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Steady plane poiseuille flows of incompressible multipolar fluids
STEADY PLANE POISEUILLE FLOWS OF INCOMPRESSIBLE MULTIPOLAR FLUIDS+ HAMID BELLOUT and FREDERICK Department
of Mathematical
Sciences, Northern
BLOOM
Illinois University.
DeKalb.
IL 601
15.U.S.A.
Abstract-A detailed study of the problem of plane equilibrium Poiseullle flow of an isothermal. incompressible. dipolar fluid exhibiting non-linear viscosity is presented. Estimates are derived for the mean velocity, friction factor and net volume flow associated with such a steady motion; rhete may be useful in computing bounds for the constitutive constants associafcd with multipolar fluids
I.
INTRODUCTION
In two recent works, Necas and Shilhavy [l] and Bellout rf al. [2] laid the foundations of a consistent continuum thermodynamical theory for the phenomenological behavior of multipolar viscous fluids and examined the consequences of the new theory in several simple cases. The work in refs [I] and [2] builds upon earlier efforts of Toupin [3] and Green and Rivlin 14, 51 and significantly extends the work of Bleustein and Green [6] on dipolar fluids so as to allow for non-linear constitutive relations among the various stress tensors and spatial derivatives of the velocity of order greater than one; it is. most importantly, a theory which is, a priori, compatible with the basic principles of continuum mechanics and thermodynamics. e.g. the principle of material frame indifference and the second law of thermodynamics as expressed by the Clausius-Duhem inequality. The next logical step in the process initiated in rcfs [I, 21 consists of checking predictions made by the new model against experimental data; for this express purpose it is desirable, at least within the context of a few simple flows. to be able to produce as explicit a dependence as possible of the qualitative predictions of the theory on the constitutive parameters which underlie the model. However, even for simple flows, such as the plane Poiseuille flow considered in the present work, it is not possible to solve. in closed form, the relevant non-linear boundary-value problem, i.e. equations (1.I 7) and (I 13). The specific purpose of this work, therefore, will be to derive explicit qualitative, order of magnitude bounds. from both above and below (in terms of the basicconstitutive parameters appearing in the model) for such basic, experimentally determinate quantities associated with plane Poiseuille flow as mean velocity, net volume flow, and friction factor or drag. For an isothermal, incompressible, dipolar viscous fluid the constitutive theory formulated in ref. [2] assumes the form
ri,k
(l.2)
=
where the quantities denoted above have the following interpretation: (ui). i = 1.1, 3 are a set of rectangular Eulerian coordinates, ~~~and Tijk are, respectively, the components of the (usual) stress tensor and the first multipolar stress tensor, p is the pressure, 6,, the Kronecker delta, &,/l”./ll > 0, and r,O < u < 1, are constitutive constants, and
are the components of the rate of deformation tensor, where the L’,, i = 1. 2. 3 are the components of the velocity vector. As is standard by now. we sum on repeated indices ’ Research supported. in part, by ONR Contributed by K. R. Rajagopal. Nlfl 28:5-B
Grant
No. NW?O!~-~I-J-ICO?.
SO?
H. BFLLOUT and F
504
BI.OOM
wherever they appear, e.g. in equation (1.1). For x = jj, = 0. and jj” = j1,). the usual viscosity coefficient. equations (1.1) and (1.2) reduce to the well-known constitutive relation which underlies the Navier-Stokes equations for an isothermal, incompressible viscous tluid; the constitutive relations (1.1) and (1.2) go beyond the Stokes theory, therefore, in three significant ways: the presence of a non-linear relation for the viscosity, i.e. jl = jLO(&+ elrlclrl) -l.z.
( 1.4)
the presence of higher-order spatial derivatives of velocity, and the presence of an explicit constitutive relation for the first multipolar stress tensor rijk which, as is shown in refs [ 1, 31. enters into the balance of energy equation but not into the relation expressing balance of momentum. In earlier work, Necas and his co-workers considered [7-lo] a linear version ofequations ( 1.I) and ( 1.2) for a compressible dipolar viscous fluid, namely sij = - pb,, + Lo6,,div v + 2j~+~,, - i., iiij Adiv v - 2j1,Aci,
(1.5)
7
(
Ti,t
=
j.oSijI'l,la
eij
2/11i’x,
+
and proved a number of existence theorems for the relevant associated boundary-value problems in bounded domains; extensions to the case of a compressible, heat conductive multipolar fluid were also made but. also, within the domain of a linear constitutive theory similar to that expressed by equations (1.5) and (1.6). When equations (1 .l) and (1.2) hold in a domain R c R3 the associated initial-boundaryvalue problem assumes the form (we take the external body force to be identically equal to zero): ,>;+vVv=
-Vp+2V.(j1e)-2j1,V+(Ae) divv = 0
in R x [0, T),
T > 0,with jr given by equation
(1.7) (1.X)
(1.4)
v(x, 0) = ro(x)
in R
(1.9)
and v = 0,
Tij~vj~,Ti
=
0,
on
?R x [O, T).
(1.10)
In equations (1.7)(1.10), p is the constant density, the non-linear viscosity is given by equation (1.4), while the first boundary condition in equation (1.10) is the familiar expression of the fact that the tluid adheres to the boundary and the second set of boundary conditions (with v the exterior unit normal to LQ) are a consequence of the principle of virtual work and express the fact that the hypertractions (moments) vanish on i?R, e.g. ref. [3]. When j1’ = jco (usual viscosity), x = j1, = 0, equations (1.7))( 1.10) reduce to the standard form of the initial-boundary-value problem for the Navier-Stokes equations in R x [IO, T). Among the special cases of equations (1.7)-( 1.10) considered in ref. [2] are the classes of plane equilibrium Poiseuille flows (between parallel plates) which have the form 1!, = t.,(_X~), L’2= 0,
L‘J= 0,
(1.1 I)
with the fixed plates located at s, = fa. for some N > 0. For the sake of simplicity of notation. and because in all that follows here we will consider, exclusively, flows of the form given by equation (1.1 I), we will set (xr ,.xz,x3) = (xY,:). and r, = I‘. so that v = [~‘(~),0,0]: it then follows quite easily that equations (I .7)-( I, IO) reduce to the non-linear boundaryvalue problem: (1.1’) j1O([‘: + :r”(_r)]-‘zL.‘(?‘))‘j’,‘:““(y) = p,, L‘(2 a) = 0,
f”( *Cl) = 0.
(1.13)
where we have absorbed a factor of two into the constant pressure gradient p, = ?p,ic’.t,. Several problems associated with the boundary-value problem of equations (I. 12) and ( 1.13) were addressed in ref. [2]. First of all. if r = j1, = 0. j1’ = jco then the second boundary
Steady plane Poiseuille
flows of incompressible
multipolar
fluids
505
(cl Fig. I.
condition
does not appear
in equation
(1.13) while equation hU”(Y)
(1.14)
= Pl
associated with Navier-Stokes; the integration yields the familiar parabolic profiles
of equation
-;[I -($I,
u(y)=
(1.12) reduces to the familiar
(1.14) subject
to u( +a) = 0
(1.15)
-a
which have the property of both blowing up and remaining parabolic as /co + O+, thus indicating the lack of utility of the Navier-Stokes in the case of vanishing viscosity and the need to implement something like the Prandtl boundary-layer theory [l l] in the laminar flow domain. In fact, for plane (laminar) Poiseuille flows it is well know experimentally, and predicted by the boundary-layer theory, that velocity profiles will flatten out in a manner similar to that depicted in Fig. 1, at least until the breakdown of laminar flow and the onset of turbulence.’ In ref. [2], we set a = a, = (n - 1)/n and integrated equations (1.12) and (1.13) in the special case in which E = p, = 0; denoting the corresponding profiles by u,(y) the relevant boundary-value problem reduces to one of the form (1.16)
2’“- ‘“*oh(y) = IL-Csgn g(y)l”g”(y),
(1.17)
U”(I!Ia) = 0, with g(y) =
y. As solutions
of equations
(1.16) and (1.17) we may exhibit, for example, y < 0,
y > 0.
(1.18)
(1.19) The profiles u.(y) exhibit, as n + + cc (i.e. as a -+ 1 -), the successive flattening (expected as the fluid becomes less viscous) which is depicted in Fig. 1. With E = pI = 0, the fourth-order boundary-value problem of equations (1.12) and (1.13) reduces to the second-order problem ~“~C~~‘2(Y)l-a’2~‘(Y)J’
= PI>
c(&a) = 0 ‘All of our work in this paper specifically assumes that the flows remain in the laminar a non-linear hydrodynamic stability theory is not relevant at this stage.
(1.20) (1.21) regime, so that
506
H
BELLOCT and F BIIX)U
and it would seem, a priori, that there is no firm basis for relating the solutions of equations (1.20) and (1.21) to those ofequations (1.12) and (1.13). i.e. that the profiles r,(y) need not be close, in any precise sense to those predicted by the full model, and that a boundary-layer approach must be employed if we wish to let !l, 4 0’ in equation (1.12) and drop the second set of boundary conditions in equation (1.13): however, this is. emphatically. not the case if we are only interested in continuous dependence up to a certain order, namely C’. so that both the velocity and velocity gradients predicted by equations (1.20) and (1.2 1) will be close to those predicted by equations (1.12) and (1.13) in the L ’ norm on [-(1. a]. In ref. [2, Section 61 it is proven that there exists a unique solution of the boundary-value problem of equations (1.I 2) and (I .13) in the space Lt’,,, = B,,,(O)
n H 'I-II, a).
( I.22)
where B,,(O). AI > 0, is the ball of radius :U in the space I’= Hi"+"(--I. u) with 0 < 6 < l/2. The existence and uniqueness result cited above is established by use of the Schauder fixed-point theorem. Moreover, through the development of appropriate sets of 3 priori estimates for the soiution of equations (1 .I 2) and (1.13) it is also proven in ref. [2. Section 61 that r(.;~,l(,)-+
rO(.)
inC’+‘.
O
asc,fl,
-+ O+,
(1.23)
where (‘(.;E,/(,) denotes the unique solution of equations (1.12) and (1.13) while I’,~(.) denotes the corresponding solution of equations (1.20) and (I .2 1). In the course of proving the result of equation (I .23), it is established that the unique solution of equations (1.12) and (1.13) satisfies estimates of the form
J --(1 and
(1.25) with C, > 0, i = 1,2 independent of both E and /l , : estimates (1.24) and (1.25) will be used in the sequel here. In this paper we continue the study of plane equilibrium Poiseuille flows of incompressible, isothermal. dipolar fluids initiated in ref. [‘_I. Through the use of dimensionai analysis applied to equation (1.12). we isolate the natural counterparts of Reynolds number associated with the Navier-Stokes theory. We then investigate, in greater detail than was done in ref. [Z], properties of the solutions c.,,(.) of equations (1.20) and (1.21) and use these solutions to compute the associated mean velocity. maximum velocity, volume tlow. and pressure drop. Finally, although the continuous dependence result for solutions of equations (1.12)and (1.13) in Cl+‘, 0 < 6 < 4. as 6, jr, 9 O’, was established in ref. [2]. precise estimates of the errors incurred by setting E = IL, = 0 and using, in place of I’(. : 6:.11,) the solutions rlo( .) of equations (1.20) and (1.21). were not presented there; such estimates arc derived here and are subsequently employed to establish the related estimates for the volume flow, etc. It is hoped (and expected) that such estimates will eventually serve as a guide in the formulation of experiments directed at the determination of the constitutive constants in the model; these estimates have also been used in recent work [12-141. involving various studies of existence uniqueness. and asymptotic stability of flows of multipolar fluids.
