SOME THEOREMS O N D I S C O N T I N U O U S PLANE FLUID M O T I O N S * By Robert Finn in Los Angeles, California, U.S.A. 1,
Introduction.
One of the classical problems of Mathematical Physics is the determination of the irrotational flow, in two dimensions, of an incompressible, non-viscous fluid past a prescribed rigid obstacle W. The usual solution, based on a theorem of Riemann, leads to the physically unsatisfactory conclusion that the component of the resultant force exerted by the fluid on W in the asymptotic direction of flow is zero. This result, known as d'Alembert's Paradox, is perhaps most strikingly illustrated in the case that W is a flat plate situated orthogonal to the flow direction (Fig. 1), and is apparent in this case from the symmetry of the flow. The difficulty may be traced to the assumption, tacit in the method, that the velocity field of the flow is continuous throughout the exterior of VII.
Figure 1. As a better approximation to reality,
we may assume the presence of
a stationary air mass, or wake, situated behind W and bounded in part by W and in part by lines of discontinuity in velocity, or free streamlines A (Fig. 2). Since the pressure in this wake would be constant, one has by Bernoulli's law that any such flow must have constant speed on A. If the * This research was sponsored in part by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command. Prepared under Contract AF 18 (600) 680. 246
DISCONTINUOUS PLANE FLUID MOTIONS
247
wake is assumed to extend infinitely far behind W, this speed must equal the speed of the undisturbed
flow at infinity. It is remarkable that in
many cases this condition determines uniquely both the shape of the free streamlines and a corresponding flow. If W is a flat plate, the flow is explicitly known.
Figure 2. Another problem which has been studied extensively is the determination of the flow of an ideal fluid which issues from a prescribed nozzle into a medium of constant pressure (Fig. 3). The mathematical formulation is evidently very similar to that of the problem just discussed, although in this case the speed of the flow on A and the direction of A at infinity are unknown quantities.
What is required is an extension of Riemann's
theorem to a case where only part of the boundary is prescribed,
the
remainder to be determined by a non-linear boundary condition.
c2
Figure 3.
Figure 4.
The study of free boundaries can be traced at least to the time of Newton
[27].
The first significant theoretical
discussion
seems due to
248
ROBERT FINN
Borda [4], who determined the contraction coefficient, or v e n a c o n t r a c t a, of a flow of fluid entering a channel (Fig. 4). A complete description o f this flow was given by Helmholtz [13]. Many particular examples
of flows appear in subsequent
literature.
W e cite especially the book of Cisotti [5], in which a large collection is given. These examples to Levi-Civita [26], streamlines
are based on a parametrization
which
utilizes
in order to remove
of the flow, due
an analytic continuation across the free these
unknown
boundaries
from
direct
consideration. This representation has been fundamental to the development of much of the modern theory. As Levi-Civita pointed out, it reduces the problem o f proving the existence of a flow past a prescribed profile or through a prescribed nozzle to the determination of an analytic function in the unit disc. In this formulation the problem was studied by Weinstein [38], who applied a suggestion of Weyl [42], that an existence theorem
could
be obtained by the method o f continuity. Weinstein established a boundary condition satisfied by an infinitesimal variation* of a flow and proved, under severe restrictions, a theorem on local uniqueness of a flow through a symmetric polygonal convex nozzle. (A polygonal nozzle is called c o n v e x if the exterior angles at the intersection points of the segments have the sense indicated in Fig. 5). Hamel
[12] derived
an integral equation satisfied by
W
r
. . . . . . . . . . . .
-
-
.
.
.
.
.
.
.
Figure 5. an infinitesimal variation and proved local uniqueness if the total curvature K of the nozzle, defined as the sum o f the exterior angles, is smaller than 110 ~ and the
vena
contracta
is unvaried. The latter condition was
removed by Weinstein [39], who also indicated that his result should lead * The infinitesimal variation of a flow is defined in Section 2.3.
DISCONTINUOUS PLANE FLUID MOTIONS
249
to an existence theorem. Weyl [42] proved local uniqueness for symmetric flows if K ~_ ~r, and for asymmetric flows if in addition the v en a c o n tr a c t a and asymptotic direction are unvaried. Weinstein [40] then showed that the continuity method leads to an existence theorem in the former case. An example of an asymmetric bifurcation given by Weyl [42] with K = u + ~, r>0,
is incorrect. (The present paper contains a proof of local uniqueness
in this case). Friedrichs [7] used a variational principle to prove, under certain restrictions, the local uniqueness of a symmetric flow among symmetric disturbances for K<2zc.
This result and the result of [39] were
used by Leray and Weinstein [24] to obtain an existence and uniqueness theorem for symmetric flows through symmetric convex polygonal nozzles with K < 2~r. A significant advance appeared with the development of fixed point theorems for mappings in function space. The theory of Leray and Schauder [23] was successfully applied to the problem of cavitational flow past an obstacle, first by Leray [21], later by Kravtchenko [17, 18], and most recently by Serrin [29]. These authors use an integral equation formulated by Villat [36]. We mention briefly some results based on variational procedures. Minimum principles were formulated by Weyl [47] and Friedrichs [9]. A variational approach due to Lavrentiev [19] has attracted much attention, although some details of his arguments remain to be clarified. Lavrentiev's comparison theorems have been simplified and developed by Gilbarg [10] and by Serrin [30, 31], who have obtained elegant proofs of uniqueness in many cases.
A conceptually simple variational principle was suggested by
Riabouchinsky [28], later rediscovered and developed by Schiffer.
This
principle has been applied to problems in plane and axially symmetric flow by Garabedian and Spencer [9], and by Garabedian, Lewy, and Schiffer [8]. The starting point for the present study is the paper of Leray and Weinstein (l. c.). We summarize here the principal results. 1. The existence and uniqueness of a flow through a convex polygonal nozzle is established without the usual assumption of symmetry (Theorem 4.1). Although flows past asymmetric obstacles have been considered by Leray and by Kravtchenko, the problem of determining the flow through a nozzle presents a new difficulty, in that the asymptotic direction of the flow is an
250
ROBERT FINN
undetermined parameter of the motion. It is shown that this difficulty can be handled by the use of a remark, due to Leray [21], on the behavior of an infinitesimal variation of a flow near the separation points. 2. It is shown that for convex symmetric nozzles the restrictions on total curvature which have appeared throughout the literature are without mathematical significance. Thus, existence theorems are obtained for multisheeted flows past boundaries which may have self-intersections. This type of result is best formulated as a theorem on conformal mapping (Theorem 4.2) and appears as an extension of Riemann's mapping theorem. It should be noted that even in the simple cases considered a class of boundaries is distinguished for which the problem admits no solution. The characterization of such boundaries in the general case appears to present serious topological difficulties. 3. The stability of a given flow through a nozzle is established in several cases not previously considered. (A flow is said to be s t a b l e
if
it cannot be imbedded locally in a one parameter family of distinct flows, all with the same prescribed boundary). Stability of a given flow is proved for a) an arbitrary concave polygonal nozzle (Theorem 3.1), b) a polygonal nozzle, not necessarily concave or convex, but otherwise restricted (Theorem 3.2),
c) an arbitrary convex symmetric polygonal nozzle (Theorem 3.3).
A feature of the method is the application to an infinitesimal variation of a flow of a form of Green's Idenuty published in another connection by Levi-Civita [25]. We remark that in all three cases stability is proved with respect to a l l
disturbances of the flow, both symmetric and asymmetric.
Previous work on asymmetric flows (cf. [7, 12, 24, 38, 39, 42]) established stability for symmetric disturbances alone. 4. Some of the results discussed above are extended to the case of a flow past a prescribed obstacle with formation of infinite wake.
Our
conclusions seem of less interest in this case but are included for the sake of completeness. Existence of a flow past a convex obstacle (Fig. 7) is demonstrated without restriction on total curvature. This flow is the only such symmetric flow and is stable among symmetric disturbances (Theorem 3.4). 5. The existence theorems for polygonal nozzles (Theorems 4.1 and 4.2) are extended to include the case of curvilinear boundaries (Theorem 5.1). The discussion involves only simple compacmeqs arguments. We are, how-
251
DISCONTINUOUS PLANE FLUID MOTIONS
ever, unable to establish uniqueness in this case. It is likely that under suitable assumptions, uniqueness can be proved by use of a more sophisticated approach (cf. [21, 23]). Finally we include some remarks that seem pertinent. Theorem 4.2 includes as special case all previous existence and uniqueness theorems on flows through prescribed nozzles. Theorem 4.1 overlaps Theorem 4.2 and the previous work, but together with Theorem 4.2 leads to the conclusion that if a nozzle is symmetric and if /~ ~ r ,
then the flow through it is
unique and symmetric. This result is not yet established if K > n . Throughout this paper the separation points of the flow are considered as prescribed data. The problem of determining these points to obtain a physically reasonable flow has been treated by various authors [1, 17, 22, 29]. I am grateful to Professor M. Morse, who suggested that I study free boundary problems. My thanks are due also to Professor A. Huber for his interest in the work and for several helpful comments. 2.
F u n d a m e n t a l Relations Governing the Flow.
In this section we collect, for later reference, various formulas which describe the flow in a parametric form first used by Levy-Civita. The significant feature of this continuation
across
representation is the introduction of an analytic
the
free streamlines of the
flow.
This
procedure
simplifies the existeLlce problem by removing the unknown boundary from direct consideration. We then find ~ in an appropriately loose sense----that there are as many flows as there are odd harmonic functions in the unit disc. 2.1. T h e L e v i - C i v i t a
Representation
[26].
Since we shall deal with the irrotational motion of a plane incompressible fluid, the flow is completely described by an analytic flow function, f(z) = ~ +i~.
