This document studies the stability of a gas cylinder embedded in a compressible liquid. It derives the dispersion relation that relates the growth rate of perturbations to wavenumbers and other parameters. For an incompressible liquid, the model is capillary unstable only for axisymmetric perturbations with small wavenumbers, and stable otherwise. Compressibility has a strong destabilizing effect and can cause the model to collapse. The capillary force has a mix of stabilizing and destabilizing effects depending on the mode of perturbation.

1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
1
STABILITY OF A GAS CYLINDER IN A COMPRESSIBLE LIQUID
Hussain E. Hussain1,2
1
Mathematics Department, Faculty of Science, Taif University, Hawia(888) Taif, Saudi Arabia
2
Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt
ABSTRACT
The stability of a gas cylinder in a compressible liquid endowed with surface tension has been
studied. The dispersion relation is derived and discussed for all axisymmetric and non-axisymmetric
perturbations.
The capillary force is destabilizing only in small domain of axisymmetric mode. While it is
capillary stabilizing in the rest domains. The compressibility has a strong destabilizing tendency and
causes collapsing the model.
1. INTRODUCTION
The stability criterion of a gas jet pervaded into an incompressible liquid endowed with surface
tension for axisymmetric perturbation is indicated for first time by Chandrasekhar (1981). See also
Drazin and Reid (1980) (p.16) where the inertia force of the liquid is paramount over that of the gas.
Cheng (1985) studied the instability of a streaming gas jet in an incompressible liquid for all
axisymmetric and non-axisymmetric modes of perturbation. However, we have to mention here that
the results given by Cheng (1985), in Eqs.(4) and (5) are incorrect in the third term. In fact the quantity
(1-m2
-k2
R2
0) must be in the numerator as it is clear from Eq. (3) there. See also equation (30) in the
present work and Drazin’s result (1980) p.16 and also Chandrasekhar’s dispersion relation P.538 and
P.540 [Eqs. (147) and (155) there]. Radwan and Elazab (1987) examined the viscosity effect on the
capillary instability of this model for axisymmetric perturbation. In (1989) Radwan identified the
stabilizing effect of the magnetic field on the stability of this model, for other topics see Radwan
(2005). The stability of different cylindrical models under the action of self gravitating force in
addition to other forces has been elaborated by Radwan and Hasan (2008) and (2009). They (2008)
studied the gravitational stability of a fluid cylinder under transverse time-dependent electric field for
axisymmetric perturbations. Hasan (2011) has discussed the stability of oscillating streaming fluid
cylinder subject to combined effect of the capillary, self gravitating and electrodynamic forces for all
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH
IN ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 6, June (2014), pp. 01-07
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2014): 7.8273 (Calculated by GISI)
www.jifactor.com
IJARET
© I A E M E

2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
2
axisymmetric and non axisymmetric perturbation modes. Hasan (2011) studied the instability of a full
fluid cylinder surrounded by self-gravitating tenuous medium pervaded by transverse varying electric
field under the combined effect of the capillary, self-gravitating and electric forces for all modes of
perturbations. He (2012) discussed the instability of a full fluid cylinder surrounded by selfgravitating
tenuous medium pervaded by transverse varying electric field under the combined effect of the
capillary, self-gravitating and electric forces for all modes of perturbations. He (2012) studied the
magnetodynamic stability of a fluid jet pervaded by transverse varying magnetic field while its
surrounding tenuous medium is penetrated by uniform magnetic field.
In all foregoing works it is assumed that the fluid moves such that the divergence of the flow
fluid vanishes. Here we, will not consider this behavior i.e. the velocity of the liquid is not solenoid
anymore, investigate the capillary instability of a gas cylinder embedded into (real) compressible
liquid. This phenomenon may occur in the geological drillings as a gas escapes from below oil layers
in the crust. For astrophysical applications, we may refer to the experimental work of Kendall (1986).
2. FORMULATION OF THE PROBLEM
Consider a gas cylinder of radius R0 pervaded into a compressible liquid.Assuming that the
inertia force of the liquid is paramount over that of the gas. The basic equations are being:
0)()( =∇⋅+⋅∇+
∂
∂
ρρ
ρ
uu
t
(1)
r
P
u
z
u
r
u
r
u
t
u
rzr
r
∂
∂
−=











∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
ϕ
ρ ϕ
(2)
ϕϕ
ρ ϕ
ϕϕ
∂
∂
−=











∂
∂
+
∂
∂
+
∂
∂
+
∂
∂ P
r
u
z
u
r
u
r
u
t
u
zr
1
(3)
z
P
u
z
u
r
u
r
u
t
u
zzr
z
∂
∂
−=











∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
ϕ
ρ ϕ
(4)
)( uPT
z
u
r
u
r
u
t
T
C zrv ⋅∇−=











∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
ϕ
ρ ϕ
(5)
TRP c
ρ= (6)
)( 1
2
1
1
−−
+= rrSPs , Nrr ⋅∇=+ −− 1
2
1
1 (7), (8)
f
f
N
∇
∇
= , 0),,,( =tzrf ϕ (9), (10)
Here ρ, u(=(ur,uϕ, uz)) and P are the polytropic liquid density, velocity vector and kinetic
pressure. T is the temperature, Rc
polytropic gas constant, Cv the specific heat at constant volume,

3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
3
S the surface tension coefficient and N the outward unit vector normal to the gas-liquid interface
pointed as r does where (r, ϕ, z) are the cylindrical coordinates with the z-axis coinciding with the
axis of the gas-liquid model. r1 and r2 are the principle radii of curvature of the gas-liquid interface.
3. PERTURBATION ANALYSIS
For small departures from the initial state, every perturbed quantity χ(r,ϕ, z, t) may be
expressed as
χ(r,ϕ, z, t) = χ0 + ε(t) χ1 (r, ϕ, z) + … (11)
Here χ(r, ϕ, z, t) stands for u, ρ, P, N and the radial distance of the gas cylinder. ε(t) is the
amplitude of the perturbation
ε(t) = ε0eσt
(12)
where ε0 (=ε at t=0) is the initial amplitude and σ is the growth rate. Based on the perturbation
technique
Q1 (r, ϕ, z) = Q1 (r) exp [i(kz + mϕ)] (13)
where k is the longitudinal wavenumber and m the azimuthal wavenumber. The radial distance of the
gas cylinder due to the perturbation is given by:
R = R0 + ε0R1 (14)
with
R1 ≈ exp (i(kz + mϕ) + σt) (15)
where ε0R1 is the elevation of the surface wave measured from the unperturbed position. By inserting
the expansion (11) into the basic equations (1) – (10) we get two systems of partial differential
equations, the unperturbed system and the perturbed one.
By solving the unperturbed system of equations with [u0 = (0, 0, 0)], we obtain






−=
0
00
R
S
PP g
(16)
where g
P0 is the gas constant pressure. If )( 00 RSPg
< , the kinetic pressure 0P of the liquid will be
negative and the model collapses.
The perturbed system of equations is given by:
1
1
0 P
t
u
−∇=





∂
∂
ρ , 1
2
1 ρaP = (17), (18)
( ) 01
1
=⋅∇+
∂
∂
u
t
oρ
ρ
(19)

4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
4






∂
∂
+
∂
∂
+= 2
1
2
2
2
1
2
121
z
R
R
R
R
R
S
P o
o
s
ϕ
(20)
where )( 00 ργPa = is the speed of sound in the liquid
By solving equations (17)-(19), we get






−=⋅∇ 2
0
1
1 )(
a
P
u
ρ
σ
(21)
and
110 Pu −∇=σρ (22)
Combining equations (21) and (22), yields
02
1
2
1
2
=−∇
a
P
P
σ
(23)
This leads, on using the space dependence (13), to
0
1
1
2
2
2
1
=





+−





P
r
m
dr
dP
r
dr
d
r
ζ (24)
With
2
2
22
a
k
σ
ζ += (25)
The non-singular solution of equation (24) is given by
( )[ ]ϕζ mkzirAKP m += exp)(1 (26)
where Km (ζr) is the second kind of modified Bessel function of order m while A is constant of
integration could be determined upon using appropriate boundary condition at r = R0: viz.,
t
R
u r
∂
∂
= 1
1 at 0Rr = (27)
From which we get
)(
00
2
yyK
R
A
m
ι
ρσ
−= (28)
where y (= ζR0) is the dimensionless longitudinal wavenumber of a compressible liquid. Also equation
(20) yields

5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
5
( ) 1
22
2
0
1 1 Rxm
R
S
Ps −−= (29)
where x(=kR0) is the ordinary dimensionless longitudinal wavenumber.
By applying the balance of the pressure across the gas-liquid interface at r=R0, following
dispersion relation is obtained
( ) )(
)(
1 22
3
2
yK
yyK
xm
R
S
m
m
oo
ι
ρ
σ −−
−
= (30)
4. DISCUSSIONS
Equation (30) is the required dispersion relation of a gas cylinder embedded into a
compressible liquid endowed with surface tension. It relates the growth rate σ with the longitudinal
dimensionless wavenumbers x and y, the azimuthal wavenumber m, the modified Bessel function
Km(y) of second kind of order m and its derivative and with other parameters S, ρ0 and R0 of the
problem.
As a → ∞ we have y → x and the liquid is incompressible here, we have
( ) )(
)(
1 22
3
2
xK
xxK
xm
R
S
m
m
oo
ι
ρ
σ −−
−
= (31)
This relation coincides with that given by Drazin and Reid (1980) p. 16, and Radwan result
(1989) as we neglect the magnetic field effect there.
As a → ∞ we have y → x and suppose m = 0, we get
( ) )(
)(
1
0
12
3
2
xK
xxK
x
R
S
oo
−=
ρ
σ (32)
where )()( 10 xKxK −=ι
. This relation coincides with that given by Chandrasekhar (1981).
In discussing equations (31) and (32), it is found that














≠∀≠<
<<=>
===
∞<<=<
000
100,0
1,0,0
1,0,0
)( 3
00
2
xasm
xm
xm
xmas
RS
K
ρ
σ
This means that the gas-liquid model as the liquid is incompressible is capillary unstable only
for m = 0 as 0 < x < 1 while it is stable for (m = 0 as 1 < x < ∞) and (m≠0, ∀ x ≠0). These results are
confirmed numerically (cf. Chandrasekhar (1981) for m=0 and Radwan (1989) and (2005) for m ≥ 0).

