.1)~(1.9
e - 8, E H(y(R),
(3.2b)
if
p -po E H;~‘m;
(3.4a)
and for all u E C;(a) /n
V8*Vvdx=
/n
k(0)lVpl’vdx-t
fvdx;
(3.4b)
= ;g(dx. cl
(3.4c)
/
and for all 5 E H;‘(0) /n
k(B)lVpl’-‘VpVedx
+ i@ds I
Remark. If 1 < r < 2, IVplrP2 becomes singular on the set where Vp = 0. We define lVpl’-2Vp
= 0 on that set. The main result of this section is the following. 3.3. Assume that (3.M3.3) hold. Then there exists a weak solution to problem (1.1)~(1.5) in the sense of definition 3.1. The proof will be carried out under a fixed point argument. We first establish some technical lemmas.
THEOREM
548
R. P. GILBERT
LEMMA 3.4. The following statements u E L”(a) and u E L P(fI) then
and P. SHI
hold. (i> For any positive
numbers
cx and p, if
where y =
uv E LY(fiR),
Moreover, II4 (ii) If p •Hl,~(fi)
LY(fI) 5 Il~II~“(~~II~lI~~~~~.
and 1
(iii> If p E H1lr(fi)
E [L”~‘“-“(s1)IIl”.
Moreover
I Ilpll~~~~~~~~~~n,ll~Pll~~~~,.
and 1
E Li(R),
where
++;)-‘. Moreover
Proof By Holder’s inequality,
it follows that
where s* = -P Y’ Y’ This proves part (i> of the lemma. Part (ii> is a consequence of part 6) and the Sobolev imbedding theorems. By setting u =p and v = IVpl’-‘, we obtain a=nr/(n -r.) and /3=r*. Therefore SC?
‘=
(
nr/(n
-1
1 -r>
+I
r* i
=y+y
n-r
r-l
-l =x’
n
) ( The proof of part (iii) is similar to that of part (ii) and we leave the detail to the reader. LEMMA 3.4. Suppose that (3.1)-(3.3)
n
hold. Suppose that 8 and p satisfy (3.4a) and (3.4~).
Then tlv E C’(a) / fl
k(B)IVpl’vdx=
kWvJVpJ’-*VpVp, / f1 -
/n
+ /n
k(e)(p
dx
-pJVpl’-“VpVudx
g(p -po)u dx -
/5
l(p -p,Jv
ds.
(3.5a)
Nonisothermal, nonNewtownian
Moreover,
there exist a polynomial
549
Hele-Shaw flows
F that is independent
of 8 and p such that (3.5b)
Prooj We first show that (3.5a) implies (3.5b), and this will be discussed for two different ranges of r. n
Case 1. 1 < r < n. We denote the four terms on the right-hand IV, respectively. Using lemma 3.3 we obtain
side of (3.5a) by I, II, III, and
I11~k~llVpllrL;(:I~II~P~ll~~~~~ll~llr.~*~n,~
(3.6)
1111I Ilp -p~ll~~~~~~~~~r~,ll~pll~;c:~,ll~~ll~~~~~,~ IVZZII IlgllL”2(n,llp -POIIL”‘/‘“~~In,ll~llr.~cn,,
(3.7) (3.8)
and
ID7 I IIuL~~cr,,IIp -p,~llL1’.~“~~““~~~~r,,ll~llL~~r,,.
(3.9)
In (3.6) [ is the nuimber defined in (iii) of lemma 3.4. In (3.8) and (3.9) p and ji are some (possibly large) positive numbers. Invoking the Sobolev imbedding theorems H’,‘( 0) L* L”‘/(“-‘)( a), H’3”(fl) -q(a), vqcm, H’-‘(R)
(3.6H3.9)
~L[(“~t)‘I/(“-‘)(r,),
and
H’.“(a)
-*Lq(r,),
vq<‘;c,
lead to
III + IHI + WI + WI I F(llpll~~~~~n,)llul~~~~~~~~, for some polynomial
F. This proves that (3.5a) implies (3.5b) for the case 1 < r < n.
Case 2. r > II. The proof in this case is similar but technically simpler than the previous case. This is due to the compact imbedding of the space H’,‘(fi) into the space Ca(a) with (Y= 1 - (n/r). We leave the details for the interested reader. We now turn to the proof of (3.5a). Setting [ = u(p -p,,) in (3.4~) we obtain / fl
k~empl’- %‘p+V(p +
/6
-pJ
+ (p -p,)Vuldx
l(p -po)u ds
= (g(p
-p,)udx. JCl This yields exactly (3.5a) after straightforward
calculations.
LEMMA 3.5. Assume that 0, -+ 0 a.e. on Kl and p, E H{‘(R)
/n
k(em)IVp,I’-Z Vp;V~dx=G(.$)
is a solution to the equation
V[ E H;VU,
(3.10)
550
R. P. GILBERT and P. SHI
where G E (H,!%‘(R))*.
Then there exist p E Hi,‘(Q)
and a subsequence {p,,} such that
weakly in Hi,‘(fin),
IP,,) +P
(3.11)
in the norm of [ Lq(R)]”
VP,, + VP
Vl
(3.12)
Remark. Lemma
3.1 is almost a consequence of theorem 3.1 in Boccardo and Murat [121, but the exact statement of the result in 1121 does not fit our setting. The proof of lemma 3.5 given below follows the idea introduced in [12]. Proof: By use of the standard theory of monotone operators, it follows from (3.10) that the H’,r(fl) norm of pm is independent of m. Thus (3.11) holds for a suitable subsequence of {pm,}. For the proof of (3.12), we choose in (3.10) the test function [=p,, -p, to obtain
/[n
k( %,)l~Pm,lr-2~Pm~ - k(B,,)lVplrP2Vp] = G(p,,
-p) -
/n
.V(p,,
k( O,,,)lVplr-%pV(p,,
-p)dx
-p)dx.
Since by assumption that 0, + 0 a.e. on R, it follows from (3.4) that k( e,?) + kte)
in Lq(fin)
(3.14)
for all 1 < q < =. Using (3.11) and (3.14) it is easy to see that all terms on the right-hand of (3.13) approach zero as j + m. Hence lim k( %;)l~Pm,l’-20p,~ I+= /I n
- k( O,,$VpI’zVp]
V(p,,
-p)dx
= 0.
side
(3.15)
Let
e, =k( ~m,)(lCp”‘,lr-zVpm,- IVPI’~~VP)~V~P,,, -PI. Then (3.15) can be written as lim e, dx = 0. +z /n Fix a number 0 < A < 1. The Holder inequality gives
(3.16)
j
(3.17) To complete the proof of (3.12) we use the inequality
(IXl’-2X - W2y)(x where a > 0 and b > 0 are certain
-y) 2
alx -ylr lx -y12 a (b + IxI+ IyP
constants. This together
if 1
Nonisothermal, nonNewtownian
Hele-Shaw flows
551
VP,, + Vp a.e. on Cn. Hence by Vitali’s theorem we obtain lim
IV(p”, -p)lrA dx = 0.
j+ m / n
n
Proof of theorem 3.3. We construct a mapping A whose fixed points will be solutions to the problem. Here we only present the proof for the case where 1 < r < n. For r > n, the same proof goes through with only slight modification. Recall that in this case u = n/(n - 1). Let z E H,‘,“(fi> + &,, and let & be the unique solution of the problem
k(z)lV&I’-*V,V(
dx + lWs= ~gSdx 1 in the above equation, we obtain a constant cr > 0 that is
4, -pa E H;wu,
/R
K$E H+‘(R). By setting (= & -p. independent of z, such that
(3.18)
IMIILyn, 2 Cl. Next we define a linear functional (F,,u)
=
F, E (H’*“*(IR))*
determined
by
k(z)ulV~,I’-*V~~.Vp,dx /R -
k(t)(& -p”)lV4zlr-2w*.V~dx /n
+
1(& -p,Ju dx + ,f” dx. g<$z -po)u dx / r1 / /n
By virtue of lemma 3.4, F, is well defined, and there exists a constant c2 > 0 independent such that
I~~,,u~l~c211uII~~~~*~1R~.
of z (3.19)
We now define the mapping A that maps H,‘, ” (a) + 8, into itself such that w, = AZ is the unique solution of the linear elliptic problem wz - 8, E H,1Wl>,
/0
Vw;Vvdx=
and
Vu E H,1,“*(0)
(Fz,v).
