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Thermally coupled magnetohydrodynamics flow
Thermally
Coupled
Magnetohydrodynamics
Flow
A. J. Meir Department
of Mathematics
Auburn University Auburn, Alabama 36849
Transmitted
by Melvin Scott
ABSTRACT This paper deals with the questions of existence and uniqueness of solutions to the equations buoyancy
of stationary,
incompressible
effects due to temperature
To that effect we couple the MHD the well-known a bounded
Boussinesq
equations
approximation.
three-dimensional
such phenomena
magnetohydrodynamics
differences
domain.
as the cooling
to the heat equation
posed on
We point out that these equations
of nuclear
when
and employ
We consider the equations reactors
fluids, continuous metal casting, crystal growth,
1.
(MHD)
in the flow cannot be neglected.
by electrically
and semiconductor
model
conducting manufacture.
INTRODUCTION In this paper
we study
magnetohydrodynamics differences
the equations
of stationary,
(MHD) when buoyancy
incompressible
effects due to temperature
in the flow cannot be neglected in the momentum
equation.
We
therefore consider the stationary MHD equations coupled to the heat equation and employ the well-known Boussinesq approximation. In this model it is assumed that the density variations are negligible, except for the body force term in the momentum equation [l, 21. We mention that these equations model the flow of electrically conducting fluids (e.g., the flow of liquid metals) in the presence of magnetic fields and of plasmas. Thus these equations have direct applications to nuclear reactor technology, magnetic propulsion devices, design of electromagnetic pumps, continuous metal casting, crystal growth, and semiconductor manufacture (see [3, 41 and the references cited therein, [5&12]). In this paper we are only interested in problems posed on threedimensional domains, the analysis for lower dimensional domains is similar and simpler. In Section 2 we introduce some notation, function spaces, APPLIED @
MATHEMATICS
AND
COMPUTATION
65:79-94
(1994)
79
Elsevier Science Inc., 1994
655 Avenue of the Americas, New York, NY 10010
0096.3003/94/$7.00
80
A. .I. MEIR
and the system of equations. Section 3 is devoted to a description of the boundary conditions and Section 4 is devoted to the description of the weak form employed. The main result of this paper, an existence and uniqueness result may be found in Section 5. In previous works [3, 5-121 the authors have been mostly interested in the modeling of this phenomenon and in obtaining explicit solutions to this problem (classical solutions); obviously this is only possible in some very special situations, i.e., two-dimensional and geometrically simple domains with simple boundary data (see also [4] and the references cited therein). In contrast here we are interested in existence and uniqueness of weak solutions and consider the problem posed on a (bounded simply connected) three-dimensional domain. Finite element analysis for this problem may be found in [4]. 2.
EQUATIONS
AND FUNCTION
SPACES
In this section we describe the equations governing the flows under consideration and introduce notation and function spaces that will be needed later. We consider the equations posed on a bounded, simply connected domain R c Iw3 which is of class C1ll or is a convex polyhedron [13, 141 (we mention
that the following
analysis
can be extended
to include
multiply
connected domains). We assume that the dissipative heating and joule heating are negligible and thus we neglect these in the energy equation (2.3), the nondimensionalized system of equations which governs the flow under consideration is (see [4] and the references cited therein):
in the domain
u.V)u+Vp-&(VxB)xB=f,
-&A,+;(
v.u=o,
x (V x B) -V
(2.1)
(2.4 (2.3)
-&AT+(u.V)'=$,
&V
R,
x (u x B) = 0,
(2.4)
and V.B=O.
(2.5)
Here u the velocity, p the pressure, B the magnetic field, and T the temperature are the unknowns. Given are f a body force and $ a heat source. These physical quantities have been nondimensionalized as follows: the velocity u by u, the pressure p by cmB21, the magnetic field B by B, the
Incompressible
81
Magnetohydrodynamics
body force f by mB2, the temperature T by t, and the heat source 1c,by Also here u, B, 1, t, and cr, are a characteristic velocity, magnetic field, length, temperature difference, and the specific heat at constant pressure, respectively. Other parameters appearing are the density p, kinematic viscosity V, heat conductivity K, electrical conductivity 0, and magnetic permeability p, we assume that these and cp remain constant (are independent of, e.g., temperature and magnetic field) in the fluid and that these are given. In these equations appear some nondimensional numbers which characterize the flow; these are: l Hartmann number Ha = Bl(a/pv)li2, l interaction parameter N = nB2(l/pu), l Reynolds number Re = ul/v, l magnetic Reynolds number Rm = paul, l Prandtl number Pr = upc,/n, and l Grashof number Gr = /3gt13/u2. Here g = (g], and g is the gravitational acceleration (for details see [2, 4, 6, 7, 15-171). We assume that the effects of gravity and buoyancy in the momenturn equation (2.1) cannot be neglected. Thus we employ the Boussinesq approximation. We assume that the variations in density are negligible, except for a contribution to the body force term in the momentum equation which is given by pg, where p is the density and g is the constant acceleration of gravity. It is assumed that the density is given by P = pr[l -P(T-%)I, where pr and T, are a reference density and temperature, respectively. Here T is the absolute temperature and /3 is the thermal expansion coefficient. The resulting density change is p - pr = -p,P(T - T,) and the resulting pressure change is p - p,, where p, is the pressure corresponding to mechanical equilibrium, i.e., p,(x) = p,g . x + c (for some constant c) [2]. Finally assuming that the thermal expansion coefficient fl is constant (and given) we get that the body force is pc,ut/l.
f = fo + p,g[l
- PG”
- T,)],
where fe is some other given body force, e.g., a Coriolis force, obviously if the only body force is that due to buoyancy fe = 0. We now drop the subscript r and let p denote the reference density, replace the pressure p by p - p,, and the temperature T by T - T,. The normalized body force (in (2.1)) is now f=fa---
Gr ET NRe2 g ’
(2.6)
82
A. J. MEIR
We partition Xl, the boundary of 0, let 80 = GUI’, where each of the rd and I?, is regular, open, with a finite number of connected components, and rd fl I’, = 8. Also let n denote the unit, outward-pointing, normal vector to 52. The boundary condition for the velocity is a homogeneous Dirichlet condition (a nonhomogeneous boundary condition is considered in [4]), i.e., there is no flow through the boundary and a no-slip condition is satisfied on the boundary, thus l&o = 0. (2.7) The magnetic field boundary conditions are with
(B. n)lan = 1
ldx=O,
(23)
= k.
(2.9)
.Ian
and
an
For the physical interpretation of these see [15, 171. Moreover, k must satisfy some additional compatibility conditions, which will be detailed later [4, 15, 171. The temperature boundary conditions are the mixed conditions (2.10) T]r,i = Td, and (VT.
n)]r,
(2.11)
= Tn.
We now introduce some function spaces and their associated norms, along with some related notation (for details see [13]). Let H”(R) (m a nonnegative integer) be the usual mth order Sobolev space equipped with the norm ](.(jm, and let Hm(S2) := (IzP(Q))~ with norm ]].]lrn,be its vectorvalued counterpart. On H’(0) we use the norm lblll
= w11~
+ IIWIC!)