6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
6
Here the effect of the capillary instability on the present model could be identified via
discussing equation (30) numerically in its general form. However, we see as above that the most
dangerous mode of instability is m=0.
In the present study since the argument of ι
mK and K`m in equation (30) (is being y(=ζR0), the
longitudinal wavenumber of the compressible liquid with ( )2
0
2222
Raxy σ+= ) includes σ2
(see
equation (25)), it is cumbersome to identify the stability conditions and states using the well-known
standard approaches. So, we have used the numerical iterative technique as m=0 for different values of
a(= 0/ Ra )=1.0, 1.5, 2.0, 5.0 and 10.0. Due to the discussing results we may see (fig.1) that the unstable
domains are decreasing vertically with increasing a-values while they are the same 0< x <1
horizontally. Keep in mind that the parameter "a" in the denominator of equation (25), we conclude
that the compressibility has strong destabilizing tendency on the present model and causes collapsing
it.
It is worthwhile to mention here that through the numerical calculations, we may have
)( 3
0
2
RS ρ
σ
> 0 or )( 3
0
2
RS ρ
σ
=0 or )( 3
0
2
RS ρ
σ
< 0 i.e. we have the cases 3
0RS ρ
σ
is real or zero or
imaginary.
As 3
0RS ρ
σ
is real, then the area under the curves σ/√S/(ρR3
0) (see equation (15) for time
dependence) represents the ordinary unstable domains. As 3
0RS ρ
σ
is imaginary we have to write
σ = iω (i=√-1, imaginary factor) with ω/2π is the oscillation frequency, then the area under the curves
(ω/√S/ρR3
0) is being the ordinary stable domains. As 3
0RS ρ
σ
=0, we have neutral (marginal) stability
states, from which one obtain the critical values of x(=xc) which separates stability domains from those
of instability: here in our problem xc=1.
REFERENCES
[1] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability “(Dover Publ., New York)
1981.
[2] L.Y. Cheng, Phys. Fluids, 28 (1985) 2614.
[3] P.G. Drazin and W. Reid, Hydrodynamic Stability, (Cambridge University Press, London)
1980, pp 16.
[4] J.M. Kendall, Phys. Fluids, 29 (1986) 2086.
[5] A.E. Radwan and S.S. Elazab, Simon Stevin, 61 (1987)29.
[6] A.E. Radwan, J. Phys. Soc. Japan, 58 (1989) 1225.
[7] A.E. Radwan, Mechanics and Mechanical Engineering, 8 (2005)127.
[8] Mohammed, A. A. and Nayyar, A. K., J. Phys. A, 3(1970) 296.
[9] Radwan, A. E., Acta Phys. Polonica A, 82 (1992) 451.
[10] Radwan, A. E., Phys. Scrpt., 76 (2007)510.
[11] Radwan, A. E. and Hussain, E.H., Int.J. Maths. & Comput., 3(2009) 91.
[12] Radwan AE and Hasan AA, Magnetohydrodynamic stability of selfgravitational fluid
[13] cylinder, Appl. Mathal Modlling, 33(4)(2009) 2121.
[14] Hasan, A. A., Journal of Physica B, 406(2), 234(2011).
[15] Hasan A. A., Boundary Value Problems, (31), 1(2011).

7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 01-07 © IAEME
7
[16] Hasan A. A., Journal of Applied Mechanics Transactions ASME, 79(2), 1(2011).
[17] Hasan A. A., Mathematical Problems in Engineering, 2012, 1(2012).
[18] Hany L. S. Ibrahim and Elsayed Esam M. Khaled, “Light Scattering from a Cluster Consists of
Different Axisymmetric Objects”, International Journal of Advanced Research in Engineering
& Technology (IJARET), Volume 4, Issue 6, 2013, pp. 203 - 215, ISSN Print: 0976-6480,
ISSN Online: 0976-6499.
[19] M. M.Izam, E. K.Makama and M.S.Ojo, “The Gravitational Potential of a Diatomic System”,
International Journal of Advanced Research in Engineering & Technology (IJARET),
Volume 4, Issue 2, 2013, pp. 206 - 209, ISSN Print: 0976-6480, ISSN Online: 0976-6499.
[20] Hussain E. Hussain and Hossam A. Ghany, “Self-Gravitating Electrodynamic Stability of
Accelerating Streaming Fluid Cylinders in Self-Gravitating Tenuous Medium”, International
Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 7,
2013, pp. 92 - 100, ISSN Print: 0976-6480, ISSN Online: 0976-6499.