(3.20)
Since the operator - A is an isomorphism between Hi%u(CI> and (H$“*(fl))*, the estimate (3.19) shows that A is well defined. Next we show that the mapping A is continuous from H$“(Ln) + 19~ into itself under the weak topology of H’, ” (G). To do so, it is sufficient to prove that the mapping z -+ F,
under the weak topology of H’,“(R)
is continuous
and (H’,“*(SZ))*,
Fz + wz
(3.21)
and that the mapping
is continuous
(3.22)
552
R. P. GILBERT
under the weak topology of (H’,“*(R))* with
and P. SHI
and H’s”(R).
To prove (3.21), we let z, E H,$” + 8,
weakly in HLZ”(fl).
z, +z z, +z
(3.23)
a.e. in LR.
(3.24)
By the estimate (3.181, we may also assume (for a subsequence of {z,} in fact) that 4 I, - 4
weakly in Z-I’zr(Q).
(3.25)
We need to show (3.26)
4= 4,. Applying lemma 3.5 to passing to the limit in the equation /n with 5 E Hk’(R)n
~(z,JIV~,~I’-*V#Q’~
C’(a)
dx + l,WS=
j+
fixed, we obtain, in light of (3.23)-(3.25),
that
dx + L@ds= Lg(dx. 1 This proves (3.26) since the solution to the last equation is unique. To proceed with the proof of (3.211, we recall /n
(~,,,v) =
k(z)lV+l’-*V&V<
k(z,,)UlV4~mlr-2V4~m.V~Odx In -
k(z,)(+zm -Po)lv4~mIr-Zv4~_.Vudx /n
+
g(4z,, -p,Ju dx I(&~ -p,,)u dx + fu dx. /n / rl /n
(3.27)
Using lemma 3.5 again together with (3.26) we may assume that 4 I, -+ 4z
V4zm+ 04,
weakly in H’sr(fi)
in the norm of [L4(filn
(3.28) Vq
(3.29)
These allow us to pass to the limit in (3.27) as n + m, to obtain (3.30) vu EH’~“*(il). lim (F,,,u> = (F,,v> n-m Here we only calculate the limit for the second term on the right-hand side of (3.27) to illustrate a justification of (3.301, and leave the calculation of other terms to interested readers. Let
(ZZm,v> =
/n
k(z,)(4,m -p0)lv4~~lr-2v4~~.V~dX~
v E War(n).
Nonisothermal, nonNewtownian
Hele-Shaw flows
553
Note that k(z,) -+k(z) in the norm of Lq(fin) Vq < ~0. We may assume that &,, --f & strongly in L”‘/(“-‘) (a). Using lemma 3.5, we may assume that
in the norm of [ Lq(Q)]”
(Zl,d
VO < q < r*. These give
+
I0
k(z)($, -p~)lV~,l’-2V~z.Vvdx~
vu E C’(i?n).
(3.31)
Since llZZ,,,ll(~~~*(~l),*IC, with c3 independent of m, and C’(a) is dense in H’,r*(fln), we have (3.31) hold for all v •H’~~*(fln). This proves (3.30). Since H1x”*(R)* is a reflexive Banach space (3.30) implies (3.21). Finally, (3.22) is obvious in light of the linear equation (3.20). We now conclude that the mapping A : H$“tfl> + 8, + Hi,“(R) + B0 is weakly continuous. It remains to show that A has a fixed point. Invoking (3.18) and (3.191, it follows that
for some constant c independent
of z. This shows that A maps the ball
B={ z; z Ezz;Jw)
II&~ “(Cl)5 c}.
+ e,,
into itself. Since H’*“(fl) is a reflexive Banach space, B is a compact set under the topology induced by the weak topology of H’,“(n). Hence A has a fixed point in B by using Tychonoffs fixed point theorem. Theorem 3.2 is proved. w 4. WEAK
SOLUTIONS
OF PROBLEM
II
In this section, we prove that problem II has a weak solution for 1 < r < IZ and n = 2. We assume that the given data for problem II satisfy the same assumptions as specified in (3.1)-(3.3). For the sake of clarity, we do not assume that these data depend on time although there is no difficulty to deal with such cases by use of the same method which we shall demonstrate in this section. In addition, we assume that the initial temperature cp satisfies pEL2(n). Definition
(4.1)
4.1. For 1
and for all v E C”(fir)
-J
07
p -p,, E L’(0, T; H;WW;
(4.2)
with u = 0 on Jfl x (0, T)U fl x IT), (Ov, + VB.Vv) dx dt =
/ nr
k(fMVpl’vdxdt
+ l,,xdt+
&v(x,O)dx;
(4.3)
554
R. P. GILBERT
and for all CE L’(0, T; Hi,‘(fl))
In
and P. SHI
and for almost all t E (0, T)
k(B)lVp(‘-2VpVS
dx + 1,1+=
~gtd~.
(4.4)
The main result of this section is the following theorem. 4.3. Assume that (3.1)-(3.3) and (4.1) hold. Then there exists a weak solution to problem (1.6M.11) in the sense of definition 4.1.
THEOREM
The proof of theorem 4.2 will be carried out by the Rothe’s method of time discretization. To this end, let the time step 6 = T/N for some large N. For each m = 1,2.. . N we consider the stationary problem for {&“, p,“]
-div{kcs,N>lOp,“I’-“Vp,N} 0,” = 0, P’PO
=g
in 0,
(4.6)
on ~90,
(4.7)
on ro,
(4.8)
-ktf3~,,vpy’~
=I
on rI.
To start the time marching, we choose @Ns 0 p.
(4.10)
By virtue of theorem 3.2, for each m problem (4.5)-(4.10) has a solution in the sense of definition 3.1. Note that for 1 < r < rz and n = 2, 0,” lies in the Hilbert space IzI’,~(C~) since g= 2. To recover a solution to the time dependent problem, we define (@,, pN) on fl, via 8,(x, t) = 8~-~e~-l[r-(m-1)S1+~~-,, PJ& LEMMA
r) =p,“,
if (m - l)Grt
4.4. There exists a constant c > 0 that is independent
(4.11) (4.12)
of N such that
~~~Nl~~2~0.~;~‘.Z~~))5 c
(4.13)
and (4.14) proof From theorem 3.2, {@,“, p,“} satisfies the following weak form 0,” - 00 E H;,2(sr),
p,” -po E H;*w);
(4.15)
Nonisothermal,
nonNewtownian
Hele-Shaw
555
flows
and for all u E C;(a) (8,“=
/ fl
B,N_JUdx+
Ve,“. Vu dx
k(8,N)lVp,Nlrv
dx +
(4.16)
fu dx; / cl
and for all [EH+‘<~I>
/n
k(B,N)IVp,NIr-‘VpmN.V5dx
Then (4.14) follows by setting obtain
(=pt
-p.
+ lWs= I
(4.17)
l@x.
in (4.17). We now use (4.14) and lemma 3.4 to
where F is a polynomial that is independent of (0,“, p,“). This allows the test functions in (4.16) to be taken from the space H,‘,‘(R). In particular, we can choose v = 0,” - 8,. This leads to (8,“-
8,“_,)(6’,N-
= /k(t?,f)lVp;lQ,f’R We now multiply
e,)dx+
Qdx
Ve,“T
+
/n
8,)dx
f
(4.19)
both sides of (4.19) by S and sum over 1 I m I N. Using the inequality 2e,“ce,” - e,“_,> 2 (8,Nf - ce,“_,,’
and the estimate (4.18) with v = 0,” - 13,, it follows that after some standard calculations 2 m=l
where c > 0 is a constant independent result over (0, T) we obtain
/%
lVe,l* dxdt=
6 ;
/n
(4.20)
lVe,“l” dx + dx I c,
of N. Squaring both sides of (1.11) and integrating
(
+ IVe,,J’+
+lVe,” - Ve,,J’).
m=l
Hence (4.13) follows from (4.20) and (4.21). The lemma is proved.
n
LEMMA 4.4. A subsequence of {0,} converges in the norm of L*(fI,).