Here q~ is the velocity potential of the flow, ~ the stream
i
i
i
i
i
W
i
S
~
I
~11'
Figure 6.
A
--
252
ROBERT FINN
function. If the flow region is simply covered by streamlines of the flow, the function f(z) will map this region in a schlicht manner onto a region bounded by lines xp = const. For the flow through a nozzle we obtain in this way an infinite strip, which we normalize to the strip - - ~r K xp ~= ~t (Fig. 6) and for the flow past an obstacle with infinite wake (Fig. 7) we find, for the image of the flow region, the open plane
S
cut by the semi-infinite slit [q0 ~ 0, V -----0]
A
(Fig. 8). In this case it is not possible to normalize
Jw
?
the total flux. The flow
equation contains, however, an undetermined multiplicative parameter M, which plays a role similar to the v e n a
z-plane
con-
t r a c t a, also an undetermined parameter, in the theory of jets. Figure 7. |
WS '
We shall develop the formulas for the jet problem and for the wake problem
A
8
S"
simultaneously; distinguished letters J designating
Figure 8.
these
formulas will be
by the adjunction of the
and
W,
respectively, to
numbers.
Formulas
the
without
this designation apply to both situations.
Following Levi-Civita, we map the q~ + i~p plane conformally onto a semi-circle ]~'1~ 1, 7/~ 0, in the ~"= ~ + i~2 plane, in such a manner that the free boundary corresponds to the segment ~----0 and the point at infinity on the free boundary maps to the point ~ ' = 0 (Fig. 9, 10). W e have
I
(t--J) (t--W)
f -- f s = 2 log 2 f--fs
~+ II~--2coso* 1 - - cos o*
-----2M [~-(~ + 1/~) -- cos o0]' - - 2M[1 + cos o0]2 .
Here e i"* corresponds to the point at infinity on the fixed walls of the nozzle and f s
is the value of f(z)
at the upper separation point S ;
the
point ei~ corresponds to the stagnation point (Fig. 7) at which the flow branches. It should be noted that these mappings are based on an intuitive conception of the structure of the flow patterns justification,
of course,
will appear
from the
to be expected.
demonstration
Their
of unique
DISCONTINUOUS PLANE FLUID MOTIONS
253
existence of flows with these structures. The constants o*, t~0, and M are undetermined parameters of the flows. W e will have to show that they are uniquely determined by the prescribed fixed boundaries and conditions at infinity.
g*
-
0
A
sLT
-o
,
~ - pt~,',+'
Figure 10.
Figure 9. Let us represent by w = u - - i v
the conjugate complex velocity of the
flow. Then w (z) = i f (z). We are given that the magnitude I w] of w(z) is constant on the free boundary A.
We
therefore introduce this quantity
explicitly by setting w (z) = I
e_iO~(z) ' to (z) = ~ + i t . Then t9 represents, IX at each point, the angle made by the streamline through that point with
the positively directed x-axis, and
e r. For any given flow, the IX parameter IX may be chosen so that z = 0 on A. We find then, for the i wl=
1
1 --~ Ix
flow past an obstacle with infinite wake, that velocity at infinity, so that ~ is a k n o w n
V,
the prescribed
constant of the motion. For
the flow through a nozzle, on the other hand, Ix is related to the asymptotic thickness I of the jet (and hence to the v e n a c o n t r a o t a) by the equation (2--3)
Ixl = 2st.
The parameter Ix is therefore not prescribed in this case. The
functions
representation the type
of
considered
f(~') the
and
flow
uniquely
Conversely,
if f(~') and
the z-plane
is completely
w(f)
define
in the z-plane. determines
w(~') a r e
known,
a parametric Every
these then
flow
of
functions. the flow
in
determined.
That f (~') and w(~') are determined by a given flow is immediate from the definitions of these functions. To show the converse, we define
254 the and
ROBERT FINN
functions
Q(~" ; ~*) and Q(~" ; cs0) by the relations
1 m
Q=
d(f--fs) d4'
4"--e~*' + ~ - e - ~ "
Q(r ; ~0) = ; - ' ( ~ -
(3--J)
z-
as
zs =
d (f - fs)
d~"
Thus
Q(~" ; ~*) = 2
W e then have, if
Q =
4"
1) ( ~ - e~"o) (~ - e-iO.).
represents the upper separation point,
f'
--Q(~
;
*
f
"
W
1
e i~(~)
~"
|
which determine f ( ~ ' ) and w (~) as functions of z. The
reason
for Levi-Civita's
function T, considered
representation
is now
as a function of ~', vanishes
apparent.
The
on the real axis and
hence can be continued analytically across this axis to a single valued odd function, harmonic in the disc I~" I < 1 except perhaps at the point
~"= O.
This proves, in particular, the analyticity of the free streamlines A throughout the finite plane. We
shall
impose
further
restriction
streamlines, in a manner immediately sentation.
We
require that
asymptotic behavior of A,
the
on
the
behavior of the free
suggested by the form o f the repre-
function
t9 +
iv,
which
remain analytic at the point
determines
the
~-----0 (i.e., at
z = oo on the streamlines). The naturalness of this requirement follows from the fact that it is sufficient to suppose that ,9 is bounded in the flow region, that is, that no streamline turns through requirement
of
this
type is certainly
an arbitrarily large angle.
Some
necessary to rule out pathological
behavior, as we shall show by example. Thus, to every (not pathological) function
/~(~) in the unit
disc, with
flow there corresponds a harmonic ~9(~') = 0(~'),
and the boundary
values of 0 on the upper half of the disc correspond, in a I ~ 1 manner, with
the
set of directions of the
tangents
to the prescribed
boundary.
Conversely, to every such harmonic function 0 corresponds a two parameter
255
DISCONTINUOUS PLANE FLUID MOTIONS
family of flows with free streamlines, obtained by varying the parameters o* and ~ (or o0 and M). The problem we face then, is to determine the function 0 and the two parameters in such a way as to obtain, under the mapping ( 3 - - J ) or ( 3 - - W ) , a flow of desired character with prescribed fixed boundary as image of the semi-circular arc [~'l= 1, ~7~ O. 2.2. C i s o t t i
Flows
[5].
O f special interest are the Cisotti flows, polygonal boundaries.
obtained by considering
In this case 9 is piecewise constant on
I~'] = 1,
and the analytic function O + iv is found explicitly by adjoining a conjugate function to a superposition of harmonic measures.
Denoting by 9 1 , . . . , 0,
the counterclockwise succession of values of 0 on the upper semicircle, by ~'1 ~ g% . . . . . r _ ~ ~ e,*,-1 the associated points of discontinuity, and by J3j, j - ~ 1..... n - - l , the expression ( , g j + l - - 0 j ) / n , we have
f~
~0 (~) = 0 + iT = i ~ , ~Sjlog i
(4)
j=l
1 --
~'~'s
"
In the context we shall add a real constant to this expression so that # = 0 at ~'= f* (or ~'= 0). Each such function c0(O, inserted in ( 3 ~ J ) or ( 3 ~ W ) , yields a two parameter family of flows. If we permit the quantities {ail and {[3j} to vary, we will then have at our disposal a 2n parameter family of flows. Now a polygonal
nozzle of n - - 1
sides (not counting the two
segments) is completely defined by n - - 1
semi-infinite
side lengths lj, n - - 1
exterior
angles [3j at the intersection points of the sides, and two coordinates x L , y L expressing the positions of the separation points relative to each other. Similarly a polygonal obstacle of n sides is completely defined by n side lengths, n - - 1
exterior angles, and a coordinate to express the orientation
of the obstacle.
We
thus obtain from equations (3--J)
and
(3-.W)
as many equations as there are unknows. The resulting problem requires the determination of only a finite number of parameters and is hence a considerable simplification of the existence problem for a curvilinear boundary, but it is in no sense trivial, for the equations obtained by these considerations are highly non-linear. The formulation of the most general conditions under which they can be solved uniquely (or even solved at all) is a problem that remains open after more than thirty years of study.
256
ROBERT FINN 2.3.
Weinstein
The
Method
of
Continuity;
the
Relations
of
[38].
The first discussions of the existence of Cisotti flows through prescribed polygonal nozzles have been based on a suggestion of Weyl [42], that the existence of a solution for the parameters of the problem could be proved by the method of continuity. In general outline the method is as follows: we start with a reference configuration for which a solution is known to exist, and define a continuous (one-parameter) deformation of this configuration into the desired boundary. In other words, a continuous path in the 2ndimensional space of parameters
{/j}, lf3j}, XL ,yL
is defined which joins
the initial configuration to the desired one. It is first shown that there is a neighborhood N of every point P on this path with the property that if there is a flow at this point, and if the point P is moved to another point in N, then the motion can be followed in a unique manner by a continuous change of the parameters of Levi-Civita's representation. This step of the continuity method involves only a proof of the non-vanishing of the Jacobian of the transformation
from the
Levi-Civita parameters
to
the
physical
parameters. The second (and final) step is to show that the neighborhoods N can be chosen with their diameters uniformly bounded from zero, so that the path can be traversed in a finite number of steps. To do this it is necessary to show that the representation remains uniformly well behaved throughout the motion, that is, the parameters laj} are bounded from each other and from the points 0, ~, and ~*, that ~ is bounded from zero and infinity, and that (~* is bounded from 0 and ~. Thus, if the diameters of the neighborhoods N should shrink to zero along a sequence of points on the path (which we may assume converges to a limit point PL), a simple compactness argument shows the existence of a flow at PL and a neighborhood NL in which a flow at any point can be deformed to a flow at any other point. If the existence of a flow is demonstrated by the continuity method, then the uniqueness of the flow follows immediately from the uniqueness of the solution for the reference configuration. For suppose there were two solutions. W e choose a path in the space of physical parameters which joins the given configuration to the reference one. This would then correspond to two distinct paths in the space of Levi-Civita parameters. Since
DISCONTINUOUS PLANE FLUID MOTIONS
257
only one set of such parameters is possible for the reference configuration, the two paths must join at some point. This contradicts the assumed existence of a neighborhood N about each point of the physical path in which continuation is unique, A quite analogous argument applies to other types of flow with polygonal boundaries. Let us assume as known a given flow with physical parameters {ls} {~j}, x L , y r . ,
j = 1..... n - - 1 . This flow can always be imbedded in a one-
parameter family of flows, obtained by introducing suitable motions of the Levi-eivita parameters. Thus, corresponding to functions
{oj(~)}, [[3j(k)l,
o* (k), 8 (k), we obtain functions {li(k)}, {[Sj(;t)}, xL (k), yL (k), f (z ; k), c0(~';k).
