International Journal of Trend in Scientific Research and Development(IJTSRD)
International Open Access Journal
ISSN:2456-6470 —www.ijtsrd.com—Volume -2—Issue-5
NONLINEAR ASYMMETRIC KELVIN-HELMHOLTZ INSTABILITY OF
CYLINDRICAL FLOW WITH MASS AND HEAT TRANSFER AND THE
VISCOUS LINEAR ANALYSIS
DOO-SUNG LEE
Department of Mathematics
College of Education, Konkuk University
120 Neungdong-Ro, Kwangjin-Gu, Seoul, Korea
e-mail address: dslee@konkuk.ac.kr
Abstract
phenomenon of boiling accompanies high heat
The nonlinear asymmetric Kelvin-Helmholtz and mass transfer rates which are significant in
stability of the cylindrical interface between the determining the flow field and the stability of the
vapor and liquid phases of a fluid is studied when system.
the phases are enclosed between two cylindriHsieh [1] presented a simplified formulation
cal surfaces coaxial with the interface, and when of interfacial flow problem with mass and heat
there is mass and heat transfer across the inter- transfer, and studied the problems of Rayleighface. The method of multiple time expansion is Taylor and Kelvin-Helmholtz stability in plane
used for the investigation. The evolution of am- geometry.
plitude is shown to be governed by a nonlinear
The mechanism of heat and mass transfer
first order differential equation. The stability cri- across an interface is important in various industerion is discussed, and the region of stability is trial applications such as design of many types of
displayed graphically. Also investigated in this contacting equipment, e.g., boilers, condensers,
paper is the viscous linear potential flow.
pipelines, chemical reactors, and nuclear reacKeywords Kelvin-Helmholtz stability, Mass tors, etc.
and heat Transfer, Cylindrical flow.0
In the nuclear reactor cooling of fuel rods by
liquid coolants, the geometry of the system in
many cases is cylindrical. We have, therefore,
1. Introduction
considered the interfacial stability problem of
In dealing with flow of two fluids divided by a cylindrical flow with mass and heat transfer.
an interface, the problem of interfacial stability is Nayak and Chakraborty[2] studied the Kelvinusually studied with the neglect of heat and mass Helmholtz stability of the cylindrical interface
transfer across the interface. However, there are between the vapor and liquid phases of a fluid,
situations when the effect of mass and heat trans- when there is a mass and heat transfer across
fer across the interface should be taken into account in stability discussions. For instance, the
0
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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT (IJTSRD)ISSN:2456-6470
the interface, while Elhefnawy[3] studied the effect of a periodic radial magnetic field on the
Kelvin-Helmholtz stability of the cylindrical interface between two magnetic fluids when there
is mass and heat transfer across the interface.
The analysis of these studies was confined within
the frame work of linear theory. They both found
that the dispersion relations are independent of
the rate of interfacial mass and heat transfer.
Hsieh[4] found that from the linearized analysis,
when the vapor region is hotter than the liquid
region, as is usually so, the effect of mass and
heat transfer tends to inhibit the growth of the
instability. Thus for the problem of film boiling, the instability would be reduced yet would
persist according to linear analysis.
It is clear that such a uniform model based
on the linear theory is inadequate to answer the
question of whether and how the effect of heat
and mass transfer would stabilize the system, but
the nonlinear analysis is needed to answer the
question.
The purpose of this paper is to investigate the
Kelvin-Helmholtz asymmetric nonlinear stability
of cylindrical interface between the vapor and
liquid phases of a fluid when there is a mass and
heat transfer across the interface.
first order theory and the linear dispersion relation are obtained in Sec.3. In Sec .4 we have
derived second order solutions. In Sec.5 a first
order nonlinear differential equation is obtained,
and the situations of the stability and instability
are summarized. In Sec.6 we investigate linear
viscous potential flow. In Sec.7 some numerical
examples are presented.
2. Formulation of the problem and basic
equations0
We shall use a cylindrical system of coordinates (r, θ, z) so that in the equilibrium state
z−axis is the axis of asymmetry of the system.
The central solid core has a radius a. In the
equilibrium state the fluid phase ”1”, of density ρ(1) , occupies the region a < r < R, and,
the fluid phase ”2”, of density ρ(2) , occupies the
region R < r < b. The inner and outer fluids
are streaming along the z axis with uniform velocities U1 and U2 , respectively. The temperatures at r = a, r = R, and r = b are taken
as T1 , T0 , and T2 respectively. The bounding
surfaces r = a, and r = b are taken as rigid.