2 l/2
.
Clearly, Ho(o) = L2(s2). A particular subspace of H1(R)-functions and two particular subspaces of H1(Q)-functions that satisfy specific boundary conditions are needed; they are
H#2) :=
{eE Hl(S1) : elrd
=
o},
HA(R) := {W E H1(R) :wlan = 0}, and HA(o)
:= {D E H’(0)
: (D . n)lan= O}.
Incompressible
Magnetohydrodynamics
83
Note that Hi(a) is a closed subspace of H’(a) under the usual H1(R)norm, and that HA(R) and HA(R) are closed subspaces of H1 (0) under the usual H1(R)-norm; thus on these subspaces we use the Hl(fl)-norm and the H1(R)-norm, respectively. We also make use of the product spaces W(R) := H1(Q) x H1(R),
We(0)
:= H;(R)
x H1(R),
and ‘Wcn(!2) := HA(R) x HA(R), which we equip with the usual product norm. We also define a subspace of LV)
[I,
141
On this subspace, and in general on subspaces, we will use norms induced by the original spaces. Certain trace spaces will also be needed (see [lS] and the references cited therein). In particular,
H1’2(dS2) := {wlan : w H-1’2(aR)
E
and its dual
Hl(cq}
:= (H1’2(as2))*,
H,‘,/“(&)
:= {w]r,l : w E wa~lr\i;;;
= o}
and its dual
(H$“(L4)*r and H1’2(dR) H-1’2(dR)
:=
(WI30
:
Wi E
H1’2(dS1),
w
=
(IU~,W~,UJ~)}
and its dual
:= (H1’2(8s2))*,
which are equipped with the usual norms. 3.
BOUNDARY
CONDITIONS
In this section we state the precise conditions (and regularity) which the boundary conditions, and right-hand sides must satisfy, in order to guarantee existence of a solution to our problem. We also explain why these boundary conditions must satisfy the compatibility conditions alluded to earlier. We assume that fo E H-‘(R) and that 1c,E Hi(R)*. We assume
84
A. J. MEIR
the magnetic
field boundary
conditions
with
1 E H1'2(Xl)
here the compatibility is solenoidal, and
condition
k E H-1/2(Xl)
and
here (. , .) denotes
duality
J’ac2
ldx=O;
(3.1)
arises from the fact that the magnetic
with
(k, 1) do = 0
1 and k satisfy
k.n=O, (k, V4lanjm
pairing;
= 0
v4 E H2(f4,
the compatibility
conditions
field
(3.2)
arise from
the fact that k is the tangential trace of an irrotational vector field (for further discussion of these conditions see [4, 15, 171. The temperature boundary conditions Td and T, are assumed to satisfy
Td E
H1”(rd),
(3.3)
and
Tn E also
if rn
=
X2
then
T,
(H;o/“(L))*,
must
satisfy
the
(34 compatibility
condition
condi-((W’r Re)G, I) aa = ($, l)n, which is a standard compatibility tion needed for Neumann-type problems. If the domain R is of class Cl>’ and the given boundary data satisfies (3.1), there exists an extension Bo E H1(R), of the essential boundary data 1 into the domain R; moreover one may choose this extension so that V . B. = 0 and V x Bo = 0. I n case the domain 0 is only a convex polyhedron we must require that the data for the magnetic field satisfy an additional condition in order to guarantee that one may find an extension Bo as above. One such condition is that In E H1/2(Xi) (this condition guarantees compatibility of the boundary data along the edges and at the vertices of the domain). The existence of these extensions can be shown using the methods of [15, 171. Also because of condition (3.3), there exists an extension To E H1(R) of the essential temperature boundary data Td into the domain 0.
Incompressible
4.
Magnetohydrodynamics
85
WEAK FORMULATION
We introduce the following forms. For (u,B), 9 E L:(R), and T, 8 E P(R)
a((~,B), (v, C)) :=
L {&Vu:Vv +&[(V + (V.B)(V
c((u, B), (v, C), (w,D))
(v, C), (w, D) E W,(n),
;(u '0)~.
:= /c, -
.C)]
1
x B)
(v x c)
dx,
w dx
R &rcv ./
x
x C)
B.w
-(VxD)xB.v]dx, b((v, C),q) F((v,
C))
G(T, (v, C))
:= -s,
qV. vdx,
:= (fo, vh
+ &
:= ./, $
e(T, ~9):=
d((u, B), T, 0) :=
(;T)
’ -VT s R PrRe
IR
(k Cl&n, vdx,
’ VB dx,
(u . V)TBdx,
and
s(e) := huh
+
&,k
eh
where: denotes the scalar product on lk3 ’ 3, denotes the scalar product on Iw3 and x denotes the vector product on Iw3. We are now prepared to introduce the weak form of the equations: find
((u,B)>T,p) E WoP) x ff’(fl) x -G#) such that B - B,, E H;(R),
(4.1)
T - To E H;(R),
(4.2)
A. 3. MEIR
86
a((u,B)>(v,C))+ c((u,BL(u,B),(v,C))
+b((v,C),p)
= F((v, C)) - GP’, (v, C)) b((u>B), 4) = 0
vv7 Cl
E
Won(Q),
(4.3) (4.4)
vqE -G(f$
and e(T) 0) f d((u, B), T, Q) = Q(Q)
ve E H;(Q).
(4.5)
PROPOSITION 4.1. Equations (4.1)-(4.5) are a weak formulation (2.1)-(2.5) with (2.6) and boundary conditions (2.7)-(2.11).
of
PROOF. This proposition is proved in a standard fashion, i.e., using integration by parts and a judicious choice of test functions (see [4, 15, 171 for details). n 5.
EXISTENCE
AND UNIQUENESS
We are now in a position to consider existence and uniqueness of a solution to problem (4.1)-(4.5). We begin by stating some preliminary results. Throughout this section we assume that fe, $J, and the boundary conditions satisfy the hypotheses and conditions stated in Section 3. We assume that the partition of dR into I’d and P, is such that Pd is of positive measure (of course if I’d = 80, Hi(Q) = HA(R)). If this is not the case, i.e., if l?d has zero measure the following analysis carries over with Hi(a) replaced by (0 E H1(R) : s, Bdx = 0). LEMMA 5.1. The forms a(. ,.), c(.,.,.), b(. ,.), F(.), G(. ,.), e(. ,.), d(. , . , .), and @(.) are continuous, i.e., there exist constants A,, A,, Xb, XF, XG, A,, Ad, and XQ (also x, and xd) such that b((u,B),(v,C))I
5 ~~II~~,~~llwll~~,~~ll~ v’(u, B), (v> C) E w,(Q),
lc((~,B),(v,C),(w,D))l
(5.1)
5 ~~ll~~,~)ll~~x~~lI(~,~)IIwIl(~,~)IIw 5 ~~ll~~~~~llwll~~,~~llwll~~,~~llw
(5.2)
Yu> B), (v, C), (w, D) E We(Q),
Ib((v,QdI 5 Wl(v1c)llwll~llo WV,(3 E ‘wo(f4, IF((v,C))i
5 ~FII(v,c)iiW
4 E G(Q)>
Yv, C) E WOP),
(5.3) (5.4)
IncompressibleMagnetohydrodynamics
IW’, (v, c))l 5
xGIITIhII(v~
c)bl
VT E Hl(fl),
(VPC) E WOP),
le(T,e)lI ~ellTll~ll~ll~ M(u,B),T,Q)l I ~~ll~~~~~ll~~x~~ll~ll~ll~ll~ I Ul(~,B>llwllTII1ll~ll1 T, 8 E H1(R), ‘J(u,B) E WOW, VT, 0 E Ill(R),
(5.5) (5.6)
(5.7)
and i*(e)1 I 414il
ve E P(R).
