Unstable capillary waves on surface of separation of two viscous fluids

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Title:
Unstable capillary waves on surface of separation of two viscous fluids = Neustoichivye kapilliarnye volny na poverkhnosti razdela dvukh vyazkikh zhidkostei by V.A. Borodin and Y.F. Dityakin
Portion of title:
Neustoichivye kapilliarnye volny na poverkhnosti razdela dvukh vyazkikh zhidkostei
Physical Description:
19 p. : ; 28 cm.
Language:
English
Creator:
Borodin, V. A
Dityakin, Y. F
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aeronautics   ( lcsh )
Nozzles -- Fluid dynamics   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographic references (p. 18).
General Note:
"Technical memorandum 1281."
General Note:
"Report date April 1951."

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University of Florida
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oclc - 76823852
sobekcm - AA00006204_00001
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Full Text
Cfr17'i ~- (2c~ I










NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1281


UNSTABLE CAPILLARY WAVES ON SURFACE OF SEPARATION

OF TWO VISCOUS FLUIDS*

By V. A. Borodin and Y. F. Dityakin


The study of the breakup of a liquid jet moving in another medium,
for example, a jet of fuel from a nozzle, shows that for sufficiently
large outflow velocities the jet breaks up into a certain number of
drops of different diameters. At still larger outflow velocities, the
continuous part of the jet practically vanishes and the jet immediately
breaks up at the nozzle into a large number of droplets of varying
diameters (the case of "atomization"). The breakup mechanism in this
case has a very complicated character and is quite irregular, with the
droplets near the nozzle forming a divergent cone.

Rayleigh (reference 1) was the first to make a theoretical study
of the jet and to establish the possibility of droplet formation. The
disturbance of a jet of an ideal fluid flowing into a vacuum and having
a wave length 4.4 times as large as the diameter of the jet is shown
to grow more rapidly than other disturbances; eventually, the jet
breaks up into droplets of the same diameter. Rayleigh succeeded in
determining theoretically the drop diameter, the value of which agrees
well with tests on jets issuing with very small velocities. Later,
the viscosity of the jet was also taken into consideration. The
viscosity is found to decrease the rate of amplitude increase of the
disturbances but the ratio of the optimal length of the wave to the
diameter of the jet remains unchanged.

Other authors that studied the conditions of the axial-symmetrical
breakup of a jet of a viscous liquid found that the ratio of the optimal
wave length to the jet diameter was somewhat greater than that computed
by Rayleigh.

In addition to the viscosity, Tomotika (reference 2) took into
account the density and viscosity of the medium surrounding the jet
and obtained good agreement with tests on jets issuing with very small
velocities for which droplets of the same diameter are formed.


*"Neustoichivye Kapilliarnye Volny na Poverkhnosti Razdela Dvukh
Vyazkikh Zhidkostei." Prikladnaya Matematika i Mekhanika. Vol. XIII,
no. 3, 1949, pp. 267-276.







NACA TM 12Zl


Neither of the aforementioned theories of the breakup of a liquid
jet provided a basis for the phenomenon for the case of breakup into
droplets of different diameter, a fact that is explained by the
idealized conditions of the problem. This idealization consisted either
in neglecting the viscosity of the jet, the density, and viscosity of
the surrounding medium, or the inertial forces. Such simplifications
were assumed in view of the complicated mathematical equation (generally
transcendental) that determines the relation between the wavelength
and the increment of the vibration amplitude.

In the present paper, an attempt is made to provide a mathematical
basis for the possibility of the appearance of droplets of different
diameters as a result of the jet breakup on the basis of the considera-
tion of unstable capillary waves on the surface of separation of two
viscous liquids.

For simplification of the solution of the problem, particularly
for obtaining the algebraic characteristic of the equation, the
lengths of the capillary waves on the surface of the liquid jet are
assumed to be so small in comparison with the jet radius that the jet
may be considered infinitely large; study of the stability of the
plane surface of separation of two infinitely extending viscous fluids
can thus be made. This assumption represents a considerable degree of
idealization but nevertheless permits a qualitative explanation of
not one but several unstable capillary waves that, in passing through
the jet, lead to the formation of droplets of differing diameters.