The interface, after a disturbance, is given by
the equation
F (r, z, t) = r − R − η(θ, z, t) = 0,
(2.1)
The nonlinear problem of Rayleigh-Taylor in- where η is the perturbation in radius of the interstability of a system in a cylindrical geometry is, face from its equilibrium value R, and for which
however, studied by the present author in (Lee[5- the outward normal vector is written as
½
µ
¶
µ ¶2 ¾−1/2
6]).
∇F
1 ∂η 2
∂η
n=
= 1+
+
The multiple time scale method is used to ob|∇F |
r ∂θ
∂z
tain a first order nonlinear differential equation,
µ
¶
1 ∂η
∂η
from which conditions for the stability and in× er −
eθ −
ez ,
(2.2)
r ∂θ
∂z
stability are determined.
In more recent years,Awashi, Asthana and
Zuddin[7] considered a problem in which a viscous potential flow theory is used to study the
nonlinear Kelvin-Helmholtz instability of the interface between two viscous ,incompressible and
thermally conducting fluids.
we assume that fluid velocity is irrotational in
the region so that velocity potentials are φ(1) and
φ(2) for fluid phases 1 and 2. In each fluid phase
∇2 φ(j) = 0. (j = 1, 2)
(2.3)
The solutions for φ(j) (j = 1, 2) have to satisfy
The basic equations with the accompanying the boundary conditions. The relevant boundboundary conditions are given in Sec.2. The ary conditions for our configuration are
(i) On the rigid boundaries r = a and r = b:
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K1 (T1 − T0 )
,
(2.8)
(R + η)(log(R + η) − log a)
and we expand it about r = R by Taylor’s ex(2.4) pansion, such as
1
S(η) = S(0) + ηS 0 (0) + η 2 S 00 (0) + · · · , (2.9)
2
(2.5)
and we take S(0) = 0, so that
The normal field velocities vanish on both central solid core and the outer bounding surface.
∂φ(1)
=0
∂r
on r = a,
∂φ(2)
=0
∂r
on r = b,
−
(ii) On the interface r = R + η(θ, z, t):
K2 (T0 − T2 )
K1 (T1 − T0 )
=
= G(say), (2.10)
R log(b/R)
R log(R/a)
(1) The conservation of mass across the interface:
indicating that in equilibrium state the heat
fluxes are equal across the interface in the two
·· µ
¶¸¸
fluids.
∂F
ρ
+ ∇φ · ∇F
= 0,
∂t
From (2.1), (2.7), and (2.9), we have
¶¸¸
·· µ
µ (1)
¶
∂φ ∂η 1 ∂η ∂φ ∂η ∂φ
∂η 1 ∂η ∂φ(1) ∂η ∂φ(1)
(1) ∂φ
or
ρ
−
−
−
= 0,
ρ
−
−
−
∂r
∂t
r ∂θ ∂θ
∂z ∂z
∂r
∂t
r ∂θ ∂θ
∂z ∂z
(2.6)
= α(η + α2 η 2 + α3 η 3 ),
(2.11)
where [[ h]] represents the difference in a quantity
where
as we cross the interface,i.e., [[ h]] = h(2) − h(1) ,
G log(b/a)
where superscripts refer to upper and lower fluα=
,
LR log(b/R) log(R/a)
ids, respectively.
µ
¶
1
3
1
1
(2) The interfacial condition for energy is
α2 =
− +
−
,
¶
µ
R
2 log(b/R) log(R/a)
∂F
·
+ ∇φ(1) · ∇F = S(η),
(2.7)
Lρ(1)
2 log(R2 /ab)
1 11
∂t
−
α3 = 2
R 6
log(b/R) log(R/a)
¸
where L is the latent heat released when the fluid
3
log (b/R) + log3 (R/a)
is transformed from phase 1 to phase 2. Phys+
.
{log(b/R) log(R/a)}2 log(b/a)
ically, the left-hand side of (2.7) represents the
latent heat released during the phase transforma(3) The conservation of momentum balance,
tion, while S(η) on the right-hand side of (2.7) by taking into account the mass transfer across
represents the net heat flux, so that the energy the interface, is
µ
¶
will be conserved.