2. GENERALIZED
REYNOLDS
NUMBERS
In this section we briefly indicate the appropriate form which a dimensionless version of the evolution equation associated with equation f 1.12) assumes, and, in the process. are led to the definition of generalized Reynolds numbers that are connected with plane Poiseuillc flows of incompressible dipolar fluids. Employing a standard analysis, we set (2.1)
Steady plane Poiseuille
flows of incompressible
multipolar
fluids
507
and (2.2) in the evolution
equation
for plane
Poiseuille
flow of an incompressible
dipolar
fluid.’ (2.3)
Here V is a measure of mean- or far-field velocity associated Elementary calculations using the chain rule show that Sl
vz ?I-
_--
---
sj-
I3 -=_21’
(2.4a)
a 3’
Z?p
with the flow and L’= u(y, t).
pV2(‘j a
(2.4b)
?_c’
V&T
(2.4~)
a l?y
and
with $
= A$_; substitution
PV -=--
v
PF
?p4
f14
s_f’
(2.4d)
of the above results into equation
(2.3) then yields
i”:j~~~~~~~+“I:~~,~+%(~~]-.;!‘)~}_~~,
which becomes,
after multiplication i?v -_= 2T
(2.5)
by 4,
and setting
1o=!$,&
-5+~~1,~~~(3’]-.~2~~~~~.
In as much as E must have the dimension
(2.6)
of a velocity
gradient
squared,
E = ca2/V2 is dimensionless; (2.6) as
using the definition
of equation
(2.7) (2.7) for .? we may now rewrite equation
where we have dropped the superposed bars from J, t,p, t‘, and E. The dimensionless version, given by equation (2.8) of the evolution equation (2.3), leads naturally to the definition of the following two generalized Reynolds numbers that are associated with plane Poiseuille flows of dipolar viscous fluids: VI+1 R”’ = R, = c. (2.9) 0 “Oa’1’ ”1 Using the definitions
(2.9), the evolution
equation
for c(y, t) assumes
St is=-$+-g{[&+(~)y2$}+$.
the form (2.10)
For tl = pr = 0, clearly Rr’ = Vu/v0 and R; t = 0, so that equation (2.10) reduces to the standard dimensionless form for plane Poiseuille flow within the context of the Navier-Stokes formulation, with Rr’ being the usual Reynolds number, if p” = /lo. ’ Note that for a = 0. JI, = 0 equation (2.3) reduces to the equation predicted by the Stokes constitutive law, while for z = I, E = 0, JI, = 0 it reduces IO the equation for channel Row of an incompressible inviscid fluid.
H.
508 3. THE
BELLOUT
POISEUILLE
F BLOOM
and
FLOW
FOR
& = /i,
= 0
In this section we study, in greater detail than was done in ref. [2]. consequences of setting I: = 11, = 0 in equations (1.12) and (1.13), i.e. we look carefully at certain aspects of the qualitative behavior of solutions to(y) of /P{ [:
L$
(y)]
-‘fZL.b(?‘))’
=
(3.1)
PI,
L‘J + a) = 0.
(3.2)
In the next section our interest will be in obtaining estimates which relate the behavior of solutions o,(y) of equations (3.1) and (3.2) to those of the solution L’(~;E,/L~) of equations (1.12) and (1.13); the quantities of particular interest to us will be the volume flow
tf.v;c,p,)dy,
(3.3)
Qo.o = Qo = ’ u,(.dd) s -*
(3.4)
which for E = jc, = 0 has the form
the mean velocity
t (3.5)
and its counterpart
for I: = /i, = 0. i.e. Lo = -
I
Q.
(3.6)
2a and the friction
factors
fo= 4r,,(f&O>O) 4PG
(3.7)
where T, 1( 2 a, c, 11,) is the shear stress at the walls located r,2(+a,F*P,)
at y = 5 a, i.e.
“‘~L”(_fa,E,~I,)-~1,t!“‘(~a,E,~~,).
= /‘O[E + hU’2(fa,E,/1,)J-
(3.8)
We begin by noting that if oo(y) is a solution of the boundary-value problem of equations (3.1) and (3.2) then so is uo(-y) and, thus, by uniqueness of solutions go(y) = am, - a I y I a; from this result it follows that L&(Y)= - ob( - y), so that t&(O) = 0. Moreover, with pt. the constant pressure gradient, negative, a first integration of equation (3.1) yields PO c:
-“21;b(y)= - [p, ly 412(Y)l
where the constant of integration vanishes From equation (3.9) it is immediate that
(3.9)
in view of the fact that 0 < z < 1 and t&(O) = 0.
t!b(y) > 0,
y E [-a,@,
t&(y) < 0,
y E ux~l.
(3.10)
In as much as t&(y) I 0, for y E (0, a), we have v;(y) = if we set c, = j1’2”‘, equation (3.9) becomes on [0, a],
J vb(y)l,
IL,’ (y)\’ -= = lp’ly 0 c,
0 I y I a, and, therefore,
(3.1 I)
or 1rb(y)I = Czy’!(’ -1’. l/l1
0 I y 5 a,
(3.12)
-1)
. We rewrite equation
(3.12) as
rb(l’) = - C,y’“‘-~‘.
0
5
I’
5
a
(3.13)
Steady plane Potseuille
and integrate
flows of incompressible
multipolar
fluids
509
from a to y obtaining (3.14)
In computing
equation (3.14) we have used, of course, the boundary for 7 in equation (3.14) and noting that L’~(J) = ~‘e(-I’),
Substituting
condition we have
~~(a) = 0.
(3.15) where (1, = c,
1-Z 2-x
( > IPII
=
($2
- !N( I - 2)
‘~~‘-=)~(2-.,,(,-*)
1- 2 2-z (4
H c,
=(s)( It is clear. from equation
!e!&J’~“-z’.
(3.16)
(3.15), that (3.12)
By direct
calculation,
the mean velocity
I CO= -2a
O0 associated
with
a _
vo(y) is obtained
as
dy)dy
s u
(3.18) Carrying
out the integration
in equation
(3.18) we are led to (3.19)
or, in view of equation
(3.16) I‘o = (!&)(Q5!&)“~‘-z’
(3.20)
We note that Cal,=
0
=! .-IPIlU2 3
/P
which is the classical result associated with Navier-Stokes. it follows that ]im rtaX = lim d, = lim I-Iz-1 1-r However.
in view of equation lim I-I-
from which
(3.21)
’
From equations PO.
(3.22)
(3.16)
ff, = *_m,[(~){$&)“““] Ii
it is clear that the critical
quantity
(3.23) in computing
lim,_,
- craK is
c _ IPI laz-m I l‘op . Specifically,
if ez > I, for I sufficiently
that we set I’=
(3.17) and (3.19)
L.oI,=~in equation
(3.24)
close to 1, then I$‘~’ --* cc as z + (2.9), so that jp,l = pV’/u,
1 -. Suppose, now.
then (3.25)
BtLI.OIT and F.
H.
510
Bi.oov
while lim I_,
R”’ _ !Y. 0 (1 I’
Thus, l.Fd’ +
%
,
asr-,
(3.26)
I-
provided
R;’
lim I_, To
emphasize
parameter the lim,,
further
the role of the criteria
r. in situations
,
rb(u). From
involving
(3.27).
small
equations
(3.15)
(3.17)
> ;j. and its connection
physical
viscosity
and (3.19),
with
the status
of the
(i.e. CYclose to I), we compute
we have
(3.28) so that
and, thus,
I l.;)(fI) = (3 - 2x) Therefore.
by virtue
of equations
= _
(3.20)
I
_
(1 - z)
(3.30)
I‘,)
and (3.24).
(,i (’ -I’=
Iin1
- -~
_
(3.3 I)
y_
*I
I
. , RI:’ > L i._, under these same conditions
if lim,
lim so that
if liml
indicated
= lim-.
r;)( -a)
,
[-r;,(u)]
(3.32)
= + x .
Rif’ > x!2._ and r is close to I, the velocity
_,
in Fig.
~.,
profile
assumes
the form
2.
In fact, not only
is V;,(O) = 0, but the rapid flattening
respect to the axis J’ = 0. US x -+ and the companion
result
for
-
I
of the profile,
_ is easily demonstrated (I I
J’ I
as follows:
depicted in Fig.
2, with
from equation
(3.29),
0. (3.33)
so that r:(O)
= 0. for all r. 0 < r <
I. Then (3.34)
so that I’;;‘(O) = 0, for I > i. A further clear
that
rn suffkiently
by an induction close to 1.
computation
argument
we may
shows show
that r,;;“(O) = 0. for r > 5. and it is that
r:‘(O)
= 0. z > 2,.
for
some
Steady plane Poiseuille flows of incompressible
From (3.4),
equation
(3.20)
it follows
directly
multipolar
that the volume
flow
fluids
511
Qo, as given by equation
is
(3.35) Also,
from
equation
(3.20).
it is a simple Ip,I
Now,
by equation
(1.20),
z
=$&
r,,(-n,O,O)
(3.7),
to compute
3-22
that
l.,-,
( I-r0
(3.36)
1
= IP~ 1~. or 0212 3= & I_r n (
s,~(-CJ,o,o) SO that, by equation
matter
the friction
factor
‘r 1
_,_l L’o
(3.37)
*
is (3.38)
Also,
in view of equation
(3.36)
and the fact that (I > 1
(3.39) estimate
:tn
which
constitutive pressure
may
parameters
gradient
be useful /lo, r,
r(y;e./~,) and
approximating
p, and the mean velocity
4. TfiE
this
In
in
based on careful
section
of equations
(1.12)
those
(1.2 I), which are missing
such as equations
(3.38)
FLOW
precise
and (l.l3).’ from
and (3.39)
experimentally,
of the
of the magnitude
of the
Co.
POISEUILLE
we provide
the values,
measurements
FOR
qualitative
relative
8, [tl
to the solution
our original
uork
quite
We begin with
useful.
\V(!‘; 2:. 11, )
#
0
estimates
for ad
in ref. [2],
the unique
and which some
solution
of equations render
notation,
i.e. we set (4.la)
= r’( !‘; I:, /iI ).
Gj?‘;c) = \v(y;r:,O), :(I’;r:./(,) Z(y;c)
(1.20) results
(4.1 b)
= I: + IW(.KE, p1)12,
= z(?‘:I:,o)
(4fc)
= 1: + I\i(y;I:))’
(4.
Id)
and Y(K) = (C + ],v12))z When
the interpretation
e.g. I(C) = Z(y;c). equation
-
[ -CI,N]
Our
ro(y)
from
quantities
boundary-value will
we will
suppress
such
problem be obtained
goal both
is
below
as QL,tiI. of equations from
to
(4.2)
the explicit
We also note that Gl(y:O) = r’(!.;O,O)
(3.29).
= r(_r;c,~~,) estimate
is obvious
2)(. = Z~‘ZIV.
estimate
dependence on y and write,
= L’;(Y)
the
which
difference
is given explicitly ~(y;e,
p,)
and above and to then use the resulting
the volume
flow
associated
with
-
by
r(y;O,O)
bounds
the solution
to
of the
( I. 12) and ( I. 13). The bounds for o(y; E, p, ) - co(y) on
a series
of subestimates
that
result
from
a succession
of
lemmas. Lemttm 4.1.
Let L’(J; E,I(~)
and rO(y) the unique such that on [-cr.
solution
be the unique of equations
classical
solution
of equations
(1.20) and (1.21). Then
3K,,
(1.12)
depending
and (1.13) only on IL,
a] )r’(J’;e,O)
-
r;(y)\
’ In future work [15] we plan to studyequations(l.12)and of singular perturbation theory.
<(I
+ K&/G.
(4.3)
(1.13) both numerically as well as from the viewpoint
H.
