Zeitschrift für Angewandte Mathematik und Mechanik: Band 60, Heft 5 Mai 1980 [Reprint 2022 ed.] 9783112649886

Citation preview

INGEN

U N T E R M I T W I R K U N G V O N E . B E C K E R • H. B E C K E R T • L . B E R G • L. B I T T N E R • L. C O L L A T Z W . F I S Z D O N • H . G A J E W S K I • H. G Ö R T L E R • J . H E I N H O L D • H . H E I N R I C H • J . H U L T A . J U . I S C H L I N S K I • R. K L Ö T Z L E R • P . H . M Ü L L E R • H . N E U B E R • W . O L S Z A K • K . O S W A T I T S C H A . S A W C Z U K • L. S C H M E T T E R E R • J . W . S C H M I D T • H . S C H U B E R T • G. G. T S C H O R N Y H. U N G E R • F. W E I D E N H A M M E R U N D F. Z I E G L E R HERAUSGEGEBEN V O N G.SCHMIDT, BERLIN C H E F R E D A K T E U R : G. S C H M I D T

RE D A K T EU R E: W. H EI N Rl C H, H. W E I N E RT

1980

B A N D 60

HEFT 5

A U S OEM I N H A L T H A U P T A U F S Ä T Z E H. F. Bauer/St. Metten/J. Siekmann : Dynamic Behavior of Distensible Fluid Lines Carrying a Pulsating Imcompressible Liquid / K. Beyer: Zur Stabilität einer ferromagnetischen Flüssigkeit in einem vertikalen Magnetfeld / Woon-Shing Y e u n g : Perturbation Solutions for the Particle Trajectories of a Gas-Solid Mixture Entering a Curved Duct / H. Ramkissoon/S. R. Maj u m d a r : Potentials and Green's Functions in Micropolar Fluid Theory / H . Friedrich/K. H e n n i g : Numerische Auswertung von Berechnungsformeln für Überschreitenswahrscheinlichkeiten stochastisch beanspruchter mechanischer Systeme K L E I N E

M I T T E I L U N G E N

B U C H B E S P R E C H U N G E N E I N G E G A N G E N E

BÜCH.ER

N A C H R I C H T E N

A K A D E M I E - V E R L A G ZAMM EVP 18,— M

Bd. 60

Nr. 5

S. 221-276

B E R L I N Berlin, Mai 1980 34115

INHALT Hauptaufsätze Seite H . F. Bauer/St. Metten/J. S i e k m a n n : Dynamic Behavior of Distensible Fluid Lines Carrying a Pulsating lmcompressible Liquid 221 K B e y e r : Z u r Stabilität einerferromagnetischen Flüssigkeit in einem vertikalen Magnetfeld 235 Woon-Shing Y e u n g : Perturbation Solutions for the Particle Trajectories of a Gas-Solid Mixture Entering a Curved Duct 241 H . Ramkissoon/S. R. M a j u m d a r : Potentials and Green's Functions in Micropolar Fluid Theory 249 H . Friedrich/K. H e n n i g : Numerische Auswertung von Berechnungsformeln für Überschreitenswahrscheinlichkeiten stochastisch beanspruchter mechanischer Systeme 257 Kleine Mitteilungen J . G . V e r w e r : On Generalized Runge-Kutta Methods Using an Exact Jacobian at a Non-Step-Point G . U . S c h u b e r t : Inversion der hyperbolischen Kepler-Gleichung mittels Laplace-Transformation J. S c h n e i d e r : Der Satz von Gerschgorin für gedämpfte Schwingungen K. M l a d e n o v : Über das dynamische Verhalten eines Druckstabes mit tangententreuer Endbelastung

263 265 266 268

Buchbesprechungen

271

Eingegangene Bücher

273

Nachrichten

274

Wir bitten, Manuskriptsendungen zweifach (Original und eine Kopie, sprachlich einwandfrei, Formeln mit Maschine oder in Druckschrift geschrieben) an folgende Anschrift zu richten: Zeitschrift für Angewandte Mathematik und Mechanik z. Hd. Herrn Prof. Dr. Günter S c h m i d t Zentralinstitut für Mathematik und Mechanik an der Akademie der Wissenschaften der D D R DDR-1080 B e r I i n, Mohrenstraße 39. Z u den Arbeiten, die als Hauptaufsätze bestimmt sind, ist auf gesondertem Blatt eine Zusammenfassung von 5 bis 10 Zeilen in englischer und (möglichst) deutscher und russischer Sprache beizufügen. Ausführliche Hinweise für die Autoren, um deren strikte Berücksichtigung gebeten wird, finden sich im Anschluß an das Inhaltsverzeichnis des Jahrganges 59 (1979). Die Autoren erhalten von den Hauptaufsätzen 75, von den Kleinen Mitteilungen 25 Sonderdrucke ohne Berechnung, darüber hinaus weitere Sonderdrucke gegen Berechnung. Der Verlag behält sich für alle Beiträge das Recht der Vervielfältigung, Vorbereitung und Übersetzung vor. Bestellungen sind zu richten — in der D D R an den Postzeitungsvertrieb; an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR-1080 Berlin, Leipziger Str. 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der B R D und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle K U N S T U N D WISSEN, Erich Bieber, 7000 Stuttgart 1, Wilhelmstraße 4—6 — in Ö s t e r r e i c h an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle K U N S T U N D WISSEN, Erich Bieber GmbH, Ch-8008 Zürich/Schweiz, Dufourstraße 51 — i m übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR-7010 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR-1080 Berlin, Leipziger Straße 3—4 ZEITSCHRIFT FÜR A N G E W A N D T E MATHEMATIK U N D M E C H A N I K Herausgeber und Chefredakteur: Prof. Dr. Günter Schmidt. Redaktion: Dr. Winfried Heinrich, Dr. Horst Weinert, Dipl.-Math. Friedhild Dudel, Helga Rühl, Zentralinstitut für Mathematik und Mechanik der Akademie der Wissenschaften der DDR. Verlag: Akademie-Verlag, DDR-1080 Berlin, LeipzigerStraße 3—4; Fernruf: 2236221 oder2236229 Telex-Nr.: 114420; Bank: Staatsbank der DDR, Berlin, Kto.-Nr.: 6836-26-20712. Anschrift der Redaktion: Zentralinstitut für Mathematik und Mechanik der Akademie der Wissenschaften, DDR-1080 Berlin, Mohrenstraße 39; Fernruf: 2000561. Veröffentlicht unter der Lizenznummer 1282 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckerei „Thomas Müntzer", 5820 Bad Langensalza. Erscheinungweise: Die Zeitschrift fUr Angewandte Mathematik und Mechanik erscheint monatlich. Die 12 Hefte eines Jahres einschließlich Tagungsheft bilden einen Band. Bezugspreis ¡e Band 360, — M zuzüglich Versandspesen (Preis für die D D R 216,— M). Bezugspreis je Heft 30, — M (Preis für die D D R 18,— M). Bestellnummer dieses Heftes: 1009/60/5. © 1980 by Akademie-Verlag Berlin • Printed in the German Democratic Republic. A N ( E D V ) 35937

ISSN 0 0 4 4 - 2 2 6 7 H . P . B a u e r / S t . M e t t e n / J . S i e k m a n n : D y n a m i c Behavior of Distensible Fluid Lines

221

ZAMM 60, 221 - 2 3 4 (1980) H . F.

