j WJY f ft - (2? II B. M. LEADON
00
^ NATIONAL ADVISORY COMMITTEE
l FOR AERONAUTICS
2
TECHNICAL MEMORANDUM 1281
UNSTABLE CAPILLARY WAVES ON SURFACE OF SEPARATION
OF TWO VISCOUS FLUIDS
By V. A. Borodin and Y. F. Dityakin
Translation
'Neustoichivye Kapilliarnye Volny na Poverkhnosti Razdela Dvukh
Vyazkikh Zhidkostei. " Prikladnaya Matematika i Mekhanika.
Vol. XIII, no. 3, 1949.
NACA
Washington
April 1951
DOCUMENTS DEPARTMENT
^^r^,w
^
NATIONAL ADVISOEY COMMITTEE FOE AERONAUTICS
TECHNICAL MEMORANDUM 1281
UNSTABLE CAPILLARY WAVES ON SUEFACE OF SEPAEATION
OF WO 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.
Eayleigh (reference l) 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. Eayleigh succeeded in
determining theoretically the drop diameter, the value of which agrees
well with teats 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 Eayleigh.
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 Eazdela Dvukh
Vyazkikh Zhidkostei." Prikladnaya Matematika i Mekhanika. Vol. XIII,
no. 3, 1949, pp. 267-276.
NACA TM 1-281
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 s mall waves and their solution. - A plane surface
of separation of two infinitely extending viscous fluids (fig. l) is
considered. The viscosity and density of the lower fluid are denoted
by i-i-L and p 1; respectively, and of the upper fluid by u 2 and p 2*
The lower fluid is a.ssumed to move with the velocity V-j. an & the upper
fluid with the velocity V 2 , the direction of motion being the same and
the velocities independent of y.
NACA TM 1281
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 surface
tension that impart to both liquids small disturbances parallel to the
x-axis is presented. The fluids shall be considered incompressible and
weightless and shall cause certain disturbances to the components of the
motion.
v x = V + ^x
V = v
y y
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.
From the Navier-Stckes equations, the following equations of the
imposed disturbances are obtained:
dv x
~5T
+ V
av x
1 dpj*;
P ox
+ uAv^
oV ovy
ot ox
P oy y
(1.1)
where u = |j/p is the kinematic viscosity.
The equation of continuity is
Sv-^
oX By
By introducing the stream function of the disturbance
(1.2)
By
ox
(1.3)
and by eliminating the pressure p* from equations (l.l), the idealized
equation is thus obtained in the Helmholtz form
oX
at
(1.4)
4 MCA TM 1281
Let the stream function of the imposed disturbance be a periodic
function of x and of the time t:
„ = f(y)e i(-et) £..*) (1 . 5)
where a is the propagated circular frequency of the vibrations (the
wave number), A. is the wavelength of the imposed disturbance,
p = P r + ip^ is the complex frequency of vibrations in time, P
is the real frequencj^ of vibration in time, and p^ 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 0j.. If Pi is
positive, there will be an increase in the wave amplitude with time;
if Pi is negative, there will be a damping of the wave amplitude;
finally, if P r = 0, there will be an aperiodic increase (0^ > 0)
or a damping (Pj < 0) of the wave amplitude. By substituting expres-
sion (1.5) in equation (1.4), the following equation is obtained:
Uf IV - (2co 2 U - 10) f " - (ipa 2 - pa 4 ) f - iVa (f " - a 2 f) = (1.6)
The problem of the characteristic values of a homogeneous system
of equations of the fourth order will be considered.
By setting f " - ccrf = tp, a system of equations of the second
order is obtained.