512
BELLOUT
and F. B~oow
Proof: We set !L, = 0, in equations (1.12) and (1.13). divide through by p” # 0, and let p: = p, 111 ‘; then ~.(y;s.O) is the solution of the boundary-value problem [(E + ILi.(~;E)IZ)-I.21;.(L);&)]’ = p:, L.( _to;c,O)
(4.4,)
-a
(4.4b)
= 0,
where we have used the definitions (4.la) and (4.lb). Now if K(~;E,A~,) is a solution of equations (1.12) and (I. 13) so is L’(- y; E. 11,) for any e, p, 2 0; by uniqueness of solutions to the boundary-value problem we must have c(y; 6,~~) = L‘(-y; ~11,) from which it follows that L”(I’; E,11,) = - r’( - y; E,11,); -a
~‘(0; E,/( r ) = 0 for all E, ,n, 2 0. Integration
of equation
(I: + I~;.(!.;E)1*)~I’2)i.(~;E) = p:y,
-U
as G*(O:I:) = t.‘(O;~.,0) = 0. It then follows from equation k(y; E) # 0, as 0 < z < I. Squaring
both sides of equation 3(y;C)-=\i’*(y;F)
We rewrite equation
(4.4a) leads, therefore,
vc 2 0,
(4.6)
(4.6) that 4’ # 0
(4.7)
(4.6) and using the definition
= p:zJJ,
(4.8) in the form [recall
(4.ld), we obtain
- a < J’ < N.
that Z(c) = 1(y;~).
(4.8)
- a 2 y 5 U]
I(E)-QII(E) - F] = p:QJ or 3(I:)‘-= - r::(c)-’ If we now differentiate
equation
- a < r’ < u.
(4.9) with respect to I: we obtain,
:(E)-‘[( where 1, = -4 ?(y;c).
= p:zy2,
I - z)Z, - 1 + &(c)-‘2,]
We now restrict our attention L!(E)= r:+ ~U’()‘;E,0~2 #O.
it follows from equation
(4.9)
after a simple calculation,
= 0,
(3.10)
to the set of all y E (--a. 0). As
ve 20,
I’E(-(1.0)
(4.1 1)
(4.10) that
(1 -r)l,-
I +ZEI(E)-l:,=O,
r:>o.
yE(-l,kO)
(4.12)
in which case we find that (4.13) As a direct consequence
of equation
(4.13) we see that
0 2 &(y;s,O) I j& Now, for y E (-a,
E20,
L’E(-u,O).
(4.14)
0) we may write that r i(y;s)
= I(y; 0) +
i,(y;i.)di..
(4.15)
I0 Combining
equations
(4.14) and (4.15). we then have Oii(y:f.)-f(y;O)~~~;r:2O.
where we have used the continuity
yE[--,O),
(4.16)
of c’(y; E,/[,) to extend the result to _V= - a. However,
1(0; 0) = I,i(O;O)l’ = L2(O; 0,O) = 0. :(O;E) = E + li(O;c)l’, = I: + /r~‘(O;E,0))2. = 6.
(4.17)
Steady plane Poiseuille
so 1(0; E) -
_;(O;O) = E < &,
Now, equation
for
flows of incompressible
multipolar
0 < z < 1, and, thus, equation
(4.16) is equivalent
fluids
513
(4.16) also holds at y = 0.
to
YEC--a.01
0 I [E + r;2(y;&)] - kZ(y;O) I & Or -&I
bv(y:&)-
i2(y;O)l
ea,
yE[-a01,
(4.18)
which, in turn, yields the two estimates GJ2(y:O) - E I &2(y;E),
(4.19a)
e
(4.19b)
G2(y:&) < k'(y:O)
+
on [--a, 01. Consider the set of all y E [--a, 0] such that G(y:O) 2 4 y in this set it follows from equation (4.19) that 0
with the upper bound
5
Q*(y:&)
holding,
<
i2(y:O) +
-!Y!- yEC--701 1 - a’ i $(y:O) 2
of course, on all of [--a,
for fixed E > 0; for
(4.20)
4’
01. Therefore,
for all y E [ -
N.
01.
such that G(y;O) L ,,& 0
2
G(y;&)
5
J-z-
G(y:O) +
(4.2 1)
l-a
and we have used the fact that equation
(4.6), with p: < 0, implies that r;(y;c) > 0, V,: # 0,
y E [-a, have
0] but k(y:O) < 4;
0). Now, suppose
that y E [-a,
h2(y:c)
with Kz = l/(1 - a). Thus, if y E [--a,
<
ae & +
-
=
1-z
then by equation
Kit,
(4.19b), we
(4.22)
0] and G(y: 0) < ,,& then ko):~) < K,&
(4.23)
and ) G( y: E) - q y;O) I I qy; E) + G( y: 0) < (1 + K,)$
or A(y:O) - (1 + K,)& for all y E [--a,
< k(y:c)
0] such that G(y;O) < 4.
< G(y;O) + (1 + K,)&
(4.24)
However,
K.=/G>Ea, so a comparison of equations (4.21) and (4.24) shows that equation (4.24) holds for all y E [-a, 01. Using the definitions of G(y;e), G(y; 0) we may rewrite equation (4.24) as lu’(y:~,O)
where K, = ~ A.
Replacing,
-
d,(y)1 < (1 + Q/k
y by -y,
y E C-a,
01,
for y E [O, a], we see that equation
(4.25) (4.25) holds for
all y, -a I y 5 u, as both u’(y: E,O) and t&(y) are odd functions on [-a. a]; this establishes the validity of the estimate (4.3) and concludes the proof of the lemma. Cl Lemma 4.1 enables us to compare C(Y:E, 0) with uO(y) on [--a, a]; our next set of lemmas are aimed at enabling us to compare u( y: E,p, ) with u( y; E,0), the first of these being stated as follows.
H BELLWT and F 81 OOM
514 Lcrnm~
Let I’( J’: K. jl,
4.2.
) be the unique classical solution of equations ( I. 12) and ( I 13) I = I.“‘( J’: t:, p, ). Then 3C + . C _ > 0. independent of X, ji, such th:,t
and set r( \‘:I:. j,,
r(!,;i:.jl,)
I
C..
!‘E
I ( j’: 1:.p, 1 2 - c - . Pr(j(J:
\i’e will
proof
establish
of equation
that
,‘(!‘:)A Jo, 1 and
differential
only
(4.26b)
equation
follows
(426a).
I‘(>‘: r:.O) are,
J’ E [o.
which
in an entirely respectively.
is all that is needed in the sequel;
to r( +o:r:.
ing equation
the solutions
the
We begin by recalling
of the
/I, ) = r”( *KJY.~~,)
(4.27b)
from
future
refcrcnce
non-linear
Ordinary
(‘( is I.‘( !‘:~.j1,
an even function:
(4.27~)
the definition
(4.2) enables
,;.‘()
cJ; +
on [
Vi:, jl,
.:,: ),‘?
with
= U, /j,“. Subtract-
respect
to J. we obtain
-LJ.
equation
of all, as
J’ E [ - (1.Cl]. but so is 1.“’ (J:
(I]
(4.X)
Il:\(“‘(!‘:f:.jJ, 1.
=
first
2 0, ,““(O;r:.ji,
us to write
) = 0. Next
(4.17a)
i:, 111 )]’ - jt: r”“(
;‘[I“(?‘:
(4.27b)
\i,( y: I:)
) = c( !‘: L jt, ).
) ~111 odd function
(327a)
p:,
and integrating
here the following: 1’;t:. j’,
in particular,
' ~r;(y:r:);'=
= r,:,
= 0 and r( &tr: i:,O) = 0. where ji:
equation
we record
j!:“““(J’;E.j1,)
\F(y:c)]
bi’(j’: 1:.ji , ) [t: + \‘.+ .:,;, j1,)]1’2 -
not only
fashion.
equations
([I:+
For
(4.Xb)
JJ].
analogous
I[(: + “.?(!.:i:.j’,)]~IZ’\.(?‘;‘:,j’,))’-
subject
(426a)
[I-lJ.01.
I:. /J,
).
while
we observe
I.“( J’: I:, ,H,
) is
that the USC of
in the form
) = pT
(4.29)
+ \V’)_ ‘1
(4.30)
r’: 1:. ;Jl
and that ;“(\v) = (1: + +~“[I
;,“(,,.)
=
_
% (,; + ,,.‘)I’ 21t I
\c
~_
i from
which
from
equation
it follows.
as 0 < x <
(4.29)
-
L
1,2(I:
%)WZ
(1 -
l: + -___
3%: (?;+ \,2)? + (2 2)
+
i: + \\.?
1
(4.3
I)
i
I, that ;s’(,t,) > 0. Vr: > 0, while
sgn Y”(,c) =
_ sgn 1,‘. Now.
with s( J’; 1:. jI,
(4.32)
) = r”( j’: I:, /I, ),
we have ;“(r,‘(Y;r:,j(,
Suppose .s(
\‘(I;r:.
that
/I,)
>
s(~‘;r:,ji,)
0. From
)).s(.Kr:.j~,)
-
S( - (1; I:, /I,
)=
takes
equation
But ;“(,.‘(Y~;E./I,))
any
> 0. while
there. Thus. positive
boundary
.s( I‘: ~,j’,
maximum
conditions
S(CJ; I:. I’,
a positive
(4.33)
(4.33)
< 0.
(4.34)
0.
at
some
y0 E
(-((1.
(J).
so
that
(4.35)
+ I/J:1 = jI:.s”(~O;i:,jI,).
s”(~~, ‘I:, jl,)
of equaiion
) =
maximum
I
0, if \‘D is interior
) cannot achieve ;t positive
of s( ~:c,jl,)
= II:
we have
;“[,.‘(.o:r:.j,,)].s(‘,,:r:,j,,)
imum
j,:r”(JYr:,j(,)
must,
(4.34)
therefore.
it follows
to [-a.
maximum
that
(J] and
at a point
.S has
j’O E
occur
at .\’=
&-a. In
there
is no positive
;I mar-
( - ~1,(J) view
;iJld
of the
max,mum
for
.S(J’; &jl,
1 anywhere on [ -cJ, ~1: thus. if the maximum of s( J’; E, ji, ) occurs at an interior point y0 E ( -(J. (I) we must have A(J~;s, jl,) < 0 in which case .S(J’; C,j’ ,
and the same result (4.35).
.s(),: J:./‘~
holds
) Cannot
) _<.S(j’“: l:, /‘, ) < 0. J’ E (- il,
if the maximum have a zero l“‘(?‘:i:.j’,)<
which
shows
that the graph
occurs
maximum
of I’( j’; E, j(,
at J’ =
~CJ.
where s vanishes.
at an interior
O;xE(--cl.(I).
c.j1,
(4.36)
(I)
point
)‘0 E
>o.
) is concave (down) on (- LI,a).
(--(I,
By equation (J).
Thus. (4.37)
Steady plane Pokseuille Rows of lncompresslblr
multipcrldr tluld\
515
Now let y E (--a. S) for any 6 I a. Then Y r”‘(kc,p,)di.
j -(1 and as y may
be chosen
(--a. a)] it follows
arbitrarily
= c”(y;c,jj,)
close
to
--a
[and
(4.38)
,““(y;~jj,)
is continuous
in J on
that V”‘(--_a;&. jj,) c”‘(U; E, jj,
since r”‘(_r;c,jj,)
< 0
is an odd function
of y on
< 0
(4.39a) (4.39b)
)> 0 From
(--a,~).
the definitions
of s( ,r:~,jj,)
and
t(.r: E. jj, ), t(~:F,jI*)=S’(1’:E,jI,), Therefore,
if we differentiate
equation
i”[r.‘(~;c,j(,)]f(_~:E,j(,)+ The
calculation
argument:
in
equations classical
[Z]
to equation
it was
(4.33) and (4.34), solution.
(4.33)
with
respect
./“[u’(v;E,~,)][~“(~:E,j(,)]~
leading
ref.
_,‘E(-0.0).
(4.41)
to y we readily -
that
subject to the additional
obtain
jc:r”(_V:E,jj,)
may be validated
demonstrated
i.e. a solution
(4.40)
by the following
the boundary-value constraint
(4.41)
= 0.
elementary
problem
given
by
I*( + (J, E, jj, ) = 0 has a unique
in C4( --(I,
a); in light of this observation,
= &:I
+ i”[L.‘()‘:E.jj,)].S(.KE.jI,))
and the definition
of ;*. S”(_Kc,jj,) is continuously argument
differentiable
shows,
is, in fact, in C
’ (-a,
We now return at ~9~E
in y on
classical
(4.41) and assume
by equation
tion
Repetition
of this problem
) achieves a positive maximum
(4.41)
it must
be true that (4.42)
IO.