B A U E R / ST. M E T T E N / J . SIEKMANN

Dynamic Behavior of Distensible Fluid Lines Carrying a Pulsating Incompressible Liquid *) An analysis of the propagation of pressure waves through a thin-walled elastic tube, filled with a streaming incompressible fluid, is presented. The outer surface of the tube is assumed to be free. Axially symmetric solutions for the linearized equations governing the motion of the (viscous or nonviscous) fluid and the (elastic) solid are obtained..The motion of the tube wall is described by the D o n n e l l shell equations. The solutions lead to a complicated transcendental dispersion equation relating wave number and frequency. This equation is studied by both analytical and numerical means in some detail. Vorliegende Arbeit befaßt sich mit der Ausbreitung von Druckwellen in einem dünnen elastischen R o h r , das mit einer strömenden, dichtebeständigen Flüssigkeit gefüllt ist. Die Außenwand des Rohres soll als spannungsfrei angenommen werden. Die axialsymmetrischen Lösungen der linearisierten Grundgleichungen für Flüssigkeit und Festkörper werden angegeben, wobei die Flüssigkeit als reibungsbehaftet bzw. reibungsfrei vorausgesetzt wird. Die Bewegung der Schalenwand werde durch die DoNNELLSchen Schalengleichungen beschrieben. Die Lösungen führen auf die sogenannte Dispersionsgleichung, welche Wellenzahl und Frequenz miteinander in Beziehung setzt. Resultate analytischer und numerischer Untersuchungen der Dispersionsgleichung werden mitgeteilt. B n a c T o n m e ñ paöoTe paceMaTpmsaeTCH p a c n p o c T p a H e m i e y n a p H U x bojih b t o h k o h y n p y r o i i T p y ß e , nepe3 KOTOpyio n p o T e n a e T í k h h k o c t I j c i i o c t o h h i i o h n j i o T H o c T b i o . B H e m H H H o S o j i o w a T p y ö a n p e ^ n o j i a r a e T C H CBOÖOHHOH. BblBOAHTCH OCeCHMMeTpimHbie peilieHHH JIHHeapH3HpOBaHHbIX OCHOBHblX ypaBHGHMÜ HJIH h í h u k o c t h h T B e p n o r o T e j í a , n p H q é M HiHHKOCTb n p e n n o j i a r a e T C H h b h 3 k o K h H e B H 3 K O ö . J U b i u k c h h g c t c h k h T p y 5 b i oiiHCbiBaeTCH y p a B H e i M e M floimeji;ia k j i h o ö o J i o ^ e K . P e m e H H e npHBOflHT k cjlOJKHOMy T p a n c u e n «eHTHOMy y p a B H e n H K ) p a c n p o c T p a H e H H H b o j i h , c o n e p w a m e i v i y BOJiHOBoe h h c j i o h i i a c T O T y . . 3 t o y p a B H e H H e fleTajibiio HCCJieayercH c anajiHTHHeci;oii h nucjienHOtí t o h g k speHHJi.

I. Introduction Currentproblems in biological modeling, fluidics, and missile fuel systems (propellant lines) have stimulated researchers to study the propagation of waves in elastic or viscoelastic tubes containing liquid at rest or filled with a flowing fluid. Considerable attention is paid, particularly on the part of physiologists concerned with blood flow and the blood pressure pulse in mammalian arteries, to the problem of wave propagation in a system in which the distensibility of the tube is of far greater importance than the compressibility of the liquid. Thus, in a first approximation, the fluid may be regarded as incompressible. There are a number of reasons why it is important to know the precise distribution of the flow velocity, the shear rates, and the mass transport in the blood circulation system. One of the most important is in relation to the onset of atheroma, a degeneration of the arterial wall which leads to the hardening of the arteries. Pathogenic investigations of arteriosclerosis show that the incidence of the disease increases with age, is more pronounced and occurs earlier in male humans, is accelerated by hypertension and prefers certain vessels and locations. Medical evidence indicates that the basic laws of hydraulics determine the behavior of the flow in the arterial tree after loss of elasticity. The network of vessels containing the blood flow exhibits rather complicated distensibility relations, which in the case of the arteries are of great importance in matching the pumping action of the heart to produce a steady perfusion of the peripheral capillaries. Therefore, the problem that has to be treated is that of pulsatile flow in an elastic vessel. E U X E R [ 1 ] in 1 7 7 5 was convinced that ventricular contraction sets up a propagation of waves through the arteries, but he did not succeed in obtaining an expression for the wave velocity. E . H . W E B E R [ 2 ] announced that he and his brother Tu. W E B E R had measured the wave velocity in elastic tubes in accordance with the predicted value. Their brother W . W E B E R [ 3 ] developed a theory of wave transmission, which appeared finally in 1 8 6 6 . Later R E S A L [ 4 ] , KORTEWEG [ 5 ] and MOENS [ 6 ] worked on the problem also and rediscovered the formula for the velocity of the pulse wave, already found by YOUNG [ 7 ] in 1 8 0 8 , namely, that the wave propagation velocity is proportional to the square root of YOUNG'S modulus of elasticity of the wall material and the thickness of the wall and inversely proportional to the square root of the liquid density and the diameter of the vessel. According to these investigations, waves of all frequencies are propagated at the same velocity and aré not attenuated. This is not true; experimental evidence has shown that the wave velocity increases considerably with increasing frequency of the disturbance. In connexion with the treatment of the problem, we must remark that the most striking feature of the pulse is its regularity. I t is maintained for long periods of time, if no changes occur due to exercise, etc.. Thus it is realistic to regard the arterial pulse as a compound wave with a fundamental frequency, namely that of the heart rate. The arterial circulation is therefore in a steady state of oscillation, and the pressure may be represented by a FOURIER series. Since the average flow velocity of blood is fairly small compared to the propagation speed, a linearization of the equations of fluid motion seems to be justified; In addition, the distension of the arteries is not very large, allowing us to employ linearized shell equations. The diameters of the blood vessels under study are not too small, hence *) Presented in p a r t a t the GAMM Annual Scientific Conference, Wiesbaden, 1 6 — 2 1 April, 1979. An a b s t r a c t „ Ü b e r Wellenbewegungen inkompressibler Flüssigkeiten in dünnen elastischen R o h r e n " appears in ZAMM. 15 -ZAJI3I, Bd. 00,11. 5