(1.7)
Hereinafter, the following notations are introduced:
J
P - vVl / - V 2 a
a 2 = m A /i — a^ = m9 (1.8)
1 v v 2
The solution of the first of equations (1.7) has the form
qp = (^e 1 ™!^ + Cge-^iy (1.9)
By substituting expression (1.9) in the second of equations (1.7),
a non-homogeneous equation is obtained for which the solution is
MCA TM 1281
-F =
.imiy
w-,2 + a2
C'l
■lnny
2 2 "2
m-, + or
Co + e a y C, + e" a y C,
(1.10)
The stream function for the lower and upper liquids according to
equation (1.5) will be
i(ox-pt)
imiy
■imiy
m-, + cc'
2 1
m-, + a'
2 C 2 + e"*C 3+ e-«*C
(1.11)
V 2 = e
i(cac-Pt)
,in»2y
-im2y
mo + a
Bio + ar
C c + e ay C-7 + e
-ay
(1.12)
The arbitrary constants C^ 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 = +<*>), finite solutions must be maintained for
^-j_ and '4< 2' Hence, the arbitrary constants of the terms with positive
exponents for ty n and with negative exponents for \|/ o. must be equated
to zero: C-, = C 3 = Cg = Cg = 0. Thus, equations (1.10) and (l.ll) will
have the form.
¥]_ = e
i (ax-(3t]
^2 = e
= i(ax-pt) f _
—2 2 C 2 + e ^ C a
m-^ + ar
(2.1)
m2 + ar
On the surface of separation, there must be no slip, that is,
( V .Tl) y=0 " M y=0
or
NA.CA TM 1281
o^
St;
ey/ y=0 \oy/ y=0
2
ox;
y=o
(2.2)
3. The tangential stresses on the surface of separation are
continuous
^ A ViV=0 = ^ A V 2 )y=0
(2.3)
4. The difference between the normal stresses p -j_ and p g
on the surface of separation is equal to the pressure brought about by
the surface tension; that is,
(-
dv.
b 2 h
" ? yl " %2 = [~ ?! + 2 ^l -£r I - I " p 2 + 2 ^2 ^& J = - T -__2 (2.4)
ay
oy
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
im]_
imc
2 , „2 C 2 ~ ^4 + „ 2 . „2 u 5 " ^7
aC 7 =
m l
m-p + a
^2 ^5
2 2 + C 4 + — 5 2 _ C 7 = °
m-^ + a.^ nu^ + ar
Similarly, the boundary condition (2.3) is obtained in the form
(2.5)
^2
C2
m-i + a
+ C A \a? + — ~ -7T C 2 + a 2 C 4
m-| 6 + ar
mo + a
+ C 7 \a? +
m 9 9
— S C q + a^C 7
2 , „2 5 7
(2.6)
mg + cc
NACA TM 1281
The pressures p^ and P2 are computed from, equations (l.l)
and (2.1). Thus
Pi = Pie
i(ox-Bt)
ae
m n
I f _± — _ — i_ + iR + m-, V-, \ C 2 +
^ + a 2 V m l m l m l J
(p + iU a 2 - V a - il)a 2 e- a y C^
(2.7)
2 = p 2 e
Pq =
(CQC-Pt)
^i m 2J / u oQ)
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
( v yiL = o =
^i
m
+ V
y= u Vdx/ y =o ot
1
Sh
ox
(2.9)
After differentiating expressions (2.1) and (2.8) and by substi-
tuting in expression (2.9), the following equation is obtained:
H =
a
aV, - 3
C„ -
(2.10)
m-i + a
By substituting equation (2.10) in (2.9) and by differentiating
equation (2.9),
cr-
o 2 h
ix 2 " "i - U 2 ,
2-^ - C/U 1 ^-^)
(2.11)
WACA TM 1281
By computing the derivatives dv ,/dy and 6v 2 /dy and substi-
tuting them simultaneously with expressions (2.7) and (2.11) in (2.4),
the following boundary condition is obtained:
a
m-i + of 1
'^l 0- + ip l V l a " i ' 3p l Ta'
+ m 1 |i 1 +
m
1
aV 1 - B
Co -
p-j_P - P^a - i2u 1 a 2 +
aY n - B,
C„ +
a
2 2
m. + ar
\±2 a ~ i p 2^2 a + ^ p 2
m-
3m 2 u 2
C 5 + ^ p 2 |3 " p 2 V 2 a + i2 M 2 a ) c 7 = °
The following nondimensional parameters are then introduced
(2.12)
Z =
U 1 a
R l
V l
V
K 2
V 2
A
U l
~ U 2
N =
Tp x
2
\i 1 a
K
. iii
(2.13)
WACA TM 1281
where c = p/cc. is the complex wave velocity.