(5.8)
PROOF. The proof follows from the definition of the forms, the conditions on the data stated in Section 3, Holders inequality, and the fact that H1(R) -c-t L4(Q (h ere +~--t denotes compact embedding); see [4, 15, 171 for details. W LEMMA 5.2. Let (u,B) E Ws(CI) with V . u = 0, the trilinear form c(. , , .) is antisymmetric with respect to its last two arguments and the trilinear form d(. , . , .) is antisymmetric with respect to its last two arguments, i.e., C((K B), (v, CL (w, D))
= -4(u,
B), (w,D)r
(v> C)),
(5.9)
and d((u,B),T,@
= -d((u,BLQ,T).
(5.10)
PROOF. The proof involves the definition of the forms and an application of the divergence theorem (see [l, 14, 15, 171 for details). Note that (5.9) and (5.10) imply that c((u, B), (v, C), (v, C)) = 0
(5.11)
d((u, B), T, T) = 0.
(5.12)
and
W LEMMA 5.3. The bilinearforms a(. , .) and e(. , .) are coercive on Won(o) and Hi (Cl), respectively, and the form b(. , .) satisfies the infsup condition (Ladyzhenskaya-BabuSa-Brezzi condition [l, 14,191) on Won(Q) x L:(R),
A. J. MEIR
88 i.e., there exist positive
constants cy,, (Y,, and 6 such that
and
PROOF. The first inequality follows from Poincare’s inequality and from the existence of a Poincare-type inequality for the functions in Hi(R) (see [15, 171 for details). The second inequality follows from the existence of Poincare’s inequality for functions in Hi (0) (as long as I’d has positive measure). The inf-sup condition (5.15) follows from the classical result that the divergence operator maps H;(R) onto L;(R), (see [l, 14, 151 for n details). The regularity of the essential boundary conditions and the compatibility conditions stated in Section 3 ensure the existence of an extension Bo E H’(0) of the boundary data 1 into the domain R, which satisfies (Bo
n)lan = 1
and
with V. B. = 0
V x B. = 0,
(5.16)
[4, 15, 171. Obviously there also exists an extension TO E H1(R) of Td into the domain R. Using these extensions (Bo and TO) of the essential boundary data into the domain we write B=Bo+i3,
where 6 E H:(R)
and T = To +T^,
where T^E Hi(O). In the sequel we assume that Td is such that it has an extension into the domain which satisfies
~ll~o1,
< a*a,
(5.17)
and let h, := Q, - $llTo]]i.
(5.18)
THEOREM 5.4. Assume that Td is such that it has an extension TO which (5.17). Then problem (4.1)-(4.5) has at least one solution.
satisfies
Incompressible
Magnetohydrodynamics
89
PROOF. First from (5.6) and (5.7) of Lemma 5.1, (5.12) of Lemma 5.2, and (5.14) of Lemma 5.3 it follows that given (u,B) E Wo(0), with is a continuous, coercive bilinear form V.u= 0 then e(.,.)+d((u,B),.,.) on H,‘(R) x Hi(O). Thus by the Lax-Milgram lemma and (5.6)-(5.8) there
exists a unique ? E Hi(Q)
which satisfies
e(?,0) + d((u, B), p, 0) = Q(B)- e(To,0) - d((u,B), To,0) ‘46’ E H;(R); then
T = T^+ To is a solution of (4.2) and (4.5). Thus we may define the
map 3 by S((u, B)) = T. In fact
iiT\\ = Il3((u,B))ll1
I 2
+ (1 + 2)
IlToll~
+ ~Il(u.B)liwll~ollI.
(5.19)
In order to conclude the stated result we must show that there exist at least one (u, B) E Wo(C!) which satisfies the boundary conditions and at least one p E L;(0) such that
4(u,B), (v>f-3) + c((w B), (u, B), (v, Cl) + b((v,‘3, P) = JY(v, C)) - G(y((u,
B)), (v, C))
v’(v, C) E Won(fl),
(5.20) b((u, B), Q) = 0
(5.21)
Vq E L;(0).
This will be shown using a fixed-point argument (the Leray-Schuader principle). Given (w, D)EWO(S~) with V . w = 0, consider the map G((w,D)) = (u,B), where ((u,B),p) E We(R) x L;(a) satisfies (5.22)
B - Bo E H;(Q),
e((u,B),
(v, C)) + b((v, C),P)
= J’((v,
C)) - G(F((w,D)),
(v, C))
- c((w, D), (w, D), (v, C)) b((u,B),q)
vJ(v,C) E wh(fl), vq E Lfj(s1). = 0
(5.23) (5.24)
The existence of such a ((u, B), p) satisfying the linear system (5.22)-(5.24) follows easily from (5.1)-(5.5), (5.13), and (5.15) [15, 171. To show that 5 has a fixed point we will show that:
A. J. MEIR
90
(i) The map 9 is completely continuous. (ii) There exists 0 < M such that for every s E [0, l] and (u, B) E Wc(s2) such that sS((u, B)) = (K B), we have that [I(u,B)ll~+ 5 M. Writing (5.23) for (u, B)r, (u, B)s and (w, D)r , (w, D)s we get
PI>
(v>C)>+ f~((v,C),PI) = F((v, Cl) - G(~((w, D)I), (v, C)) - c((w,D)l> (w,D)l, (v, Cl) (5.25) v’(v,C) E won(~),
d(u,B)2,(v,C))
+b((v,C),~z)
=
JY(v,C)) -
- G(?(w,D)z),(v,C))
c((w,D)~~(w,D)~,(v,C))
Yv,C)
(5.26)
E Won(Q);
setting (v, C) = (u,B)r - (u,B) 2, subtracting (5.26) from (5.25), and taking into account (5.24) we get
4(u,B)l =
-
(u,B)z,(u,B)l
--GW(w,Dh)
- (u,B)2)
- 3((w,D)2>,(u,B)l
-
c((w,Dh
-
c((w,D)~,(w,D)~,-(w,D)~,(u,B)~
-
-
(w,D)~,(w,D)I,(u,B)I
(u,B)2) - (u,B)2) -
(u,B)2).