222

H. P. BAUER / ST. METTEN / J. SIEKMANN : Dynamic Behavior of Distensible Fluid Lines

the blood may be considered as a NEWTONian liquid. This is> quite a good approximation for the range of higher shear rates which predominate in the larger blood vessels. Regarding biomechanics in general and flow and pressure in the arterial system in particular, excellent reviews of-the state of the art have been given recently [8 through 21]. I t is the purpose of the present investigation to make available analytical and numerical results obtained from an extensive study of the dispersion equation [22]. Only the flow in a single tube has been dealt with. This, however, is the basic element out of which, finally, a successful model of the circulatory system will be formulated. 2. Formulation of the problem Let us consider an infinitely long circular cylindrical tube of radius a and wall thickness h, filled with a streaming homogeneous, viscous and incompressible liquid (Fig. 1). The tube is assumed to be of an elastic and isotropic

v////////////////////y//////,

Fig. 1. Fluid flowing through an elastic cylindrical tube, geometry and notations

material. Restricting our attention to a HAGEN-POISEUILLE flow in axial direction, superimposed by small amplitude motions of the liquid and the tube wall, the equations of motion as well as the boundary conditions can be linearized. The small amplitude motion is due to some periodic pressure disturbances. Then the NAVIER-STOKES equations, governing the axisymmetric flow of a viscous NEWTONian fluid, can be put into the following form: du

1 8p

dt

q 8r

d2u 8r 2

1 du r 8r

u r2

82w (1)

8a2

for the radial direction, and dio

1

d2w

J 8s

~dt~

ST2"'

1 Qw r 8r

d2w 8z2

(2)

for the axial direction. We further have to add the continuity equation 8m u dw T, I 1" ' = dr r "8a

0.

(3)

I n these equations, r and z are cylindrical polar coordinates, with r measured in radial direction from the centerline of the tube, and z measured in axial direction along the centerline. The velocity components in direction of increasing r and z, respectively, are denoted by u and w, t is the time. Moreover, we denote the density of the liquid by g, the kinematic viscosity by v ( = /liIq, with y, as the dynamic viscosity), and the pressure by p. The motion of the thin-walled tube {hi a

, | \

j = 1, 2 , ^

(18) (19)

H. F . B a u e r / S t . M e t t e n / J . S i e k m a n n : Dynamic Behavior of Distensible Fluid Lines

224

and (20) o

r

with 2_ %n —

2

tCn

Ï71CÛ

(21)

V

and J1 as the B e s s e L function of first kind and first order. The solution of equation (20) with W0(a) = 0 reads (22)

This is, however the velocity distribution of the well-known H a g e n - P o i s e u i l l e flow. Equation (19) yields Wjn{r) = AjnJ0(xnr)

kn ± —-

PfnJa{iknr)

j = 1 j = 2

,

(23)

with Apt as an integration constant. Differentiation of equation (16) with respect to r and substitution of the expression so obtained into equation (18) allows the determination of Ujn, namely Ujn(r) = ± -— AjnJ^r) Kn

-f-

ik n

~

1

UQOJ

yfnJ\{ìknr)

fihn

j

I \

(24)

2

Next we turn to the solution of equations (4) and (5), governing the behavior of the shell. Defining the flexural rigidity of the shell by Eh3

D =

12(1 -

(25)

v2)

there results, employing equations (14) and (15) as well as (22), (23) und (24), after a few standard manipulations d^, dz*

+

Eh Da2

_

2 - v (QPo . — -1\— •2 dz I

Po

io

const ,

«

(26)

and .

+

il

(1

h* Ï2 " "2) Eh

1-v».i_ E Sn

I

Mi^a)

-r. t ivkn . [n2co2 T hn — + (1 ±

(1 — V2) ßj^ iknJJ

(ikna)

M

-

I itcnv „ 2(1 - v2) ., r , v . =F Çjn ± ST iknfiJ Mna) Ajn j a hin, 2

X„ LQNW

V2) Q -

2

kn Cjn -i

ikn /ifxkn x'i \(>na>

^ j

-

L] J[{ikna)\ J

(l-V2) ßj£ikn gnat

J

+ -(27)

= 0 ,

n I

i*l-kl\ K l M

Ain

± (28)

i?n.= 0 .

In these equations the upper sign belongs to j = 1 (waves travelling in the positive z-direction), while the lower sign belongs to / = 2 (waves travelling in the negative «-direction). Equation (26) is recognized as the governing differential equation occurring in the general (bending) theory of cylindrical shells loaded symmetrically with respect to its axis. Since solutions are discussed at length by Timoshenko and W o i n o w s k y - K r i e g e r [24], further details will be discarded. They are irrelevant for the problem under consideration. The boundary conditions (6) and (7) furnish the relations . 1 y -r- îkn j , iwnÇjn + —Jy\x.na) Ajn X»

iconÇjn — J0(xna)

ì-kn 5 juxnincoo

k Ajn ^ - n -J0(ikna) ona>

r 1 J-^iknü) Pfn = 0 ,

(29)

— Pf„ = 0 .

Equations (27) through (30) constitute four linear homogeneous algebraic equations for the unknowns and Pfn. A nontrivial solution of this system exists only, if the determinant of the coefficients vanishes.

(30)

'Qpl,

A..jn

225

H. F. BAUER / ST. METTEN / J. SIEKMANN : Dynamic Behavior of Distensible Fluid Lines

This postulate, t h e so-called dispersion relation, leads to a transcendental equation, which may have an infinite number of solutions kn. In the problem under study, however, only the real values of kn are of interest, since they represent propagating waves. For pure imaginary or complex kn-values, the waves are attenuated and decay, depending on the magnitude of Im(/; a ). These waves are non-propagating waves and are of less interest in t h e present investigation. From the above equations (28) through (30), the ratios ZjnjPfn, CjnJPfn and A j ^ P f n can be obtained and introduced into equations (12), (13), (14) and (15). Hence t h e flow velocities and t h e wall motion are determined for a given pressure (cf. equation (11)) a t z = 0. I n many practical cases the pressure can only be measured as the. mean pressure over the cross-section. Therefore a

2>mean(z, t) = ^ j"rp(r, z, t) dr = p0 - ¡ ^ j

z +

o 00

2

1

z ?r h(kna) d n=l ten

H

-

cos

{[^ln

=

{not — knz) -f- Pln sm(na>t — knz)]

+

(31)

+ [i*>2n cos (nmt + knz) + P«n sin (ncot + knz)]} .

This expression yields at the location z = 0 the value °° 1

2

-

t) = Vo + — 27 7 « 11 = 1 Kn

{(Pm

-

=

=

+ Pin) cos ncot + (Pln

+ P2n) sin nut}

(32)

,

while 8?>mean(0, t) dz

fyo H E Ii{kna) {{Pin — Pin) cos nwt — {P2n — Pm) sinwwi} . dz 2=0 ® «=!

(33)

7 t denotes t h e modified BESSEL function of the first kind and first order.

4. Discussion of the Dispersion Equation The dispersion relation, can be p u t into the form A = Det (a^p) = 0 ,

«,

(34)

= 1, 2, 3, 4

where elements of the determinant are given by «11 =

h2

1

+ Y2

1 — v2

kn

&

e—

1 - v2

i1

iknv «21= T a

a

Eh

3i = inw ,

«34 = — — t

iknJ-^iknà)

l) )

>3 =

±

Eh

¡xiknJ¿x„a)

,

J[(ikna) 11

, o ,i q niwi — k n ,

ikn lifikl

'

2(1 - v2)

a

a23 = —

A

— vi 2

—

v 2 — /K r / «tt

Jx\xna)

\

,

fit, onw

Kn \Qnc0

,

Ji(ik n a) ,

¡J,Xn \Qnto

«41 = 0 ,

i2 = T —

ikn «33 = T — JMna)

a32 = 0 ,

— —

iknv

— vif -

a„ = —-— E

1 - v2 ±

'

J, 1 {ikna) + ~ Xn \Qna>

Eh

«24 =

a

j

ati = into ,

ai3 = —JJx„a)

,

a., — T

kn JJikna) Qnco

.