(4.3 I)
< 0, y~(-an,
a), while
~0 E (0. (I) then by equation ,‘o E
that r(y; s. jt,
+ Y”[V’(J o:c,jj,)][L.“[.“;1:,jj,)]2
sgn~“[L(yO;c,jf,)] ,“‘(.,‘;c,jj,)
holds.
of the boundary-value
5 0; then by equation
;“[~“(?‘o;c,j1,)]l(yo;c,jI,)
But
(4.41)
solution
(I).
to equation
(-cl, (I) so that 1“ (yo;c,jj,)
However,
a) and equation
(-0,
in fact. that the unique
(4.43).
= -
r’(O;c,jj,)
we must
sgnr’(yO;>:,
jj,).
= 0, so r’(y;r:,jj,)
(4.43) < 0 for y~(O,n).
have y”[c’(y,:~:,jj,
Thus,
)] > 0. contradicting
if
equa-
(4.42).
This means, of course, that if f(y;c,jj, ) achieves a positive maximum at N, (I) then, in fact. y. E (--a, 0); note that t(O;c,jj,) = 0 as ~““(y;c,jj, ) is odd on (--a,
( -
N). At such a _ro E (-a,
0) we will
have, by virtue
t(yo;E, j(I) I Now, a t _rO, t’(~,;c,jj,)
-
of equation
~“Cr;‘(yo;E,jII)][~“(yO;~,j~,)]Z ;“Cr!‘(yo;E,jl,
= s”(y,;c,jj,)
(4.42)
= 0, in which case it follows
;‘[u’(y,;c,j(,)]v”(L’o;E,
jj,)
(4.44)
,]
from equation
(4 .33) that
= p: < 0
(4.45)
so that c”(y,;c,jj,) Substituting
from
equation
(4.46)
into
= p:/;“[l:‘(Yo;E,j(,)]. equation
t(y,;E,p*) I Employing follows
the pointwise
from
equations
bound (4.47)
we obtain
;“‘[c’(.vo;E,jI) 1-J ;.‘3[r’(_vo;c,jt1)]
-
of equation
and (4.39a)
(4.44)
(4.46)
(4.47)
pT’.
(1.25) of ref. [2] relative to 1L.‘(J; E,p, ) 1it now
that 3C+
> 0. independent
of both I: and jj,,
such
that f(‘;E,j(,)~f(~O;E,j(,)IC+, If r(y; c, jil ) I equation
(4.48)
0 on (-a, holds
0) so that no positive
t/C+ > 0. An analogous
(4.38)
JE[-KO]. maximum argument,
exists which
on [-(I,
0] then certainly
begins with
the assumption
516
H. BELLOUT and
F.
BLCOM
that t(y;e,l~,) has a negtive minimum on (--a, a). can be used, as above, existence of a C. > 0. independent of both E and ii,, such that t(y;c,p,)> but we omit the details.
the
(4.49)
-C-,yE[O,a],
0
The next lemma provides us with an upper specifically the following result.
bound
for c(y;~,/l,
Lcmn~o 4.3. Let L’(p; E,it, ) be the unique classical solution R. 11, > 0. and all ye [--(~,a],
UC+
L’(y;E,p,) - C(y;E,O)I
) - c(y;e, 0); we have,
of (1.12) and (1.13). Then for all
(4 + c,)lil:,
j-y
where C,, CL, independent of both E and /i,. are the positive constants tively, in equations (4.26a) and (1.25). Proc$
to establish
From equation
(4.28) and the definitions
appearing.
of y( .) and r(y;s,p,)
respec-
we have (4.5 1)
However, with k(y;c) for each fixed y E [-a,
I w(y;E,/(,)
I \V(y;c,jc,)
(4.52)
a], E, /1, > 0. Thus, \1.(I’;
)-
E, p,
I’:f(y;Gfc,)
k( y; E) =
(4.53)
fc4y;LP,)lFrom equation
(4.30) f(w)
=
-r)w2
E+(I (E
+
,vz)‘+(~!21’
so I
(,; +
---= y’(W)
,72)1
+lzI2)
(E +
W’)’
+(x,2) = J-(E
i: +
(I - sc)G2
<
(I
-
a)c
+
(1 -
+
\,2)I’2
r)G2
I &(,:;:+lW&(~+c2)= by virtue of equations (1.25) and (4.52). Employing this last estimate for [y’(E)] _ r, as well as equation (426a). in equation (4.53) we are led to the upper bound w(y;c,p,) - \t(y;&) 5
G(& +c,y,
)‘E[-(LO).
Choosing y E (-u, 01, and integrating both sides of equation (4.54) from -a to y, we have. in view of the definitions of w.(y; E,11,) and k( y; E) and the fact that r( -a; E,11,) = 0. for t: > 0 and /L, 2 0: c(y;E,/1,)
- c(y;c,O) I
2
(JE
+ c,)z/1:.
y 6 C-a,
However, r.(y;E,fl,)= c(-yy;~,~,), for y~[--,a] so we see that equation (4.55) is, in fact, valid for all y, -(I I y I II. Our next lemma is as follows. ,!_ernrtrcr4.4. Let L.(y; 6, I(, ) be the unique Thenfor --N<~
classical
- rl(y;E,o)l I
__
solution
01.
the upper
of equations
(4.55) bound
in
(I .I 2) and (1.I 3).
(4.56)
Steady plane Poiseullle Rows of incomprewblc
Prooj.
We return
to equation
(4.53); for YE [ -
multipohr
[w(%;E, pI) - k(i;s)]di
1
_~ ~‘[W(j.;s.~,)]
-a
517
a, 0) it follows that Y
Y
fluids
t(A;c,/i,)dj.
(4.57)
from which, we obtain It’(y;E,P,) by again bounding
- c(y;E,O)I I $(&
[r’(G)]-’
3
(4.58)
+ C,Y lY It(E.;e,/c,)ldl --(1
from above and using the definitions
of w and G. However,
However, by virtue of the estimate (1.24) of ref. [2], there exists a positive constant, independent of both E and p,, which we will denote by C, (even though we have replaced I’, by 11: = /(,/ll”) such that
and we are led to the bound 11:
’ ItIdE. I (a~c:Cr)“*. -*
(4.59)
Use of the estimate (4.59) in equation (4.58) now yields equation (4.56) for y E [--a, 0] and the fact that c(y;~,/l,) is an even function of y on [--a, a], for all E,P, 2 0, then establishes the validity of equation (4.56) for all y, --a < y 2 a. 0 By combining fol!ows.
the estimates
of Lemmas
4.3 and 4.4 we may state our final lemma
as
Lenrrnu 4.5. Let ~l(y;c,p,) be the unique classical solution of equations (1.12) and (1.13). The 3C+. C,, C2, all positive and independent of both E and ,u,, such that for all y. -u_
-J’;rc, , _ r (J;: + c2Jw”z
5
c(y;E,/II)
- u’(Y;E,o)
I
(4.60)
Now, by virtue of Lemma 4.1, we have -(I
+ K,)&
forallyE[-u,u],whereK,=(! y E [-a, 0) we find that
< u’(y;e,O) - l&(y) < (1 + K,)d --a)-
- (1 + K&,/i
‘/‘. By integrating
equation
(4.61) (4.61) from
-u
toy, for
< ~‘(y;c,O) - co(y) < (1 + K,,u&
(4.62)
with this last result holding for all y E C-u, N], as t:(y; E, 0). tlo(y) are both even functions y. Since V(FE,/1,) - co(y) = [r(y;E*/I,) - 4y;GO)l + C4y;E.O)- L*,(Y)] by combining the estimates following theorem.
(4.60) and (4.61) we see that
we have, in fact, proved
of
the
Theorem 4.1. If r(y; E, 11~) is the unique classical solution of the boundary-value problem of equations (1.12) and (1.13). while uo(y) is the corresponding solution of the boundaryvalue problem of equations (1.20) and (I.2 I), then 3C+, C, , Cz, all positive and independent
SIX of
H.
&LLOI’T
and
‘r
BI.OO~I
both E and AL,,’ such that for all y, --(l I >’ I u. Hlth K, = I I - xF’ - (I + K,)u,Ji:
-
v’uc,
- 1(\‘J:
5
From
the above theorem
Theoren~
difference difference
-
+
c‘2yjl:’ 2 5 L.(y:i:.jI,)
UC, (1 + K,)(l \s,c + ~ 1- A
two other estimates
‘:
- I’(](!‘)
(4.63)
‘E + cz)ljl:.
now follow which we record
as follows.
4.2. Under the same conditions as those which prevail in Theorem 4.1 the of the mean velocities. i.e. L;,,, - P,, also satisfies the estimate (1.63). while the of the net volume flows QF.II, i Q. satisfies
(4.64) In particular. bounds
by virtue of equation
(3.35). for example.
we may now exhibit
the explicit
(4.65) and
Similar estimates may be developed for the friction factor ji.,, of equation (3.7). by employing the bounds for CC.,,, if we simply note that equation (1.12) is equivalent to ?/?J [~,~(y;~,p,)] = p,, so that for all I:, /I, 2 0. rlz( fn;~:,jit) = & Ip, 1~1. REFERENCES I
J. Necas and M. Shilhavy. Multipolar viscous fluids. Quwr. rlppl. ,Iforh. (to appear) Bellout. F. Bloom and J Nccaq;. Phenomcnological hch.lvlor of multipol.tr wscouc fluldh Qwrr Appl. hluih. (IO appear). R. A. Touoin. Theortes of elasticttv with coupic stress. :t R UA 17. 85 i I Z ( 196-l). A. E. Circe; and R. S. Rlvlin. Slmpl; force andstress multlpalcs Arch. Rationul .\fcch. And 16. 325-353 (IY64). A. E. Green and R. S. Rivlin. Multipolar continuum mcchanlc< Arc,h. Rutionul Afrch. Anal 17. I 13-147 (1964). J. L. Bleustcin and A. E. Green. Dipolar fluids. /!lf. J. Ent/!~g &.I. 5. 323- 340 (1967). 1. Necas. A. Novotry and M. Shilhavy. Global solution to the compressible isothermal multipolar tluid. J Afurh. And. Appl. (IO appear). J. News, A. Novotry and M. Shdhavy, Global solution lo the Ideal compresslbls multipolar hear-conductlvc Ruid. Comtnrr~f. ,\ftrrh. Unir. Carolrnur 30. 55 t --%At 1989). 1. Necas and M. Shilhavy. Some qualitative propcrtxs of the viscous compressible heat-conductive multipolar fluid. Comnt. PDE (IO appear) J. Nccas. A. Novotry and M. Shllhavy. Global solutions to the ~ISC‘OUIcompressible barotropic multipolar gas, preprint. S. Goldstejn, khlodrm kwlnpmcnf rn fiuin Dl’nn~~ic’s. Vol\ 1 and II. H. Bellout and F. Bloom. Stabiltty and uniquencsc of flouz of incomprc>siblc multipolar flutds I” bounded domains (submitted). H. Bellout and F. Bloom. On the uniqueness and stablilt> elf plant equilibrium wlutions of the equalrons of incompressible multipolar flulds (submltted). H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent Poisuillc flows of isothermal dipolar fluids (submitted). H. Bellout and F. Bloom. Numerical and singular pcrturbdtion aspects of the boundar? \alue problems of plane Poisculllc flow of dipolar lluids (forthcomlnp).
2 H 3. 4. 5. 6. 7. 8. 9. IO. II. I’. I3 14. 15.
‘Estimates such as equation (463) provide a qualitative b.lrl\ for initiating an cyperimrntal study of the behavior of dipolar t7ulds; only tuch enperimentat studies can wrbc the aim of eztlmating the con\trslnts Cc, C,. and Cz
of Mathematical
Sciences, Northern
BLOOM
Illinois University.