The existence of several unstable capillary waves is demonstrated
that can lead to the breakaway of several infinitely long strings of
different dimensions from the partition surface. The problem investi-
gated gives a rough approximation of the disintegration pattern of a
liquid jet in another medium and does not pretend to explain the com-
plicated mechanism of the limiting form of the disintegration of a
jet, namely, atomization. Nevertheless, one of the peculiarities of
atomization, the appearance of a dimension spectrum of the droplets,
begins to appear even for the given idealized consideration of the
stability of the partition surface.

1. Equations of small waves and their solution. A plane surface
of separation of two infinitely extending viscous fluids (fig. 1) is
considered. The viscosity and density of the lower fluid are denoted
by tl and pi, respectively, and of the upper fluid by L2 and P2.
The lower fluid is assumed to move with the velocity V1 and the upper
fluid with the velocity V2, the direction of motion being the same and
the velocities independent of y.







TFACA 'T i281a 3


A study of the character of the equilibrium of the surface of sep-
aration under the action of the viscous forces and the forces of silrface
tension that impsrt to both liquids small disturbances parallel to the
x-axis is presented. The fluids shall be o-nsidered incompressible and
weightless and shall cause certain disturbances to the components of the
motion.

X = V + v

V = v

p = P + p*


It is further assumed that the velocities of the imposed disturb-
ances and their derivatives up to the third inclusive are small and that
the magnitudes of the second-order smallness may be neglected.

Fromn the Navior-Stct:es equations, the following equaLions of the
imposed disturbances are obtained:

OVx Vx -1
+ V =---+ =--
ot ox P x x
(1.1)
S+V v = I+ Iv,
ot ox P oy

where u = p/p is the cinematic viscosity.

The equation of continuity is

OVv OV,,
ovx ovyI
---+ 0 ?
x 3y

By introducing the stream function of the disturbance


v v =- (1.3)
oy ox

and by eliminating the pressure p* from equations (1.1), the idealized
equation is thus obtained in the Helmholtz form

0 (1.4)
6x 6t






NACA TM 1281


Let the stream function of the imposed disturbance be a periodic
function of x and of the time t:

Sf(y)ei(ax-t) (1



where a is the propagated circular frequency of the vibrations (the
wave number), X is the wavelength of the imposed disturbance,
P = Pr + iPi is the complex frequency of vibrations in time, pr
is the real frequency of vibration in time, and Pi is the increment
of the growth of vibration or the decrement of damping.

The character of the wave motion on the surface of separation
after the imparting of disturbances to both surfaces will thus depend
on the sign of the imaginary part of the frequency .i* If Pi is
positive, there will be an increase in the wave amplitude with time;
if Bi is negative, there will be a damping of the wave amplitude;
finally, if Pr = 0, there will be an periodic increase (Si > 0)
or a damping (Pi < 0) of the wave amplitude. By substituting expres-
sion (1.5) in equation (1.4), the following equation is obtained:

fIV (2, ip) f" (ipa2 a4) f iVa (f" a2f) = 0 (

The problem of the characteristic values of a homogeneous system
of equations of the fourth order will be considered.

By setting f" a2f = p, a system of equations of the second
order is obtained.

p" + Pi 2 = 0 f- a2f= (l

Hereinafter, the following notations are introduced:

SP- VlM P Va (
i i2=ul i =2 (1
D1 V2 M


The solution of the first of equations (1.7) has the form


S= CleiY + C2e-im1Y (1

By substituting expression (1.9) in the second of equations (1.7),
a non-homogeneous equation is obtained for which the solution is







NACA TI 1281


f = Cl C2 + e' C, + e-a C4 (1.10)
m2j + -2 2 .+ a'

The streacu furctinn for the lower and upper liquids according to
equation (1.5) will be

i ^(-t) e/ -iLmly
S2i- + a m1 + a2 4'


S e (a-.-t) ( .em -im2y -ay \
S2 = ei(ax-t) ( 2 C2 5 e'2 2 C6 + eay 7 + e C (1.12)
m2 + a. m22 + a2 /

The arbitrary constants Ci must be determined from the conditions
on the surface of separation and at infinity.