With J[(xna)

= J„(xna)

— J^XnO.) ,

J[(ik„a)

= J0(ikna)

—

J^ik^a)

and in(xua)

.= — -, Ji\xna)

Gn(ikna)

:=

—•-—Jj(t kna)

(35)

226

H. F. BAUER / ST. METTEN / J . SIEKMANN: Dynamic Behavior of Distensible Fluid Lines

there follows after straightforward but tediuns calculations 1 h2 « i + Ì2

A = Det (a„ß) =

1 - V2 _ E Gn2°J''

4

fl - V l

E

t K„onoi

X

+

. (1 - V2) Ehxl

X

kn lipkl U^co

\ J

en2a>2

|t

H

nco a« + * 12

5- X . 1 - v2 kn

(1 - y2) icl tikn libici Ehxn \xl [gnoj

\ )

Qn^oj3 ) x

ftknY qnwJ

nmvkn Ehx„ Ti

\

axn) \gnw

2Ä/7j, (%jxìcfi | Gn(ikna) H - I O xn \Qno) 1

, ( 1- - v1) K

/

, n{ikna)

— TT 1 -il iknaj)

nwvkn Ehv.non

am3/2

(|É,| < 1, | M < 1) , g^m) rt + 0{K),

(|m| >\kl\, ]x„a\
) and g(w) are given by /(a>) = 8(1 — p) Qn2a>2(4:iafi + 8ihvQ — oha2noj) , g(a>) = (1 - i 2 ) hQQaWco3 ^ 2(H(1 - v^ h^aVw2

(38 a) + 4(1 - v2) yh2a3n3(o3 - 8E e ha 2 nw -

- 80(1 - v ) aju nco - 4i(l - v ) QfjtaVm - 64iEfiv + 32(1 - v2) Qfivhm + 2

2

+ 8(UEhfi - 4Eh2aQnto .

2

2

(38b)

H. P.

B a u e r

/ St.

M e t t e s

/ J.

S i e k m a n n

: Dynamic Behavior of Distensible Fluid Lines

227

'-IM

Fig. 2. Plot of

iOn{ihnd), Iv(kna)

and

I,(kna)

vs.

k*a

A plot of the real function iOn(ikna) vs. kna (kn real) is shown in Fig. 2. Note that iGn(ikna) iGn{ik.na) —> 0 for kna —• oo. For small values of a and h, equation (37) reduces to 2(1 — v2) gan2a>2 Eh(5 - iv)

1/2

oo for kna -> 0, and

(39)

With that we obtain for the phase velocity 5 — 4v 2m2

1/2

/h

(40)

Cl :

a

where E (1 — V2) Q

cL

1-/2

(41)

denotes the (finite) wavespeed of longitudinal waves in the tube wall. If o)

[yjn) • \kn\, there results from the dispersion equation (36) gh + oa + otm" 1 / 2 ) g ah

v 4n

(42)

and therefore c ~

gh + ga pah

4 nkn

(43)

kn\ — Re {kn

Figures 3 — 6 represent results of the numerical calculations employing equation (37) and the following typical values of the parameters: fj, — 0.04 g c m - 1 s - 1 , v = 0,038 cm 2 s~ ~g = 1,1 g c m - 3 , E = 157 • 10 s g c m - 1 s~2, v = 0,5, a — 0..1 cm, h = 0,019 cm and n = 10. The graphs labelled by F correspond to h]a — 0,19; n = 10; v = 0,5 and g/g = 0,95, while the other curves indicate ,the altered parameter, the others remaining unchanged. The angular frequency a> was made dimensionless by a>0 — [E/ga2( 1 — v 2 )] 1 ' 2 . For very small values of u>, damping of the waves

P-03 £2

0.5 1,0 10ewluua —-

1

Fig. 3. r i o t of 10 s | R e ( i ; ! ) | a vs. 10" . In case of very small w, we observe a constant phase velocity (Fig. 5). Figure 6 (a — d), show the different amplitude ratios AjnIPjn, CjnIPjn and vs. mja)0, where Pjn = Pfn. In particular we have (44 a) fjn -p p ~ + •*•T jn

H(kn,a>)

2i

H{kn,a>)

3/2

{.no)'

±

(44 b)

gn2w2~

™J12

v

'

ikn

—

' 2i

(44 c)

gan^ai-'

K

with j = 1, 2 and 2 ih H{kn,

a>) =

±

\

(1 — v2) gn2a>2 \ ah (1 -

i?

E

g v2)gn2(a2

\ 12gv _

/u

igana>\

~E

7 \~a

K

4

+ o{kl)

(45)

.

The solid lines correspond to R e H{kn, a>), the dashed lines to Im H{kn, a>). Next let us consider the limiting case of a rigid tube (E ->• oo). The dispersion equation (36) reads h%

1 '

12

L'

k,i -f-

Vkl

K n : — Gn{ikna) nw

i (ivkl - vkn \ na)

1

Fn{xna) =

0.

(46)

H. F. Bauer / St. Metten / J. Siekmann: Dynamic Behavior of Distensible Fluid Lines

229

This relation is satisfied if one of the brackets vanishes. The first bracket is independent of the kinematic -viscosity v and hence yields solutions for a nonviscous fluid, namely (cf. B ) kn = 0

and

ki — —

12(1 - v2

(47)

I n both cases no wave propagation occurs. A detailled study of the second bracket reveals that waves are not propagated. This result is confirmed also b y numerical analysis-[22]. On the other hand it has been demonstrated by R u b i n o w and K e l l e k [17] that in a rigid tube, filled by-a compressible nonviscous fluid: waves occur due to compressibility. Finally, the special cases of waves in a fluid contained in a negligibly thin.tube (h = 0) and waves in an empty •thin tube (q = 0) deserve mentioning. I n the first case the situation is the same as wave propagation in a fluid jet. There exist waves, however, the propagating modes are strongly attenuated. For waves in the tube alone the dispersion equation (36) can be written as 1 a*

+

1 — v2 _ OW2ft)2

v2

E

12 * *

— Gn(ikna) ma

+

E

— (—- — 1 ) vkn \ nco '

Fn(xna)

ik n

—r

Q

(48)

y.n

One solution of this equation is given by kn = 0, since (knjnco). On(ikna) remains finite for kn - » 0. W i t h that c ->• oo. T h e expression in the second bracket is identical with the second bracket of the dispersion equation for the limiting case of a rigid tube (cf. eq. (46)). As shown there, waves do not propagate along the tube. The first term of (48) is independent of v; for n = 1, the bracket is identical with the dispersion equation for an inviscid fluid in an " e m p t y " tube and will be studied to some extend in the next section. I t turns out that for real values of k, the values of cd are real too and positive. For n oo we have to = 0 and therefore c = 0. The solutions for 1 < n < oo are o>± = [kiK)

± (fl(K)

-

(49)

gi(^))1/2]1/2,

where 1 + — hïaïkn - f aïkn fi(K)

=

1 -

r2

9i(k„) a2gn

=

(I

(50a),

v2 _

-

(50b)

Plots of kna vs. co/co0, and cjcL vs. co/co0, based on equation (49), are shown in Figures 7 and 8. T h e y exhibit in particular the role of n, and also of the P o i s s o n number (standard value v = 0,5).