We may of course choose the imbedding so that all functions
that occur are continuously differentiable. We define the i n f i n i t e s i m a 1 v a r i a t i o n s of the above functions by the relations
8~j =
dcs~0.) dk
8li =
ah ( ~)
' "'
dk
~f = ~0 [f (z; x)],
... '
'
~,o = - ~0 [o(~'; O]
Note that the variation of f is defined for fixed z and the variation of o is defined for fixed ~. We have
l Olj
81j -- ,~-_t Oli
dl s
Olj
k:=l
j= r
(,-j)
n--t
Ox
[sxL- S ~
8f3k]=
~OxL -
-
1,...,n--1
o OxL OxL oak + 80" + 8~
k~l
-
"-' o
.-t
~
OyL 8o~ + ~ ,
OyL
8o* + - - -
,,
.
_
k=l
This is a system of n + l
k=t
linear equations for the n + l
quantities
18c~j), 8o ~ 8bt. It follows that there will be a neighborhood of the initial configuration in which every motion of the walls can be followed uniquely by a change in the {oil, o*, and Ix provided that the determinant of the system does not vanish for this configuration. It is of particular interest that the partial derivatives with respect to the {[3~} do not appear in the determinant. For the flow past an obstacle we obtain the system
238
ROBERT
FINN
,,-t Olj Olj Ols . ~] = ~,~6k+~-o~OO+TM-lVl,
- ~ k=l
j = t,...,n,
k=l
"-~ O
OxL ~
- ~ 8M .
~X L
Here xL represents the x-coordinate of the separation point S relative to the point S'. Differentiating ( 3 - ' J ) with respect to ~ for fixed z, .we obtain
o : --~ f Q~(~ ; ") +
(6,--J)
--wt oQ 6"1o6. (s ; ~.'ds 1
r
- f
~5 log w Q (s ; ~s*)ds
l
with
Q(~. ~ . ) _ a U - f s ) '
d~
Now
f
l w
,~ 7A~s
OQ 06* as = !H(C.w
~__,,~n ( s . , 6")as
,6*) + j
1
1
where w ' = dw]d~ and
.)~
H (s .6") = '
dQ (t ;,')
__ 1
d~
dr-- O(f--fs)
From the relation ( l - - J ) we find
~(f--fs)--
d ( f - - f s ) ~ + O(f--fs) &~, Or " 0~*
= Q ( ( ; 6")5( + H ( ( ; 0')~6" . Combining the above relations, g
0 = - - 8 ( f - - fs) + ~6"
_w_=H (s ; 6 ~ ds - - f Wz "
W
1
~ IOgww Q (s ; ~')ds.
I
Thus, (7-j)
~ ( f - f s) = w f
~ lOgww (?(s; 6 * ) a s - w&r f
1
1
~w" i H(s ; o*) ds
DISCONTINUOUS PLANE FLUID MOTIONS
259
with
(s-J)
Q (~" ; o') = 2
(9--J)
n (~ ; o') = 2
~-
e~" + ~ ' - e - ~ ~
[ --id '~ These equations
-
ie-~~
sin o ~ | 1 ----~so'J '
~. _ e~o, + ~. __ e_,.~
generalize a relation of Weinstein [38].
It is important to
note that the function H(~" ; o*) is not singular at ~" = O, and has a second order zero at ~ - - - 1 . In a quite analogous
manner we find,
for the flow past an obstacle
the relation
(7--W)
8(f--fs)= Mwf 81og~ Q(s'oo)as ml
- -
8M H (s ; Oo) ds - - ~v --m--
wSo0
f
~w" U - I s ) ds
~t
with
w' =
dw/d~,
(8--W)
and
Q (~" ; o0) = ~'-s (~"2- - 1) (~" - - ei%) (~" - - e - ' % )
(9.-.W)
H ( ~ ' ", Oo) = 2 M (~" + 1)~ sin o0.
W e shall need also the expression for 8 log w for fixed It~J/" W e have (10--J)
8 1 o g w ~ - ~ log~t - i 8 o = -
81og~ + iE
~
8oj,
and similarly
O0-w)
8 log w = i ~ .
~i
(~" -- ~ ) (~" _
~-j)
8oj
Q--I
+ i[ (~--e,o.)(~--e"~%) ] 80~ " Finally, we derive a boundary
8(f--fs)
condition
on the free boundaries A. Consider
nozzle. W e have from ( 7 - J )
satisfied by the
variation
first the flow through a
ROBERT FINN
260
dw f dlogw df w Q ds +
d5 ( f -- f s)
af
Q5 log w d~"
1
1
dw d--f 5o* ~
--
dlogw
w
d~
H ds
-- w - ~d~f - o%s* - I- - - -dw n w 2 d~
1
from which (11 --J)
d8 ( f - - f s)
d log w
df
df
--
[Sf - - HSo*] + 8 log w .
On the free boundary A, log w = - - log ~t -- iO, Further,
dlog w _
af
dO
aq~
H ( r ; o*) is real on A. Taking the real part of (l l - - J ) on A,
we find (I 2 .---J)
dS~p d~p
d0 drp 5 , ------ - (3log ~ .
This boundary condition was first stated by Weinstein [39]. Its geometrical significance has been pointed out by Friedrichs [7]. For the flow past an obstacle we obtain the simpler condition (12.-W)
d~p
d~9 ~p = o .
a,
3.
The
Non-Vanishing
d,
of the D e t e r m i n a n t s of
the S y s t e m s
( 5 - - J ) and ( 5 - - W ) . We are now prepared to discuss conditions under which the determinants of ( 5 ~ J )
and of ( 5 ~ W )
do not vanish. We
are unable to present a
complete analysis of this question, but shall content ourselves with a discussion of several particular cases.
Our results, however, include as special
cases all known results of this type for jets, and significantly extend the known work on wakes. We shall consider an asymmetric polygonal nozzle with a restriction on the
range of directions of the boundary segments,
polygonal concave nozzle without
such restriction,
an asymmetric
a symmetric convex
polygonal nozzle without such restriction (we shall here permit asymmetric infinitesimal motions), and finally a symmetric polygonal convex obstacle. By a concave boundary we mean one for which all exterior angles {~#} are non-positive, by a convex boundary one for which all I~3j} are non-negative.
261
DISCONTINUOUS PLANE FLUID MOTIONS
It is hoped that this collection of isolated results will serve at least to indicate the type of general theorem that should be expected.
~(f--fs).
3.1. T h e V a r i a t i o n
W e prove here a preliminary lemma which applies to all cases considered in this section. We
formulate this lemma first for the case of a nozzle,
through which we assume the existence of a flow defined by a flow function
f(z). This flow may then be imbedded in a one-parameter family of flows f(z;~)
by introducing appropriate variations of the parameters appearing
in the Levi-Civita representation.
We
assume throughout
that
~5~j= O,
j=l, ...,n--l. Definition : Set sin o ~ 1--coso ~ '
~. =
~j ~
sin oj l+cosoj
j=l,...,n--l.
'
An infinitesimal variation of the Levi-Civita parameters will be called a singular
variation
provided that
Ej=l /~S~J=O, ~3J=O,
and
~ioj=~.~
~ j=
~Vt----O,
1 ..... n - - 1 .
A variation which is not
singular in this sense will be called a n o n - s i n g u l a r
variation.
Note that concave walls and convex walls do not admit singular variations. Lemma
3.1. I f
and ~oj=O, j=l Civita
~(f--fs)-~O, t h e n
..... n - - l ,
parameters
for any singular
is
either
(13-j)
~o#=0,
or else
the variation
of the Levi-
singular.
Conversely,
8(f--fs)=O
variation.
The proof is based, on the representation ( 7 J J ) (8.--J), ( 9 J J ) ,
~t=O,
and (4). From ~ ( f - - f s ) = 0 = - - ~ log ~t
-
-
and on the formulas
0 we thus find
"~'+1 ~.
i E f~j
8oj
(~-[- 1)2(~ " - 1)
( : - 6) ( r "--~
j:,
2sin o i
(r
sin o ~ 1--COSO ~ " ( ~ - 1 ) "
262
ROBERT FINN
Since the second and third terms of ( 1 3 . - J ) both vanish at . f = 1, we immediately find j = 1 ..... n -- 1.
81oglx-- 8IX=O.
If ~o~
Otherwise we let ~" ->- ~
then
dearly
~ioj--O,
and find ~oj ~ ),j )," 8o*, which
together with (13.--J) implies EI3i2Lj ~ - o . j=l by insertion of these expressions in (7---J).
The converse follows directly
The non-vanishing of the determinant of the system (5 ~ J ) will follow if it is known that the associated homogeneous system admits only the trivial solution, j ~ - I .... , n - - l ,
that
is,
if the relations
~/i ---- 813j = ~xL = ~Yc -----0 ,
imply the relations ~ o j = / 5 o * - - - ~ i x = O ,
j=
If then, for a given flow, it is known that Y. 1 3 j ~ O , the
above
lemma
that
it
is
sufficient
to
prove
1.... , n - - 1 .
it follows from that the relations
~/j = B[3j~ 8xc = ~YL ~ 0 have as consequence the relation ~ ( f -- f s ) ~ O. This is the central problem considered in this section. The case of singular variations will then be disposed of by showing that (~xc)2 + (~yL)2~ 0 for any such variation. Before proceeding to this discussion we insert the analogue, for an infinite wake, of Lemma 3.1.