(5.27)
Also writing (4.5) for (w, D)r and (w, D)s we get e(3((w,
D)I), 6’) + d((w, D)1,3((w,
e(3((w, D)2), 0) + d((w, D)2,3((w,
D)r), 0) = Q(Q) VQ E H;(R), D)2),
e) =
(5.28)
e(e) ve
E ff;(cq;
(5.29)
setting 0 = 3((w, D)r) - 3((w, D)2) and subtracting (5.29) from (5.28) we get that e(3((w,
D)r) - 3((w, D)s.), 3((w, D)I) - 3((w, D)z))
= - d((w, D)l - (w, D)z, 3((w, D)r), 3((w, D)I) - 3((w,D)z)) - 3((w,D)s)). - 3((w,D)z),3((w,D)r) - d((w,D)2,3((w,D)r) (5.30)
Incompressible Using
(5.7),
Magnetohydrodynamics (5.12),
(5.14),
Il3((w,D)1) I
XT! pllWh
I
k[$+
-
91
(5.19), and (5.30) we find that
3((w7DMll1 - (w,D)211~4
.L~II~((w,D)~)II~
(1+~)llT,ll1
II(w,D)l -
+ $llWWll~]
(w,D)211~4
x~4.
(5.31)
Now if (w, D) E Wo(s2) and V. u = 0, using arguments similar to the ones used in Lemma 5.1 and Lemma 5.2, we get that there exists a constant z A, such that
Ic((u,B),(v,C),(w,D))I
<~~II(~~~~ll~ll~~,~)ll~4~~4ll(~,~)IIw.
(5.32)
Let X, = max{X,, ZC}, combining (5.2), (5.5), (5.13), (5.27), (5.31), and (5.32) yields 5
ii
- (u,‘3)2llw
I
<[ll(wD)~llw + ll(w,D)2llw]
+ ~ll(wD,~llwll~oll~]} x ll(~~D)l
- (w,D)~IIL~
x ~4,
and since H1 (0) -+-+ L4(0) we get that 9 is completely continuous. Now if s = 0 then (u, B) = 0 and ]](u, B)]]w = 0, so let s E (O,l], if
sS((u,B)) = (u, B); then
A. J.MEIR
92
-
c((u,B), (~1W, (u, &I
- b((u,@,P)
and b((u, B), 4) = 0; using (5.1), (5.2), (5.4), (5.5), (5.11), (5.13), and (5.16), (remembering that s E (0, 11) we get that
using the triangle inequality, and (5.17)-(5.19)
we get that
So indeed we have the required boundedness and we conclude the existence of a fixed point and thus the existence of at least one (u, B) E ‘Wo(s2) satisfying the boundary conditions (5.20) and (5.21) (and thus (4.1), (4.3), and (4.4)) for some p E L:(R). Now given (u,B), (5.15) implies the existence of a unique p E L;(R) satisfying (4.3). Therefore there exists at least one solution ((u,B),T,p) E Wo(Ci) x H’(Q) x L;(R) satisfying (4.1)-(4.5). H We now give a global uniqueness criteria. THEOREM
[
5.5. There exists at most one solution to (4.1)-(4.5)
A, + +r,ll~]
ll(~,B)llu)+ $
[$ + (1 f $) ir,lll]
with
< aa.
PROOF. Assume there are two different solutions ((u, B), T, p)~ and ((u, B), T,p)z satisfying the above condition and (4.1))(4.5). Writing (5.20) for the two solutions, setting (v, C) = (u, B)l - (u, B)z, subtracting the two equations, taking into account (5.11) and (5.21) we get
a((u,B)I - (u, Bh (u, Bh - (u, B)2) = -G(~((u,B)I) - 3((u, BM, (u, B)l - (u, BM - c((u,B)l-
(u,B)~,(u,B)~,(u,B)~
-
(u,B)2).
Incompressible
93
Magnetohydrodynamics
And using (5.2), (5.5), (5.7), (5.12)-(5.14), wzIl(~,Bh
-
I
xGlIy((u,
(u,B)2llw B)1)
-
s((u,
B>2)111
+~~II(~,~)~llwll(~,~)1 5
(5.19), and (5.30) we get that
{ %
[$’
+
(I+
+oll~
II(u,B)l
+~cII(u,B)~llw < ~alI(~,Bh
-
(u,%llw + ~ll(u;B)~llvllT,,]
-
(u,J3)2llw
(~,J3)2llw~
which is a contradiction; thus there exists only one such solution. In particular if the boundary data 1 and Td are such that their extensions into the domain Bo and To satisfy
m
there exists at most one solution.
We remark that the conditions for existence and uniqueness in Theorem 5.4 and Theorem 5.5 are conditions on the size (smallness) of the data (boundary conditions, body force, and heat source) relative to the parameters (nondimensional numbers, flow parameters, and constants depending on the domain) characterizing the flow. We emphasize that for fixed flow parameters the boundary data must be such as to have an extension into the domain TO with sufficiently small norm llTol[l in order to have existence of a solution. In particular if Td = 0, there always exists at least one solution. REFERENCES 1
M.
D.
Flow, 2
Gunzburger, Academic
L. D. Landau 1987.
Finite
Press,
Element
Boston,
and E. M. Lifshitz,
Methods
for
Viscous
Incompressible
1989. Fluid
Mechanics,
Pergamon
Press,
Oxford,
94
3
8
9
10 11
12 13 14 15
16 17
18 19
A. J. MEIR J. P. Garandet, T. Alboussiere, and R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Internat. J. Heat Mass Transfer 35(4):741-748 (1992). A. J. Meir, Thermally coupled, stationary, incompressible MHD flow; Existence, uniqueness, and finite element approximation, preprint. J. F. Osterle and F. J. Young, Natural convection between heated vertical plates in horizontal magnetic field, J. Fluid Mech. 11(4):512-518 (1961). G. Poots, Laminar natural convection flow in magneto-hydrodynamics, Internet. J. Heat Mass Trunsfer 3:1-25 (1961). E. L. Resler, Jr. and W. R. Sears, The prospects for magneto-aerodynamics, J. Aero. Sci. 15(4):235-245, 258 (1958). K. R. Singh and T. G. Cowling, Thermal convection in magnetohydrodynamics I. Boundary layer flow up a hot vertical plate, Quart. J. Mech. Appl. Math. 16(1):1-15 (1963). K. R. Singh and T. G. Cowling Thermal convection in magnetohydrodynamits II. Flow in a rectangular box, Quart. J. Mech. Appl. Math., 16(1):17-31 (1963). E. M. Sparrow and R. D. Cess, The effect of a magnetic field on free convection heat transfer, ht. J. Heat Muss Transfer, 3:267-274 (1961). R. Viskanta, Effect of transverse magnetic field on heat transfer to an electrically conducting and thermal radiating fluid flowing in a parallel-plate channel, ZAMP, 14(5):353-369 (1963). R. Viskanta, Some consideration of radiation in magnetohydrodynamic Couette flow, ZAMP 15(3):227-236 (1964). R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. M. D. Gunzburger, A. J. Meir, and J. S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp. 56(194):523563 (1991). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960. A. J. Meir, Existence, Uniqueness and Finite Element Approximation of Solutions of the Equations of Stationary, Incompressible MHD, Ph.D. Thesis, Carnegie Mellon Univ., Pittsburgh, 1989. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Wiley, Chichester, 1984. J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. 2, Finite Element Methods (Part l), North Holland, Amsterdam, 1991, pp. 523-639.