CICL 1000 100

0,5

10

01 F i g . 7. P l o t of (

kna

vs.

iolw0

) : o>\ solution, ( —

based on E q . (49) f o r d i f f e r e n t values of

v

and n.

— — ) : (»--solution

B. I n v i s c i d i n c o m p r e s s i b l e f l u i d i n an e l a s t i c

0,Z 0,3 —

Oi

c/c£

F i g . 8. P l o t of vs. co/îo0 based on E q . ( 4 9 ) f o r d i f f e r e n t values of n

tube

T h e next part of the discussion deals with the behavior of an ideal (nonviscous) fluid in an elastic tube. I n the absence of waves, the tube is a circular cylindrical shell of constant thickness, and the fluid is at rest. W i t h negligible. v, the N a v i e k - S t o k e s equations are replaced by the E u l e r equations, while the continuity equation remains unchanged. The D o n n e l l equations are simplified also, in the first one, the term 2/j, I ^ J

is zero, while in the

second equation, the right hand side vanishes. Again, let us seek a solution of the linearized equations of motion of the fluid and the shell which is harmonic in time and in the longitudinal space direction: [p, ii, w,

f ] = [¿5(0, u(r), £•„, Z0] exp (— kz + cot).

(51)

230

H. F . BAUER / ST. METTEN / J. SIEKMANN: Dynamic Behavior of Distensible Fluid Lines

Here S0 and Z0 are constants. Substitution in equations j>(r) = PQJ0(ikr) ,

where P0 is an undetermined constant. From the /1 ^! \a

v a

'

JQ

(9)

and the

EULEB

equations yields (52)

equations there follows

v2

~12~

+

DONNELL

and

_ 1 îi>(r) — P^J^ikr) , 00)

1 — _ „\ _ 1 — v2, • P0J0(ika) ---QU)2 50 = E E

h2ifi

+

- k2Z0 - -ik a

(8)

k u{r) = — PyJ^ilcr) , QÜ)

(53)

,

—ëw2Z0 = 0 .

(54)

Substitution of equation (51) into the boundary condition 6f

for r = a ,

et

(55)

and of the resulting relation into equation (52) yields k

—

„ icoE0.

P0/1(I'AA) =

(56)

Now equations (53), (54), and (56) are a linear homogeneous, algebraic system of equations for the unknowns P 0 , S0, and Z0. In order to obtain a nontrivial solution, the determinant of the matrix of the coefficients must vanish. This leads to the dispersion equation 1 - v2 -Y U J0(ika)

+

E

7

U JMa)

E

7 \a2

1

CO2

.

1

hk ^

hW 12

1

'

+

g J0(ika) i j> J^ika) hk

\

h2ki

=

(57)

0

The solution of this equation can be put into the form -Ti

(58)

{f«{k) ± [fl(k) - fir0(&)]1/2}1/2 , Q Ia(ku) 1

- m

=

2 (i \

+

4

(59a)

1

g Ix{ka) hk (59 b)

9o(k) = a2k2 (l - v2 + ~ h2a2Jc4j 1 + 1-

1

g I-^ka) hk

From equations (54) and (56), we find for the ratios S0jZ0 and PajB'„ the relations a[(l - v2) qo)2 - Ek2] ivEk

S0

P0 _

Qto2 klx{ka)

(60)

Numerical results according to equations (58) and (60) are plotted in Figures 9—13. Figures 9 and 10 show that (co/ft>0)+ is nearly independent of the parameters, (hja), (o/o), and v. Furthermore, (a>/co0)~ increases for decreasing

le) ka

0,1 0,Z 0,3 0,1- Oß Oß 0,7 Oß Oß 1,0 w!wô— . Note also that the different values of v in Fig. 18 c are labelled in the wrong order.

232

H . F . B A U E R / S T . M E T T E N / J . SIEKMANN: D y n a m i c B e h a v i o r of D i s t e n s i b l e F l u i d L i n e s

F o r E —> 0 (or q -> oo, respectively) we find co ->• 0, while for E • oo, we have k and co = [Ej{l — v2) g] 1/2 k, while q = 0 (empty tube) leads to

0. W i t h h = 0, we obtain co = 0

'¿V/2

co

(62)

and thus to the constant phase velocity c

~

(1

-

v )

'

l

2

c

(63)

.

L

Of interest is also the case of a massless tube, yielding (64)

'

k

2 g a )

and therewith _ c

( E h y i *

(65)

v

~

2 g a J

where c v denotes Y O U N G ' S velocity. W e see t h a t this velocity depends on both the tube and t h e fluid properties. I t is this velocity which is generally observed in measurements of t h e pulse velocity in mammalian arteries. If all t h e parameters are real it turns out t h a t t h e inner b r a c k e t of equation (61)< is real too. Hence for real and positive values of k, t h e values of co are real and positive, and a> increases almost' linearly with k. In case of (A/a) 1 there follows from t h e solution (61), neglecting the terms o(...), E

1-

q{

1/2

. h

v2)

-l-o I —

(66)

- o f / A y *

k ,

.«

Since both co and k are supposed to,.remain small, we have to postujate t h a t the'slope becomes not too large. This condition is satisfied for o r , while for a> Y O U N G ' S modulus ( E ) should be not too large, or the density of t h e t u b e material {g) not too.small. Then c + approaches the constant wave speed c L .+

Although the dispersion equation (57), together with (68), has only the trivial solution for t h e problem under consideration, it seems worthwile to study the amplitude ratios. T h e y are for small values of A a good approximation to t h e e x a c t solutions, which converges toward t h e zero-solution for k —• 0. I f (o/p) < (1/2) (h/a) 1, we can discard t h e second term in the inner b r a c k e t of (61), as compared with the first term. Hence "1

E

1

J

l o

5(1 -

a

1/2 o

k ,

v2)

I

k

(67)

and with t h a t 1 J

E

! l '

'

o

1/2 CfY,

a

(68)

o

where 1/2

E h

l_2j)(l -

(69)

V2) a

is the well known water hammer formula of hydraulics. W i t h equation (67) we find . a k • i — v

/ ,

1

1— — \ 2

h

p

' ~ o ( k ) ,

a

h E

a2(l -

v

2

^ ) = const . -o/

'

)

(70a)

q

(70b)

Finally, a crude estimate of equation (61) allows to set 1 1

E h V

\ 2 o q

1/2

E_ J

+

/

]

k

,

(71)

i.e., in case of small k we recognize t h a t the curves for large and small w exhibit a similar behavior. The larger ( q I q ) , as compared with 2 a h k (where a k 1), the more accure the approximate formula (71). Furthermore, for very small h t h e approximation (71) simplifies to (66), except t h e o-term. W i t h (71), we have 2