Since we shall deal only with symmetric
configurations and symmetric disturbances, we restrict consideration to this simpler case. Lemma
3.2.
of a symmetric
If ~(f--fs)=:o flow past
for
a symmetric
an obstacle,
~M=8o i=o,
variation
then
j = 1..... n - - 1 .
In this case the relation ~ ( f - - f s ) = 0 implies
(.c- G)( c --1) 2 n--I
sin Oj
_~_~1 ]
and the result follows by first letting ~--~ i, which shows that ~M ~-O, and then letting ~->-~j,
j~l,..,n~l.
We shall now discuss particular cases in which the determinants of the systems ( 5 ~ J ) and ( 5 ~ W ) sufficient to
do not vanish. As already noted, it is
show that the relations
15lj ---- ~ j ~ ~SxL~-- ~y~ ~ 0
(or
263
DISCONTINUOUS PLANE FLUID MOTIONS 81# ---- ~ # = ~xL = O)
f
imply
that
8 ( f - - f s ) -~ O, and that these relations
cannot occur for a singular variation. 3.2. T h e C o n c a v e
W
Nozzle.
We consider first the free boundary flow of a perfect fluid through a concave polygonal nozzle (Fig. 11).
Theorem 3.1. L e t f(z)=q~+iq~
from
the
those
a flow
concave
polygonal
Then
Figure 11. through
represent
same
nozzle,
of the given
there
is
no
mally
neighboring
having
streamlines
through
a
nozzle. infinitesiflow distinct
flow.
The theorem may be interpreted as an assertion on the stability of a given flow. Our problem, of course, is to prove that 8 ( f - - f s )
vanishes
whenever the walls are unvaried. The proof depends heavily on the structure of the representation for ~ ( f - - f s ) . (7--J)
8 (f - fs) = w
f
81ogw -
W
Q ( s . ~~ ds
1
w~a~ w1 dlogw d~
H (s ; ~*) ds .
1
The following lemma is due to Leray [21], who applied it to the theory of the flow past an obstacle. L e m m a 3.3. D e f i n e
b y [3(x , y ) t h e e x p r e s s i o n
8 (~p -- q~s) + (q~ -- •s) 8 log ~. Then
there
region
from
is
a level
curve
the separation
~=0 point
which
enters
the flow
S.
Since ~ ( x , y) is harmonic in the z-plane, it is harmonic when considered as a function of ~, and since level curves are transformed by invariance, it is sufficient to consider the behavior of this function near the point ~'-~ 1. This is a particular convenience, since ~ can be extended
264
ROBERT FINN
harmonically across the boundary
I(]=
1 near this point, but cannot be
extended across the corresponding boundary in the z-plane. Consider the analytic function
B -- 8 ( f - - fs) + ( f - - f s ) ~ log l~ of
which ~ is the imaginary part Clearly B (~') vanishes at ~ ~ 1. Now dB (~, d~" = Q " ~1768 log w
dlogw d~"
H(r;~')So* g
dw +-a~f
8 log w - Q (s ; ,s')ds - ~dw f - ~ ~ ,, f,~
w
u1
d lao:g w H ( s - o' ' ) d s
1
I
+ Q(~ ; ~') ~log ~. By reference to the formulas ( 8 - - J ) and ( 9 ~ J )
we see that all terms but
the second and last have second order zeros at ~ ' = 1. But Q~ log w + Q~ log bt -- - iQ~ro and by ( 8 - - J ) and ( I o J J ) we see that Q and ~co both have first order zeros d2B at r Therefore ~ vanishes at ~ ' = 1, that is t~ (~) has the form B ( O = a3 (~" -
1) 3 +
....
It follows that there are at least three distinct level curves ~ = 0 which meet at ~ ' = 1,
making equal angles with each other. At least one
of these must therefore enter the upper semicircle. The lemma is proved. The proof of Theorem 3.1 is based on consideration of the behavior of such a level curve ~ - - 0
in the f = q ~ + i ~ p
plane (Fig. 6). The curve
may of course branch, but it creates in the strip - - ~ < ~ < ~
two open
components Ow and OA in which ~ has opposite sign and which have on their boundaries segments of the images W and A*) of the fixed wall and free boundary respectively, the segments being connected at the image S of the upper separation point.
W e note also that the assumption that the
prescribed walls are unvaried implies that ~ = 0 on W, except perhaps at the images f i
of the vertices of W. But an examination of the formulas
(7---J) and (4) shows that w behaves like r~j near f ~ , distance to f i , these points.
where r represents
and the integrals appearing in ( 7 ~ J ) a r e It follows then
from the
bounded near
Phragm6n-LindelSf theorem on
harmonic functions [34] that ~ vanishes on all of W and can be continued harmonically across this line in the f-plane. * The same symbols will be used in all planes.
DISCONTINUOUS PLANE FLUID MOTIONS
265
W e distinguish the several possibilities. C a s e 1: OA contains arcs extending to q~ = + eo and arcs extending to q ~ = u
oo .
In this case Ow lies in a left half strip.
If Ow lies also in a right
half strip, then it will be bounded by a finite number o f analytic arcs Fw on each o f which [3 = O. By the maximum principle [3 = 0 throughout Ow and
hence
(*-
[3--=0.
~ (q~-- tps) = - - ( ~ - Ws) S log [~.
Therefore
Since
~Ps)= 0 on the lower fixed boundary we find 8 log ~t = 0 and hence
(xp-- *s) = ~ ( f - - f s )
= O. I f Ow contains arcs extending to ~ = - - ~
we
obtain the result from an estimation o f the behavior o f [~ near this point, i. e., near the point ~" = ~'*. At ~'~ t9 = 0 and 81ogw = - - i~0 + Slog [w] = O, since the total flux, and the direction o f flow and thickness
o f the jet at
x - - - - - - o % are assumed unvaried. Also w(~') is analytic at this point, since 0---- 0 on an arc o f ]~'I= 1 containing ~'~ From the relations ( 7 - - J ) , (8.--J), and ( 9 J J),
indicating by an affix* the evaluation o f a
we thus obtain,
quantity at ~"= ~'*,
a(f -
(~" -- ~') Pt (~') w
fs) = w f
P2 (~')
-
1
-1
( ~ - - U ) 8o ~
where
the
symbols
of the
form
1
PS(~)
represent
functions
Thus
+ (:) (:- :')] [ -;:. 9 [-2ir
f
a (,-r
~.s')
9 8a ~
1
from which ( 1 4 - - 3)
8 ( f -- f s ) = P9 ( 0 + P~o (~') (~ - - (*) log ( ( - - (*) -
dT ~
2~
log ( ~ - C~ 9 ~G'.
regular
"]
at
266
ROBERT FINN
Set ~"- - ~'* = Re '~ . Then
8 (f-fs)
(I 5 - - J )
+ (f-fs)
~ log ~t = - - 2 ~dr
*[log p +ia]~So ~
+ bounded terms so that
de.
[3 = -
20t do
86* + bounded terms,
and a bound on [3 follows from the fact that ct is bounded in the circle 1~'1< 1. (Note that the function From
the bound
on
conjugate to (3 is unbounded at ~ ' = ~'*).
~ follows immediately from the Phragm6n-Lindel6f
principle that (3, and hence also ~ ( f - - f s ) , Case
2:
vanishes identically.
OA lies in a right half strip and possesses no boundary
points on the lower boundary of the strip except the point q~ = + oo. Case
2a:
OA is a bounded
consists of a finite number
region.
Then
the boundary
F^ o f OA
of analytic arcs on which (3-= 0, and a finite
number of segments T~ of A on which, by (12-~J) and the definition of ~, d~ dt~ a-~- = d--~- [3
(16--J)
Applying a classical form of Green's identity,
dO
2
~-
0,,
L,
From the formulas (4) and ( l ~ J )
[32 d s .
T,,
we find that on A do)
dq:
d ( f - - f s) d~
= 0
so that
(is---j)
dq~
_
By assumption,
_ U--
[
n--t ,/,
I
~3j< 0, j = 1, ..., n - - 1 .
j-----t
on TA. By ( 1 7 J J ) ,
--
"
Since the part of A on the upper
boundary o f the strip corresponds to the segment dO --<0 dq~
2~-O~OS sinoj oj + 1] ~
O < ~ : < 1,
we find
therefore, (3-----0.
C a s e 2b : O~ extends to c9 ~-- + ~ .
W e restrict consideration to a
DISCONTINUOUS PLANE FLUID MOTIONS
267
finite domain by introducing a vertical segment q9 = c. We shall then let c-->-oo. It is thus
sufficient to
show that
~-~-~0
as q ~ + o o .
We accordingly inspect this function near ~"= 0. From (7--J), (8--J), ( 9 - . J ) and the definition of B (~'),
B (~) --- 8 ( f - - f s ) + ( f - - f s) 8 log
1
where O0 represents the asymptotic direction of the stream and the Pi (~') represent functions regular at ~ = 0 . Setting ~ ' = p e ~ and using the relation (1 --J), B (~) = [8 log p + i8~0] [log,o + io] + ~'P4 (~') log ( + Ps (~) + [--log p -- io] 8 log ~t so that (19--J)
B(~') = 8Oo[--o+ilogp] + ~P4 (r162
+ Ps(~).