To prove this lemma, we need a compactness result due to Simon [13].
the
(4.21)
556
R. P. GILBERT
and P. SHI
LEMMA 4.5. (Simon) Let X, B, and Y be Banach spaces with Xc B c Y. X is compactly imbedded in B. Let 1 I 9 < cc and F be a bounded subset of Lq(O, T; x). Moreover, the set
is bounded in L’(0, T; Y ), where the partial derivative is the distributional valued functions. Then F is compact in Lq(O, T; B).
derivative for vector
Proof of lemma 4.4. Equation (3.16) can be rewritten as: for each m = 1, 2.. . N and for each t such that Cm - 1)6
VB,NVvdx=
k(B,“‘)(Vp;lrvdx+
fvdx /R
(4.22)
for all v E C?[O, T]; H$*(fi)). The test functions in (4.22) can be chosen in such a way since the domain of all integrations is independent of t and all the integrands do not involve the time derivative of the test functions. With such choices of the test function we now integrate (4.22) over each I, = ((m - l)S, ma>, and then sum the result over 1 I m IN, to obtain zvdxdt+
g m=l
V8,NVvdxdt //I,
11
N = =I/ m=l
k(B,N)IVp/I’vdxdt Im 11
+
fv dx dt.
(4.23)
Next, we show that there exists a constant c > 0 that is independent
of N such that
Using the Cauchy’s inequalities respectively, we obtain
and for vectors in RN
VO,NVvdxdt
for square integrable
1< m=l : [(
s j,
~,VO;,zdx]l”(
( l,V0,$,‘dx)ii2fi(
functions
~IVv,2dx)“2dt
i@,2dxd(i”’
(4.25)
Nonisothermal, nonNewtownian
where in the last inequality (4.18) we obtain
557
Hele-Shaw flows
we have used (4.20). Similarly,
using the Cauchy’s inequality
and
(4.26) k(8,N)IVp,Nlru dxdt I cll~ll~~(o,~;~,‘~~(n)). 5 I m=l 11Im fl Therefore, (4.24) follows from (4.231, (4.25) and (4.26). Note that C?[O, T]; H,‘12(0n)) is dense in L2(0, T; IIZ,‘*~(~~)). Hence d0z/dt can be extended uniquely as a bounded linear functional of L2(0, T; H,‘.2(fl)>. Using the notation of the duality pairing between L’(O, T; H,‘s2(fl))* and L2(0, T; H,‘.2(fi>), (4.24) implies a*, I cllvll L”(0.T;H;qIl)) I( dt’v )I We are now ready to apply Simon’s lemma. Let x= H;,2(n),
B =L2(fl)
(4.27)
Vu E L2(0 3T-7H’-2(fl)). 0
Y=H-‘,2(fl).
and
Let F=
{e, - e,;N=
1,2,3...).
Then from (4.20) to (4.21), it follows that F is a bounded subset of L*(O, T; X). Moreover, from (4.271, it follows that dF/dt is a bounded subset of L2(0, T; Y). Therefore, lemma 4.4 is proved. n LEMMA
4.6. Let 3, be the function defined by
(x,t) E R,.
if(m--1)6It
i&xx, t> = 0,“(x)
Let tI~L~(flr) and let (Nj} be a sequence of natural numbers. Then {e,) converges to 8 strongly in L2(n),) if and only if {ON,} converges to 8 strongly in L2(Ln,). Proof Setting v = 0,” - O,“_, in (4.16), we obtain
(4.28) = /ktt’;)lV~:l’te,h. - B,N_Jdx + f(O,“- B;mJdx. /n R We now multiply both sides of (4.28) by a2 and sum over m = 1, 2.. . N. Using the Cauchy’s inequality and the estimates (4.18), (4.20), it is straightforward to show that
N 6 (e,“- e/~J2dx=o(v%). =/ m=l n Using (4.11), it is easy to establish the following relations:
/ fh
(8,,-
O)‘dxdt=
f
f
6 /(f?;-
m=l
n
(4.29)
‘-’ (3, - 0)‘dxdt / 0 / bl
8,N)2dx+
[t
- (m - l)S](e,N_,
- r3)dxdt.
(4.30)
R. P. GILBERT
558
Moroever, right-hand
by use of the Cauchy’s inequality, side of (4.30) is less or equal to (O~-ee,N?d
and P. SHI
the absolute value of the last term on the
x]“‘[
Therefore, lemma 4.6 follows (4.29) to (4.31).
iT-‘j,(i&-
(4.31)
O)?dxdj’-I.
n
Proof of theorem 4.3. First, we extend 8, to 1Rx [ - 6, T] by setting e, = P if - 6
(4.32)
pN -p. E L2(0, T; H’~TYn>);
with u = 0 on JQ X (0, T)U fi X {T]
1
[(B,(x,t)-B,(x,I-6))]udxdt+ s / Sl7 = and for all ~EH~*‘(fi,>
k(&v)lVp,l’u / QT
I SIT
dx dt +
V&.Vvdxdt
(4.33)
fvdxdr, / fb
and 0 < TS T Vp,*Vtdxdt+
k( ,)l,lr-* / 07
/ r,x(O.r)
l
where .R, = 1Rx (0,~). By making a change of variable in the term In, 8,(x, have 1 s / [(&,(x,~) fb
- &,,(x,t
(4.34)
g5 dx dt, /% t
- S)dx dt we
- 6)]udxdr
i$.udxdl-
pv(x, t + S> dx dt
+
T-8
(4.35)
8,(U(x,t)--U(x,t+~))dxdt. n Using the equation (4.34) we obtain an anolog of lemma 3.4: ++
/ 0
k( &,)IVp,J’u
dxdt =
I4
/
k(e,)uiVpsl’-‘V~,~.Vpo
+I
dx dt
k(ii,)(p,-p,)lVp,I’~‘Vp,.Vvdxdt
g(pN -p,)vdxdt %
-
l(p, / l-,X(0.7)
-p&
ds dt.
(4.36)
Nonisothermal,
nonNewtownian
Hele-Shaw
flows
559
Using lemma 4.3, lemma 4.4, lemma 4.6 and expressions (4.20)-(4.211, we may assume that, without loss of generality s, -+ e
weakly in
e, + 8
L2(0,T;H"2(fk)),
strongly in
(4.37) (4.38)
L2($).
Moreover, we may further assume that weakly in L'(O,T;H','(fk>>. (4.39) +P In order to pass to the limit in (4.32)-(4.35), the convergence in (4.37FC4.39) are not sufficient. Here we shall need stronger convergence for VP,. A careful examination of the proof of lemma 3.5 shows that the following can be proved for (pN) PN
VP, -y!?