E h 2à@

E

1/2

(72)

q

and, for large values of o, c

~

c

w

•

(73)

233

H. F. BATJEB / ST. METTEN / J . SIEKMANN : Dynamic Behavior of Distensible Fluid Lines

This is in agreement with the c + -formula of (68), but terms of higher order. Together with (71) the ratios

SQIZ0 and PJ30 are S0 _ .k I Z~0 l j ( a

kg + 2ag\ 2Q /'

P„ E{hv + 2a) h E0 ~ 2ga{l - 5») / # „ ) '

(74a), (74b)

A numerical evalution of the approximate ( +)-solution -,-,

e(i

valid for /¡fc

E

-2,

fc,

- " )

(75)

1, and originating from the dispersion equation (57) for large values of co,

i 3i 3i 63 Gn{ikna) ~ i — — K1 — ^ fc«2 — g^g A:«3 — 12gfl4

^

21i —

Ki + o(k¡¿6),

1 and

lcna > 1

(76)

shows excellent agreement with the exact numerical solution of the dispersion equation. Practically both curves coincide. The same is true regarding the approximate co + -solution (66). 5. Conclusions

A theoretical analysis of the-propagation of pressure waves through an infinitely long, thin-walle'd, elastic tube of circular cross-section is presented. The tube is filled with a streaming incompressible viscous liquid or with an inviscid incompressible liquid, the latter being at rest in the absence of waves. The analysis is restricted to waves whose amplitudes are small. Of special physiological interest (pulsating blood flow through an artery) are waves whose wavelengths are large compared to the radius of the tube. They are discussed in some detail. Analytical expressions are derived and graphs are given which show the dependence of the phase velocity on the frequency for various sets of parameter values. The most important results are summarized as follows: I n case of a'rigid tube (K - > oo) there are no waves, regardless whether the liquid is nonviscous or viscous, while in the opposite case ( E = 0) there exist waves in an inviscid fluid for certain constant values of the wave number (k n ), such that the wave speed increases linearly with the frequency (co). In a viscous fluid there are waves too, they are, however, strongly attenuated. The same behavior is observed for a viscous liquid jet (h = 0), while there are no waves in a nonviscous fluid. Concerning the case of an empty tube (Q = 0) we find that waves are propagated along the tube wall. With increasing frequency the phase velocity of these waves approaches either (inviscid fluid) the longitudinal wavespeed (c L ), or (viscous fluid) becomes infinite. For small frequencies the complete problem yields a constant phase velocity for both nonviscous or viscous fluids. For increasing values of the frequency damping of the waves in a viscous fluid increases also while in an inviscid fluid waves are propagated without attenuation. References 1 EITLER, L., Principia pro motu sanguinis per arterias determinando, Opera posthuma mathematica et physica anno 1844 detecta, ediderunt P. H. Fuss et N. Fuss, Petropoli, Apud Eggers et socios, Vol. 2, pp. 814—823 (1862). 2 WEBER, E. H., Uber die Anwendung der Wellenlehre auf die Lehre vom Kreisläufe des Blutes und insbesondere auf die Pulslehre, Ber. Verh. Kgl. Sachs. Ges. Wiss., Math.-Phys. Kl. 1850. 3 WEBER, W., Theorie der durch Wasser oder andere incompressible Flüssigkeiten in elastischen Köhren fortgepflanzte Wellen, Ber. Verh. Sachs. Ges. Wiss., Math.-Phys. Kl. 18, p. 3 5 3 - 3 5 7 (1866). 4 RESAL, H., Sur les petits mouvements d'un fluid incompressible dans un tuyau clastique, C.R. Acad. Sei. 82, 698—699 (1876). 5 KORTEWEG, D. J . , Über die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Röhren, Anm. Phys. Chem. (Neue Folge 5, 5 2 5 - 5 4 2 (1878).

6 MOENS, A. I., Die Pulskurve, E . J . Brill, NV, Leyden, 1878. 7 YOUNG, TH., Hydraulic investigations, subservient to an intended Croonian lecture on the motion of the blood, Phil. Trans. Roy. Sei. London 88, 1 6 4 - 1 8 6 (1808). 8 WojUEKSLEy, J . R., An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, Wright Air Development Center Tech. Rep. no. 56—684. 9 ATTINGER, E. O. (ed.), Pulsatile Blood Flow, Mo Graw Hill, New York, 1964. 10 Mc DONALD, D. A., Blood Flow in Arteries, Second Edition 1974, Williams and Wilkins, Baltimore, Maryland, Edward Arnold (Publishers) Ltd., London. 11 FUNG, Y. C. (ed.), Biomechanics, Am. Soc. Mech. Eng., New York, 1964. 12 ASME — Biomedical Fluid Mechanics Symposium, Am. Soc. Mech. Eng., New York, 1966. 13 FUNG, Y . C., Biomechanics, its scope, history and some problems of continuum mechanics in physiology, Appl. Mech. Rev. 21, 1—20 (1968). 14 COPELEY, A. L. (ed.), Hemorheology, Pergamon Press, Oxford, 1968. 15 FUNG, Y. C. et al. (ed.), Biomechanics, Its Foundations and Objectives, Prentice Hall, Englewood Cliffs., New Jessey, 1971. 16 FUNG, Y. C., Biomechanics: A Survey of the Blood Flow Problem, in Advances in Applied Mechanics, edited by C. S. YIH, Vol. 11, 1971, Academic Press. 17 RUBINOV, S. I.; KELLER, J . B., Wave Propagation in a fluid-filled tube, J . Acoust. Soc, Am. 50, 198—223 (1971). . 18 LIGHTHILL, J., Physiological Fluid Mechanics, CISM, Courses and Lectures, No. I l l , TJdine 1971, Springer, Wien 1971. 19 LIEBERSTEIN, H. M., Mathematical Physiology (Blood Flow and Electrically Active Cells), American Elsevier Publishing Company, Ini., New York, 1973.

234

H . F. BAUER / ST. METTEN / J . SIEKMANN: Dynamic Behavior of Distensible Fluid Lines

20 LIGHTHILL, J . , Mathematical Biofluiddynamics, Chapters 10, 12, 13, S1AM 1975. 21 RUBINOV, S. I . ; KELLER, J . B., Wave propagation in a viscoelastic tube containing a viscous fluid, J . Fluid Mech. Vol. 88, part, 1, p p . 1 8 1 - 2 0 3 ( 1 9 7 8 ) .