If 8~0:~ 0 then the imaginary part [3 of this expression would become unbounded uniformly in ~p as q~--~ + oo. But under the assumptions of the present case there is a level curve [3 = 0 extending to q0 = + oo. Therefore 830 = O. Since the other two terms of (19--J)
approach limits as .~ + O,
we obtain that [3--~ 0 uniformly as q)->-+ co. Now 2
for some constant C, as ~ ' ~ 0 .
sc
_Cllogpl.p 013 Thus - ~ - - - ~
O.
(We have incidentally
shown that the gradient of ~ approaches zero as q3--~ + oo). integrals in (17--J)
Thus the
can be extended over the entire region O~ and we are
led to a contradiction. Case
3:
O~ possesses a boundary point on the lower part of the
image W of the prescribed boundary. In this case Ow lies in a left half strip and the consideration of Case 1 apply. C a s e 4 : O~ has a finite boundary point on the lower part of A but
268
ROBERT FINN
no boundary point on the lower part of W. Then the point S will be joined to the lower part of A by a piecewise analytic arc T on which 13= 0. Define the function ~ = 8 (~ -- hbs,) -b (~ -- Vs,) 8 log ~. By reasoning entirely analogous to the reasoning we have applied to the function ~5, we find the existence of a level curve ~
0 which enters the flow region
from the point S 1. We define the regions Ow and O~ by analogy with the above definitions of Ow and Oh. Let TA~ represent the boundary of (~A. Case
4a: TA contains a point of T other than S. Then ~-----13 at
this point, hence 8 log b t - 0, ~----~ and Case 1 applies. Case
4b: T~ contains no point of T other thanS. Then Ow lies in
a left half plane and the considerations of Case I apply. The reader will note that all possibilities are now exhausted. Theorem 3.1 is proved. 3.3. T h e g e n e r a l
asymmetric
nozzle.
As a second example, we consider an asymmetric polygonal nozzle without restriction on the signs of the exterior angles {[Bj}. We shall have to pay for this liberty by introducing a hypothesis on the range of directions assumed by the boundary segments. H y p o t h e s i s A : All vectors from the origin, parallel to the boundary segments and directed in the sense of motion along the segments toward the separation points, lie interior to a common half-plane. T h e o r e m 3.2. L e t a polygonal
nozzle
is n o i n f i n i t e s i m a l l y nozzle
having
f(z)=q~+i~
represent
satisfying
Hypothesis
neighboring
streamlines
flow
distinct
a flow through
from
through
A. T h e n
there
the same
those
of
the
given
flow. The proof of Theorem 3.2 is identical to that given for Theorem 3.1. d,9 except for Case 2. Since the sign of ~ will in general no longer be negative on the upper part of A - - f o r convex nozzles it will be positive---
we are unable to obtain an immediate contradiction from the relation (17--J). We have recourse to a lemma of Friedrichs [7], related to Jacobi's Principle of the Calculus of Variations. L e m m a 3.4. S u p p o s e harmonic
there
and non-vanishing
is
a function
in the strip
U(q~,~p),
--sr"~,~r,
such
DISCONTINUOUS PLANE FLUID MOTIONS
that as
OU
dO
0~
dq~
. . . . ~fl++oo.
U on A and such
Then
~ vanishes
that
OU
~-+0
269
and
]U[
identically.
We need only consider the configuration of Case 2. Set i3 = Uz~. Then, using the notation of Vector Analysis,
I vp !2 -- u21 v,~ I' + n21 w 12 + 2u~ ( v n . v v ) . But since U is harmonic,
On ds 0..,
0,,.
FA
-~n ds. A
Hence from
OA and the assumed boundary condition on U, we obtain
w l v ~ j 2 a , a , = - .~ OA
=
A
A
which proves the lemma. Friedrichs observed that if Hypothesis A is satisfied, then the projection of the velocity vector on a normal to a line defining the half-plane satisfies all the requirements for the function
U (q~, q)). For it follows immediately
from the maximum principle, applied to the harmonic function 0, that this projection does not vanish, and since ~ = 0 on A,
dw _ df on A. Further,
~-
1 d ~ df (e-R~ -=--
-d7
do.) iw d / -, which ~ 0
d,9 iW--dq~-as q 0 - ~ + e ~ .
Simple estimates show that the singularities of w at the separation points and at the intersection points of the segments do not affect the proof. Case 2 is thus an immediate consequence of Friedrichs' lemma. Theorem 3.2 follows directly. 3.4. T h e
convex
symmetric
nozzle.
As a third example, we show that if the nozzle is convex and symmetric, it is unnecessary to introduce any restriction on the range of directions of the boundary segments.
270
ROBERT
Theorem flow
3.3.
through
there
is
having
f(z)=q)+i~
represent
a symmetric
symmetric
polygonal
nozzle.
a convex
no
symmetric
Let
FINN
infinitesimally
or
neighboring
asymmetric,
streamlines
through
distinct
from
the
flow, same
those
Then either
nozzle
of the given
and flow.
The proof that we give for this theorem is based on a form of Green's Identity, published by Levi-Civita [25] in 1905, which seems to have been generally forgotten. We include here a statement and proof. L e m m a 3.5. bounded ously
by a curve
turning
differentiable
be
harmonic
F. I t is a s s u m e d
in
Let ~ represent
DkJF.
that
normal
a domain
D
F has a continu-
h(x,y)
and
directed
in
that
tangent
by the exterior directed
h(x,y)
Let
is c o n t i n u o u s l y the
to F with
angle
made
the positively
x-axis.
Then
(20)
o~ on~ds = 1.
where directed
y1
+ )cos as, 1.
Oh~On r e p r e s e n t s
differentiation
along
the exterior
normal.
We give Levi-Civita's proof, which could hardly be simpler.
On the other hand,
= 2
ff h~ (hx, + hyy)dx dy + 2 ~ h,
ds.
To prove Theorem 3.3 we apply this identity to the function ~ in the q~-t-i~2 plane. In order to do so, however, we shall require some preliminary estimates. Lemma plane,
3.6.
Denote
respectively,
b y A u a n d AL t h e i m a g e s
of the upper
and lower
for a flow through a symmetric 020 ~. ~: 020 0cp2 < 0 o n A v a n d ~ > 0 o n Az
convex
in the f-
free boundaries nozzle.
Then
DISCONTINUOUS PLANE FLUID MOTIONS
271
The proof, of course, starts with the relation (4) n--1
o(~') = 0 + i z -----i E ~ j l o g i 1 -- ~'G " i=l It is sufficient to consider one term for which we use the subscript 1. On the line ~-----0, 00 = -- 2~1 sincq
[ (~_e%)(~_e_,OO 1 ]
where by convexity ~t > 0 . On 22= 0, 00 0--4 - =
00
o~
o~
o,
-
oO
d."
o~
a-/- = 2 - d g - t ~ ~ - t l '
where we have set ~'*= 1 in ( t - J ) . (2t,-J)
0~ = 4~lsinat
O0 |
Thus,
[ (~_ei%) $(~2+ 1) ]' (~_e~%)(1_~2)
Consider first A v, corresponding to 0 < $ < 1 . ~,<0.
Thus, since
~+ll
0~=0~096:,,
From (8--J)
we find
it is sufficient to prove
0~og>O.
The product of positive increasing functions is a positive increasing function. ~2+t The function 1 - ~2 is clearly positive and increasing. And d d$
$ l+~2--2~cosat
t - - ~2 l+~:2--2~cosal
>0.
The corresponding property of AL follows by symmetry. We enumerate, with brief proofs, the further estimates that we shall need. 1. As q0 -->. + o% ~0/->- 0 uniformly in V. Proof : ~ = Im [B~] = Im liB A = Im [iB~ ~'j]. By (19,-J), B (~') = i800 log ~"+ ~'P (~) log ~"+ P2 (~) i50o B~ = - f + P ( O log ~ + .... By (8.-J),
B~ ~'/-- -- i50o + ~P (~') log ~ + ....
Thus, ~q ->- o. (Note that ~ -~ 800). 2. As q~ ~ Proof: From (15--J),
o% ~r + 0 uniformly in xp.
~=
Im[B/]=Im[B~l]. dl: /* B = -- 2 8 6 " - ~alog(~'--i) +
(f--fs)
5 log~t + ....
272
ROBERT FINN
For a symmetric flow, ~ dl: * = Thus
B g ~ ' / = 81ogbt + ...,
d1: I,~=i = 0 do
i
q.e.d.
3. Near the points S and S' in the f-plane,
IV131
constant C, where r represents distance to S.
Proof:
[V13[ = IB:[ = ]B~G].
In the proof of Lemma 3.3 we have
shown that near ~" = 1,
B ( 0 = az ( ~ - 1) 3 +
....
Hence
IB~GI < C,I ;'--11 <
C,':.
A similar discussion establishes the estimate at S'. 4. Near
the points S and S ' in the f-plane
[8~]<
Cr
for some
constant C. This follows from 3.
dO
5. Near the points S and S ' in the f-plane - ~
< C r - I I 2 for some
constant C.
<
Proof :
"=
d~
df
< Cr'll2 "
Consider now 9 rectangle bounded by the lines lp = + :~ and by the lines E~:
q ~ = - - c 2 and E2: q g = c 2, c > 0 .
Since 13(q~, ap) is continuously
differentiable up to the boundary we can apply to it Levi-Civita's identity. Thus,
2Afu 13,13d-2 Afc ,13j C2--~ CX~
et
es
by the estimates 1, 2, and 3, above. On A ~ and on A L,
dO 8
i" ao 8 , AU
f AL
or
Au
dS~p
13,----=~-~--~ ~2, 13~v= dqo " Hence
AL
ao
273
DISCONTINUOUS PLANE FLUID MOTIONS Integrating by parts and using the estimates 4. and 5., we find
f
f
AU
AL
o,
which contradicts Lemma 3.6 unless ~p vanishes on the free boundaries. But if 8~ vanishes on the free boundaries ~-----~+0p--Xps)~log~
would,
then the harmonic function
by ( 1 2 - J ) ,
vanish on the upper free
boundary together with its normal derivative, hence would vanish identically. The proof of Theorem 3.3 is complete. 3.5. T h e
convex
symmetric
obstacle.