2. Boundary conditions. The boundary conditions of the problem
will be as follows:

1. At infinity (y = +c), finite solutions must be maintained for
4lf and 29. Hence, the arbitrary constants of the terms with positive
exponents for 1 and with negative exponents for W 2 must be equated
to zero: C1 = C3 = C6 = Cg = C. Thus, equations (1.10) and (1.11) will
have the form

ei(cam-t) e C2 + e-ay C4
m12 + aOL 4)

(2.1)
/ im\7
w2=ei(-a't) f- emi2 C+ + e Co7)
y e Cr +
I2 = .m92 + a? 7

2. On the surface of separation, there must be no slip, that is,


(Vl) Y = ( 2)y=0


(Vy71) y= (v2) y=0






KNACA TM 1281


y =0 y )yy=o


Ox- =0 y=O x/y=0


(2.2)


3. The tangential stresses on the surface of separation are
continuous


l( Al)y=O = A2(A2)y=0


(2.3)


4. The difference between the normal stresses pyl and py2
on the surface of separation is equal to the pressure brought about by
the surface tension; that is,


I(V y2 (22 y2
yl Py2 = P + 2 P2 + 2


o2h
= T -2
ox


where T is the capillary constant of one liquid relative to the other
and h is the rise in the surface of separation at the point x.

By using equations (2.1), the boundary conditions (2.2) are obtained
in the form


iml im2
m12 2 2- +4 + 2 2 C5 aC7
m._ + a" m"


= 0


(2.5)


C2
mi + Ma


C
F5
+ C4+ 2 2 C7 = 0
2 + a


Similarly, the boundary condition (2.3) is obtained in the form


C2
m12 + M2


+ C4)a.2



C 7)M2 +
7 C .2


m2 25
ig22 + a2


S12
ml2 + a2


C2 + a2C4


2
i2 + 5
m2 +


(2.4)


(2 .C)







iAlCA TPIM 1281


The pressures jl and p2 are computed from equations (i.1)
and (2.1). Thus


PL PI( I --'
i =; + "'2


V mIC


iiv CL
i lt+
i:i} !lii


f + i la2 VaM iula o e-)e C4] (2.7)


p2 = P2ei(ax-t) a '7+ m2 C5 ( 2) e y C,
L 22 + a2 2 + m + 2


The rise of a point on the surface is a periodic function of x
and t.


h = Hei(x-t) (2.8)

where H is the maximal rise of a point on the surface of separation.

The velocity of the raised point on the surface of separation is


(2.9)


oh oil
+ V1
ot ox
'7=u


After differenttiatin expressions (2.1) and (2.6) and b.r substi-
tuLing in expression (2.9), the following equation is obtained:


H C2
H a C+ C2
ml- + 9


(2.1'0)


By substituting equation (2.10) in (2.9) and by differentiating
equation (2.9),


,2
oh
62
ox


ca C2
Sav1 P + a2


(2.11)


m"l C? +


- (v ) = ---
l^ (


c4 ei(ax-Pt)







NACA TM 1281


By computing the derivatives 8Vyl/By and 6vy2/Sy and
tuting them simultaneously with expressions (2.7) and (2.11)
the following boundary condition is obtained:


substi-
in (2.4),


a -[La2 + iP1Via iPP
ml2 + a2 m


Ta2
+ ml 1 + 0V ] C2 -


plVlao i2jla2 + T
S" V-P


C4 + M
m22 + a2


'2 -M


ip2V2co + iPP2
D+
m2


3m2p2 ]


C5 + (P2P P2V2 + i22aa2) C7 = 0


(2.12)


The following nondimensional parameters are then introduced:


Z = -


vl


V2
A 1
2






N-
2




K 1
2


(2.13)







NACA TM 1281 9


where c = V/a is the complex wave velocity.

Equations (1.8) can then be represented in the forms


m = a/i(Z R) 1


m2 = a /i(ZA R2) 1

Equations (2.5), (2.6), and (2.12) are represented in nondimen-
sional parameters. The following notations are first introduced:


i 2 N
a1 1 z- /i(z-R) 1 Z la


b* = Z-R -2 + 1- = a bl
"1 RCLP a 1


2(1 2R2 + 2AZ)
CL* = 2 2=a2c,
(ZA--P,2) (Z-P2)- ;2


a 1 i a3
a3* 2 -
2 Z-R1 a.2
(2.14)
d6* = ,.2 (ZA-PR + 2i) = 2a.2d


1 i c3




a2* = = -


S4 i(ZA-R) 1 c2
2 a ZA-R2 a,

where al, a2, a3, bl, cl, c2, c3, and dl are likewise nondimen-
sional magnitudes.