DeKalb.
IL 601
15.U.S.A.
Abstract-A detailed study of the problem of plane equilibrium Poiseullle flow of an isothermal. incompressible. dipolar fluid exhibiting non-linear viscosity is presented. Estimates are derived for the mean velocity, friction factor and net volume flow associated with such a steady motion; rhete may be useful in computing bounds for the constitutive constants associafcd with multipolar fluids
I.
INTRODUCTION
In two recent works, Necas and Shilhavy [l] and Bellout rf al. [2] laid the foundations of a consistent continuum thermodynamical theory for the phenomenological behavior of multipolar viscous fluids and examined the consequences of the new theory in several simple cases. The work in refs [I] and [2] builds upon earlier efforts of Toupin [3] and Green and Rivlin 14, 51 and significantly extends the work of Bleustein and Green [6] on dipolar fluids so as to allow for non-linear constitutive relations among the various stress tensors and spatial derivatives of the velocity of order greater than one; it is. most importantly, a theory which is, a priori, compatible with the basic principles of continuum mechanics and thermodynamics. e.g. the principle of material frame indifference and the second law of thermodynamics as expressed by the Clausius-Duhem inequality. The next logical step in the process initiated in rcfs [I, 21 consists of checking predictions made by the new model against experimental data; for this express purpose it is desirable, at least within the context of a few simple flows. to be able to produce as explicit a dependence as possible of the qualitative predictions of the theory on the constitutive parameters which underlie the model. However, even for simple flows, such as the plane Poiseuille flow considered in the present work, it is not possible to solve. in closed form, the relevant non-linear boundary-value problem, i.e. equations (1.I 7) and (I 13). The specific purpose of this work, therefore, will be to derive explicit qualitative, order of magnitude bounds. from both above and below (in terms of the basicconstitutive parameters appearing in the model) for such basic, experimentally determinate quantities associated with plane Poiseuille flow as mean velocity, net volume flow, and friction factor or drag. For an isothermal, incompressible, dipolar viscous fluid the constitutive theory formulated in ref. [2] assumes the form
ri,k
(l.2)
=
where the quantities denoted above have the following interpretation: (ui). i = 1.1, 3 are a set of rectangular Eulerian coordinates, ~~~and Tijk are, respectively, the components of the (usual) stress tensor and the first multipolar stress tensor, p is the pressure, 6,, the Kronecker delta, &,/l”./ll > 0, and r,O < u < 1, are constitutive constants, and
are the components of the rate of deformation tensor, where the L’,, i = 1. 2. 3 are the components of the velocity vector. As is standard by now. we sum on repeated indices ’ Research supported. in part, by ONR Contributed by K. R. Rajagopal. Nlfl 28:5-B
Grant
No. NW?O!~-~I-J-ICO?.
SO?
H. BFLLOUT and F
504
BI.OOM
wherever they appear, e.g. in equation (1.1). For x = jj, = 0. and jj” = j1,). the usual viscosity coefficient. equations (1.1) and (1.2) reduce to the well-known constitutive relation which underlies the Navier-Stokes equations for an isothermal, incompressible viscous tluid; the constitutive relations (1.1) and (1.2) go beyond the Stokes theory, therefore, in three significant ways: the presence of a non-linear relation for the viscosity, i.e. jl = jLO(&+ elrlclrl) -l.z.
( 1.4)
the presence of higher-order spatial derivatives of velocity, and the presence of an explicit constitutive relation for the first multipolar stress tensor rijk which, as is shown in refs [ 1, 31. enters into the balance of energy equation but not into the relation expressing balance of momentum. In earlier work, Necas and his co-workers considered [7-lo] a linear version ofequations ( 1.I) and ( 1.2) for a compressible dipolar viscous fluid, namely sij = - pb,, + Lo6,,div v + 2j~+~,, - i., iiij Adiv v - 2j1,Aci,
(1.5)
7
(
Ti,t
=
j.oSijI'l,la
eij
2/11i’x,
+
and proved a number of existence theorems for the relevant associated boundary-value problems in bounded domains; extensions to the case of a compressible, heat conductive multipolar fluid were also made but. also, within the domain of a linear constitutive theory similar to that expressed by equations (1.5) and (1.6). When equations (1 .l) and (1.2) hold in a domain R c R3 the associated initial-boundaryvalue problem assumes the form (we take the external body force to be identically equal to zero): ,>;+vVv=
-Vp+2V.(j1e)-2j1,V+(Ae) divv = 0
in R x [0, T),
T > 0,with jr given by equation
(1.7) (1.X)
(1.4)
v(x, 0) = ro(x)
in R
(1.9)
and v = 0,
Tij~vj~,Ti
=
0,
on
?R x [O, T).
(1.10)
In equations (1.7)(1.10), p is the constant density, the non-linear viscosity is given by equation (1.4), while the first boundary condition in equation (1.10) is the familiar expression of the fact that the tluid adheres to the boundary and the second set of boundary conditions (with v the exterior unit normal to LQ) are a consequence of the principle of virtual work and express the fact that the hypertractions (moments) vanish on i?R, e.g. ref. [3]. When j1’ = jco (usual viscosity), x = j1, = 0, equations (1.7))( 1.10) reduce to the standard form of the initial-boundary-value problem for the Navier-Stokes equations in R x [IO, T). Among the special cases of equations (1.7)-( 1.10) considered in ref. [2] are the classes of plane equilibrium Poiseuille flows (between parallel plates) which have the form 1!, = t.,(_X~), L’2= 0,
L‘J= 0,
(1.1 I)
with the fixed plates located at s, = fa. for some N > 0. For the sake of simplicity of notation. and because in all that follows here we will consider, exclusively, flows of the form given by equation (1.1 I), we will set (xr ,.xz,x3) = (xY,:). and r, = I‘. so that v = [~‘(~),0,0]: it then follows quite easily that equations (I .7)-( I, IO) reduce to the non-linear boundaryvalue problem: (1.1’) j1O([‘: + :r”(_r)]-‘zL.‘(?‘))‘j’,‘:““(y) = p,, L‘(2 a) = 0,
f”( *Cl) = 0.
(1.13)
where we have absorbed a factor of two into the constant pressure gradient p, = ?p,ic’.t,. Several problems associated with the boundary-value problem of equations (I. 12) and ( 1.13) were addressed in ref. [2]. First of all. if r = j1, = 0. j1’ = jco then the second boundary
Steady plane Poiseuille
flows of incompressible
multipolar
fluids
505
(cl Fig. I.
condition
does not appear
in equation
(1.13) while equation hU”(Y)
(1.14)
= Pl
associated with Navier-Stokes; the integration yields the familiar parabolic profiles
of equation
-;[I -($I,
u(y)=
(1.12) reduces to the familiar
(1.14) subject
to u( +a) = 0
(1.15)
-a
which have the property of both blowing up and remaining parabolic as /co + O+, thus indicating the lack of utility of the Navier-Stokes in the case of vanishing viscosity and the need to implement something like the Prandtl boundary-layer theory [l l] in the laminar flow domain. In fact, for plane (laminar) Poiseuille flows it is well know experimentally, and predicted by the boundary-layer theory, that velocity profiles will flatten out in a manner similar to that depicted in Fig. 1, at least until the breakdown of laminar flow and the onset of turbulence.’ In ref. [2], we set a = a, = (n - 1)/n and integrated equations (1.12) and (1.13) in the special case in which E = p, = 0; denoting the corresponding profiles by u,(y) the relevant boundary-value problem reduces to one of the form (1.16)
2’“- ‘“*oh(y) = IL-Csgn g(y)l”g”(y),
(1.17)
U”(I!Ia) = 0, with g(y) =
y. As solutions
of equations
(1.16) and (1.17) we may exhibit, for example, y < 0,
y > 0.
(1.18)
(1.19) The profiles u.(y) exhibit, as n + + cc (i.e. as a -+ 1 -), the successive flattening (expected as the fluid becomes less viscous) which is depicted in Fig. 1. With E = pI = 0, the fourth-order boundary-value problem of equations (1.12) and (1.13) reduces to the second-order problem ~“~C~~‘2(Y)l-a’2~‘(Y)J’
= PI>
c(&a) = 0 ‘All of our work in this paper specifically assumes that the flows remain in the laminar a non-linear hydrodynamic stability theory is not relevant at this stage.
(1.20) (1.21) regime, so that
506
H
BELLOCT and F BIIX)U
and it would seem, a priori, that there is no firm basis for relating the solutions of equations (1.20) and (1.21) to those ofequations (1.12) and (1.13). i.e. that the profiles r,(y) need not be close, in any precise sense to those predicted by the full model, and that a boundary-layer approach must be employed if we wish to let !l, 4 0’ in equation (1.12) and drop the second set of boundary conditions in equation (1.13): however, this is. emphatically. not the case if we are only interested in continuous dependence up to a certain order, namely C’. so that both the velocity and velocity gradients predicted by equations (1.20) and (1.2 1) will be close to those predicted by equations (1.12) and (1.13) in the L ’ norm on [-(1. a]. In ref. [2, Section 61 it is proven that there exists a unique solution of the boundary-value problem of equations (1.I 2) and (I .13) in the space Lt’,,, = B,,,(O)
n H 'I-II, a).
( I.22)
where B,,(O). AI > 0, is the ball of radius :U in the space I’= Hi"+"(--I. u) with 0 < 6 < l/2. The existence and uniqueness result cited above is established by use of the Schauder fixed-point theorem. Moreover, through the development of appropriate sets of 3 priori estimates for the soiution of equations (1 .I 2) and (1.13) it is also proven in ref. [2. Section 61 that r(.;~,l(,)-+
rO(.)
inC’+‘.
O
asc,fl,
-+ O+,
(1.23)
where (‘(.;E,/(,) denotes the unique solution of equations (1.12) and (1.13) while I’,~(.) denotes the corresponding solution of equations (1.20) and (I .2 1). In the course of proving the result of equation (I .23), it is established that the unique solution of equations (1.12) and (1.13) satisfies estimates of the form
J --(1 and
(1.25) with C, > 0, i = 1,2 independent of both E and /l , : estimates (1.24) and (1.25) will be used in the sequel here. In this paper we continue the study of plane equilibrium Poiseuille flows of incompressible, isothermal. dipolar fluids initiated in ref. [‘_I. Through the use of dimensionai analysis applied to equation (1.12). we isolate the natural counterparts of Reynolds number associated with the Navier-Stokes theory. We then investigate, in greater detail than was done in ref. [Z], properties of the solutions c.,,(.) of equations (1.20) and (1.21) and use these solutions to compute the associated mean velocity. maximum velocity, volume tlow. and pressure drop. Finally, although the continuous dependence result for solutions of equations (1.12)and (1.13) in Cl+‘, 0 < 6 < 4. as 6, jr, 9 O’, was established in ref. [2]. precise estimates of the errors incurred by setting E = IL, = 0 and using, in place of I’(. : 6:.11,) the solutions rlo( .) of equations (1.20) and (1.21). were not presented there; such estimates arc derived here and are subsequently employed to establish the related estimates for the volume flow, etc. It is hoped (and expected) that such estimates will eventually serve as a guide in the formulation of experiments directed at the determination of the constitutive constants in the model; these estimates have also been used in recent work [12-141. involving various studies of existence uniqueness. and asymptotic stability of flows of multipolar fluids.