Equations (1.8) can then he represented in the forms
mi = claJ±(Z - E 1 ) - 1
= a/^i(ZA - E 2 ) - 1
Equations (2.5), (2.6), and (2.12) are represented in nondimen-
sional parameters. The following notations are first introduced:
&1 * = |
1 Z-E-,
N
A/i (Z-K-, ) - 1 Z ~ E 1
= h a
l"!
■: = a\ ( Z-B 1 -21 + -2-\ = cc 2 ^
'1.
R 1" Z >
2(i - 2E 2 + 2AZ) „
C-i* = Uo ; ■ = UoCrCn
a 3 *=--^ j
a 2 Z_E 1 a 2
d x * = u 2 a 2 (ZA-E 2 + 2i) = u 2 a 2 d 1
1 i
a
2 ZA-E
2 a?
,* = i a/i(Z-R, ) - 1 = —
2 a V v 1' a
c ? * = -
i aJUz^Bz) - i c 2
ZA-S.
a
> (2.14)
where a-^, a 2 , aj, b-j_, c-j_, c 2 , c-j, and d^ are likewise nondimen-
sional magnitudes.
10
NACA TM 1281
The following system of equations is then obtained for the con-
stants C 2 , C 4 , C 3 , and C^:
a 1 *C 2 - b 1 *C 4 + c-j*C 5 + d 1 *C 7 =
-i
a 2* C 2 "
aC 4 + Cg*C 5 - a
c 7 =
ag*C 2 +
C 4 + c 3 *C 5 -
c 7 =
K C 2
=
(2.15)
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(a 1 + c 1 ) + (d x - Kb 1 )(a 2 + Kc 2 ) + (Kb ] _ + d x ) (Kc 3 - a 3 ) =
By solving this equation for Z, the following wave equation of
the 18th degree with complex coefficients is obtained:
r 18 Z 18 + (r in + is,,) Z 1 '
,17
17
17'
..* + {r 1 + is 1 ) Z + (r Q + is Q ) =
(2.16)
The real and imaginary parts of the coefficients depend on the
five nondimensional parameters: R-j_, R 2 , A, N, and K.
5. 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 (,Bj > 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 $±. 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 1281 11
In the case considered, the growth in the amplitudes of the
oscillations will lead to breakaway of infinitely long strings
from the surface of separation, similar to the formation and
breakaway of wave crests. The separation will take place for such
values of a or wavelengths k for which p^ has the maximal
value .
If a spectrum of small-period disturbances that can be developed
into a Fourier series can be assumed to be imposed on both liquids,
the harmonics with the wavelengths equal to the wavelengths of
the maximal unstable disturbances bring about a separation of
infinitely long strings from, the partition surface. Because the
characteristic dimension (for example, the diameter of the transverse
string) is connected with the length of maximal unstable disturbance,
strings of different dimensions will break away from the surface of
separation. In figure ?, , the scheme of formation of such strings
for three successive instants of time is shown.
Investigation of the roots of the simplest particular case of
equation (2.16) is presented.