NACA TM 1281


The following system of equations is then obtained for the con-
stants C2, C4, C5, and C7:


al*C2 bl*C4 + cl*C5 + dl*C7 = 0

a2*C2 aC4 + c2*C5 a C7 = 0
(2.15)
-a%*C2 + C4 + c3*C5 C7 = 0

K C2 C5 = 0

This system of homogeneous equations has solutions different
from zero if its determinant is equal to zero. By setting up the
determinant and expanding

2K(al + cl) + (d1 Kbl)(a2 + Kc2) + (Kb1 + dl)(Kc3 a3) = 0

By solving this equation for Z, the following wave equation of
the 18th degree with complex coefficients is obtained:

rlgZl8 + (rl7 + isl7) Z17 + ..: + (rl + isl) Z + (r0 + is0) = 0
(2.16)
The real and imaginary parts of the coefficients depend on the
five nondimensional parameters: R1, R2, A, N, and K.

3. Investigation of roots of characteristic.equation. The
increase in oscillation, that is, the loss of stability of the sur-
face of separation, arises from those waves for which the imaginary
part of the frequency is positive (Pi > 0). Hence, the investiga-
tion of the roots of equation (2.16) should determine those ranges
of the parameter N or the wave number a in which the complex
roots of the equation lie in the upper half-plane.

By the Rayleigh hypothesis, the further development of an
unstable deformation, that is, the form and dimensions of the parts
breaking away, is determined by the critical (or optimal) disturb-
ances. The critical disturbances may be defined as those that
develop more rapidly than the others or that correspond to the
maximum increment of the growth Pi. This principle of deter-
mining the character of the unstable deformations by the character
of the maximum unstable disturbance has been experimentally
confirmed by a number of investigators (reference 3).








NACA TM 12:1


In the case considered, the growth in the amplitudes of the
oscillations will lead tc breakaway of infinitely long strings
from the surface of 3eoaration. similar to the formation and
brea':awa.y of wave crests. The separation will take place for such
values of a or wavelengtl.- A for which i has the maximal
value.

If a spectrum of small-period disturbances that cen be developed
into a Fou'rier series can be assumed to be imposed on both liquids,
the. harm.onics w;i-h the wavee-ngths equal to the wselengths of
the maximal, unstable disturbances bring about a separation of
infinitely lon- strings from the partition surface. Because the
cnaracterietic dimension (for example, the diameter of the transverse
string) is connected with the length of re:-:.,iisl unstable disturbance,
str.nges of different dimensions will breai: awash from the surface of
separation. In fi'.ure 2, the scheme of formation of such strings
for three successive instants of time is shown.

Investi..jatorn of the roots o.f the simplest particular case of
equation (2.16) is presented.

Let bc.th fluids be stationary and their kinetic viscosities
the same. In this case, V V = 0 P1 = U2, mi = m2, A = 1,
P, = F2 = 0, and equation (2.16) goes over into an equation of the
8th degree whose coefficients depend only on the two parameters
K and N:


A 8 + (A iB)Z + iB)Z6 + ( + iB3)5 (A + B)4 +
A (A + E)2 (A+ i)Z + 4

(A5 + iB)Z (A6 + iB)Z2 + (A7 + iB-)Z + A8 = 0 (7.1)






12 NACA TM 1281


where

A0 = (1 K)2

A2 = 2K (K 1) N K4 + 2K3 4K2 + 6K + 13

A1 = 2K (K 1)2

A3 4K2 (K 1) N 2K (3K2 + 13)

A7 = 2K3N2

A4 = K2N2 + 2 (K4 K3 + 3K2 + 5K) N 12K3 + 26K2 10K 9.