2. GENERALIZED
REYNOLDS
NUMBERS
In this section we briefly indicate the appropriate form which a dimensionless version of the evolution equation associated with equation f 1.12) assumes, and, in the process. are led to the definition of generalized Reynolds numbers that are connected with plane Poiseuillc flows of incompressible dipolar fluids. Employing a standard analysis, we set (2.1)
Steady plane Poiseuille
flows of incompressible
multipolar
fluids
507
and (2.2) in the evolution
equation
for plane
Poiseuille
flow of an incompressible
dipolar
fluid.’ (2.3)
Here V is a measure of mean- or far-field velocity associated Elementary calculations using the chain rule show that Sl
vz ?I-
_--
---
sj-
I3 -=_21’
(2.4a)
a 3’
Z?p
with the flow and L’= u(y, t).
pV2(‘j a
(2.4b)
?_c’
V&T
(2.4~)
a l?y
and
with $
= A$_; substitution
PV -=--
v
PF
?p4
f14
s_f’
(2.4d)
of the above results into equation
(2.3) then yields
i”:j~~~~~~~+“I:~~,~+%(~~]-.;!‘)~}_~~,
which becomes,
after multiplication i?v -_= 2T
(2.5)
by 4,
and setting
1o=!$,&
-5+~~1,~~~(3’]-.~2~~~~~.
In as much as E must have the dimension
(2.6)
of a velocity
gradient
squared,
E = ca2/V2 is dimensionless; (2.6) as
using the definition
of equation
(2.7) (2.7) for .? we may now rewrite equation
where we have dropped the superposed bars from J, t,p, t‘, and E. The dimensionless version, given by equation (2.8) of the evolution equation (2.3), leads naturally to the definition of the following two generalized Reynolds numbers that are associated with plane Poiseuille flows of dipolar viscous fluids: VI+1 R”’ = R, = c. (2.9) 0 “Oa’1’ ”1 Using the definitions
(2.9), the evolution
equation
for c(y, t) assumes
St is=-$+-g{[&+(~)y2$}+$.
the form (2.10)
For tl = pr = 0, clearly Rr’ = Vu/v0 and R; t = 0, so that equation (2.10) reduces to the standard dimensionless form for plane Poiseuille flow within the context of the Navier-Stokes formulation, with Rr’ being the usual Reynolds number, if p” = /lo. ’ Note that for a = 0. JI, = 0 equation (2.3) reduces to the equation predicted by the Stokes constitutive law, while for z = I, E = 0, JI, = 0 it reduces IO the equation for channel Row of an incompressible inviscid fluid.
H.
508 3. THE
BELLOUT
POISEUILLE
F BLOOM
and
FLOW
FOR
& = /i,
= 0
In this section we study, in greater detail than was done in ref. [2]. consequences of setting I: = 11, = 0 in equations (1.12) and (1.13), i.e. we look carefully at certain aspects of the qualitative behavior of solutions to(y) of /P{ [:
L$
(y)]
-‘fZL.b(?‘))’
=
(3.1)
PI,
L‘J + a) = 0.
(3.2)
In the next section our interest will be in obtaining estimates which relate the behavior of solutions o,(y) of equations (3.1) and (3.2) to those of the solution L’(~;E,/L~) of equations (1.12) and (1.13); the quantities of particular interest to us will be the volume flow
tf.v;c,p,)dy,
(3.3)
Qo.o = Qo = ’ u,(.dd) s -*
(3.4)
which for E = jc, = 0 has the form
the mean velocity
t (3.5)
and its counterpart
for I: = /i, = 0. i.e. Lo = -
I
Q.
(3.6)
2a and the friction
factors
fo= 4r,,(f&O>O) 4PG
(3.7)
where T, 1( 2 a, c, 11,) is the shear stress at the walls located r,2(+a,F*P,)
at y = 5 a, i.e.
“‘~L”(_fa,E,~I,)-~1,t!“‘(~a,E,~~,).
= /‘O[E + hU’2(fa,E,/1,)J-
(3.8)
We begin by noting that if oo(y) is a solution of the boundary-value problem of equations (3.1) and (3.2) then so is uo(-y) and, thus, by uniqueness of solutions go(y) = am, - a I y I a; from this result it follows that L&(Y)= - ob( - y), so that t&(O) = 0. Moreover, with pt. the constant pressure gradient, negative, a first integration of equation (3.1) yields PO c:
-“21;b(y)= - [p, ly 412(Y)l
where the constant of integration vanishes From equation (3.9) it is immediate that
(3.9)
in view of the fact that 0 < z < 1 and t&(O) = 0.
t!b(y) > 0,
y E [-a,@,
t&(y) < 0,
y E ux~l.
(3.10)
In as much as t&(y) I 0, for y E (0, a), we have v;(y) = if we set c, = j1’2”‘, equation (3.9) becomes on [0, a],
J vb(y)l,
IL,’ (y)\’ -= = lp’ly 0 c,
0 I y I a, and, therefore,
(3.1 I)
or 1rb(y)I = Czy’!(’ -1’. l/l1
0 I y 5 a,
(3.12)
-1)
. We rewrite equation
(3.12) as
rb(l’) = - C,y’“‘-~‘.
0
5
I’
5
a
(3.13)
Steady plane Potseuille
and integrate
flows of incompressible
multipolar
fluids
509
from a to y obtaining (3.14)
In computing
equation (3.14) we have used, of course, the boundary for 7 in equation (3.14) and noting that L’~(J) = ~‘e(-I’),
Substituting
condition we have
~~(a) = 0.
(3.15) where (1, = c,
1-Z 2-x
( > IPII
=
($2
- !N( I - 2)
‘~~‘-=)~(2-.,,(,-*)
1- 2 2-z (4
H c,
=(s)( It is clear. from equation
!e!&J’~“-z’.
(3.16)
(3.15), that (3.12)
By direct
calculation,
the mean velocity
I CO= -2a
O0 associated
with
a _
vo(y) is obtained
as
dy)dy
s u
(3.18) Carrying
out the integration
in equation
(3.18) we are led to (3.19)
or, in view of equation
(3.16) I‘o = (!&)(Q5!&)“~‘-z’
(3.20)
We note that Cal,=
0
=! .-IPIlU2 3
/P
which is the classical result associated with Navier-Stokes. it follows that ]im rtaX = lim d, = lim I-Iz-1 1-r However.
in view of equation lim I-I-
from which
(3.21)
’
From equations PO.
(3.22)
(3.16)
ff, = *_m,[(~){$&)“““] Ii
it is clear that the critical
quantity
(3.23) in computing
lim,_,
- craK is
c _ IPI laz-m I l‘op . Specifically,
if ez > I, for I sufficiently
that we set I’=
(3.17) and (3.19)
L.oI,=~in equation
(3.24)
close to 1, then I$‘~’ --* cc as z + (2.9), so that jp,l = pV’/u,
1 -. Suppose, now.
then (3.25)
BtLI.OIT and F.
H.
510
Bi.oov
while lim I_,
R”’ _ !Y. 0 (1 I’
Thus, l.Fd’ +
%
,
asr-,
(3.26)
I-
provided
R;’
lim I_, To
emphasize
parameter the lim,,
further
the role of the criteria
r. in situations
,
rb(u). From
involving
(3.27).
small
equations
(3.15)
(3.17)
> ;j. and its connection
physical
viscosity
and (3.19),
with
the status
of the
(i.e. CYclose to I), we compute
we have
(3.28) so that
and, thus,
I l.;)(fI) = (3 - 2x) Therefore.
by virtue
of equations
= _
(3.20)
I
_
(1 - z)
(3.30)
I‘,)
and (3.24).
(,i (’ -I’=
Iin1
- -~
_
(3.3 I)
y_
*I
I
. , RI:’ > L i._, under these same conditions
if lim,
lim so that
if liml
indicated
= lim-.
r;)( -a)
,
[-r;,(u)]
(3.32)
= + x .
Rif’ > x!2._ and r is close to I, the velocity
_,
in Fig.
~.,
profile
assumes
the form
2.
In fact, not only
is V;,(O) = 0, but the rapid flattening
respect to the axis J’ = 0. US x -+ and the companion
result
for
-
I
of the profile,
_ is easily demonstrated (I I
J’ I
as follows:
depicted in Fig.
2, with
from equation
(3.29),
0. (3.33)
so that r:(O)
= 0. for all r. 0 < r <
I. Then (3.34)
so that I’;;‘(O) = 0, for I > i. A further clear
that
rn suffkiently
by an induction close to 1.
computation
argument
we may
shows show
that r,;;“(O) = 0. for r > 5. and it is that
r:‘(O)
= 0. z > 2,.
for
some
Steady plane Poiseuille flows of incompressible
From (3.4),
equation
(3.20)
it follows
directly
multipolar
that the volume
flow
fluids
511
Qo, as given by equation
is
(3.35) Also,
from
equation
(3.20).
it is a simple Ip,I
Now,
by equation
(1.20),
z
=$&
r,,(-n,O,O)
(3.7),
to compute
3-22
that
l.,-,
( I-r0
(3.36)
1
= IP~ 1~. or 0212 3= & I_r n (
s,~(-CJ,o,o) SO that, by equation
matter
the friction
factor
‘r 1
_,_l L’o
(3.37)
*
is (3.38)
Also,
in view of equation
(3.36)
and the fact that (I > 1
(3.39) estimate
:tn
which
constitutive pressure
may
parameters
gradient
be useful /lo, r,
r(y;e./~,) and
approximating
p, and the mean velocity
4. TfiE
this
In
in
based on careful
section
of equations
(1.12)
those
(1.2 I), which are missing
such as equations
(3.38)
FLOW
precise
and (l.l3).’ from
and (3.39)
experimentally,
of the
of the magnitude
of the
Co.
POISEUILLE
we provide
the values,
measurements
FOR
qualitative
relative
8, [tl
to the solution
our original
uork
quite
We begin with
useful.
\V(!‘; 2:. 11, )
#
0
estimates
for ad
in ref. [2],
the unique
and which some
solution
of equations render
notation,
i.e. we set (4.la)
= r’( !‘; I:, /iI ).
Gj?‘;c) = \v(y;r:,O), :(I’;r:./(,) Z(y;c)
(1.20) results
(4.1 b)
= I: + IW(.KE, p1)12,
= z(?‘:I:,o)
(4fc)
= 1: + I\i(y;I:))’
(4.
Id)
and Y(K) = (C + ],v12))z When
the interpretation
e.g. I(C) = Z(y;c). equation
-
[ -CI,N]
Our
ro(y)
from
quantities
boundary-value will
we will
suppress
such
problem be obtained
goal both
is
below
as QL,tiI. of equations from
to
(4.2)
the explicit
We also note that Gl(y:O) = r’(!.;O,O)
(3.29).
= r(_r;c,~~,) estimate
is obvious
2)(. = Z~‘ZIV.
estimate
dependence on y and write,
= L’;(Y)
the
which
difference
is given explicitly ~(y;e,
p,)
and above and to then use the resulting
the volume
flow
associated
with
-
by
r(y;O,O)
bounds
the solution
to
of the
( I. 12) and ( I. 13). The bounds for o(y; E, p, ) - co(y) on
a series
of subestimates
that
result
from
a succession
of
lemmas. Lemttm 4.1.
Let L’(J; E,I(~)
and rO(y) the unique such that on [-cr.
solution
be the unique of equations
classical
solution
of equations
(1.20) and (1.21). Then
3K,,
(1.12)
depending
and (1.13) only on IL,
a] )r’(J’;e,O)
-
r;(y)\
’ In future work [15] we plan to studyequations(l.12)and of singular perturbation theory.
<(I
+ K&/G.
(4.3)
(1.13) both numerically as well as from the viewpoint
H.