Our final example concerns the symmetric flow past a symmetric convex polygonal obstacle and forming a wake which extends to infinity. Theorem
3.4.
Let
free-boundary
flow
obstacle.
there
Then
symmetric lines
flow
distinct
past
from
f(z)---q~+i~p
past
a symmetric
is no the
represent convex
infinitesimally
same obstacle
those
of the given
a symmetric polygonal neighboring
and having
stream-
flow.
In this case the image of the flow under the mapping f ( z )
consists
of the entire complex plane, cut by the semi-infinite segment OS extended. By reasoning analogous to that used in the proof of Theorem 3.2, we see that the singularities of ~ f on the segment OS are removable (except, of course, for the singularity at S). Thus we need only consider the hehavior of ~f
on the
image A of the
free streamlines,
and by symmetry all
considerations can be restricted to the upper half plane. W e shall again use Levi-Civita's identity (20); however, its application is not so evident here as in the case previously considered. It is, of course, necessary to introduce a limiting process starting with a bounded region, but if we choose for this region a semicircle (or square) with center at S, we find that the integrals taken along the outer boundaries cannot a priori be expected to approach zero as these boundaries expand to infinity. It might be possible to show that they have a definite sign, but this would require more effort than seems worth while at this time. Instead, we shall use the fact that these integrals are not invariant under conformal mapping in order to obtain new outer boundaries which produce, in the limit, no contribution.
Indeed, we need only observe that a transformation which
274
ROBERT FINN
stretches a boundary curve can be expected to decrease each of the integrals occurring. In our case. we choose the transformation F - ~ ( f - - f s ) 2. This makes the singularity of ~f at S somewhat worse, but from our experience with nozzles we can expect to have some freedom in this matter. We then integrate over the region indicated in (Fig. 12). Simple estimates based on (I~W).
( 7 ~ W ) . (8---W). and (IO.-.W) show that near ~'= O.
s~w
"
Figure 12. log w = o ( 0 ,
Q = o (8_3),
: = o(~-~),
~f = o (~_1),
a~f = o if-2) d~ where the expression a = 0 (b) indicates that the ratio
d~f
f , V (8,) l' , dF , = : 1 r
d,
~ d(
[dr
__~f_f 2 ,
a[b
is bounded. Thus,
12 lay1
r
p
--olf
}
P
which approaches zero as F expands to infinity. Similarly, near ~"----- 1,
d6f d~
= o [(r + 1)2],
d (f ~--. fs)
-- o (~ + 1),
DISCONTINUOUS PLANE FLUID MOTIONS
d,:
:s, dF
d S ( fdF --fs)
Thus,
275
(~" + 1) 2 "
= O(IFt_t/4) near this point, and
=
Y
f Y
Y
which approaches zero as y shrinks to the point S in the F-plane. Thus the identity (20) gives us, in the F = q ) + i ~ plane,
2f (sv),~dr
d~
2f (sv),2~-(~V)d't'
It.
t*
= f[(~,12]~
an
&
.... ~ d~
= (o,)
~-I
l~
- j
~ ' ( 8 r 2 d20 d-d-O-~ q) = o ,
A
where we have used the relation ( 1 2 ~ W ) .
Elementary estimates show that
near f ---- 0 , ~
= o(tel-'"),
I~*1 = o c l F ? ~ ' ) ,
and near F = on,
I ~ , 1 = o (I F I,~'), d~
Therefore the term (Sap)2 - ~
d~
-~-
= o (I F l-':') 9
vanishes at both limits.
To complete the proof, we show that for a convex obstacle,
,ga~ has
constant sign. Exactly as in the proof o f Lemma 3.6, we find 0% =
4~, sin al I M2 (~_eiO)(~_e-iO)
(~2
Es 1)3(~z + 1) "
Thus M2 4~, sin ol
0# = f (~) 9 g ($) 9 h ($)
where
We
have ( f g h ) ' - ~ f ' g h + f h g ' + f g h ' ,
From this O#~ < O.
of which each term is positive.
and the formulas o f transformation, we find immediately that
276
ROBERT
3,6. T h e n o n - v a n i s h i n g
FINN
of the Jacobian.
Lemmas 3.1 and 3.2, together with the theorems of this section, show that the Jacobians of the systems (5,~J) and ( 5 ~ W )
do not vanish for
any of the cases considered except perhaps for that of a channel which is neither concave nor convex. In this last case we have yet to dispose of singular variations of the Levi-Civita parameters. To do this, we consider the variation of z for fixed ~" corresponding to a motion of the parameters. By (3--J), Z--
y'
ZS
1
~ l~ w ~d~ + ~o" ~ I
5 (z -- zs) = -- f
W
I
Oq d~ 00*
1
g
= _ f
~ log - -w
Qd~
+
&s*_ H (~" ; 0*) zv
w I
+ go" ~ 1 9 w
d l o g w Hd~ d~
!
so that, using (7--J),
~
(z -
zs)
-
1 b,(f--fs)
w
+ -~~176H ( ( " 0") w
where we have distinguished variations for fixed ( and for fixed z by the subscripts ( and z, respectively. If the variation is singular,
5g (z -- Zs) =
80"
H (~ ; o*) =
--~o*
w
w
sino* t-coso"
2(~--1) 2 (~ - e i~
(~-
e- ' ~ ' )
Letting ~" -~ -- 1,
~ (zs,- zs) =
~O*
sin o*
4
w
1 m cos o*
I -- cos o*
from which we see that for any non-vanishing singular variation, the variation of the aperture of the nozzle is also non-vanishing. The main result of this section is therefore proved. 4.
Existence T h e o r e m s .
In the method of continuity as applied to free boundary flows, we have already noted that two distinct steps are involved: first, to show that
277
DISCONTINUOUS PLANE FLUID MOTIONS
for a given flow a small motion of the prescribed boundary can be followed uniquely by a motion of the flow parameters; second, to show that a large deformation of the boundary can be followed in its entirety. In Section 3 we have verified the possibility of the first step in certain cases. This section is devoted to the second. W e shall prove the existence of flows, under varying additional hypotheses, through
an asymmetric convex nozzle
and a symmetric convex nozzle, and of a flow past a symmetric convex obstacle. For a symmetric convex nozzle with a restriction on the total curvature
K ~ defined
as st E
~i--the
necessary estimates
are quite
simple [40]. The more general cases considered here will require a more extensive discussion. 4.1. T h e We
asymmetric
convex
nozzle.
prove here the existence of a unique flow through
a convex
polygonal nozzle of total curvature smaller than s~, subject to the following additional assumption. Hypothesis
B:
The separation points S and S' can be separated
by a line which does not intersect the channel walls and which is parallel to the two semi-infinite segments (Fig. 5). Our task is to bound ~t from zero and infinity, to bound o" from zero and u, and to bound the angles {aS} from zero, from ~, and from each other. These estimates are to be uniform for all flows through the nozzles of a suitable one-parameter deformation which begins with parallel walls and ends with the prescribed channel. W e may clearly assume that throughout this deformation the point S and S' do not touch the line required by Hypothesis B (and hence remain larger than a fixed distance apart) that the number of polygonal segments remains unchanged, and that all side lengths remain bounded and bounded from zero. Lemma curvature there length
4.1:
not larger
is
a positive
L of the
S a n d S') a n d o n normal axis,
If
to such
a convex than
number
K made
with
for any flow
nozzle Hypothesis
depending
(defined
angle
th'e aperture that
st s a t i s f i e s
aperture the
polygonal
as t h e
only segment
by the exterior
the
trough
positively the nozzle,
of
total
B, t h e n on
the
joining directed
directed
x-
278
ROBERT ~INN
1
~-<
!~< K.
The lower bound follows immediately from a formula for the aperture L, due to Levi-Civita [25]. W e may suppose the nozzle situated as indicated in Fig. 5. "If the relation (20) and the similarly derived relation
~ Oh Oh ds= 1 ~ ~ 2 Oy 0~ --~ (h, + hy) sin o ds 1'
F
are applied to this configuration, we obtain without difficulty the relations L cos (ct) ---- 2~ut (cos T - - ~tu) - - Ix2 (22--J)
flWl cos (n, s) as
w L sin (ct) ---- 4~tn sin y - - la . ~ IWl2 sin ( n , s) ds W
where ( n , s) denotes the angle between the exterior directed normal and the positively directed x-axis. Since cos ( n , s) will be non-negative if K <= st, the lower bound on It follows from the first of these formulae and from the assumptions of the lemma. To obtain the upper bound we observe that z (() as obtained from (4) is non-positive for a convex nozzle, But, using the assumed normalizations, e~(~.)
1 1 2 - -l
2~ which yields the desired bound. Lernma the
additional
{/1} o f from
4.2.
the
the
conditions
assumption
sides
0 and
Under
from
is
that
bounded,
the
of sum
the parameter
Lemma of
the
4.1
and
lengths
o r is b o u n d e d
~ for any flow.
It is sufficient to show that o" is bounded from zero. Let r represent the largest angle of the {oi} which is smaller than o ~ Then
I sin o j + o ~li j=~'+ 1
=
f j~-I
*r+1
] __ 2___ sin ~ - - o
~j sinodo t cos o - - cos o* l
2 sin o cos o" 4- 1 I cos o - - cos o" i do = ~Llog cos o" - - cos o,+t
":JT+ 1
DISCONTINUOUS
279
PLANE FLUID M O T I O N S
By lemma 4.1, ~t is bounded below. Thus all oj with j ~ r + 1 are bounded from o ~ even if o * - > - 0 . W e consider the first r sides. r
or
sin2 ~ o ~
IS> g j~.l
coso--coso ~ do=
Ix log
. I . . sm -f (o + o,) sm ~ (o*
0
I f o ~ is bounded
from zero then clearly on is bounded from o'.