AS = 2K3N2 + 12K3N 8K3 8K2

A8 = K2 (1 + 2K) H2

A6 = (1 K2) K2N2 + (4K4 10K3 4K2 SK) N 8K3 + 12K2 8K + 4

B1 = 2 (K2 + 2K 3)

B3 = 3E4 14K3 + 13K2 + 18K + 13- 8KN

B2= 4K (K 1)2

B4 = 8K3- 42 20K 2K3N

B= 4K2 (1 -K)

B5 = 2K2N2 + [2K (1 + K) (1 + K K2) + 83 + 4K2 + 4K]N +

4(K 1 K2) (1 + K K2) + 20K3 4(K 1 K2)2

B7 = K2 (1+ K)2 2K4 N2 + [4K4 + 4K (1+ K) (K 1- K2) N
(3.2)







NACA TM 1281


The characteristic equation (3.1) is a polynomial whose
coefficients depend nonlinearly on the two parameters K and N.
Each pair of values of the parameters E and N' or each point
of the plane Ki correspond to the completely defined polynomial
(3.1), that is, completely determined values of the eight roots
of the nolyinomial. In the plane KI, it is evidently possible to
find a curve, each point of which corresponds to the polynomial
(3.1), that has at least one root located on the real axis so that
only in crossing this curve is a crou.iig of tre roots through the
real axis possible. This curve breaks up the plane Kn into
regions, the points of which each correspond to polynomials (3.1),
that have the same number of roots with positive imaginary part.

These curves are constructed by making use of the method of
Y. I. Neimark (reference 4) that permits a breakup of the plane
of the parameters for the roots of the polynomial lying in the left
or right half-plane.

The substitution Z = -i( is made. The upper half-plane of
the roots of equation (3.1) is transformed into the left half-plane
of the roots of the equation

- A0 r i(A1 + iB1) 7 + (A2 + iB2) 6 i(A3 + iB3)5 (A4 + iB4) +

i(A5 + iB5)( + (A6 -+ iB6)2 (A7 + iB7) + A8 = 0
(3.5)

By substituting = it/1 in the preceding equation and multi-
plying the result by 98, equation (3.3) is reduced to the form

F(;,f) + iG(C,T) = 0 (3.4)

where

F(,q) = AOE + A Al7 + A2 6 + As53 +- A44q4 + A5~3q5 +

A6g2q6 + A7 67 + AT8 (3.5)


G(t,q) = B~7r + B262 + B3 53 + B4 44 + B5 3r5 + B62+ 6 + B7+ 7






NACA TM 1281


If R2n is the space of complex polynomials of degree n and
D(k,n k) is the manifold of polynomials R2n having k roots to
the left and n k roots to the right of the imaginary axis of the
complex sphere, then by setting up the following table:

A0 Al A2 A3 A4 A5 A6 A7 Ag
0 B B2 B3 B4 B B6 B7 B8J

and by making the transformation

AO + X1B AI + XB2 A2 + lB3 A3 + lB4 ...Ag

S 0 B B2 B3 ...0 (3.7)

table (3.7) is found to correspond to a polynomial of the same type
with respect to the distribution of the roots relative to the imaginary
axis, as in equation (3.4).

From table (3.6), an inequality is obtained that defines the
region in the plane KN corresponding to the presence of the first
root of equation (3.1) in the upper half-plane:

A 0B < 0 (3.8)

By setting X1 = AO/B1 in table (3.7)

AB1 A B2)/B1 (AZB1 AOB3)/B1 (A3B1 AOB4)/B1 ..A7 A

B1 2 B3 ...B7 0
(3.9)

Because AIB1 AoB2 = 16K(K 1)3< 0 for K>1, by multi-
plying the elements of the first rows of (3.9) by B12/(A1B1 A B2)
and changing signs in the second row

SI_ D1 D2 D3 D4 D5 D6 D71
(3.10)
B2 B3 B4 B5 .- B B7 0







NACA Til 1231 15


where


E1(A B A Bj)



BI(A3B1 AC24)
AiBi A 2


Ei(A4B ABD5)
AiB1 AOB2

BI(A5Bi AeB6)
____ = D4 = (3.11)
A1Bl A 72

Bi(A6B1 AB7)

AlB1 AOB2
B A,
D



A1Bi AOB2

The first row of table (3.10) is left unchanged but to the
second row is added the first row. Thus

i D1 D2 D3 D4 D5 D6 D

0 D1 B2 D2 B3 D B4 D4 B5 D5 B6 D6 B7 C
(3.12)

From the preceding calculations, an inequality is obtained that
defines the region in the plane IK that corresponds to the presence
of the second root of equation (3.1) in the upper half-plane