512
BELLOUT
and F. B~oow
Proof: We set !L, = 0, in equations (1.12) and (1.13). divide through by p” # 0, and let p: = p, 111 ‘; then ~.(y;s.O) is the solution of the boundary-value problem [(E + ILi.(~;E)IZ)-I.21;.(L);&)]’ = p:, L.( _to;c,O)
(4.4,)
-a
(4.4b)
= 0,
where we have used the definitions (4.la) and (4.lb). Now if K(~;E,A~,) is a solution of equations (1.12) and (I. 13) so is L’(- y; E. 11,) for any e, p, 2 0; by uniqueness of solutions to the boundary-value problem we must have c(y; 6,~~) = L‘(-y; ~11,) from which it follows that L”(I’; E,11,) = - r’( - y; E,11,); -a
~‘(0; E,/( r ) = 0 for all E, ,n, 2 0. Integration
of equation
(I: + I~;.(!.;E)1*)~I’2)i.(~;E) = p:y,
-U
as G*(O:I:) = t.‘(O;~.,0) = 0. It then follows from equation k(y; E) # 0, as 0 < z < I. Squaring
both sides of equation 3(y;C)-=\i’*(y;F)
We rewrite equation
(4.4a) leads, therefore,
vc 2 0,
(4.6)
(4.6) that 4’ # 0
(4.7)
(4.6) and using the definition
= p:zJJ,
(4.8) in the form [recall
(4.ld), we obtain
- a < J’ < N.
that Z(c) = 1(y;~).
(4.8)
- a 2 y 5 U]
I(E)-QII(E) - F] = p:QJ or 3(I:)‘-= - r::(c)-’ If we now differentiate
equation
- a < r’ < u.
(4.9) with respect to I: we obtain,
:(E)-‘[( where 1, = -4 ?(y;c).
= p:zy2,
I - z)Z, - 1 + &(c)-‘2,]
We now restrict our attention L!(E)= r:+ ~U’()‘;E,0~2 #O.
it follows from equation
(4.9)
after a simple calculation,
= 0,
(3.10)
to the set of all y E (--a. 0). As
ve 20,
I’E(-(1.0)
(4.1 1)
(4.10) that
(1 -r)l,-
I +ZEI(E)-l:,=O,
r:>o.
yE(-l,kO)
(4.12)
in which case we find that (4.13) As a direct consequence
of equation
(4.13) we see that
0 2 &(y;s,O) I j& Now, for y E (-a,
E20,
L’E(-u,O).
(4.14)
0) we may write that r i(y;s)
= I(y; 0) +
i,(y;i.)di..
(4.15)
I0 Combining
equations
(4.14) and (4.15). we then have Oii(y:f.)-f(y;O)~~~;r:2O.
where we have used the continuity
yE[--,O),
(4.16)
of c’(y; E,/[,) to extend the result to _V= - a. However,
1(0; 0) = I,i(O;O)l’ = L2(O; 0,O) = 0. :(O;E) = E + li(O;c)l’, = I: + /r~‘(O;E,0))2. = 6.
(4.17)
Steady plane Poiseuille
so 1(0; E) -
_;(O;O) = E < &,
Now, equation
for
flows of incompressible
multipolar
0 < z < 1, and, thus, equation
(4.16) is equivalent
fluids
513
(4.16) also holds at y = 0.
to
YEC--a.01
0 I [E + r;2(y;&)] - kZ(y;O) I & Or -&I
bv(y:&)-
i2(y;O)l
ea,
yE[-a01,
(4.18)
which, in turn, yields the two estimates GJ2(y:O) - E I &2(y;E),
(4.19a)
e
(4.19b)
G2(y:&) < k'(y:O)
+
on [--a, 01. Consider the set of all y E [--a, 0] such that G(y:O) 2 4 y in this set it follows from equation (4.19) that 0
with the upper bound
5
Q*(y:&)
holding,
<
i2(y:O) +
-!Y!- yEC--701 1 - a’ i $(y:O) 2
of course, on all of [--a,
for fixed E > 0; for
(4.20)
4’
01. Therefore,
for all y E [ -
N.
01.
such that G(y;O) L ,,& 0
2
G(y;&)
5
J-z-
G(y:O) +
(4.2 1)
l-a
and we have used the fact that equation
(4.6), with p: < 0, implies that r;(y;c) > 0, V,: # 0,
y E [-a, have
0] but k(y:O) < 4;
0). Now, suppose
that y E [-a,
h2(y:c)
with Kz = l/(1 - a). Thus, if y E [--a,
<
ae & +
-
=
1-z
then by equation
Kit,
(4.19b), we
(4.22)
0] and G(y: 0) < ,,& then ko):~) < K,&
(4.23)
and ) G( y: E) - q y;O) I I qy; E) + G( y: 0) < (1 + K,)$
or A(y:O) - (1 + K,)& for all y E [--a,
< k(y:c)
0] such that G(y;O) < 4.
< G(y;O) + (1 + K,)&
(4.24)
However,
K.=/G>Ea, so a comparison of equations (4.21) and (4.24) shows that equation (4.24) holds for all y E [-a, 01. Using the definitions of G(y;e), G(y; 0) we may rewrite equation (4.24) as lu’(y:~,O)
where K, = ~ A.
Replacing,
-
d,(y)1 < (1 + Q/k
y by -y,
y E C-a,
01,
for y E [O, a], we see that equation
(4.25) (4.25) holds for
all y, -a I y 5 u, as both u’(y: E,O) and t&(y) are odd functions on [-a. a]; this establishes the validity of the estimate (4.3) and concludes the proof of the lemma. Cl Lemma 4.1 enables us to compare C(Y:E, 0) with uO(y) on [--a, a]; our next set of lemmas are aimed at enabling us to compare u( y: E,p, ) with u( y; E,0), the first of these being stated as follows.
H BELLWT and F 81 OOM
514 Lcrnm~
Let I’( J’: K. jl,
4.2.
) be the unique classical solution of equations ( I. 12) and ( I 13) I = I.“‘( J’: t:, p, ). Then 3C + . C _ > 0. independent of X, ji, such th:,t
and set r( \‘:I:. j,,
r(!,;i:.jl,)
I
C..
!‘E
I ( j’: 1:.p, 1 2 - c - . Pr(j(J:
\i’e will
proof
establish
of equation
that
,‘(!‘:)A Jo, 1 and
differential
only
(4.26b)
equation
follows
(426a).
I‘(>‘: r:.O) are,
J’ E [o.
which
in an entirely respectively.
is all that is needed in the sequel;
to r( +o:r:.
ing equation
the solutions
the
We begin by recalling
of the
/I, ) = r”( *KJY.~~,)
(4.27b)
from
future
refcrcnce
non-linear
Ordinary
(‘( is I.‘( !‘:~.j1,
an even function:
(4.27~)
the definition
(4.2) enables
,;.‘()
cJ; +
on [
Vi:, jl,
.:,: ),‘?
with
= U, /j,“. Subtract-
respect
to J. we obtain
-LJ.
equation
of all, as
J’ E [ - (1.Cl]. but so is 1.“’ (J:
(I]
(4.X)
Il:\(“‘(!‘:f:.jJ, 1.
=
first
2 0, ,““(O;r:.ji,
us to write
) = 0. Next
(4.17a)
i:, 111 )]’ - jt: r”“(
;‘[I“(?‘:
(4.27b)
\i,( y: I:)
) = c( !‘: L jt, ).
) ~111 odd function
(327a)
p:,
and integrating
here the following: 1’;t:. j’,
in particular,
' ~r;(y:r:);'=
= r,:,
= 0 and r( &tr: i:,O) = 0. where ji:
equation
we record
j!:“““(J’;E.j1,)
\F(y:c)]
bi’(j’: 1:.ji , ) [t: + \‘.+ .:,;, j1,)]1’2 -
not only
fashion.
equations
([I:+
For
(4.Xb)
JJ].
analogous
I[(: + “.?(!.:i:.j’,)]~IZ’\.(?‘;‘:,j’,))’-
subject
(426a)
[I-lJ.01.
I:. /J,
).
while
we observe
I.“( J’: I:, ,H,
) is
that the USC of
in the form
) = pT
(4.29)
+ \V’)_ ‘1
(4.30)
r’: 1:. ;Jl
and that ;“(\v) = (1: + +~“[I
;,“(,,.)
=
_
% (,; + ,,.‘)I’ 21t I
\c
~_
i from
which
from
equation
it follows.
as 0 < x <
(4.29)
-
L
1,2(I:
%)WZ
(1 -
l: + -___
3%: (?;+ \,2)? + (2 2)
+
i: + \\.?
1
(4.3
I)
i
I, that ;s’(,t,) > 0. Vr: > 0, while
sgn Y”(,c) =
_ sgn 1,‘. Now.
with s( J’; 1:. jI,
(4.32)
) = r”( j’: I:, /I, ),
we have ;“(r,‘(Y;r:,j(,
Suppose .s(
\‘(I;r:.
that
/I,)
>
s(~‘;r:,ji,)
0. From
)).s(.Kr:.j~,)
-
S( - (1; I:, /I,
)=
takes
equation
But ;“(,.‘(Y~;E./I,))
any
> 0. while
there. Thus. positive
boundary
.s( I‘: ~,j’,
maximum
conditions
S(CJ; I:. I’,
a positive
(4.33)
(4.33)
< 0.
(4.34)
0.
at
some
y0 E
(-((1.
(J).
so
that
(4.35)
+ I/J:1 = jI:.s”(~O;i:,jI,).
s”(~~, ‘I:, jl,)
of equaiion
) =
maximum
I
0, if \‘D is interior
) cannot achieve ;t positive
of s( ~:c,jl,)
= II:
we have
;“[,.‘(.o:r:.j,,)].s(‘,,:r:,j,,)
imum
j,:r”(JYr:,j(,)
must,
(4.34)
therefore.
it follows
to [-a.
maximum
that
(J] and
at a point
.S has
j’O E
occur
at .\’=
&-a. In
there
is no positive
;I mar-
( - ~1,(J) view
;iJld
of the
max,mum
for
.S(J’; &jl,
1 anywhere on [ -cJ, ~1: thus. if the maximum of s( J’; E, ji, ) occurs at an interior point y0 E ( -(J. (I) we must have A(J~;s, jl,) < 0 in which case .S(J’; C,j’ ,
and the same result (4.35).
.s(),: J:./‘~
holds
) Cannot
) _<.S(j’“: l:, /‘, ) < 0. J’ E (- il,
if the maximum have a zero l“‘(?‘:i:.j’,)<
which
shows
that the graph
occurs
maximum
of I’( j’; E, j(,
at J’ =
~CJ.
where s vanishes.
at an interior
O;xE(--cl.(I).
c.j1,
(4.36)
(I)
point
)‘0 E
>o.
) is concave (down) on (- LI,a).
(--(I,
By equation (J).
Thus. (4.37)
Steady plane Pokseuille Rows of lncompresslblr
multipcrldr tluld\
515
Now let y E (--a. S) for any 6 I a. Then Y r”‘(kc,p,)di.
j -(1 and as y may
be chosen
(--a. a)] it follows
arbitrarily
= c”(y;c,jj,)
close
to
--a
[and
(4.38)
,““(y;~jj,)
is continuous
in J on
that V”‘(--_a;&. jj,) c”‘(U; E, jj,
since r”‘(_r;c,jj,)
< 0
is an odd function
of y on
< 0
(4.39a) (4.39b)
)> 0 From
(--a,~).
the definitions
of s( ,r:~,jj,)
and
t(.r: E. jj, ), t(~:F,jI*)=S’(1’:E,jI,), Therefore,
if we differentiate
equation
i”[r.‘(~;c,j(,)]f(_~:E,j(,)+ The
calculation
argument:
in
equations classical
[Z]
to equation
it was
(4.33) and (4.34), solution.
(4.33)
with
respect
./“[u’(v;E,~,)][~“(~:E,j(,)]~
leading
ref.
_,‘E(-0.0).
(4.41)
to y we readily -
that
subject to the additional
obtain
jc:r”(_V:E,jj,)
may be validated
demonstrated
i.e. a solution
(4.40)
by the following
the boundary-value constraint
(4.41)
= 0.
elementary
problem
given
by
I*( + (J, E, jj, ) = 0 has a unique
in C4( --(I,
a); in light of this observation,
= &:I
+ i”[L.‘()‘:E.jj,)].S(.KE.jI,))
and the definition
of ;*. S”(_Kc,jj,) is continuously argument
differentiable
shows,
is, in fact, in C
’ (-a,
We now return at ~9~E
in y on
classical
(4.41) and assume
by equation
tion
Repetition
of this problem
) achieves a positive maximum
(4.41)
it must
be true that (4.42)
IO.