strongly in
Lq(O,T;[H',q(R)ln)
Vq
(4.40)
The proof of (4.40) is only a slight variation of the proof of lemma 3.5 in that (i) instead of equation (3.101, we begin with (4.34) with T= T, (ii) integrations and inequalities are considered in R, instead of R. We leave the details for interested readers. By virtue of (4.37)-(4.401, we can now pass to the limit as N + ~0 in (4.32lFC4.35) and conclude that the limit funcitons (0, p} satisfy definition 4.1. Theorem 4.2 is proved. n REFERENCES 1. GILBERT R. P. & SHI P., Nonisothermal, NonNewtonian HeleeShaw flows, Part I: Mathematical formulation, in Proceedings on Transport Phenomenon, pp. 1067-1088. Technomic, Lancaster (1992). 2. ACHESON D. J., Elementary fluid mechanics. Clarendon Press, Oxford (1990). 3. EISELE U., Introduction ro Polymer Ph@cs. Springer-Verlag (1990). 4. SUBBIAH S., TRAFFORD D. L. & GUCERI, S. I., Nonisothermal flow of polymers into two dimensional, thin cavity molds: a numerical grid generation approach, Inr. L Heat Muss 7’runsfer 32, 415-434 (1989). 5. CIALET P. G. & DESTUYNDER P., A justification of the two dimensional plate model, L Mecanique 18, 3155344 (1979). 6. ANTONCAEV S. N. & CHIPOT M., The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal. 25, N.4, 113881156 (1994). 7. CHEN X. & FRIEDMAN A., The thermistor problem for conductivity which vanishes at large temperature, Quart. Appl. Math. 51, N.l, 101-115 (1993). 8. CIMATTI G., The stationary thermistor problem with a current limiting device, Proc. Royal Sot. Edinburgh 116A, 79-84 (1990). 9. CIMA’ITI G., Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions, QUUI?. Appl. Math. 47, 117-121 (1989). 10. SHI P., SHILLOR M. & XU X., Existence of a solution to the Stefan Problem with Joule’s heating, .I. DifSerentiul Equutions 105, 239-263 (1993). 11. XI0 H. & ALLEGRETTO W., Solution of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SOLML Math. Anul. 22, N.6, 1491-1499 (1993). 12. BOCARDO L. & MURAT F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19, N.6, 581-597 (1992). 13. SIMON J., Compact sets in the space LP(O,T; B), Ann. Math. Puru. Appl. 146, 65-96 (1987).
Pergamon
7’hemy, Methods
&Applicamm,
Vol. 27, No. 5, pp. 539-559, 1996 Copyright 0 1996 Elsevier Saence Ltd Printed in Great Britain. All rights resewed 0362-546X/96 SlS.00 + 0.00
NONISOTHERMAL, NONNEWTONLAN HELE-SHAW FLOWS, PART II: ASYMPTOTICS AND EXISTENCE OF WEAK SOLUTIONS R. P. GILBERTt
and P. SHIS
tDepartment of Mathematical Science, University of Delaware Newark, DE 19716, U.S.A.; and *Department of Mathematical Science, Oakland University Rochester, MI 48309, U.S.A. (Received Key words andphrases:
9 November
1994; received forpublication
10 February
1995)
Hele-Shaw flows, moving boundary problems, asymptotics, weak solutions.
1. INTRODUCTION
In this paper we first give a formal derivation of equations for the injection moulding starting from the basic equations for nonisothermal, nonNewtonian flow in a three-dimensional domain. Let IR, = fl x ( - E, E) be a three-dimensional mold, where R is a bounded domain in R2 with C’ boundary, and 2~ represents the thickness of the mold. Using the method of asymptotic expansions about E we arrive at a two-dimensional model that is independent of E. The second part of the paper is devoted to the study of weak solutions to the resulting model. The present paper is a continuation of our earlier work [l]. We begin with the balance of linear momentum and the energy equations in R,: ,Dv’ =diva’+p’f’ p Dt
and (1.2) where Dv’/Dt is the material derivative of the velocity field vc = Cut, v;, vi>, v ’ = (0;;) is the Cauchy stress tensor, sc = (si;> is the deviatoric part of (T E, p’ is the density of the fluid, f’ is the volume force density, c’ is the specific heat, K’ is the thermal conductivity, and d’=
(d:,)
withd:,=i(z+g)
(1.3)
denotes the strain rate tensor. Summations with repeated indicies are used. Note that the last term on the right hand side of (1.2) represents the heat generated by the deformation of the fluid under the action of the shear forces, the so called dissipation term. The physical 539
540
R. P. GILBERT
and P. SHI
meanings of other terms appearing in (l.l)-(1.2) are self evident. The reader is referred to standard references on fluid mechanics for a more detailed explanation of these equations, for example, see Archeson [2]. We further assume that the fluid is incompressible: div vE = 0. The stress tensor is given by the constitutive Power Law (Eisle [3], Subbiah et al. [4]): with s,; = k’(T’)j,“-‘d17,
q; = -pYijj + s,;
(1.4) (1.5)
where pE is the pressure, +e is the strain rate given by $ = 2J(d:,d,;), n is the Power Law index, and k’ is a given positive function. The product qE = k’(T’&‘‘/2 is known as the viscosity of the fluid. We remark that when n = 1 the fluid is called Newtonian, and when k’ is a constant function the fluid is called isothermal. Moreover, for a Newtonian and isothermal flow, where qE is independent of the temperature T’ and the shear rate +,, the equations (1.1) and (1.3)-(1.5) exactly lead to the standard Navier-Stokes equations. In Section 2 of the paper, we derive the two-dimensional governing equations for the injection molding by use of asymptotic expansions. We scale variables in a manner reminiscent of that used by Ciarlet and Destuynder [5] for the plate model. In Sections 3 and 4 we prove the existence of weak solutions to certain initial boundary value problems associated with the resulting equations. Although the physical models are two dimensional, we shall carry out our proofs in a more general setting. The statements of these problems are given below. For a given time interval (0, T), let fl, = s1 x (0, T). The outward unit normal of aR is denoted by v. We assume that an is decomposed as JR = r;, U r,, where r, and rl are C’ manifolds with r,, 17 r1 = 0. Problem I. Find functions
13and p defined fi such that -Ar3=k(HlVpl’+f
in IR,
-div{k(8)IVpl’-‘Vp}
=g on JR,
(1.8)
P =po
on r,,
(1.9)
=I
r,.
on
8 and p defined in flT such that e,-Ae=k(B)IVpl’+f
in fiT,
-div{k(8)IVp(‘-*VP}
=g,
o= 8,
on JlR x (0, T),
O=p P =po -k(O,lVpl’-‘g
(1.7)
6= 00
-k(O)lVpl’-‘z Problem ZZ. Find functions
in R,
(1.6)
in fiT,
onflx{O], on r. x (OJ), =I
on rI x @,T).
(1.10)
Nonisothermal,
nonNewtownian
Hele-Shaw
541
flows
Here f, g, 13,, p,,, 1, cpand k are given functions, while r is a given positive constant related to the power law index n; p is the pressure of the flow and 8 is the temperature. Problem I is a model for a stationary flow and problem II is a model for the time-dependent flow. In both problems, however, moving boundaries are not taken into account. The principle difficulties of the proof lies in overcoming the critical growth in the gradient of p that is present in (1.6) and (1.11). It is interesting to note that when r = 2 equations (l.lM.2) and (1.6)-(1.7) are mathematically identical with the governing equations for thermistor problems which have received extensive study recently [6-111. 2. DERIVATION
OF
THE
EQUATIONS
As is usual, we assume that the channel is narrow with the width 2~. Indicies with Greek letters range from 1 to 2 while indicies with Roman letters range from 1 to 3. For example, we use CC,> := (x,, x2) to designate two coordinates and (xi) := (xi, x2, x3) to designate three coordinates. We assume that the flow is quasistatic; hence, the inertial term is neglected from the balance of linear momentum. This assumption has been widely used in the engineering literature (see [3,4]). Therefore, in the absence of body forces (1.1) and (1.5) become (2.1) the quasistatic form of the balance of the linear momentum. according to the rule
We stretch the x3 coordinate
x3 = Eyj.
(2.2)
At the same time we assume that the physical parameters remain unchanged K’=K,
k’=k,
CE= c,
PE = p,
K’, k’, c’, pE and the time t and
The velocity field U’ :=
;p”+p1+p2c+
(2.3)
T’ transfer accord-
a = 1,2,
... . ..
T’ = To + T’E + T2e2 + . . . We shall not notationally the notation
t=t.
(2.4)
distinguish a function in different variables. For example, we employ
542
R. P. GILBERT and P. SHI
We approximate
the shear rate by computing
4d;dl;
4dimdim _ z,“; s = ’ avi avi (I (I E2dY3JY3
I z?
i: a
p
I ’ av; ‘“,’ E dx, dY3
(2.5)
Hence, from (1.5) we approximate
the stress $I; as (n-*)/2 df, + O(
Using this and the above calculations a a dxs,; = dx,S$ I
l )d,,
.