22 METTEN, ST., Über pulsierende Strömungen in elastischen Röhren, Diplomarbeit T H Darmstadt, 1978 (unpublished). 23 HAMMEL, J . , Grundgleichungen der Kreiszylinderschale, Z. Flugwiss. Weltraumf. 1 (1977) H e f t 5, S. 353—358. 24 TIMOSHENKO, S. P., WOINOWSKY-KRIEGER, S., Theory of Plates and Shells, Second Edition, 1959, Mc Graw Hill, New York. Eingereicht am 27. 7. 1979 Anschriften:

Prof. Dr. rer. nat. H. F. BAUER, Hochschule der Bundeswehr München, Fachbereich Luft- und Raumfahrttechnik, Werner-Heisenberg-Weg 39, D 8014 Neubiberg — Dipl.-Ing. ST. METTEN, I m Klarenpesch 13, D 502 Frechen — Prof. Dr.-Ing. J . SIEKMANN, Universität Essen — Gesamthochschule, Fachbereich 12 — Mechanik, Schützenbahn 70, D 4300 Esssen 1 - B R D

K. B e y e r : Stabilität einer ferromagnetischen Flüssigkeit

235

ZAMM 60, 235 - 2 4 0 (1980) K . BEYER

Zur Stabilität einer ferromagnetischen Flüssigkeit in einem vertikalen Magnetfeld Herrn Prof. H e r b e r t B e c k u m zum 60. Geburtstag Im Rahmen der Variationsrechnung untersuchen wir die Stabilität einer ferromagnetischen Flüssigkeit in einem äußeren Magnetfeld. Für die zweite Variation der Oesamtenergie im ungestörten Ausgangszustand wird eine Ungleichung vom Gärdingschen Typ hergeleitet, die eine gute Einsicht in den Mechanismus des Stabilitätsverlusts bietet. Für unsere Betrachtungen besonders wichtig ist die Tatsache, daß sich die zweite Variation der Oesamtenergie stets in überschaubarer Weise aus dem Systemzustand berechnen läßt. In this paper we describe a variational approach to the stability theory of a ferromagnetic fluid in an exterior magnetic field. For the second variation of total energy in the unperturbed initial state a Gdrding-type inequality is derived, which gives insight in the loss-of-stability mechanism. For our considerations it is essential to have an easy possibility for constructing the second variation of total energy from the underlying state. B paMKax BapHaqHOHHoro h c ^ h c j i g h m h H3yiaeTCH ycToüHHBOCTt (JieppoMarHHTHOfi j k h a k o c t h b o bhgiiihbm MarHHTHOM none, fljin BTopoit BapwauHH nojiHoft 3HeprHH b HeB03MymeHH0M h c x o h h o m c o c t o h h h h b m b o r h t c h HepaBeHCTBO rana TopHHHra, woTopoe AaeT npeHCTaBJiemie o MexaHHSMe n o T e p n ycToiwiiBOCTH. JUjih nauinx paccMOTpeiiHü ocoSeimo Barnen t o t $aKT, h t o BTopan BapwauHH iiojihoK DneprHH m o j k c t ¿ m t b p a c c i H T a H a b o 6 o 3 p h m o h iJ>opMe H3 c o c t o h i i h h c h c t c m w .

In dieser Arbeit untersuchen wir die Stabilitätsverhältnisse in einer ferromagnetischen Flüssigkeit unter dem Einfluß eines vertikalen äußeren Magnetfeldes. Neben dem ungestörten horizontalen Gleichgewichtszustand der Flüssigkeit existieren bei Überschreiten einer gewissen kritischen Feldstärke noch weitere Gleichgewichtskonfigurationen, die längs der Oberfläche rechteckige bzw. hexagonale Muster bilden. Klassisches Beispiel einer derartigen Erscheinung ist das BÉNARD-Problem (s. [5] und die dort zitierte Literatur). Für eine unendlich tiefe Flüssigkeit mit magnetischer Permeabilität fj, — 1 wurden die Amplituden der rechteckigen und hexagonalen Wellen in [1] näherungsweise auf der Grundlage eines Variationsprinzips (minimale potentielle Energie) berechnet. Wie schon von LUKE bemerkt (s. [4]), lassen sich auch permanente Schwerewellen längs einer inkompressiblen Flüssigkeit durch ein ähnliches Variationsprinzip beschreiben. Wir werden uns hier im wesentlichen mit den linearisierten Variationsgleichungen zur potentiellen Energie befassen und für ihre zweite Variation im ungestörten Zustand eine Ungleichung vom G A R D i N G S c h e n Typ herleiten, die einen guten Einblick in den Mechanismus des Stabilitätsverlusts gestattet. Besonders wichtig für unseren Weg ist es, daß sich die zweite Variation der Gesamtenergie in übersichtlicher Weise stets aus dem Systemzustand berechnen läßt. Dieser Umstand soll an anderer Stelle zur tatsächlichen Lösungskonstruktion benutzt werden. 1. Wir betrachten die Trennfläche F zwischen einer ruhenden inkompressiblen Flüssigkeitsschicht' der Dichte Q und einem Vakuum unter dem Einfluß eines vertikalen Magnetfelds 3€ und des Schwerefelds (0, 0, — g). Ist U€ = 0, so ist die horizontale Trennebene einziger Gleichgewichtszustand. Bei wachsender Feldstärke treten hierzu weitere Gleichgewichtskonfigurationen 7 1 : s = Ç(x, y) ; ihre Form ist aus der längs F zu fordernden Bedingung (1.1)

zu bestimmen, ß > 0 ist hierbei der Koeffizient der Oberflächenspannung der Flüssigkeit und (i > 1 ihre magnetische Permeabilität, 36T bzw. Xn bezeichnen die tangentiale bzw. normale Komponente des Magnetfelds in der Flüssigkeit. Dieser Schreibweise entsprechend werden wir die sich auf das Vakuum (fi = 1) beziehenden Feldgrößen stets mit dem oberen Index ,,+" kennzeichnen. (1.1) ergibt sich aus den längs r gültigen Gleichgewichtsbedingungen (ff

f

-

a") « + ß AÇn =

01)

—n ist der Normalenvektor an F und a = (au)

der Spannungstensor des Magnetfelds — bei Beachtung der Stetigkeit der tangentiellen Komponente von X und der normalen der magnetischen Induktion. Im übrigen wird sich (1.1) aus dem zu stellenden Variationsproblem in natürlicher Weise ergeben. *) Vgl. [2], S. 167.

236

K. BEYER: Stabilität einer ferromagnetischen Flüssigkeit

In dieser Arbeit beschränken wir uns auf die Betrachtung periodischer Lösungen der obigen Aufgabe. Wir werden demgemäß von den Lösungen 2?r-Periodizität in x- und ¿/-Richtung fordern und die folgenden Überlegungen auf den Quader Ü = Q X {z| - 2 „ < 2 < % } ,

Q:0

=

j

(

l^fri2

j

+

p

m

j

h d x d y .

r

Aus (2.2) läßt sich nun die zweite Variation von Em bestimmen. Wir notieren ihren Wert hier nur für £ = 0. In diesem Fall ist u+ — 0, u~ = 0, demzufolge wird (E'm(0)

h, h>

H2

=

f vth *

d y .

dx

(2.3)

J Q

Es ist besonders wichtig, daß sich die in die zweite Variation eingehenden Punktionen v*1 bei Kenntnis von u L durch ein weiteres quadratisches Variationsproblem kennzeichnen lassen. Zu seiner Herleitung differenzieren wir die Variationsgleichungen (1.12) nach t , wobei wir in (1.12) (p* = (p\.Q>, cpr = d F + ¡x f V«~ \J