Coupled
Magnetohydrodynamics
Flow
A. J. Meir Department
of Mathematics
Auburn University Auburn, Alabama 36849
Transmitted
by Melvin Scott
ABSTRACT This paper deals with the questions of existence and uniqueness of solutions to the equations buoyancy
of stationary,
incompressible
effects due to temperature
To that effect we couple the MHD the well-known a bounded
Boussinesq
equations
approximation.
three-dimensional
such phenomena
magnetohydrodynamics
differences
domain.
as the cooling
to the heat equation
posed on
We point out that these equations
of nuclear
when
and employ
We consider the equations reactors
fluids, continuous metal casting, crystal growth,
1.
(MHD)
in the flow cannot be neglected.
by electrically
and semiconductor
model
conducting manufacture.
INTRODUCTION In this paper
we study
magnetohydrodynamics differences
the equations
of stationary,
(MHD) when buoyancy
incompressible
effects due to temperature
in the flow cannot be neglected in the momentum
equation.
We
therefore consider the stationary MHD equations coupled to the heat equation and employ the well-known Boussinesq approximation. In this model it is assumed that the density variations are negligible, except for the body force term in the momentum equation [l, 21. We mention that these equations model the flow of electrically conducting fluids (e.g., the flow of liquid metals) in the presence of magnetic fields and of plasmas. Thus these equations have direct applications to nuclear reactor technology, magnetic propulsion devices, design of electromagnetic pumps, continuous metal casting, crystal growth, and semiconductor manufacture (see [3, 41 and the references cited therein, [5&12]). In this paper we are only interested in problems posed on threedimensional domains, the analysis for lower dimensional domains is similar and simpler. In Section 2 we introduce some notation, function spaces, APPLIED @
MATHEMATICS
AND
COMPUTATION
65:79-94
(1994)
79
Elsevier Science Inc., 1994
655 Avenue of the Americas, New York, NY 10010
0096.3003/94/$7.00
80
A. .I. MEIR
and the system of equations. Section 3 is devoted to a description of the boundary conditions and Section 4 is devoted to the description of the weak form employed. The main result of this paper, an existence and uniqueness result may be found in Section 5. In previous works [3, 5-121 the authors have been mostly interested in the modeling of this phenomenon and in obtaining explicit solutions to this problem (classical solutions); obviously this is only possible in some very special situations, i.e., two-dimensional and geometrically simple domains with simple boundary data (see also [4] and the references cited therein). In contrast here we are interested in existence and uniqueness of weak solutions and consider the problem posed on a (bounded simply connected) three-dimensional domain. Finite element analysis for this problem may be found in [4]. 2.
EQUATIONS
AND FUNCTION
SPACES
In this section we describe the equations governing the flows under consideration and introduce notation and function spaces that will be needed later. We consider the equations posed on a bounded, simply connected domain R c Iw3 which is of class C1ll or is a convex polyhedron [13, 141 (we mention
that the following
analysis
can be extended
to include
multiply
connected domains). We assume that the dissipative heating and joule heating are negligible and thus we neglect these in the energy equation (2.3), the nondimensionalized system of equations which governs the flow under consideration is (see [4] and the references cited therein):
in the domain
u.V)u+Vp-&(VxB)xB=f,
-&A,+;(
v.u=o,
x (V x B) -V
(2.1)
(2.4 (2.3)
-&AT+(u.V)'=$,
&V
R,
x (u x B) = 0,
(2.4)
and V.B=O.
(2.5)
Here u the velocity, p the pressure, B the magnetic field, and T the temperature are the unknowns. Given are f a body force and $ a heat source. These physical quantities have been nondimensionalized as follows: the velocity u by u, the pressure p by cmB21, the magnetic field B by B, the
Incompressible
81
Magnetohydrodynamics
body force f by mB2, the temperature T by t, and the heat source 1c,by Also here u, B, 1, t, and cr, are a characteristic velocity, magnetic field, length, temperature difference, and the specific heat at constant pressure, respectively. Other parameters appearing are the density p, kinematic viscosity V, heat conductivity K, electrical conductivity 0, and magnetic permeability p, we assume that these and cp remain constant (are independent of, e.g., temperature and magnetic field) in the fluid and that these are given. In these equations appear some nondimensional numbers which characterize the flow; these are: l Hartmann number Ha = Bl(a/pv)li2, l interaction parameter N = nB2(l/pu), l Reynolds number Re = ul/v, l magnetic Reynolds number Rm = paul, l Prandtl number Pr = upc,/n, and l Grashof number Gr = /3gt13/u2. Here g = (g], and g is the gravitational acceleration (for details see [2, 4, 6, 7, 15-171). We assume that the effects of gravity and buoyancy in the momenturn equation (2.1) cannot be neglected. Thus we employ the Boussinesq approximation. We assume that the variations in density are negligible, except for a contribution to the body force term in the momentum equation which is given by pg, where p is the density and g is the constant acceleration of gravity. It is assumed that the density is given by P = pr[l -P(T-%)I, where pr and T, are a reference density and temperature, respectively. Here T is the absolute temperature and /3 is the thermal expansion coefficient. The resulting density change is p - pr = -p,P(T - T,) and the resulting pressure change is p - p,, where p, is the pressure corresponding to mechanical equilibrium, i.e., p,(x) = p,g . x + c (for some constant c) [2]. Finally assuming that the thermal expansion coefficient fl is constant (and given) we get that the body force is pc,ut/l.
f = fo + p,g[l
- PG”
- T,)],
where fe is some other given body force, e.g., a Coriolis force, obviously if the only body force is that due to buoyancy fe = 0. We now drop the subscript r and let p denote the reference density, replace the pressure p by p - p,, and the temperature T by T - T,. The normalized body force (in (2.1)) is now f=fa---
Gr ET NRe2 g ’
(2.6)
82
A. J. MEIR
We partition Xl, the boundary of 0, let 80 = GUI’, where each of the rd and I?, is regular, open, with a finite number of connected components, and rd fl I’, = 8. Also let n denote the unit, outward-pointing, normal vector to 52. The boundary condition for the velocity is a homogeneous Dirichlet condition (a nonhomogeneous boundary condition is considered in [4]), i.e., there is no flow through the boundary and a no-slip condition is satisfied on the boundary, thus l&o = 0. (2.7) The magnetic field boundary conditions are with
(B. n)lan = 1
ldx=O,
(23)
= k.
(2.9)
.Ian
and
an
For the physical interpretation of these see [15, 171. Moreover, k must satisfy some additional compatibility conditions, which will be detailed later [4, 15, 171. The temperature boundary conditions are the mixed conditions (2.10) T]r,i = Td, and (VT.
n)]r,
(2.11)
= Tn.
We now introduce some function spaces and their associated norms, along with some related notation (for details see [13]). Let H”(R) (m a nonnegative integer) be the usual mth order Sobolev space equipped with the norm ](.(jm, and let Hm(S2) := (IzP(Q))~ with norm ]].]lrn,be its vectorvalued counterpart. On H’(0) we use the norm lblll
= w11~
+ IIWIC!)
2 l/2
.