-
-
o,) If o*->-0
the last expression behaves as (o') 2 (o'--o,) > ~log
~tl~
1 1--o,]o ~
which shows the existence o f a constant ~/, 0 < ? / < 1 such that a i ' ~ ~7o*, .7"= 1 ..... r for all configurations considered. W e now apply the second o f the formulas ( 2 2 . - J ) : (23~J)
L sin (c0 - - 4tzz sin y
,
~ j= 2
I sin o-os
-_
%+1
s=l
sin a - - a i ]=1
~i
sin ~ + aL
ar
sin
2
sin a
fss
sin o do COS G ~ C O S O~
do
COS O - - c o g O#
2
where we have set a o = O, a n = st, and tbe integral on the right is to be considered as a Cauchy Principal Value. Suppose we are given a sequence o f flows with o*--~-O. Clearly the first two terms on the left remain uniformly bounded in magnitude. So will the third term, for
sin--~ oj I~i 2 1 ~i-I
aj--t
y=l
j=l
o-{- o t
sin-
sin o
coso~coso
do ~
2
sin o - - oj I--[3s 2 I sin o do -- ls" sin o + oj cos o -- cos o ~ 2
280
ROBERT FINN
We shall show that the improper integral on the right of ( 2 3 - J ) becomes unbounded as o*-~ 0. C a s e 1:
~l----- ~ 2 = . . . = ~,---- 0.
We have shown that if o ~
or remains bounded above zero. Therefore there is a 8 > 0 such that n-1
sin o - - oj [13j 2
H r
sin o + o i 2
l
is bounded from zero if Iol< 8.
Denote this function by F(o). Then if ~s~ f
F
(0)
ocos~
sin o cos o~ do
is uniformly bounded in magnitude, since [ F ( o ) I < 1, and we need only consider
( 2 4 J J)
f
sino F (o) cos o -- cos o"
do----f F(o') cos osino -- cos o ~
0
do
0
+ fG(o)
(o -- o') sin o do C O S (~ - -
COS if~
0
where G ( o ) =
F(o)--F(~176 (y~
F'(~')
$" between 6 and o ~
The first
O~
term on the right clearly becomes unbounded as o ~
0. The second term
can be written -
f 0
G(o)
(o o ~ sin 6 do 2sin u (o -- o*) sin -~ (o + o*) -
. ,
-
which, if 8 is sufficiently small, is dominated by 4ML5, where M is a majorant for G(o). C a s e 2: Not all [Sj vanish, j ~ r .
We have shown the existence of
an ~i, 0 < ~ / < 1 , such that oj<~/o ~ j = I ..... r. Therefore l
c5-- c5~ ~j
i~
sin
j=l
sin - ~ f -
2
--
is bounded above zero for 0 > 0 ~. Further, this is a positive increasing
DISCONTINUOUS PLANE FLUID MOTIONS
281
function between o, and ~, non-negative and smaller than unity for o between 0 and o,. Therefore it reinforces the singularity and increases the magnitude of the integral ( 2 4 ~ J ) . Lemma 4.2 is proved. The following lemma is now obvious. Lemma
4.3.
Under
the
assumptions
a n d 4.2, a l l loj} a r e b o u n d e d
away from
of
Lemmas
4.1
o'.
It is now simple to prove that for total curvature smaller than st, the loj} do not coalesce among themselves [40]. We shall later, however, need a stronger result which permits the total curvature to exceed ~r. W e prove first a preliminary lemma, also to be used later. Lemma from are
4.4.
I f ~t is
0 and from bounded
pectively,
away (except,
they
belong),
The
estimate
lengths on the
of
then
from
the
i f o* is points
upper
and lower
for
the segments
a l l {oS} a r e b o u n d e d only
segments
sum of the
above,
separation
of course,
depends the
on the signs
bounded
u, a n d i f t h e
on
from
a lower
corresponding
{I{~jl}; i n p a r t i c u l a r
bounded S a n d S'
walls,
res-
to which
0 and from bound
for
~. the
t o S a n d S', a n d it does
not depend
o f t h e {[3jl.
Suppose the angles al . . . . . 6s converge to zero, the others being bounded from zero. Let us draw, in the ~-plane, a circular arc F of radius 2as about the point ~"= 1 and interior to the circle I'-"l = 1. (Fig. 13). On the part of F for which ~ ~ 0 we have the following estimates, for j < s, when as is sufficiently I'(
t~ 3
~ z~s
small.
as, since a circle of radius
~.i
a~ about ~j will lie interior to the circle
~s
of radius 2as about ~"= 1 ;
>2as
Figure 13. Thus the ratios
[f--~jI~
~" -- ~'il < I ~'-- 1 [ + ]~i - - 1 I <~ 3as. and
"~-
are both uniformly bounded
282
ROBERT FINN
on r. Further, these ratios have unit magnitude on the line zl = O. Consider now the path r" defined by adjoining the segment [ I -
2cs~~ ~ ~ 1, ~l = O]
to the upper half o f F. Denote by Az the difference between the values o f z at the end points o f this path. By ( 3 ~ J ) and the assumptions on ~* and F, there is a constant M for which
F By the above estimates, / Az ] < M . 3xll~Jt (2 + ~t) e~ which tends to zero as as tends to zero.
This means,
if S > l ,
that the
separation point S tends to some point on the upper wall which belongs to a segment other than that of S, and if s-~ 1, that the segment to which S belongs tends in length to zero. Both these possibilities are excluded by the hypotheses of the lemma. It is o f interest to note that for concave nozzles Lemma 4.4 remains true without the assumption on the behavior o f the separation points. in this case if r
For
0 the length o f the first segment would tend to zero.
This is in agreement with what should be expected, for if a concave nozzle wall curves around and back into the boundary, there is no reason to expect the flow to become singular, region of the plane.
even though it may lie on a multi-sheeted
For a convex nozzle the situation is quite different,
for if the wall curves back to the boundary we would have ~p-----0 on the boundary of a simply connected region iying on a single sheet o f the flow, and hence ~p = O. Lemma the there
4.5.
Under
additional is a 8 > 0
the
hypothesis such
that
hypotheses that
of
the
Lemma
nozzle
is
4.4 a n d convex
i f j=#k.
]~li--~sk I > 8
By Lemma 4.4 we know that all {~j} are bounded from zero and ~. Suppose the points ~ss. . . . . (st converge together, the other angles remaining bounded from these.
We
may clearly assume that they all converge to a t
c o m m o n angle unequal to 0 or ~. Let
~= E
~"
W e distinguish
three
./'=$
cases, the first o f which has
already been encountered by Weinstein [40].
DISCONTINUOUS PLANE FLUID MOTIONS Case
1:
3'
283
throughout the process. From (3--3)
and the
assumptions of the lemma, we find the existence of a constant M such that %+~
is+, < M f
t
I I t ,-6I- ,do j=s
as Os+ 1
< M f
I ~ - - r , I- :~ '13. ,16,+, - - ~ I-~,+~ a o
% where the symbol Y/ indicates that [~s+t is omitted from the sum. zi=(~s+l--~ , T=(6--Os)/~?.
Set
Then 1
ls+x < M~I - m ~i .f T- :~ '~J (1
--
"[7)~3s+ 1
d"[
o
which tends to zero as rl--> O. Case
2:7<1
throughout
case there is a constant m > o ~s
the
motion
y+l.
In this
with
t
Cs
Y 11 Io,-o1-io ~slz
but
f Io,-ot-,-
j=s
Ost2
which clearly becomes unbounded if (~ + ~rs and 7-'>" 1. Case We
3: y ~ 1 throughout the motion. This case is now obvious. are now prepared to state an existence theorem on asymmetric
nozzles. Theorem nozzle
of
thesis and
4.1.
Let
total
W
represent
curvature
K<~,
a
polygonal
which
convex
satisfies
Hypo-
B. T h e n
(i) t h e r e
is a s c h l i c h t
in
by
part
separation schlicht which
points flow
the
analytic
bounded prescribed continua
flows there
W and
for
all
A and such
that
at
regions the
to
attach infinity,
A as s t r e a m l i n e s ,
magnitude
whose is
in part
A which
extend
has constant
all
argument, flux,
and
D having
velocity
D bounded
continua
S,S'
in
(ii) a m o n g
region
velocity most
one
D bounded
velocity
by W to the and
a
for
o n A. vectors such
flow
have with
by W and by
approaches
a con-
284 stant
ROBERT FINN in m a g n i t u d e
velocity from
vector
a given
a constant cities
as z a p p r o a c h e s has
one
factor,
but not the
bounded
by
A. E v e r y
argument
multiplying
thereby
the
changing
can
be
flow the
flow
whose
obtained
function
flux
and
by velo-
streamlines.