(2.6)
the left-hand side of (2.1) becomes
1 d + ; dx,G
(2.7)
Nonisothermal, nonNewtownian
Hele-Shaw flows
543
We now distinguish the index i as being 1 or 2 in the following computation.
(2.8) On the other hand when i = 3 we get
Let us now consider the right-hand
side of (2.1). From (2.4), (2.8) and the equation ape -= dx,
d ax,6
Nonisothermal, nonNewtownian
where in the last inequality (4.18) we obtain
557
Hele-Shaw flows
we have used (4.20). Similarly,
using the Cauchy’s inequality
and
(4.26) k(8,N)IVp,Nlru dxdt I cll~ll~~(o,~;~,‘~~(n)). 5 I m=l 11Im fl Therefore, (4.24) follows from (4.231, (4.25) and (4.26). Note that C?[O, T]; H,‘12(0n)) is dense in L2(0, T; IIZ,‘*~(~~)). Hence d0z/dt can be extended uniquely as a bounded linear functional of L2(0, T; H,‘.2(fl)>. Using the notation of the duality pairing between L’(O, T; H,‘s2(fl))* and L2(0, T; H,‘.2(fi>), (4.24) implies a*, I cllvll L”(0.T;H;qIl)) I( dt’v )I We are now ready to apply Simon’s lemma. Let x= H;,2(n),
B =L2(fl)
(4.27)
Vu E L2(0 3T-7H’-2(fl)). 0
Y=H-‘,2(fl).
and
Let F=
{e, - e,;N=
1,2,3...).
Then from (4.20) to (4.21), it follows that F is a bounded subset of L*(O, T; X). Moreover, from (4.271, it follows that dF/dt is a bounded subset of L2(0, T; Y). Therefore, lemma 4.4 is proved. n LEMMA
4.6. Let 3, be the function defined by
(x,t) E R,.
if(m--1)6It
i&xx, t> = 0,“(x)
Let tI~L~(flr) and let (Nj} be a sequence of natural numbers. Then {e,) converges to 8 strongly in L2(n),) if and only if {ON,} converges to 8 strongly in L2(Ln,). Proof Setting v = 0,” - O,“_, in (4.16), we obtain
(4.28) = /ktt’;)lV~:l’te,h. - B,N_Jdx + f(O,“- B;mJdx. /n R We now multiply both sides of (4.28) by a2 and sum over m = 1, 2.. . N. Using the Cauchy’s inequality and the estimates (4.18), (4.20), it is straightforward to show that
N 6 (e,“- e/~J2dx=o(v%). =/ m=l n Using (4.11), it is easy to establish the following relations:
/ fh
(8,,-
O)‘dxdt=
f
f
6 /(f?;-
m=l
n
(4.29)
‘-’ (3, - 0)‘dxdt / 0 / bl
8,N)2dx+
[t
- (m - l)S](e,N_,
- r3)dxdt.
(4.30)
Nonisothermal, nonNewtownian
Hele-Shaw flows
545
where (2.13) and we have replaced To by its average To over the interval (0, E). It is interesting to note that (2.12) and (2.13) constitute a nonlinear Darcy’s law, which follows from our much more basic assumptions (2.1)-(2.4X Using the second part of the assumption (2.111, the incompressibility condition (1.4) averages to asjo L=() dx,
’
which with (2.12), (2.13) yields a pressure equation (2.14) This can be improved upon using (2.10*). First we sum the square of both sides of (2.10) to obtain 2 c?pO
ape
au;
0
4y3dx,c)x,=kz(T
)
i
dvp"
dy,2jq
n-1 I
av,O
au;
dY3 dY3
or (2.15) Hence
(!&y”-;
= ( &,vpo,)“li(,
from which follows that s = I’[,‘(
&)“n,Vpo,l’n-l
dJ’dy,
= ~vpo~‘/“-‘m(~o)
(2.16)
where m(T”> Substituting
= [JG’(
&)-“‘didy3=
[k~~“~l-l’”
[JliC21)1”‘diilg,.
this s(x~) into the equation (2.14) yields (2.17)
which is the standard pressure equation such as used by Guceri et al.
546
R. P. GILBERT
and P. SHI
We now turn to the energy equation. From (2.6) and a further estimate on the term dz it is clear that (2.18)
+ O(E)
so that (1.2) becomes dT”
PC
x+ex+...
dT’
+E(u,O+EU;+...)
+~(u:t2+u:E3+
dT’ +
. ..) +
-
E dy,
*-.
+ O(E).
In light of (2.15), it is straightforward
(2.19)
to show that
(n+1)/2 =
C?y,)
(n+l)/n[k(T(I)]-(‘/n)IVpol(n+l)/~,
(2.20)
We now assume that the temperature profile is symmetric over the mold thickness and that on the surface of the mold it is insulated. This amounts to saying that JT’ -=
at
0
x3 =o,
+E.
3x3
Taking the average on both sides of (2.19) and making desired two-dimensional model for the energy equation:
use of (2.20)-(2.21)
aT0 a aT0 pcat=ay, ( K- dy, 1 + Am(?;“>lvpop+
1)/n,
we obtain the
(2.22)
where A is the constant that equates 1
(2[)(‘+1”nd(=
[k(T”>]m’l’“’
Am@‘).
(2.23)
/0 Equations molding. The r = (n + 1)/n, (2.221, with f
(2.17) and (2.22) are the form of the equations are it is easy to see that (1.11) and g added as appropriate
desired two-dimensional model for the injection independent of the mold thickness 2~. By setting and (1.12) are nondimensional forms of (2.17) and source terms.
Nonisothermal, 3. WEAK
nonNewtownian SOLUTIONS
Hele-Shaw OF
547
flows
PROBLEM
I
In this section, we assume that R is a bounded C’ domain of R”, where n is a natural number that is greater or equal to 2. For s > 1, let Hi*”
= {v; v E Zf’~‘(fi>,
v = 0 on I,}
denote the usual Sobolev space equipped with the standard norm. Let r* = r/(r ifl
- 11,
n/n-l
Cl=
8, E H’,“(Q)
and
I can be extended to functions
po E H1xT(fln),
(3.1)
where T is a fixed number that is greater than r. We further assume that gELqR)
fe L.“1(s1),
and
(3.2a)
I E L~~(r,>,
where a,, i = 1, 2, 3 satisfy * u, > !!2’
t-1.Y,
*2>
and
3
if 1
otherwise, we assume that and if r>n. I dug, f, ga51(RL Finally, we assume that there exist positive numbers k, > k, > 0 such that k,
V8ER1.
(3.31
3.1. We say that (0,~) is a weak solution to (1.1)~(1.9
e - 8, E H(y(R),
(3.2b)
if
p -po E H;~‘m;
(3.4a)
and for all u E C;(a) /n
V8*Vvdx=
/n
k(0)lVpl’vdx-t
fvdx;
(3.4b)
= ;g(dx. cl
(3.4c)
/
and for all 5 E H;‘(0) /n
k(B)lVpl’-‘VpVedx
+ i@ds I
Remark. If 1 < r < 2, IVplrP2 becomes singular on the set where Vp = 0. We define lVpl’-2Vp
= 0 on that set. The main result of this section is the following. 3.3. Assume that (3.M3.3) hold. Then there exists a weak solution to problem (1.1)~(1.5) in the sense of definition 3.1. The proof will be carried out under a fixed point argument. We first establish some technical lemmas.
THEOREM
548
R. P. GILBERT
LEMMA 3.4. The following statements u E L”(a) and u E L P(fI) then
and P. SHI
hold. (i> For any positive
numbers
cx and p, if
where y =
uv E LY(fiR),
Moreover, II4 (ii) If p •Hl,~(fi)
LY(fI) 5 Il~II~“(~~II~lI~~~~~.
and 1
(iii> If p E H1lr(fi)
E [L”~‘“-“(s1)IIl”.