Clearly, Ho(o) = L2(s2). A particular subspace of H1(R)-functions and two particular subspaces of H1(Q)-functions that satisfy specific boundary conditions are needed; they are
H#2) :=
{eE Hl(S1) : elrd
=
o},
HA(R) := {W E H1(R) :wlan = 0}, and HA(o)
:= {D E H’(0)
: (D . n)lan= O}.
Incompressible
Magnetohydrodynamics
83
Note that Hi(a) is a closed subspace of H’(a) under the usual H1(R)norm, and that HA(R) and HA(R) are closed subspaces of H1 (0) under the usual H1(R)-norm; thus on these subspaces we use the Hl(fl)-norm and the H1(R)-norm, respectively. We also make use of the product spaces W(R) := H1(Q) x H1(R),
We(0)
:= H;(R)
x H1(R),
and ‘Wcn(!2) := HA(R) x HA(R), which we equip with the usual product norm. We also define a subspace of LV)
[I,
141
On this subspace, and in general on subspaces, we will use norms induced by the original spaces. Certain trace spaces will also be needed (see [lS] and the references cited therein). In particular,
H1’2(dS2) := {wlan : w H-1’2(aR)
E
and its dual
Hl(cq}
:= (H1’2(as2))*,
H,‘,/“(&)
:= {w]r,l : w E wa~lr\i;;;
= o}
and its dual
(H$“(L4)*r and H1’2(dR) H-1’2(dR)
:=
(WI30
:
Wi E
H1’2(dS1),
w
=
(IU~,W~,UJ~)}
and its dual
:= (H1’2(8s2))*,
which are equipped with the usual norms. 3.
BOUNDARY
CONDITIONS
In this section we state the precise conditions (and regularity) which the boundary conditions, and right-hand sides must satisfy, in order to guarantee existence of a solution to our problem. We also explain why these boundary conditions must satisfy the compatibility conditions alluded to earlier. We assume that fo E H-‘(R) and that 1c,E Hi(R)*. We assume
84
A. J. MEIR
the magnetic
field boundary
conditions
with
1 E H1'2(Xl)
here the compatibility is solenoidal, and
condition
k E H-1/2(Xl)
and
here (. , .) denotes
duality
J’ac2
ldx=O;
(3.1)
arises from the fact that the magnetic
with
(k, 1) do = 0
1 and k satisfy
k.n=O, (k, V4lanjm
pairing;
= 0
v4 E H2(f4,
the compatibility
conditions
field
(3.2)
arise from
the fact that k is the tangential trace of an irrotational vector field (for further discussion of these conditions see [4, 15, 171. The temperature boundary conditions Td and T, are assumed to satisfy
Td E
H1”(rd),
(3.3)
and
Tn E also
if rn
=
X2
then
T,
(H;o/“(L))*,
must
satisfy
the
(34 compatibility
condition
condi-((W’r Re)G, I) aa = ($, l)n, which is a standard compatibility tion needed for Neumann-type problems. If the domain R is of class Cl>’ and the given boundary data satisfies (3.1), there exists an extension Bo E H1(R), of the essential boundary data 1 into the domain R; moreover one may choose this extension so that V . B. = 0 and V x Bo = 0. I n case the domain 0 is only a convex polyhedron we must require that the data for the magnetic field satisfy an additional condition in order to guarantee that one may find an extension Bo as above. One such condition is that In E H1/2(Xi) (this condition guarantees compatibility of the boundary data along the edges and at the vertices of the domain). The existence of these extensions can be shown using the methods of [15, 171. Also because of condition (3.3), there exists an extension To E H1(R) of the essential temperature boundary data Td into the domain 0.
Incompressible
4.
Magnetohydrodynamics
85
WEAK FORMULATION
We introduce the following forms. For (u,B), 9 E L:(R), and T, 8 E P(R)
a((~,B), (v, C)) :=
L {&Vu:Vv +&[(V + (V.B)(V
c((u, B), (v, C), (w,D))
(v, C), (w, D) E W,(n),
;(u '0)~.
:= /c, -
.C)]
1
x B)
(v x c)
dx,
w dx
R &rcv ./
x
x C)
B.w
-(VxD)xB.v]dx, b((v, C),q) F((v,
C))
G(T, (v, C))
:= -s,
qV. vdx,
:= (fo, vh
+ &
:= ./, $
e(T, ~9):=
d((u, B), T, 0) :=
(;T)
’ -VT s R PrRe
IR
(k Cl&n, vdx,
’ VB dx,
(u . V)TBdx,
and
s(e) := huh
+
&,k
eh
where: denotes the scalar product on lk3 ’ 3, denotes the scalar product on Iw3 and x denotes the vector product on Iw3. We are now prepared to introduce the weak form of the equations: find
((u,B)>T,p) E WoP) x ff’(fl) x -G#) such that B - B,, E H;(R),
(4.1)
T - To E H;(R),
(4.2)
A. 3. MEIR
86
a((u,B)>(v,C))+ c((u,BL(u,B),(v,C))
+b((v,C),p)
= F((v, C)) - GP’, (v, C)) b((u>B), 4) = 0
vv7 Cl
E
Won(Q),
(4.3) (4.4)
vqE -G(f$
and e(T) 0) f d((u, B), T, Q) = Q(Q)
ve E H;(Q).
(4.5)
PROPOSITION 4.1. Equations (4.1)-(4.5) are a weak formulation (2.1)-(2.5) with (2.6) and boundary conditions (2.7)-(2.11).
of
PROOF. This proposition is proved in a standard fashion, i.e., using integration by parts and a judicious choice of test functions (see [4, 15, 171 for details). n 5.
EXISTENCE
AND UNIQUENESS
We are now in a position to consider existence and uniqueness of a solution to problem (4.1)-(4.5). We begin by stating some preliminary results. Throughout this section we assume that fe, $J, and the boundary conditions satisfy the hypotheses and conditions stated in Section 3. We assume that the partition of dR into I’d and P, is such that Pd is of positive measure (of course if I’d = 80, Hi(Q) = HA(R)). If this is not the case, i.e., if l?d has zero measure the following analysis carries over with Hi(a) replaced by (0 E H1(R) : s, Bdx = 0). LEMMA 5.1. The forms a(. ,.), c(.,.,.), b(. ,.), F(.), G(. ,.), e(. ,.), d(. , . , .), and @(.) are continuous, i.e., there exist constants A,, A,, Xb, XF, XG, A,, Ad, and XQ (also x, and xd) such that b((u,B),(v,C))I
5 ~~II~~,~~llwll~~,~~ll~ v’(u, B), (v> C) E w,(Q),
lc((~,B),(v,C),(w,D))l
(5.1)
5 ~~ll~~,~)ll~~x~~lI(~,~)IIwIl(~,~)IIw 5 ~~ll~~~~~llwll~~,~~llwll~~,~~llw
(5.2)
Yu> B), (v, C), (w, D) E We(Q),
Ib((v,QdI 5 Wl(v1c)llwll~llo WV,(3 E ‘wo(f4, IF((v,C))i
5 ~FII(v,c)iiW
4 E G(Q)>
Yv, C) E WOP),
(5.3) (5.4)
IncompressibleMagnetohydrodynamics
IW’, (v, c))l 5
xGIITIhII(v~
c)bl
VT E Hl(fl),
(VPC) E WOP),
le(T,e)lI ~ellTll~ll~ll~ M(u,B),T,Q)l I ~~ll~~~~~ll~~x~~ll~ll~ll~ll~ I Ul(~,B>llwllTII1ll~ll1 T, 8 E H1(R), ‘J(u,B) E WOW, VT, 0 E Ill(R),
(5.5) (5.6)
(5.7)
and i*(e)1 I 414il
ve E P(R).