The requirement that the velocity vectors have bounded
argument
serves to exclude spiral motion. We shall later show by example that it is necessary. Suppose the curvature of W is smaller than st. W e define a deformation path Ilk(t)}, {~j(t), of W with the properties:
XL(t), yL(t),
a) for t = O ,
~j=O,
t = 1 the coordinates are those of the nozzle, and ~ j < l
throughout the motion,
0 ~t~
d)the
j~l
1, of the coordinates ..... n - - l ,
b) for
c) ~j-->_0, j = 1, ..., n - - l , lengths {/j} remain bounded
above and below, and Hypothesis B remains satisfied throughout the motion. From the Levi-Civita representation,
we find immediately that for t = 0
there is one and only one flow among all flows with prescribed flux whose velocity vectors have bounded argument. By Theorem 3.2 there is a segment 0 <__t <_ to in which the motion of the walls can be followd uniquely by a motion of the parameters in the ~-plane. Let T0 represent the upper bound of such numbers to. If T0 ~ 1 the existence of a flow is proved. If T0< 1 there will be a sequence of numbers
t(n)-->To, t (~)
for which the
corresponding parameters {~t(n)}, {o*(")} and {oj(")I converge to finite limits. By the lemmas of this section and classical theorems on harmonic functions we find that these limiting values of the parameters, together with the values {~i(z0)} define a flow through the nozzle at the point t0 on the path. By Theorem 3.2 there is a neighborhood of To in which the motion on the path can be uniquely followed. This establishes the existence of a flow through W. To prove that D is schlicht, observe that for a convex nozzle, the harmonic function 0 changes monotonically on the free streamlines. From this follows immediately that the upper (or l o w e r ) f r e e streamline cannot intersect the upper (or lower) boundary.
Neither
streamline can
intersect the opposite boundary, since this would imply an infinite speed at the point of intersection.
Since 0~ does not vanish on the line r1 = 0
in the ~ plane, the mapping of A to its image in the ~'-plane is D--1. An inspection of ( 3 - - J )
shows that the mapping of W is also 1 - - 1 .
That the map is
in the upper
1--1
semicircle now follows from the
DISCONTINUOUS PLANE FLUID MOTIONS
285
principle of the argument for analytic functions. From this we obtain the schlicht character of the flow function f (z), for this function is schlicht as a function of ~'. Uniqueness is proved by observing that two flows through a given nozzle could both be deformed, following a single prescribed path of the nozzle coordinates, into the unique uniform flow through parallel walls. The two paths in the parameter space would then join at some point, in contradiction to Theorem 3.2. 4.2. T h e We
symmetric
convex
nozzle.
restrict our attention now to nozzles which are required to be
symmetric; we shall, however, make no restriction on the total curvature, which need merely be finite. Self-intersections of a certain type are seen to be permissible. The flows considered may correspondingly be situated on multiply covered regions of the plane. Lemma any convex
The
The
constant
~ is
bounded
above
for
nozzle.
Lemma bounded
4.1a. 4.6.
below bound
Under
the
by a positive from
conditions
of Lemma
4.4,
bt i s
number.
above follows
from the argument
given under
Lemma 4.1. To obtain the bound from below, observe that 1
--1
/=1
where the path of integration may be chosen as an arc in the upper semicircle containing no boundary points other than { = 9 1.
By Lemma 4.4
the integrand remains uniformly bounded on this arc. Therefore ~t -->-0 would imply
ZswZs
,
--~ 0, i.e., that the aperture length would shrink to zero.
Thus, if a nozzle is convex, ~t is bounded above; if in addition the nozzle is symmetric, o * = r and the conditions of Lemma 4.4
are satisfied
(for appropriate walls). Therefore by Lemma 4.6, ~t is bounded below. Definition:
A polygonal, symmetric, convex nozzle wall W will be
called a d m i s s i b I e provided that it admits a continuous deformation into a channel consisting of two straight parallel walls in such a way that (i) the deformation is symmetric with respect to the axis of symmetry of W, (ii) there is a fixed e > O such that all [~j} remain non-negative and smaller than ~ - - e ,
(iii) the lengths of all wall
segments remain
bounded and
286
ROBERT FINN
bounded from zero, the number of segments remaining unchanged,
and
(iv) the separation points do not meet the channel walls during the deformation. A channel wall that is not admissible will be termed i n a d m i s s i b l e . Examples of admissible and of inadmissible nozzles are shown in Fig. 14. Only the upper wall and line of symmetry are shown. It is clear
(a) admissible walls
(b) inadmissible walls Figure 14. that for the first configuration of Fig. 14b there cannot exist a flow of the type desired, for by the monotonicity of the angle 0 on the free streamline it follows that for any flow the wall would bound a region on a single sheet of the flow, therefore h~ would vanish on the boundary of this region and hence vanish identically. For the second configuration of Fig. 14b it is not so obvious that there is no flow, but any flow could be deformed (according to the lemmas of this section) into a flow through the former nozzle. Therefore there is no flow in this case either. Similarly we see that there is no flow through the third configuration. We formulate the existence theorem as a theorem on conformal mapping. Theorem walls
W,
perhaps W
and
4.2.
there
For is
multiply in
part
by
joining
to the
i n D, s u c h
manner
onto
that
a strip
domain points
same
finite
nozzle
unramified,
D bounded analytic
(but
in part
D
width, value
in a and 1/~t
by
continuum
o f W, a n d a f u n c t i o n
f(z) m a p s
of unit
admissible
connected,
a symmetric,
separation
the
prescribed
covered)
defined
approaches
any
a simply
1~1 such as
A
f(z)
conformal that
z~A
[f'(z) l in
any
DISCONTINUOUS PLANE FLUID MOTIONS manner. which
If attention
f'(z)
has
be uniquely
bounded
determined
all symmetric For this
is
to
argument,
the
joining
f'(z)
A, t h e f u n c t i o n
f(z)
functions continuum
by the prescribed
continua
~) is u n i q u e l y
restricted
287
to the
walls
A will W among
separation
(and hence
also
for
points.
the constant
determined.
The proof is quite parallel to the proof of Theorem 4.1, Lemma 4.6 being substituted for Lemma 4.1, and Theorem 3.3 for Theorem 3.2. 4.4. T h e
symmetric
convex
obstacle.
The procedure for estimating the parameters is very similar to that used in the preceding example. The existence theorem is also quite analogous and will not be formulated explicitly. 4.5. A c o u n t e r - e x a m p l e . In Theorems 4.1
and 4.2
we have used an assumption
mat the
argument of f ' (z) remain bounded in magnitude. Some assumption oi this type is necessary for the uniqueness theorem, as we now show by example. Set o (~') ----6~ + i t = ~"- - 1 / ~'. Then ~ 0
for ]~']= 1, so that the jet issues from parallel walls, and
= 0 for ~ = 0 so that the speed is constant on the bounding streamlines of the jet. Obviously ~ is not constant on these streamlines. W e
may
trace them by the relation
f" ir
~ 2 _ 1 d~
Z--Zs= j e
~+1
~ '
1
where we have arranged symmetry for the flow. Neglecting an absolutely convergent term, this expression becomes t
f 1
e - ~'
d~ ~
=-
f e _ i t d tt,
t= - Y1
'
1
a well known convergent integral. As $ -->-0, z converges to a point on the axis of symmetry, as is easily demonstrated. sheeted. Fig. 15 is a crude illustration.
Figure 15.
The flow is infinitely multi-
288
ROBERT FINN
Curvilinear B o u n d a r i e s .
S,
The
existence theorems
of Section 4
extend without
difficulty to
curvilinear boundaries. We define a curvilinear boundary W to be c o n v e x provided that every inscribed polygon with sufficiently small side lengths is convex; we shall say that W is a d m i s s i b l e
if every inscribed poly-
gonal boundary with sufficiently small side lengths and the same separation points is admissible. The t o t a 1 c u r v a t u r e is defined as the upper bound for the total curvatures of all inscribed polygonal boundaries. The proof we present differs in some respects from that given by Weinstein in a simpler case [40], and shows convergence of a subsequence of the boundary mappings. Theorem through larger least
5.1.
any than
~r w h i c h
one flow
boundary
There
convex through
is
at
nozzle
least
W of
satisfies
one total
schlicht
flow
curvature
K not
Hypothesis
any convex
B. T h e r e
symmetric
is a t
admissible
W.
Consider first a convex nozzle W with K ~ zc. Let us approximate W by inscribed polygonal nozzles WN with the same separation points S , S' and the same distance between semi-infinite segments. Clearly K < ~ r for each WN,
and by Theorem 4.1 there is a unique flow through WN.
By
Lemma 4.1, ~N is uniformly bounded above and below, and by Lemma 4.2, o~ is bounded from 0 and 3r. Further, the functions {0N{ form a uniformly bounded family of harmonic functions in the disc [~l< 1. By the bounds on btN and the known monotonicity of ON on the free streamlines, we find that there is a fixed region R in the z-plane which contains the entire flow for every N, and whose closure does not cover the plane.
Let us map R,
by a fixed mapping, interior to a circle in the
X + i Y = Z plane. From the family of functions
1
f
g "
1
df2v
we obtain mappings ZN(~) which are each schlicht in the upper semicircle, and which map the arc [/~l = I, ~ => O] onto a convergent sequence of arcs. Since all image domains lie interior to a circle, the Dirichlet integral,
289
DISCONTINUOUS PLANE FLUID MOTIONS
D(Z~)= f
t d~Z ~
J
z d~d~
I~t < 1 ~>0 remains uniformly bounded Using a well-known l e m m a
(it is equal to the area o f the image domain). which goes back to Lebesgue [20,6],
we find
that the boundary values ZN(~') are equicontinuous on the .arc [l~'l = 1 ,
~l~>o].
Thus we can choose a subsequence o f the {N} (we may assume them reindexed) for which
~tlv+ ~t, +~v--~o*, for which the
uniformly on every compact subset o f
[l~'l< 1], and for which the {zN(()}
converge uniformly on every compact semicircle which excludes the points z ((),
{wN(~')} converge
subset o f the
~" = 0
and
closure o f the upper
~" ~ i.
as the non-constant limit o f schlicht functions,
The limit function is schlicht. Together
with the function, w (~') -----lira wN (~'), and f (~') = lira f N (~),
it represents
a flow through the prescribed curvilinear nozzle W. For admissible
convex
symmetric
nozzles
and obstacles the
proof
proceeds in an analogous way and is therefore not given explicitly. REFERENCES [1] B e r g m a n ,
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