Moreover
I Ilpll~~~~~~~~~~n,ll~Pll~~~~,.
and 1
E Li(R),
where
++;)-‘. Moreover
Proof By Holder’s inequality,
it follows that
where s* = -P Y’ Y’ This proves part (i> of the lemma. Part (ii> is a consequence of part 6) and the Sobolev imbedding theorems. By setting u =p and v = IVpl’-‘, we obtain a=nr/(n -r.) and /3=r*. Therefore SC?
‘=
(
nr/(n
-1
1 -r>
+I
r* i
=y+y
n-r
r-l
-l =x’
n
) ( The proof of part (iii) is similar to that of part (ii) and we leave the detail to the reader. LEMMA 3.4. Suppose that (3.1)-(3.3)
n
hold. Suppose that 8 and p satisfy (3.4a) and (3.4~).
Then tlv E C’(a) / fl
k(B)IVpl’vdx=
kWvJVpJ’-*VpVp, / f1 -
/n
+ /n
k(e)(p
dx
-pJVpl’-“VpVudx
g(p -po)u dx -
/5
l(p -p,Jv
ds.
(3.5a)
Nonisothermal, nonNewtownian
Moreover,
there exist a polynomial
549
Hele-Shaw flows
F that is independent
of 8 and p such that (3.5b)
Prooj We first show that (3.5a) implies (3.5b), and this will be discussed for two different ranges of r. n
Case 1. 1 < r < n. We denote the four terms on the right-hand IV, respectively. Using lemma 3.3 we obtain
side of (3.5a) by I, II, III, and
I11~k~llVpllrL;(:I~II~P~ll~~~~~ll~llr.~*~n,~
(3.6)
1111I Ilp -p~ll~~~~~~~~~r~,ll~pll~;c:~,ll~~ll~~~~~,~ IVZZII IlgllL”2(n,llp -POIIL”‘/‘“~~In,ll~llr.~cn,,
(3.7) (3.8)
and
ID7 I IIuL~~cr,,IIp -p,~llL1’.~“~~““~~~~r,,ll~llL~~r,,.
(3.9)
In (3.6) [ is the nuimber defined in (iii) of lemma 3.4. In (3.8) and (3.9) p and ji are some (possibly large) positive numbers. Invoking the Sobolev imbedding theorems H’,‘( 0) L* L”‘/(“-‘)( a), H’3”(fl) -q(a), vqcm, H’-‘(R)
(3.6H3.9)
~L[(“~t)‘I/(“-‘)(r,),
and
H’.“(a)
-*Lq(r,),
vq<‘;c,
lead to
III + IHI + WI + WI I F(llpll~~~~~n,)llul~~~~~~~~, for some polynomial
F. This proves that (3.5a) implies (3.5b) for the case 1 < r < n.
Case 2. r > II. The proof in this case is similar but technically simpler than the previous case. This is due to the compact imbedding of the space H’,‘(fi) into the space Ca(a) with (Y= 1 - (n/r). We leave the details for the interested reader. We now turn to the proof of (3.5a). Setting [ = u(p -p,,) in (3.4~) we obtain / fl
k~empl’- %‘p+V(p +
/6
-pJ
+ (p -p,)Vuldx
l(p -po)u ds
= (g(p
-p,)udx. JCl This yields exactly (3.5a) after straightforward
calculations.
LEMMA 3.5. Assume that 0, -+ 0 a.e. on Kl and p, E H{‘(R)
/n
k(em)IVp,I’-Z Vp;V~dx=G(.$)
is a solution to the equation
V[ E H;VU,
(3.10)
550
R. P. GILBERT and P. SHI
where G E (H,!%‘(R))*.
Then there exist p E Hi,‘(Q)
and a subsequence {p,,} such that
weakly in Hi,‘(fin),
IP,,) +P
(3.11)
in the norm of [ Lq(R)]”
VP,, + VP
Vl
(3.12)
Remark. Lemma
3.1 is almost a consequence of theorem 3.1 in Boccardo and Murat [121, but the exact statement of the result in 1121 does not fit our setting. The proof of lemma 3.5 given below follows the idea introduced in [12]. Proof: By use of the standard theory of monotone operators, it follows from (3.10) that the H’,r(fl) norm of pm is independent of m. Thus (3.11) holds for a suitable subsequence of {pm,}. For the proof of (3.12), we choose in (3.10) the test function [=p,, -p, to obtain
/[n
k( %,)l~Pm,lr-2~Pm~ - k(B,,)lVplrP2Vp] = G(p,,
-p) -
/n
.V(p,,
k( O,,,)lVplr-%pV(p,,
-p)dx
-p)dx.
Since by assumption that 0, + 0 a.e. on R, it follows from (3.4) that k( e,?) + kte)
in Lq(fin)
(3.14)
for all 1 < q < =. Using (3.11) and (3.14) it is easy to see that all terms on the right-hand of (3.13) approach zero as j + m. Hence lim k( %;)l~Pm,l’-20p,~ I+= /I n
- k( O,,$VpI’zVp]
V(p,,
-p)dx
= 0.
side
(3.15)
Let
e, =k( ~m,)(lCp”‘,lr-zVpm,- IVPI’~~VP)~V~P,,, -PI. Then (3.15) can be written as lim e, dx = 0. +z /n Fix a number 0 < A < 1. The Holder inequality gives
(3.16)
j
(3.17) To complete the proof of (3.12) we use the inequality
(IXl’-2X - W2y)(x where a > 0 and b > 0 are certain
-y) 2
alx -ylr lx -y12 a (b + IxI+ IyP
constants. This together
if 1
Nonisothermal, nonNewtownian
Hele-Shaw flows
551
VP,, + Vp a.e. on Cn. Hence by Vitali’s theorem we obtain lim
IV(p”, -p)lrA dx = 0.
j+ m / n
n
Proof of theorem 3.3. We construct a mapping A whose fixed points will be solutions to the problem. Here we only present the proof for the case where 1 < r < n. For r > n, the same proof goes through with only slight modification. Recall that in this case u = n/(n - 1). Let z E H,‘,“(fi> + &,, and let & be the unique solution of the problem
k(z)lV&I’-*V,V(
dx + lWs= ~gSdx 1 in the above equation, we obtain a constant cr > 0 that is
4, -pa E H;wu,
/R
K$E H+‘(R). By setting (= & -p. independent of z, such that
(3.18)
IMIILyn, 2 Cl. Next we define a linear functional (F,,u)
=
F, E (H’*“*(IR))*
determined
by
k(z)ulV~,I’-*V~~.Vp,dx /R -
k(t)(& -p”)lV4zlr-2w*.V~dx /n
+
1(& -p,Ju dx + ,f” dx. g<$z -po)u dx / r1 / /n
By virtue of lemma 3.4, F, is well defined, and there exists a constant c2 > 0 independent such that
I~~,,u~l~c211uII~~~~*~1R~.
of z (3.19)
We now define the mapping A that maps H,‘, ” (a) + 8, into itself such that w, = AZ is the unique solution of the linear elliptic problem wz - 8, E H,1Wl>,
/0
Vw;Vvdx=
and
Vu E H,1,“*(0)
(Fz,v).
(3.20)
Since the operator - A is an isomorphism between Hi%u(CI> and (H$“*(fl))*, the estimate (3.19) shows that A is well defined. Next we show that the mapping A is continuous from H$“(Ln) + 19~ into itself under the weak topology of H’, ” (G). To do so, it is sufficient to prove that the mapping z -+ F,
under the weak topology of H’,“(R)
is continuous
and (H’,“*(SZ))*,
Fz + wz
(3.21)
and that the mapping
is continuous
(3.22)
552
R. P. GILBERT
under the weak topology of (H’,“*(R))* with
and P. SHI
and H’s”(R).
To prove (3.21), we let z, E H,$” + 8,
weakly in HLZ”(fl).
z, +z z, +z
(3.23)
a.e. in LR.
(3.24)
By the estimate (3.181, we may also assume (for a subsequence of {z,} in fact) that 4 I, - 4
weakly in Z-I’zr(Q).