(5.8)
PROOF. The proof follows from the definition of the forms, the conditions on the data stated in Section 3, Holders inequality, and the fact that H1(R) -c-t L4(Q (h ere +~--t denotes compact embedding); see [4, 15, 171 for details. W LEMMA 5.2. Let (u,B) E Ws(CI) with V . u = 0, the trilinear form c(. , , .) is antisymmetric with respect to its last two arguments and the trilinear form d(. , . , .) is antisymmetric with respect to its last two arguments, i.e., C((K B), (v, CL (w, D))
= -4(u,
B), (w,D)r
(v> C)),
(5.9)
and d((u,B),T,@
= -d((u,BLQ,T).
(5.10)
PROOF. The proof involves the definition of the forms and an application of the divergence theorem (see [l, 14, 15, 171 for details). Note that (5.9) and (5.10) imply that c((u, B), (v, C), (v, C)) = 0
(5.11)
d((u, B), T, T) = 0.
(5.12)
and
W LEMMA 5.3. The bilinearforms a(. , .) and e(. , .) are coercive on Won(o) and Hi (Cl), respectively, and the form b(. , .) satisfies the infsup condition (Ladyzhenskaya-BabuSa-Brezzi condition [l, 14,191) on Won(Q) x L:(R),
A. J. MEIR
88 i.e., there exist positive
constants cy,, (Y,, and 6 such that
and
PROOF. The first inequality follows from Poincare’s inequality and from the existence of a Poincare-type inequality for the functions in Hi(R) (see [15, 171 for details). The second inequality follows from the existence of Poincare’s inequality for functions in Hi (0) (as long as I’d has positive measure). The inf-sup condition (5.15) follows from the classical result that the divergence operator maps H;(R) onto L;(R), (see [l, 14, 151 for n details). The regularity of the essential boundary conditions and the compatibility conditions stated in Section 3 ensure the existence of an extension Bo E H’(0) of the boundary data 1 into the domain R, which satisfies (Bo
n)lan = 1
and
with V. B. = 0
V x B. = 0,
(5.16)
[4, 15, 171. Obviously there also exists an extension TO E H1(R) of Td into the domain R. Using these extensions (Bo and TO) of the essential boundary data into the domain we write B=Bo+i3,
where 6 E H:(R)
and T = To +T^,
where T^E Hi(O). In the sequel we assume that Td is such that it has an extension into the domain which satisfies
~ll~o1,
< a*a,
(5.17)
and let h, := Q, - $llTo]]i.
(5.18)
THEOREM 5.4. Assume that Td is such that it has an extension TO which (5.17). Then problem (4.1)-(4.5) has at least one solution.
satisfies
Incompressible
Magnetohydrodynamics
89
PROOF. First from (5.6) and (5.7) of Lemma 5.1, (5.12) of Lemma 5.2, and (5.14) of Lemma 5.3 it follows that given (u,B) E Wo(0), with is a continuous, coercive bilinear form V.u= 0 then e(.,.)+d((u,B),.,.) on H,‘(R) x Hi(O). Thus by the Lax-Milgram lemma and (5.6)-(5.8) there
exists a unique ? E Hi(Q)
which satisfies
e(?,0) + d((u, B), p, 0) = Q(B)- e(To,0) - d((u,B), To,0) ‘46’ E H;(R); then
T = T^+ To is a solution of (4.2) and (4.5). Thus we may define the
map 3 by S((u, B)) = T. In fact
iiT\\ = Il3((u,B))ll1
I 2
+ (1 + 2)
IlToll~
+ ~Il(u.B)liwll~ollI.
(5.19)
In order to conclude the stated result we must show that there exist at least one (u, B) E Wo(C!) which satisfies the boundary conditions and at least one p E L;(0) such that
4(u,B), (v>f-3) + c((w B), (u, B), (v, Cl) + b((v,‘3, P) = JY(v, C)) - G(y((u,
B)), (v, C))
v’(v, C) E Won(fl),
(5.20) b((u, B), Q) = 0
(5.21)
Vq E L;(0).
This will be shown using a fixed-point argument (the Leray-Schuader principle). Given (w, D)EWO(S~) with V . w = 0, consider the map G((w,D)) = (u,B), where ((u,B),p) E We(R) x L;(a) satisfies (5.22)
B - Bo E H;(Q),
e((u,B),
(v, C)) + b((v, C),P)
= J’((v,
C)) - G(F((w,D)),
(v, C))
- c((w, D), (w, D), (v, C)) b((u,B),q)
vJ(v,C) E wh(fl), vq E Lfj(s1). = 0
(5.23) (5.24)
The existence of such a ((u, B), p) satisfying the linear system (5.22)-(5.24) follows easily from (5.1)-(5.5), (5.13), and (5.15) [15, 171. To show that 5 has a fixed point we will show that:
A. J. MEIR
90
(i) The map 9 is completely continuous. (ii) There exists 0 < M such that for every s E [0, l] and (u, B) E Wc(s2) such that sS((u, B)) = (K B), we have that [I(u,B)ll~+ 5 M. Writing (5.23) for (u, B)r, (u, B)s and (w, D)r , (w, D)s we get
PI>
(v>C)>+ f~((v,C),PI) = F((v, Cl) - G(~((w, D)I), (v, C)) - c((w,D)l> (w,D)l, (v, Cl) (5.25) v’(v,C) E won(~),
d(u,B)2,(v,C))
+b((v,C),~z)
=
JY(v,C)) -
- G(?(w,D)z),(v,C))
c((w,D)~~(w,D)~,(v,C))
Yv,C)
(5.26)
E Won(Q);
setting (v, C) = (u,B)r - (u,B) 2, subtracting (5.26) from (5.25), and taking into account (5.24) we get
4(u,B)l =
-
(u,B)z,(u,B)l
--GW(w,Dh)
- (u,B)2)
- 3((w,D)2>,(u,B)l
-
c((w,Dh
-
c((w,D)~,(w,D)~,-(w,D)~,(u,B)~
-
-
(w,D)~,(w,D)I,(u,B)I
(u,B)2) - (u,B)2) -
(u,B)2).
