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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