H2

=

Cdv+

/

239

H2

C

- t r ) da: dy = - —

(v+

q

/ [Vf+|2

d V

-

^

H2

- -

O'

j

r

|Vtr|» dV

^ 0.

n-

Berechnen wir zunächst das Spektrum des Operators E',' (0). Sei IV der SoBOLEwraum der 2?r-periodischen Funktionen h{x, y) mit über Q quadratisch summierbaren ersten Ableitungen, außerdem gelte n

/ h{x, y) dx dy = 0 9 (vgl. (1.17)) für die Funktionen von W. Wegen W c # 1/2 (Q) und v±[x, y, 0) e # - 1 / 2 ( 6 ) (s. [3]) läßt sich die Bilinearform (2.7) stetig über W X' W fortsetzen. Entwickeln wir die Funktionen h e W in Fourierreihen y)

=

&ikx

Ea-k k^O-

.

oc-k

=

ü.k

— wir benutzen die Multiindexschreibweise: k = k2) ganzzahlig, \k\2 = k\ + k\, kx = k±x + k2y. Nach den sich an (2.5) anschließenden Überlegungen und (2.7) ist das Bildfunktional ¿>Ä(0) k? 0

E'm(0)h=

wie folgt zu berechnen: Man löse zunächst das Randwertproblem (1.4) —(1.10), — (1.6) ist durch vt{x,

0) —

y,

vj(x,

0) =

y,

/1

zu ersetzen —, für die Funktionen E'm(Q)

h

=

2

H

^

+ const.

eikx

und setze anschließend

Z

V> 0) •

0Z

Man prüft leicht nach, daß v%

=

e i f a sinh |£| (z

A \

A\ = {/n — 1) (/u cosh Aj = -

,

vi

sinh |/fc| (z + z0) ,

A l eikx

z1 + cosh \k\ zt sinh |i| z 0 ) - 1 cosh |i| z0 ,

z0 sinh

(/u cosh |£| z0 sinh

/1

=

z1 -f cosh

zL sinh \k\ «¿) -1 cosh |i|

,

also 6»t , {x, 8z

y,

0) =

( p - 1) |*| e ikx /j, tanh |£| z1 + tanh |A| z0

zutrifft. Gleichzeitig ist nachgewiesen, daß die Funktionen =

E'm(0)

,(m) Ä

\k\

=

— "

I

P

e'kx

Eigenfunktionen zu

E'm(0)

eite ,

^

sind: (2.8)

(1 — i«)2 l&l 4n[x {(i tanh \k\zx + tanh \ k\ z0)

Man beachte, daß die Höhen z0, zL infolge unserer speziellen Periodenwahl als dimensionslos anzusehen sind. Wir haben damit einen vollständigen Überblick über die quadratischen Anteile an der Magnetfeldenergie gewonnen. Fassen wir zusammen: Nach (2.8) gilt

=

± n

2

H

2

I«*l2 ä 4 t z 2 H 2 ^

E

k* 0

£

>¡#0

|£| | v) LUik(x, y) — Ltjtk(x, y) M4(y)] A + / [m i t «(v) 0>ik(x, y) -

Lmn

k{x,

dA y

y) Vl(y)} n, dA v

+ f Ui(y) LUI\X, y) + h{y) LVik(x, y)] AVv . v C o r o l l a r y 1. An.integral repi-esentation for the pressure can be obtained if we substitute the expressions derived into equation (6).

(21)

just

C o r o l l a r y 2. In the absence of couple stresses and body couples, equation (20) reduces to the form FUk(x, y) — Pt,ik(x, y) u^y)] UjdAy + / fi(y) Fuf(x, y) dF„ .

uk{x) = / [^(M) A

V

This is the classical result given by L A D Y Z H E N S K A Y A [10]. Explicit expressions for the fundamental singular solutions have been obtained by the authors [9] and are given by: xek x {ek • r) , 2« - y y) = fP1^, y) —

y),

(52)

and Oij{x, y) =

L

Ui\x, y) — Lgi1{x, y) ,

L

JI^x,

y) = lV^x, y) — Ji^x,

y) ,

(53), (54)

with lQAx, y) =

p (x, y) -

L 1

q\x, y) ,

(55)

L

where MMi'> mVi, mV1(M = F, L) are the fundamental solutions given by (22) — (27), and ug^, MhJ, as the solutions of the STOKES problem: • *Usr',

1

MQ

)

=

o,

D(M91,

UH>)

= 0 ,

v • M91

=

0 ,

q' are defined

M

(56), (57), (58)

and satisfy certain boundary conditions specified for each boundary value problem. On applying the reciprocal theorem to the system (u,v, p) and ( M g 1 , mM, m1') associated respectively with (11)—(13) and (56)—(68), we obtain: / [tfM, v) M9k(x, y) - Mtjix, y) « h> ••• >

h> — >

258

H . FRIEDRICH / K . HENNIG: A u s w e r t u n g v o n B e r e c h n u n g s f o r m e l n f ü r

Überschreitenswahrscheinlichkeiten

zu ermitteln. F ü r einen ergodischen stochastischen Prozeß Z(t) wird die numerische B e s t i m m u n g von fzz(ZQ, 0, 0) z2. \ 0 ) T > 0 , T) und von pL bzw. p2(r) und fzzzz{Zü, Za, angeführt. D a m i t lassen sich von ( 2 ) ausgehend Näherungen 1. und 2. Ordnung (s. FRIEDBICH-HENNIG [4]) bestimmen. D a s beschriebene Vorgehen kann ohne größere Schwierigkeiten auf allgemeinere Fälle übertragen werden. E s sei vorausgesetzt, daß der ergodische stochastische Prozeß Z(t) und seine Ableitung Z(t) stetige Verteilungsdichtefunktionen

0, 0) und

fzz(zL,

fzzzz{zL,

z2,

zv

\ \ 0, r , 0, T) besitzen. D a n n kann für A x - » 0, A 2 —• 0,




Z0,

z(i,-_i) i S

Zl < Z i , i ,

Z0

und i i | i t _ i ^

k=l,...,'L

1

z*{tt)

< z i > / ; für

.

(6)

Auf ähnliche Weise werden die positive z^Achse in die I n t e r v a l l e [zlk z liA .), k = 1, ... , L1 und die positive z 2 -Achse in die I n t e r v a l l e [z2,!-i, z2j), l = 1, ... , L2 unterteilt. Mit r = s • A r , A r = n • Af, s = 1, 2, ... bezeichne H'z[(k, l) die absolute Häufigkeit der Meßpunkte mit z( Z0, z ( i ; _ i ) ' ^ Z0 und z(tt + r ) > z(ti+1 + r) Zg, sowie mit z i , t _ i < z*(i 4 ) < z i , j t u n d z 2 , ; _ i fS z*(tt + r ) I 2 , ; für ^ = 1, — , M. D a n n folgt für hinreichend großes M fzzzz{Z0,

Z0,

für

Hf,{k,l)

21; z2; 0 , r , 0 , T) :

(7)

M • (Zl,* — Z l . i - l ) (z 2 ,i — Z 2 , i - l )

Z i , i _ i ^ «! < z i , t ; Z2,z-1 ^ Z2