Table Of ContentTHB MSTENCG AND UNT.UBN^SS OF FL0..3
SOLVING FOUR FRDS BOUNDARY PROBLEMS
by
James B. Serrin, Jr.
G
Submitted to the Faculty of the Graduate School
in partial fulfillment of the requirements
for the degree, Doctor of Philosophy,
in the Department of Mathematics,
Indiana University
June, 1951
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ACKNOlU LEDGMJ T
I wish to express n$r sincere thanks to Professor
DavldL Gilbarg for his continuing interest in this work,
and for his invaluable help and encouragement. Also, I
wish to thank my mother for her aid in preparing the
finished manuscript.
TABLE OF CONTENTS
Acknowledgment
I. INTRODUCTION
1. The plane wake problem • • * ....................• • • . 1
2. The Brillouin condition ......................................... 4
3. Methods and results . . . . . . ........................ 6
II. DERIVATION OF THE CENTRAL FUNCTIONAL EQUATIONS
4* Preliminaries . . . . . . . . ............................. 8
$. The Helmholtz problem . 10
6. The jet problem .................................15
7. The finite cavity problem • • • ...................... 21
III. UNIQUENESS theorems
8. A corollary of Juliafs theorem..........................27
9. Applications to hydrodynamics . . . . . . . . 28
10. Preliminary lemmas .................... . 37
11. Uniqueness theorem s..............................................41
12. Existence of finite wake flow s..........................47
IV. THE LIMITATION OF THE FUNCTION l(s)
13. Prelim inaries...........................................................50
14. The behavior of the free streamlines . . . . . 57
15. Domination of h .................................... 65
16. Domination of M * 69
17. Domination of dl/ds • . . . .............................71
18. Domination of d . ........................ 78
V. EXISTENCE THEOREMS
19. Preliminaries . . . . . . . . . 81
20. The Helmholtz problem . . . . . . . . . . . 82
21. The schlicWt cavity problem . . . . . . . . 84
22. The schlicHt cavity problem: Continuous
dependence theorems* Detachment en proue . 87
23* The jet problem . ...................................................93
APPENDIX A ....................................... 98
APPENDIX B ................................................................................ . 107
FOOTNOTES . . . . . . . . . . . . . . . . ill
BIBLIOGRAPHY.................................................................................U7
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" I
F I^URE 3
- 1 -
I, INTRODUCTION
1. The
Consider a steady, plane» irrotational flow of an ideal fluid.
There is a fixed obstacle BC (not necessarily symmetric) immersed in
the flow* which is assumed to have a given uniform velocity at in
finity (that is, the complex velocity u + iv approaches a definite
limit as one approaches the point at infinity).
In order to avoid d'Alembert's paradox and in order to simu
late a realistic situation, we imagine that a whke or cavity is
produced behind the obstacle. This wake is a region of constant
pressure bounded by a streamline 2 ^ detaching from 3 and a
streamline 2 2 detaching froi> C l21J. It is supposed that Z ^
and Z 2 have continuously turning tangents. The pressure
being assumed continuous across the free streamlines 2 ^ and 2
it follows from Bernoulli's law that the flow must have a
constant speed on 2 ^ and 2 3 sPee<* will be referred to
2
as the cavity speed* For such a flow there will be a stagnation
point 0 on the obstacle.
The usual Helmholtz wake problem l1] now states that % 1 ,
2 extend downstream to infinity, IS , and requires us to determine
2
the shape of 2 ^ and % » the location of 0, and the particulars
2
of the flow, (Fig.l). It is obvious that for such a flow the speed
on the free streamlines must be identical to that at infinity.
Another possible behavior of 2 ± and S’ 2 has been pro
posed by Schmieden t6] and treated by Kolseher l3-3J » namely, that
S’ ^ and 2 2 meet and end at a point E behind the obstacle;
continuity requires that the region enclosed by the free stream
lines shall have a cusp at E (Fig. 2). The x>oint at infinity, I,
is an interior point of the flow. The problem of determining a
flow of this kind will be called the finite cavity problem. It
must be mentioned that the d'Alembert paradox is not avoided in
the finite cavity model: this is balanced, perhaps, by the ad
vantage that no stagnation point forms behind the obstacle, and
that the flow is only slightly disturbed far from the obstacle.
The cavity speed is again not arbitrary, a fact which is not
obvious; for symmetric flows it is less than that of the incident
flow (Theorem 3*6).
Now it turns out that the finite cavity problem cannot
always be solved, However, we may state a third problem which
relates the Helmholtz problem and the finite cavity problem in
a neat way. The free streamlines 2 ^ and % 2 are either to meet
at some point E behind the obstacle, or to extend to infinity and be
non-overlapping sufficiently distant from the obstacle.
In the theory of the Helmholtz problem it is well known that
a necessary and sufficient condition for the free streamlines to
be non-overlapping at infinity is that the quantity c*j'(Q)
appearing in the analytic solution be non-positive. Hence the
problem above can be stated: find either a solution of the finite
-3-
cavity problem or a solution of the Helmholtz problem in which
co »(0) $ 0* It turns out that in the symmetric case of this
problem, that the free streamlines must be simple non-intersect
ing curves (§ 11), and I have therefore named this the schlietit
cavity problem* To be convinced that this is a well set problem,
note that there is one and only one symmetric solution (Theorem 5*2).
In the problems just proposed the cavity speed is never
greater than the speed of the incident flow. This fact is not
borne out by experiments in cavitation phenomena ^ , and
hagner has proposed a third possible behavior of the free
streamlines which avoids this difficulty* In his flow model
the free streamlines turn back, forming a jet (Fig.3),* Such a jet
would strike either the rear of the obstacle or the free stream
lines thus causing an unsteady flow. If, however, the non-physieal
assumption is made that the jet is not interrupted or, more pre
cisely, that the jet forms a second sheet of the flow plane, the steaiy
flow is still mathematically possible* The point at infinity in
the main flow plane, denoted by I, is an interior point of the flow,
while the point at infinity on the jet, denoted by B , is a
boundary point. This problem will be called the jet problem, and,
as before, we are to determine a flow of the type described.
In this model, the cavity speed, as well as that of the
incident flow, may be arbitrarily pre-assigned, with the sole
restriction that the former be greater. This restriction is
dictated by cavitation phenomena as indicated. However, from a
- 4 **
mathematical standpoint there is no reason to retain this con
dition, and we shall actually not use it. In fact, the condition
is not even a necessary one for such flow to exist, he shall
show (Theorem 5*6) that, for a large class of obstacles,
symmetric flows of the type postulated exist if and only if the
cavity speed is greater than a certain value. This value de
pends on the obstacle in the following remarkable way: it is just
the cavity speed in the (unique) symmetric solution of the schlicht
cavity problem for the obstacle.
2* The Briilouin Conditions.
In the Helmholtz free boundary theory there are two well-
known conditions which should be met by any flow which we assert
to solve this problem* Mamelys
A. The free streamlines must never intersect themselves or
the obstacle.
B* The cavity speed must exceed the speed at any interior
point of the flow.
The second condition reflects through Bernoulli's Law that the
pressure in the body of the fluid never falls to zero. The
validity of these conditions when applied to the other models must
be examined, and analogous conditions set up for them.
For the jet model, the first condition must obviously be
modified to account for the jet passing through the obstacle. The
modification is made by dividing each streamline Into two arcs by