(3.13)


B1(DI B2)<0 0







NACA TM 1281


By carrying out a transformation, similar to (3.7) of table (3.12),


+ X2(D1 B2)
0


D1 + 2(D2 B3)
D1 B2


D2 + 2(D3 B4) .."D )
D2 B3 ...C


By setting; 2 = B1/(D1 B2) and substituting in (3.14)


D1(D1 B2) B1(D2 B3)
D1 B2
D1 B2
D3(Dl B2) B1
D1 B2
D3 B4


D2(DI B2 B(D B4)
D B2
D2 B3 (3.15)
(D4 B5)


Because DI(D B2) B1 (D2 B3) > 0
plying the elements of the first row of (3.15)
[DI(D B2) BI(D2 B3)]


for K>1, by multi-
by (DI B2)2/


SB 2(Dl B2) Bl(D3 B4) (D B2)
D-1 2 D1(DI B2) B1(D2 B)

DI B2 D2~ B3


(3.16)


The elements of the first row are subtracted from the elements
of the second row of table (3.16).


[(D2 (D1 B2) B(D3 B4) (D B2)


0 (D2 B3)


D2(D1 B- B Bl(D3 B)] (DI B2)
D1(D1 B2) B1(D2 B3)


From the preceding table, an inequality is obtained that defines
the region in the plane of the parameters KN that corresponds to the
presence of the third root of equation (3.1) in the upper half-plane.


(D2 B3) -


[D2(D1 B2) B1(D3 B)] (D B2
DI(D1 B2) B1(D2 B3)
(3.18)


(D1 B2)


(3.17)


*






NACA TM 1281


Similar conditions can be obtained for all the remaining roots
of equation (3.1). This investigation has been limited to the three
conditions that are sufficient for proving the existence of several
unstable waves.

By replacing inequalities (3.8), (3.13), and (3.18) by equations,
the equations of the curves determining the breakup of the FN plane
into regions are obtained. The most interesting case of large
K = pj/p2>>1 is considered. From inequalities (3.8), (3.13),
and (3.18) and by considering equations (3.2) and (3.11) and neg-
lecting small powers of K, the following equations are obtained:

2(K l)3(K 4 3) = 0

eC3 + elN2 + e2N + e3 = 0

4(K2 1)(K + 3)N + K(K l)(K3 + 17K2 96K + 99) = 0

where (3.19)

e = 128 (K4 K5 23K2 39K 18)

el = 592K(K5 + 8.4K4 + 3.18K3 96K2 20.3K + 0.98)


e2 = 9K2(6 + 8.4K5 97.3K4 2045K3 + 1700K2 + 390K + 363)

e3 = 24K5(K5 + 12.3K4 + 306K3 4100K2 + 12,300K 7000)

By plotting the curves (3.19) in the KN plane and separating
by hatched 14nes the regions corresponding to the signs of the
inequalities (3.8), (3.13), and (3.18), the diagram shown in
figure 3 is obtained. This diagram shows that for K>0 and NT>0
a region of values of K and I exists that corresponds to the
presence of three roots with positive imaginary part, that is,
of three unstable waves on the surface of separation.

The division of the KN plane for the remaining roots could
establish regions with a still greater number of roots with posi-
tive imaginary part. The given incomplete diagram already shows,
however, the existence of several unstable waves. In the presence
of a maximum Pi or ci, several infinitely long strings will






NACA TM 1281


break away from the surface of separation, the cross-sectional
dimensions of which will depend on the wavelength of the critical
disturbance.


Translated by S. Reiss,
National Advisory Committee
for Aeronautics.


REFERENCES

1. Rayleigh: The Theory of Sound. Dover Pub., 2d ed., 1945.


2. Tomotika, S.: On the Instability of a
Viscous Liquid Surrounded by Another
Roy. Soc. London, vol. CL, no. A870,
pp. 322-337.


Cylindrical Thread of a
Viscous Fluid. Proc.
ser. A, June 1935,


3. Petrov, G. I.: On the Stability of Turbulent Layers. Rep.
No. 304, CAHI, 1937.

4. Neimark, Y. I.: On the Problem of the Distribution of the Roots
of Polynomials. DAN, T. 58, No. 3, 1947.






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