(4.3 I)
< 0, y~(-an,
a), while
~0 E (0. (I) then by equation ,‘o E
that r(y; s. jt,
+ Y”[V’(J o:c,jj,)][L.“[.“;1:,jj,)]2
sgn~“[L(yO;c,jf,)] ,“‘(.,‘;c,jj,)
holds.
of the boundary-value
5 0; then by equation
;“[~“(?‘o;c,j1,)]l(yo;c,jI,)
But
(4.41)
solution
(I).
to equation
(-cl, (I) so that 1“ (yo;c,jj,)
However,
a) and equation
(-0,
in fact. that the unique
(4.43).
= -
r’(O;c,jj,)
we must
sgnr’(yO;>:,
jj,).
= 0, so r’(y;r:,jj,)
(4.43) < 0 for y~(O,n).
have y”[c’(y,:~:,jj,
Thus,
)] > 0. contradicting
if
equa-
(4.42).
This means, of course, that if f(y;c,jj, ) achieves a positive maximum at N, (I) then, in fact. y. E (--a, 0); note that t(O;c,jj,) = 0 as ~““(y;c,jj, ) is odd on (--a,
( -
N). At such a _ro E (-a,
0) we will
have, by virtue
t(yo;E, j(I) I Now, a t _rO, t’(~,;c,jj,)
-
of equation
~“Cr;‘(yo;E,jII)][~“(yO;~,j~,)]Z ;“Cr!‘(yo;E,jl,
= s”(y,;c,jj,)
(4.42)
= 0, in which case it follows
;‘[u’(y,;c,j(,)]v”(L’o;E,
jj,)
(4.44)
,]
from equation
(4 .33) that
= p: < 0
(4.45)
so that c”(y,;c,jj,) Substituting
from
equation
(4.46)
into
= p:/;“[l:‘(Yo;E,j(,)]. equation
t(y,;E,p*) I Employing follows
the pointwise
from
equations
bound (4.47)
we obtain
;“‘[c’(.vo;E,jI) 1-J ;.‘3[r’(_vo;c,jt1)]
-
of equation
and (4.39a)
(4.44)
(4.46)
(4.47)
pT’.
(1.25) of ref. [2] relative to 1L.‘(J; E,p, ) 1it now
that 3C+
> 0. independent
of both I: and jj,,
such
that f(‘;E,j(,)~f(~O;E,j(,)IC+, If r(y; c, jil ) I equation
(4.48)
0 on (-a, holds
0) so that no positive
t/C+ > 0. An analogous
(4.38)
JE[-KO]. maximum argument,
exists which
on [-(I,
0] then certainly
begins with
the assumption
516
H. BELLOUT and
F.
BLCOM
that t(y;e,l~,) has a negtive minimum on (--a, a). can be used, as above, existence of a C. > 0. independent of both E and ii,, such that t(y;c,p,)> but we omit the details.
the
(4.49)
-C-,yE[O,a],
0
The next lemma provides us with an upper specifically the following result.
bound
for c(y;~,/l,
Lcmn~o 4.3. Let L’(p; E,it, ) be the unique classical solution R. 11, > 0. and all ye [--(~,a],
UC+
L’(y;E,p,) - C(y;E,O)I
) - c(y;e, 0); we have,
of (1.12) and (1.13). Then for all
(4 + c,)lil:,
j-y
where C,, CL, independent of both E and /i,. are the positive constants tively, in equations (4.26a) and (1.25). Proc$
to establish
From equation
(4.28) and the definitions
appearing.
of y( .) and r(y;s,p,)
respec-
we have (4.5 1)
However, with k(y;c) for each fixed y E [-a,
I w(y;E,/(,)
I \V(y;c,jc,)
(4.52)
a], E, /1, > 0. Thus, \1.(I’;
)-
E, p,
I’:f(y;Gfc,)
k( y; E) =
(4.53)
fc4y;LP,)lFrom equation
(4.30) f(w)
=
-r)w2
E+(I (E
+
,vz)‘+(~!21’
so I
(,; +
---= y’(W)
,72)1
+lzI2)
(E +
W’)’
+(x,2) = J-(E
i: +
(I - sc)G2
<
(I
-
a)c
+
(1 -
+
\,2)I’2
r)G2
I &(,:;:+lW&(~+c2)= by virtue of equations (1.25) and (4.52). Employing this last estimate for [y’(E)] _ r, as well as equation (426a). in equation (4.53) we are led to the upper bound w(y;c,p,) - \t(y;&) 5
G(& +c,y,
)‘E[-(LO).
Choosing y E (-u, 01, and integrating both sides of equation (4.54) from -a to y, we have. in view of the definitions of w.(y; E,11,) and k( y; E) and the fact that r( -a; E,11,) = 0. for t: > 0 and /L, 2 0: c(y;E,/1,)
- c(y;c,O) I
2
(JE
+ c,)z/1:.
y 6 C-a,
However, r.(y;E,fl,)= c(-yy;~,~,), for y~[--,a] so we see that equation (4.55) is, in fact, valid for all y, -(I I y I II. Our next lemma is as follows. ,!_ernrtrcr4.4. Let L.(y; 6, I(, ) be the unique Thenfor --N<~
classical
- rl(y;E,o)l I
__
solution
01.
the upper
of equations
(4.55) bound
in
(I .I 2) and (1.I 3).
(4.56)
Steady plane Poiseullle Rows of incomprewblc
Prooj.
We return
to equation
(4.53); for YE [ -
multipohr
[w(%;E, pI) - k(i;s)]di
1
_~ ~‘[W(j.;s.~,)]
-a
517
a, 0) it follows that Y
Y
fluids
t(A;c,/i,)dj.
(4.57)
from which, we obtain It’(y;E,P,) by again bounding
- c(y;E,O)I I $(&
[r’(G)]-’
3
(4.58)
+ C,Y lY It(E.;e,/c,)ldl --(1
from above and using the definitions
of w and G. However,
However, by virtue of the estimate (1.24) of ref. [2], there exists a positive constant, independent of both E and p,, which we will denote by C, (even though we have replaced I’, by 11: = /(,/ll”) such that
and we are led to the bound 11:
’ ItIdE. I (a~c:Cr)“*. -*
(4.59)
Use of the estimate (4.59) in equation (4.58) now yields equation (4.56) for y E [--a, 0] and the fact that c(y;~,/l,) is an even function of y on [--a, a], for all E,P, 2 0, then establishes the validity of equation (4.56) for all y, --a < y 2 a. 0 By combining fol!ows.
the estimates
of Lemmas
4.3 and 4.4 we may state our final lemma
as
Lenrrnu 4.5. Let ~l(y;c,p,) be the unique classical solution of equations (1.12) and (1.13). The 3C+. C,, C2, all positive and independent of both E and ,u,, such that for all y. -u_
-J’;rc, , _ r (J;: + c2Jw”z
5
c(y;E,/II)
- u’(Y;E,o)
I
(4.60)
Now, by virtue of Lemma 4.1, we have -(I
+ K,)&
forallyE[-u,u],whereK,=(! y E [-a, 0) we find that
< u’(y;e,O) - l&(y) < (1 + K,)d --a)-
- (1 + K&,/i
‘/‘. By integrating
equation
(4.61) (4.61) from
-u
toy, for
< ~‘(y;c,O) - co(y) < (1 + K,,u&
(4.62)
with this last result holding for all y E C-u, N], as t:(y; E, 0). tlo(y) are both even functions y. Since V(FE,/1,) - co(y) = [r(y;E*/I,) - 4y;GO)l + C4y;E.O)- L*,(Y)] by combining the estimates following theorem.
(4.60) and (4.61) we see that
we have, in fact, proved
of
the
Theorem 4.1. If r(y; E, 11~) is the unique classical solution of the boundary-value problem of equations (1.12) and (1.13). while uo(y) is the corresponding solution of the boundaryvalue problem of equations (1.20) and (I.2 I), then 3C+, C, , Cz, all positive and independent
SIX of
H.
&LLOI’T
and
‘r
BI.OO~I
both E and AL,,’ such that for all y, --(l I >’ I u. Hlth K, = I I - xF’ - (I + K,)u,Ji:
-
v’uc,
- 1(\‘J:
5
From
the above theorem
Theoren~
difference difference
-
+
c‘2yjl:’ 2 5 L.(y:i:.jI,)
UC, (1 + K,)(l \s,c + ~ 1- A
two other estimates
‘:
- I’(](!‘)
(4.63)
‘E + cz)ljl:.
now follow which we record
as follows.
4.2. Under the same conditions as those which prevail in Theorem 4.1 the of the mean velocities. i.e. L;,,, - P,, also satisfies the estimate (1.63). while the of the net volume flows QF.II, i Q. satisfies
(4.64) In particular. bounds
by virtue of equation
(3.35). for example.
we may now exhibit
the explicit
(4.65) and
Similar estimates may be developed for the friction factor ji.,, of equation (3.7). by employing the bounds for CC.,,, if we simply note that equation (1.12) is equivalent to ?/?J [~,~(y;~,p,)] = p,, so that for all I:, /I, 2 0. rlz( fn;~:,jit) = & Ip, 1~1. REFERENCES I
J. Necas and M. Shilhavy. Multipolar viscous fluids. Quwr. rlppl. ,Iforh. (to appear) Bellout. F. Bloom and J Nccaq;. Phenomcnological hch.lvlor of multipol.tr wscouc fluldh Qwrr Appl. hluih. (IO appear). R. A. Touoin. Theortes of elasticttv with coupic stress. :t R UA 17. 85 i I Z ( 196-l). A. E. Circe; and R. S. Rlvlin. Slmpl; force andstress multlpalcs Arch. Rationul .\fcch. And 16. 325-353 (IY64). A. E. Green and R. S. Rivlin. Multipolar continuum mcchanlc< Arc,h. Rutionul Afrch. Anal 17. I 13-147 (1964). J. L. Bleustcin and A. E. Green. Dipolar fluids. /!lf. J. Ent/!~g &.I. 5. 323- 340 (1967). 1. Necas. A. Novotry and M. Shilhavy. Global solution to the compressible isothermal multipolar tluid. J Afurh. And. Appl. (IO appear). J. News, A. Novotry and M. Shdhavy, Global solution lo the Ideal compresslbls multipolar hear-conductlvc Ruid. Comtnrr~f. ,\ftrrh. Unir. Carolrnur 30. 55 t --%At 1989). 1. Necas and M. Shilhavy. Some qualitative propcrtxs of the viscous compressible heat-conductive multipolar fluid. Comnt. PDE (IO appear) J. Nccas. A. Novotry and M. Shllhavy. Global solutions to the ~ISC‘OUIcompressible barotropic multipolar gas, preprint. S. Goldstejn, khlodrm kwlnpmcnf rn fiuin Dl’nn~~ic’s. Vol\ 1 and II. H. Bellout and F. Bloom. Stabiltty and uniquencsc of flouz of incomprc>siblc multipolar flutds I” bounded domains (submitted). H. Bellout and F. Bloom. On the uniqueness and stablilt> elf plant equilibrium wlutions of the equalrons of incompressible multipolar flulds (submltted). H. Bellout and F. Bloom, Existence and asymptotic stability of time-dependent Poisuillc flows of isothermal dipolar fluids (submitted). H. Bellout and F. Bloom. Numerical and singular pcrturbdtion aspects of the boundar? \alue problems of plane Poisculllc flow of dipolar lluids (forthcomlnp).
2 H 3. 4. 5. 6. 7. 8. 9. IO. II. I’. I3 14. 15.
‘Estimates such as equation (463) provide a qualitative b.lrl\ for initiating an cyperimrntal study of the behavior of dipolar t7ulds; only tuch enperimentat studies can wrbc the aim of eztlmating the con\trslnts Cc, C,. and Cz