(5.27)
Also writing (4.5) for (w, D)r and (w, D)s we get e(3((w,
D)I), 6’) + d((w, D)1,3((w,
e(3((w, D)2), 0) + d((w, D)2,3((w,
D)r), 0) = Q(Q) VQ E H;(R), D)2),
e) =
(5.28)
e(e) ve
E ff;(cq;
(5.29)
setting 0 = 3((w, D)r) - 3((w, D)2) and subtracting (5.29) from (5.28) we get that e(3((w,
D)r) - 3((w, D)s.), 3((w, D)I) - 3((w, D)z))
= - d((w, D)l - (w, D)z, 3((w, D)r), 3((w, D)I) - 3((w,D)z)) - 3((w,D)s)). - 3((w,D)z),3((w,D)r) - d((w,D)2,3((w,D)r) (5.30)
Incompressible Using
(5.7),
Magnetohydrodynamics (5.12),
(5.14),
Il3((w,D)1) I
XT! pllWh
I
k[$+
-
91
(5.19), and (5.30) we find that
3((w7DMll1 - (w,D)211~4
.L~II~((w,D)~)II~
(1+~)llT,ll1
II(w,D)l -
+ $llWWll~]
(w,D)211~4
x~4.
(5.31)
Now if (w, D) E Wo(s2) and V. u = 0, using arguments similar to the ones used in Lemma 5.1 and Lemma 5.2, we get that there exists a constant z A, such that
Ic((u,B),(v,C),(w,D))I
<~~II(~~~~ll~ll~~,~)ll~4~~4ll(~,~)IIw.
(5.32)
Let X, = max{X,, ZC}, combining (5.2), (5.5), (5.13), (5.27), (5.31), and (5.32) yields 5
ii
- (u,‘3)2llw
I
<[ll(wD)~llw + ll(w,D)2llw]
+ ~ll(wD,~llwll~oll~]} x ll(~~D)l
- (w,D)~IIL~
x ~4,
and since H1 (0) -+-+ L4(0) we get that 9 is completely continuous. Now if s = 0 then (u, B) = 0 and ]](u, B)]]w = 0, so let s E (O,l], if
sS((u,B)) = (u, B); then
A. J.MEIR
92
-
c((u,B), (~1W, (u, &I
- b((u,@,P)
and b((u, B), 4) = 0; using (5.1), (5.2), (5.4), (5.5), (5.11), (5.13), and (5.16), (remembering that s E (0, 11) we get that
using the triangle inequality, and (5.17)-(5.19)
we get that
So indeed we have the required boundedness and we conclude the existence of a fixed point and thus the existence of at least one (u, B) E ‘Wo(s2) satisfying the boundary conditions (5.20) and (5.21) (and thus (4.1), (4.3), and (4.4)) for some p E L:(R). Now given (u,B), (5.15) implies the existence of a unique p E L;(R) satisfying (4.3). Therefore there exists at least one solution ((u,B),T,p) E Wo(Ci) x H’(Q) x L;(R) satisfying (4.1)-(4.5). H We now give a global uniqueness criteria. THEOREM
[
5.5. There exists at most one solution to (4.1)-(4.5)
A, + +r,ll~]
ll(~,B)llu)+ $
[$ + (1 f $) ir,lll]
with
< aa.
PROOF. Assume there are two different solutions ((u, B), T, p)~ and ((u, B), T,p)z satisfying the above condition and (4.1))(4.5). Writing (5.20) for the two solutions, setting (v, C) = (u, B)l - (u, B)z, subtracting the two equations, taking into account (5.11) and (5.21) we get
a((u,B)I - (u, Bh (u, Bh - (u, B)2) = -G(~((u,B)I) - 3((u, BM, (u, B)l - (u, BM - c((u,B)l-
(u,B)~,(u,B)~,(u,B)~
-
(u,B)2).
Incompressible
93
Magnetohydrodynamics
And using (5.2), (5.5), (5.7), (5.12)-(5.14), wzIl(~,Bh
-
I
xGlIy((u,
(u,B)2llw B)1)
-
s((u,
B>2)111
+~~II(~,~)~llwll(~,~)1 5
(5.19), and (5.30) we get that
{ %
[$’
+
(I+
+oll~
II(u,B)l
+~cII(u,B)~llw < ~alI(~,Bh
-
(u,%llw + ~ll(u;B)~llvllT,,]
-
(u,J3)2llw
(~,J3)2llw~
which is a contradiction; thus there exists only one such solution. In particular if the boundary data 1 and Td are such that their extensions into the domain Bo and To satisfy
m
there exists at most one solution.
We remark that the conditions for existence and uniqueness in Theorem 5.4 and Theorem 5.5 are conditions on the size (smallness) of the data (boundary conditions, body force, and heat source) relative to the parameters (nondimensional numbers, flow parameters, and constants depending on the domain) characterizing the flow. We emphasize that for fixed flow parameters the boundary data must be such as to have an extension into the domain TO with sufficiently small norm llTol[l in order to have existence of a solution. In particular if Td = 0, there always exists at least one solution. REFERENCES 1
M.
D.
Flow, 2
Gunzburger, Academic
L. D. Landau 1987.
Finite
Press,
Element
Boston,
and E. M. Lifshitz,
Methods
for
Viscous
Incompressible
1989. Fluid
Mechanics,
Pergamon
Press,
Oxford,
94
3
8
9
10 11
12 13 14 15
16 17
18 19
A. J. MEIR J. P. Garandet, T. Alboussiere, and R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Internat. J. Heat Mass Transfer 35(4):741-748 (1992). A. J. Meir, Thermally coupled, stationary, incompressible MHD flow; Existence, uniqueness, and finite element approximation, preprint. J. F. Osterle and F. J. Young, Natural convection between heated vertical plates in horizontal magnetic field, J. Fluid Mech. 11(4):512-518 (1961). G. Poots, Laminar natural convection flow in magneto-hydrodynamics, Internet. J. Heat Mass Trunsfer 3:1-25 (1961). E. L. Resler, Jr. and W. R. Sears, The prospects for magneto-aerodynamics, J. Aero. Sci. 15(4):235-245, 258 (1958). K. R. Singh and T. G. Cowling, Thermal convection in magnetohydrodynamics I. Boundary layer flow up a hot vertical plate, Quart. J. Mech. Appl. Math. 16(1):1-15 (1963). K. R. Singh and T. G. Cowling Thermal convection in magnetohydrodynamits II. Flow in a rectangular box, Quart. J. Mech. Appl. Math., 16(1):17-31 (1963). E. M. Sparrow and R. D. Cess, The effect of a magnetic field on free convection heat transfer, ht. J. Heat Muss Transfer, 3:267-274 (1961). R. Viskanta, Effect of transverse magnetic field on heat transfer to an electrically conducting and thermal radiating fluid flowing in a parallel-plate channel, ZAMP, 14(5):353-369 (1963). R. Viskanta, Some consideration of radiation in magnetohydrodynamic Couette flow, ZAMP 15(3):227-236 (1964). R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. M. D. Gunzburger, A. J. Meir, and J. S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp. 56(194):523563 (1991). L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960. A. J. Meir, Existence, Uniqueness and Finite Element Approximation of Solutions of the Equations of Stationary, Incompressible MHD, Ph.D. Thesis, Carnegie Mellon Univ., Pittsburgh, 1989. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Wiley, Chichester, 1984. J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, Vol. 2, Finite Element Methods (Part l), North Holland, Amsterdam, 1991, pp. 523-639.