(3.25)
We need to show (3.26)
4= 4,. Applying lemma 3.5 to passing to the limit in the equation /n with 5 E Hk’(R)n
~(z,JIV~,~I’-*V#Q’~
C’(a)
dx + l,WS=
j+
fixed, we obtain, in light of (3.23)-(3.25),
that
dx + L@ds= Lg(dx. 1 This proves (3.26) since the solution to the last equation is unique. To proceed with the proof of (3.211, we recall /n
(~,,,v) =
k(z)lV+l’-*V&V<
k(z,,)UlV4~mlr-2V4~m.V~Odx In -
k(z,)(+zm -Po)lv4~mIr-Zv4~_.Vudx /n
+
g(4z,, -p,Ju dx I(&~ -p,,)u dx + fu dx. /n / rl /n
(3.27)
Using lemma 3.5 again together with (3.26) we may assume that 4 I, -+ 4z
V4zm+ 04,
weakly in H’sr(fi)
in the norm of [L4(filn
(3.28) Vq
(3.29)
These allow us to pass to the limit in (3.27) as n + m, to obtain (3.30) vu EH’~“*(il). lim (F,,,u> = (F,,v> n-m Here we only calculate the limit for the second term on the right-hand side of (3.27) to illustrate a justification of (3.301, and leave the calculation of other terms to interested readers. Let
(ZZm,v> =
/n
k(z,)(4,m -p0)lv4~~lr-2v4~~.V~dX~
v E War(n).
Nonisothermal, nonNewtownian
Hele-Shaw flows
553
Note that k(z,) -+k(z) in the norm of Lq(fin) Vq < ~0. We may assume that &,, --f & strongly in L”‘/(“-‘) (a). Using lemma 3.5, we may assume that
in the norm of [ Lq(Q)]”
(Zl,d
VO < q < r*. These give
+
I0
k(z)($, -p~)lV~,l’-2V~z.Vvdx~
vu E C’(i?n).
(3.31)
Since llZZ,,,ll(~~~*(~l),*IC, with c3 independent of m, and C’(a) is dense in H’,r*(fln), we have (3.31) hold for all v •H’~~*(fln). This proves (3.30). Since H1x”*(R)* is a reflexive Banach space (3.30) implies (3.21). Finally, (3.22) is obvious in light of the linear equation (3.20). We now conclude that the mapping A : H$“tfl> + 8, + Hi,“(R) + B0 is weakly continuous. It remains to show that A has a fixed point. Invoking (3.18) and (3.191, it follows that
for some constant c independent
of z. This shows that A maps the ball
B={ z; z Ezz;Jw)
II&~ “(Cl)5 c}.
+ e,,
into itself. Since H’*“(fl) is a reflexive Banach space, B is a compact set under the topology induced by the weak topology of H’,“(n). Hence A has a fixed point in B by using Tychonoffs fixed point theorem. Theorem 3.2 is proved. w 4. WEAK
SOLUTIONS
OF PROBLEM
II
In this section, we prove that problem II has a weak solution for 1 < r < IZ and n = 2. We assume that the given data for problem II satisfy the same assumptions as specified in (3.1)-(3.3). For the sake of clarity, we do not assume that these data depend on time although there is no difficulty to deal with such cases by use of the same method which we shall demonstrate in this section. In addition, we assume that the initial temperature cp satisfies pEL2(n). Definition
(4.1)
4.1. For 1
and for all v E C”(fir)
-J
07
p -p,, E L’(0, T; H;WW;
(4.2)
with u = 0 on Jfl x (0, T)U fl x IT), (Ov, + VB.Vv) dx dt =
/ nr
k(fMVpl’vdxdt
+ l,,xdt+
&v(x,O)dx;
(4.3)
554
R. P. GILBERT
and for all CE L’(0, T; Hi,‘(fl))
In
and P. SHI
and for almost all t E (0, T)
k(B)lVp(‘-2VpVS
dx + 1,1+=
~gtd~.
(4.4)
The main result of this section is the following theorem. 4.3. Assume that (3.1)-(3.3) and (4.1) hold. Then there exists a weak solution to problem (1.6M.11) in the sense of definition 4.1.
THEOREM
The proof of theorem 4.2 will be carried out by the Rothe’s method of time discretization. To this end, let the time step 6 = T/N for some large N. For each m = 1,2.. . N we consider the stationary problem for {&“, p,“]
-div{kcs,N>lOp,“I’-“Vp,N} 0,” = 0, P’PO
=g
in 0,
(4.6)
on ~90,
(4.7)
on ro,
(4.8)
-ktf3~,,vpy’~
=I
on rI.
To start the time marching, we choose @Ns 0 p.
(4.10)
By virtue of theorem 3.2, for each m problem (4.5)-(4.10) has a solution in the sense of definition 3.1. Note that for 1 < r < rz and n = 2, 0,” lies in the Hilbert space IzI’,~(C~) since g= 2. To recover a solution to the time dependent problem, we define (@,, pN) on fl, via 8,(x, t) = 8~-~e~-l[r-(m-1)S1+~~-,, PJ& LEMMA
r) =p,“,
if (m - l)Grt
4.4. There exists a constant c > 0 that is independent
(4.11) (4.12)
of N such that
~~~Nl~~2~0.~;~‘.Z~~))5 c
(4.13)
and (4.14) proof From theorem 3.2, {@,“, p,“} satisfies the following weak form 0,” - 00 E H;,2(sr),
p,” -po E H;*w);
(4.15)
Nonisothermal,
nonNewtownian
Hele-Shaw
555
flows
and for all u E C;(a) (8,“=
/ fl
B,N_JUdx+
Ve,“. Vu dx
k(8,N)lVp,Nlrv
dx +
(4.16)
fu dx; / cl
and for all [EH+‘<~I>
/n
k(B,N)IVp,NIr-‘VpmN.V5dx
Then (4.14) follows by setting obtain
(=pt
-p.
+ lWs= I
(4.17)
l@x.
in (4.17). We now use (4.14) and lemma 3.4 to
where F is a polynomial that is independent of (0,“, p,“). This allows the test functions in (4.16) to be taken from the space H,‘,‘(R). In particular, we can choose v = 0,” - 8,. This leads to (8,“-
8,“_,)(6’,N-
= /k(t?,f)lVp;lQ,f’R We now multiply
e,)dx+
Qdx
Ve,“T
+
/n
8,)dx
f
(4.19)
both sides of (4.19) by S and sum over 1 I m I N. Using the inequality 2e,“ce,” - e,“_,> 2 (8,Nf - ce,“_,,’
and the estimate (4.18) with v = 0,” - 13,, it follows that after some standard calculations 2 m=l
where c > 0 is a constant independent result over (0, T) we obtain
/%
lVe,l* dxdt=
6 ;
/n
(4.20)
lVe,“l” dx + dx I c,
of N. Squaring both sides of (1.11) and integrating
(
+ IVe,,J’+
+lVe,” - Ve,,J’).
m=l
Hence (4.13) follows from (4.20) and (4.21). The lemma is proved.
n
LEMMA 4.4. A subsequence of {0,} converges in the norm of L*(fI,).
To prove this lemma, we need a compactness result due to Simon [13].
the
(4.21)
556
R. P. GILBERT
and P. SHI
LEMMA 4.5. (Simon) Let X, B, and Y be Banach spaces with Xc B c Y. X is compactly imbedded in B. Let 1 I 9 < cc and F be a bounded subset of Lq(O, T; x). Moreover, the set
is bounded in L’(0, T; Y ), where the partial derivative is the distributional valued functions. Then F is compact in Lq(O, T; B).
derivative for vector
Proof of lemma 4.4. Equation (3.16) can be rewritten as: for each m = 1, 2.. . N and for each t such that Cm - 1)6
VB,NVvdx=
k(B,“‘)(Vp;lrvdx+
fvdx /R
(4.22)
for all v E C?[O, T]; H$*(fi)). The test functions in (4.22) can be chosen in such a way since the domain of all integrations is independent of t and all the integrands do not involve the time derivative of the test functions. With such choices of the test function we now integrate (4.22) over each I, = ((m - l)S, ma>, and then sum the result over 1 I m IN, to obtain zvdxdt+
g m=l
V8,NVvdxdt //I,
11
N = =I/ m=l
k(B,N)IVp/I’vdxdt Im 11
+
fv dx dt.