Let both fluids be stationary and their kinetic viscosities
the same. In this case, V-j_ = Vg = 0, u^ = v^ } m.-. = nig, A = 1,
R-j_ = Kg = 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 Q Z 8 + (A ] + iB 1 )Z 7 + (A 2 + iB 2 )Z 6 + (A + iB 3 )Z 5 + (A + iB 4 )Z 4 +
(A 5 + iB 5 )Z 3 + (A 6 + iB 6 )Z 2 + (A 7 + iB ? )Z + A 8 = (3.1)
12 RACA TM 12 SI
where
A Q = - (1 - K) 2
A 2 = 2E (K - 1) N - E 4 + 2E 3 - 4K 2 + 6K + 13
A-, = - 2K (K - I) 2
A 3 = 4E 2 (K - 1) W - 2K (3K 2 + 13)
A 7 = 2E 3 K 2
A 4 = - E 2 E 2 +2 (E 4 - E 3 + 5E 2 + 5K) W - 12K 3 + 26K 2 - 10K - 9
A 5 = - 2E 3 H 2 + 12E 3 N - 8K 3 - 8K 2
A 8 = - K 2 (1 + 2K) N 2
= (1 - K 2 ) E 2 N 2 + (4E 4 - 10E 3 - 4K 2 - 6K) N - SE 3 + 12E 2 - 8E + 4
S-l = 2 (E 2 + 2E - 3)
B
o
3E 4 - 14E 3 + 13E 2 + 18E + 13 - SEN
B 2 = 4E (E - I) 2
B 4 = 3K 3 - 4E 2 - 20E - 2E 3 K
B 6 = 4E 2 (1 - E) IT
B 5 = - 2E% 2 + [2E (1 + K) (1 + E - E 2 ) + SE 3 + 4E 2 + 4E]n +
4(E - 1 - E 2 ) (l + E - E 2 ) + 4E 4 - 20E 3 - 4(K - 1 - E 2 ) 2
B 7 = [E 2 (1 + E) 2 - 2E 4 ]n 2 + [4E 4 + 4E (l + E) (E - 1 - E 2 )]
(3.2)
NACA TM 1281 13
The charact eristic equation (3.1) is a polynomial vhose
coefficients depend nonlinearly on the two parameters K and N.
Each pair of values of the parameters K and K or each point
of the plane EN correspond to the completely defined polynomial
(3.1), that is, completely determined values of the eight roots
of the polynomial. In the plane Etv, 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 crossing of the roots through the
real a,xis possible. This curve "breaks up the plane EN 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 = -it, is made. The upper half -plane of
the roots of equation (o.l) is transformed into the left half -plane
of the roots of the equation
- A Q g + i(A x + iB^C 7 + (A 2 + iB 2 )t 6 - i(A 3 + iB 3 )$ 5 - (A 4 + iB 4 )£ 4 +
i(A 5 + iB 5 )£ 3 + (A 6 + iB 6 )C 2 - (A 7 + iB 7 )C + A 8 =
(3.3)
By substituting £ = i£Al in the preceding equation and multi-
plying the result by tjS, equation (3.3) is reduced to the form
F(£,ti) + iG(£,n) = . (3.4)
where
F(t,n) = A e 8 + A x e 7 Tl + A 2 | 6 n 2 + A 3 £V + A 4 | 4 n 4 + A 5 | 3 n 5 +
A 6 e 2 T! 6 + A-^ 7 + A 6 ti 8 ( 3 . 5 )
G(U) - ^fn + B 2 e 6 ri 2 + B 3 ^3 + B 4 eS 4 + B 5 e 3 n 5 + B^ 6 + B^r, 7
14
WACA TM 1281
If K 2n is the space of complex polynomials of degree n and
D(k,n - k) is the manifold of polynomials R 2n 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:
A Q A ± A 2 A 3 A 4 A 5 A 6 A ? A f
B l B 2 B 3 B
B 5 B 6 B 7 B 6
(3.6)
and by making the transformation
% + \ lBl A 1 + \ lB2 A 2 + \b 3 A 3 + \ lB4 . . .A~
B l B 2 B^ ...C
(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:
AqB-l < (3.8)
B y setting \^ = - Aq/B-j_ in table (3.7)
(A 1 B 1 " Vz^l (A 2 E 1 " A B 3 } / B 1 (A 3 B 1 " A B 4 ) / B 1 — \ A 8 "l
B-,
B,
B,
..B 7
(3.9)
Because A 1 B ] _ - A^B 2 = " 16K(K - l) 3 < for K>1, by multi-
plying the elements of the first rows of (3.9) by B - L 2 /(A 1 B 1 - ApBg)
and changing signs in the second row
B-
- Bn
D l
B 2
- B 3 - B 4
D„
B c
D c
D £
B 6 - B 7
(3.10)
NACA TM 1281
15
wnere
B l (3.15)
B l " B 2
D 3 - B 4
Because D 1 (D 1 - B 2 ) - B x (D 2 - B,) > for K>1, by multi-
plying the elements of the first row of (3.15) by (D-j_ - B 2 ) /
[D^Di - B 2 ) - B^Dg - B 3 )]
(
% ~ B 2
[P 2 (D 1 - B 2 ) - B 1 (D 3 - B 4 )] (D x - B 2 )
DiCD-L - B 2 ) - B 1 (D 2 - B 3 )
L D 1 " B 2
D 2 - B 3
.]
(3.16)
The elements of the first row are subtracted from the elements
of the second row of table (3.16).
D-l - B 2
[(D 2 (D x - B 2 ) - B 1 (D 3 - B 4 )] (D 1 - B 2 )
(D 2 - B 3 )
[DgCDi - B 2 ) - B 1 (D 3 - B 4 )] (D x - B 2 )
1
D 1 (D 1 - B 2 ) - B 1 (D 2 - B 3 ) "J
(3.17)
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.
(Di - Bo)
(Dg - B 3 ) -
[D 2 (D X - B 2 ) - B!(D 3 - B 4 )] (Di - B 2 )
D 1 (D 1 - B 2 ) - B 1 (D 2 - B 3 )
<0
(3.18)
NACA TM 1281 17
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 KN plane
into regions are obtained. The most interesting case of large
K = n-)_/(i2^>l 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 + 3) =
egW 3 + e-]_N 2 + e 2 N + e 3 =
4(K 2 - 1)(K + 3)N + K(K - l)(K 3 + 17K 2 - 96K + 99) =
(3.19)
where
e l
e = 128 (K 4 - K 3 - 23K 2 - 39K - 18)
= 592K(K 5 + 8.4K 4 + 3.18K 3 - 96K 2 - 20. 3K + 0.98)
e 2 = 9K 2 (K 6 + 8.4K 5 - 97. 3K 4 - 2045K 3 + 1700K 2 + 390K + 363)
e 3 = 24K 5 (K 5 + 12. 3K 4 + 306K 3 - 4100K 2 + 12,300K - 7000)
By plotting the curves (3.19) in the KN plane and separating
by hatched lines 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> and N>0
a region of values of K and N 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 B^ or c^, several infinitely long strings will
18 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., 1S45.
2. Tomotika, S.: On the Instability of a Cylindrical Thread of a
Viscous Liquid Surrounded by Another Viscous Fluid. Proc.
Roy. Soc. London, vol. CL, no. A870, ser. A, June 1935,
pp. 322-337.
3. Petrov, G. I.: On the Stability of Turbulent Layers. Rep.
Wo. 304, CART, 1937.
4. Neimark, Y. I.: On the Problem of the Distribution of the Roots
of Polynomials. DAN, T. 58, No. 3, 1947.
NACA TM 1281
19
v t.Pi.N
mm.
WW////M
Figure 1.
Figure 2.
NACA-Langley - 4-16-51 - 875
(O
to
C
•
o
H
•H
•
-P
<-\
NACA^
es on Surface of Separa
•H
-P
•H
n
•
/ >
•
>H
r &
CO
f >
T3
-O
>j
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