(4.23)
Next, we show that there exists a constant c > 0 that is independent
of N such that
Using the Cauchy’s inequalities respectively, we obtain
and for vectors in RN
VO,NVvdxdt
for square integrable
1< m=l : [(
s j,
~,VO;,zdx]l”(
( l,V0,$,‘dx)ii2fi(
functions
~IVv,2dx)“2dt
i@,2dxd(i”’
(4.25)
Nonisothermal, nonNewtownian
where in the last inequality (4.18) we obtain
557
Hele-Shaw flows
we have used (4.20). Similarly,
using the Cauchy’s inequality
and
(4.26) k(8,N)IVp,Nlru dxdt I cll~ll~~(o,~;~,‘~~(n)). 5 I m=l 11Im fl Therefore, (4.24) follows from (4.231, (4.25) and (4.26). Note that C?[O, T]; H,‘12(0n)) is dense in L2(0, T; IIZ,‘*~(~~)). Hence d0z/dt can be extended uniquely as a bounded linear functional of L2(0, T; H,‘.2(fl)>. Using the notation of the duality pairing between L’(O, T; H,‘s2(fl))* and L2(0, T; H,‘.2(fi>), (4.24) implies a*, I cllvll L”(0.T;H;qIl)) I( dt’v )I We are now ready to apply Simon’s lemma. Let x= H;,2(n),
B =L2(fl)
(4.27)
Vu E L2(0 3T-7H’-2(fl)). 0
Y=H-‘,2(fl).
and
Let F=
{e, - e,;N=
1,2,3...).
Then from (4.20) to (4.21), it follows that F is a bounded subset of L*(O, T; X). Moreover, from (4.271, it follows that dF/dt is a bounded subset of L2(0, T; Y). Therefore, lemma 4.4 is proved. n LEMMA
4.6. Let 3, be the function defined by
(x,t) E R,.
if(m--1)6It
i&xx, t> = 0,“(x)
Let tI~L~(flr) and let (Nj} be a sequence of natural numbers. Then {e,) converges to 8 strongly in L2(n),) if and only if {ON,} converges to 8 strongly in L2(Ln,). Proof Setting v = 0,” - O,“_, in (4.16), we obtain
(4.28) = /ktt’;)lV~:l’te,h. - B,N_Jdx + f(O,“- B;mJdx. /n R We now multiply both sides of (4.28) by a2 and sum over m = 1, 2.. . N. Using the Cauchy’s inequality and the estimates (4.18), (4.20), it is straightforward to show that
N 6 (e,“- e/~J2dx=o(v%). =/ m=l n Using (4.11), it is easy to establish the following relations:
/ fh
(8,,-
O)‘dxdt=
f
f
6 /(f?;-
m=l
n
(4.29)
‘-’ (3, - 0)‘dxdt / 0 / bl
8,N)2dx+
[t
- (m - l)S](e,N_,
- r3)dxdt.
(4.30)
R. P. GILBERT
558
Moroever, right-hand
by use of the Cauchy’s inequality, side of (4.30) is less or equal to (O~-ee,N?d
and P. SHI
the absolute value of the last term on the
x]“‘[
Therefore, lemma 4.6 follows (4.29) to (4.31).
iT-‘j,(i&-
(4.31)
O)?dxdj’-I.
n
Proof of theorem 4.3. First, we extend 8, to 1Rx [ - 6, T] by setting e, = P if - 6
(4.32)
pN -p. E L2(0, T; H’~TYn>);
with u = 0 on JQ X (0, T)U fi X {T]
1
[(B,(x,t)-B,(x,I-6))]udxdt+ s / Sl7 = and for all ~EH~*‘(fi,>
k(&v)lVp,l’u / QT
I SIT
dx dt +
V&.Vvdxdt
(4.33)
fvdxdr, / fb
and 0 < TS T Vp,*Vtdxdt+
k( ,)l,lr-* / 07
/ r,x(O.r)
l
where .R, = 1Rx (0,~). By making a change of variable in the term In, 8,(x, have 1 s / [(&,(x,~) fb
- &,,(x,t
(4.34)
g5 dx dt, /% t
- S)dx dt we
- 6)]udxdr
i$.udxdl-
pv(x, t + S> dx dt
+
T-8
(4.35)
8,(U(x,t)--U(x,t+~))dxdt. n Using the equation (4.34) we obtain an anolog of lemma 3.4: ++
/ 0
k( &,)IVp,J’u
dxdt =
I4
/
k(e,)uiVpsl’-‘V~,~.Vpo
+I
dx dt
k(ii,)(p,-p,)lVp,I’~‘Vp,.Vvdxdt
g(pN -p,)vdxdt %
-
l(p, / l-,X(0.7)
-p&
ds dt.
(4.36)
Nonisothermal,
nonNewtownian
Hele-Shaw
flows
559
Using lemma 4.3, lemma 4.4, lemma 4.6 and expressions (4.20)-(4.211, we may assume that, without loss of generality s, -+ e
weakly in
e, + 8
L2(0,T;H"2(fk)),
strongly in
(4.37) (4.38)
L2($).
Moreover, we may further assume that weakly in L'(O,T;H','(fk>>. (4.39) +P In order to pass to the limit in (4.32)-(4.35), the convergence in (4.37FC4.39) are not sufficient. Here we shall need stronger convergence for VP,. A careful examination of the proof of lemma 3.5 shows that the following can be proved for (pN) PN
VP, -y!?
strongly in
Lq(O,T;[H',q(R)ln)
Vq
(4.40)
The proof of (4.40) is only a slight variation of the proof of lemma 3.5 in that (i) instead of equation (3.101, we begin with (4.34) with T= T, (ii) integrations and inequalities are considered in R, instead of R. We leave the details for interested readers. By virtue of (4.37)-(4.401, we can now pass to the limit as N + ~0 in (4.32lFC4.35) and conclude that the limit funcitons (0, p} satisfy definition 4.1. Theorem 4.2 is proved. n REFERENCES 1. GILBERT R. P. & SHI P., Nonisothermal, NonNewtonian HeleeShaw flows, Part I: Mathematical formulation, in Proceedings on Transport Phenomenon, pp. 1067-1088. Technomic, Lancaster (1992). 2. ACHESON D. J., Elementary fluid mechanics. Clarendon Press, Oxford (1990). 3. EISELE U., Introduction ro Polymer Ph@cs. Springer-Verlag (1990). 4. SUBBIAH S., TRAFFORD D. L. & GUCERI, S. I., Nonisothermal flow of polymers into two dimensional, thin cavity molds: a numerical grid generation approach, Inr. L Heat Muss 7’runsfer 32, 415-434 (1989). 5. CIALET P. G. & DESTUYNDER P., A justification of the two dimensional plate model, L Mecanique 18, 3155344 (1979). 6. ANTONCAEV S. N. & CHIPOT M., The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal. 25, N.4, 113881156 (1994). 7. CHEN X. & FRIEDMAN A., The thermistor problem for conductivity which vanishes at large temperature, Quart. Appl. Math. 51, N.l, 101-115 (1993). 8. CIMATTI G., The stationary thermistor problem with a current limiting device, Proc. Royal Sot. Edinburgh 116A, 79-84 (1990). 9. CIMA’ITI G., Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions, QUUI?. Appl. Math. 47, 117-121 (1989). 10. SHI P., SHILLOR M. & XU X., Existence of a solution to the Stefan Problem with Joule’s heating, .I. DifSerentiul Equutions 105, 239-263 (1993). 11. XI0 H. & ALLEGRETTO W., Solution of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SOLML Math. Anul. 22, N.6, 1491-1499 (1993). 12. BOCARDO L. & MURAT F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19, N.6, 581-597 (1992). 13. SIMON J., Compact sets in the space LP(O,T; B), Ann. Math. Puru. Appl. 146, 65-96 (1987).