∂F
(1)
(1)
(1)
ρ (∇φ · ∇F )
+ ∇φ · ∇F
∂t
In the equilibrium state, the heat fluxes in
µ
¶
∂F
the direction of r increasing in the fluid phase
(2)
(2)
(2)
= ρ (∇φ · ∇F )
+ ∇φ · ∇F
1 and 2 are −K1 (T1 − T0 )/R log(a/R) and
∂t
−K2 (T0 − T2 )/R log(R/b), where K1 and K2 are
+(p2 − p1 + σ∇ · n)|∇F |2 ,
(2.12)
the heat conductivities of the two fluids. As in
where p is the pressure and σ is the surface tenHsieh(1978), we denote
sion coefficient, respectively.0 By eliminating the
K2 (T0 − T2 )
pressure by Bernoulli’s equation we can rewrite
S(η) =
the above condition (2.12) as
(R + η)(log b − log(R + η))
———————————————————
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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT (IJTSRD)ISSN:2456-6470
½
µ
µ ¶2 ¾−1
·· ½
µ ¶
µ
¶
µ ¶
¶
1 ∂η 2
∂η
∂φ 1 ∂φ 2 1 1 ∂φ 2 1 ∂φ 2
ρ
+
+
+
− 1+
+
∂t
2 ∂r
2 r ∂θ
2 ∂z
r ∂θ
∂z
µ
¶µ
¶¾¸¸
∂φ ∂η
1 ∂φ ∂η ∂φ
∂η ∂φ ∂η
1 ∂φ ∂η ∂φ
×
+
−
+
+
−
∂z ∂z r2 ∂θ ∂θ
∂r
∂t
∂z ∂z r2 ∂θ ∂θ
∂r
½
µ
¶2
¾
σ
1 ∂η
2
=
1+
(R + η)|∇F |
r ∂θ |∇F |2
µ
µ ¶2 ¾¸
· 2 ½
¶ ¾
½
σ
∂ η
1 ∂η 2
2 ∂η ∂ 2 η ∂η
1 ∂2η
∂η
−
1+
− 2
+ 2 2 1+
.
3
2
|∇F | ∂z
r ∂θ
r ∂θ ∂θ∂z ∂z r ∂θ
∂z
————————————————————————————–
(2.13)
When the interface is perturbed from the equiIt is clear from the above inequality that the
librium η = 0 to η = A exp[i(kz + mθ − ωt)], the streaming has a destabilizing effect on the stabil(2)
dispersion relation for the linearized problem is ity of a cylindrical interface, because Em
is always negative from the properties of Bessel functions. (ii) when α 6= 0, we find that necessary
D(ω, k, m) = a0 ω 2 + (a1 + ib1 )ω + a2 + ib2 = 0, and sufficient stability conditions for (2.14) are
(2.14) [3]
where
b1 > 0,
(2.18)
(1)
(2)
a0 = ρ(1) Em
− ρ(2) Em
,
a1 =
b1 =
a2 =
and
(2)
(1)
2k{ρ Em
U2 − ρ(1) Em
U1 },
(1)
(2)
α{Em − Em },
(1) 2
(2) 2
k 2 {ρ(1) Em
U1 − ρ(2) Em
U2 }
σ
2 2
2
− 2 (R k + m − 1),
R
(2)
(1)
b2 = αk{Em
U2 − Em
U1 },
where for the simplicity of notation, we used
(j)
(j)
Em
= Em
(k, R),
(j = 1, 2)
a0 ω 2 + a1 ω + a2 = 0.
(2.15)
Therefore the system is stable if
a21 − 4a0 a2 > 0,
(2.16)
since a0 is always positive.
Putting the values of a0 , a1 , a2 , b1 and b2
from(2.14) into(2.18) and( 2.19) we notice that
the condition (2.18) is trivially satisfied since α
is always positive, and from properties of Bessel
(2)
functions Em is always negative. From (2.19), it
can be shown that the condition for the stability
of the system is
(1)
(1)
ρ(1) Em
−
(2)
ρ(2) Em
·
¸
(1) (2)
Em Em (ρ(1) − ρ(2) )2
× 1−
> 0.
(1)
(2)
(Em − Em )2 ρ(1) ρ(2)
(2.20)
The stability condition (2.20) differs from (2.17)
by the additional last term:
(1)
(2)
ρ(1) ρ(2) Em Em (U2 − U1 )2
(2)
(1) (2)
2
σ
2 2
2
2 ρ ρ Em Em (U2 − U1 )
(R
k
+m
−1)+k
(1)
(2)
R2
ρ(1) Em − ρ(2) Em
(2)
(1)
(2)
Em Em (ρ(1) − ρ(2) )2 /[ρ(1) ρ(2) (Em − Em )2 ].
Thus the condition (2.20) is valid for infinitesimal α and when α = 0 the last term is absent.
σ
(R2 k 2 + m2 − 1)
R2
+k 2
(2.19)
0
(1)
(j)
where Em (k, R), (j = 1, 2) are explained by
(3.4)-(3.5). (i) When α = 0, (2.14) reduces to
or
a0 b22 − a1 b1 b2 + a2 b21 < 0,
(2)
> 0.
(2.17)
We now employ multiscale expansion near the
critical wave number. The critical wave number
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is attained when a2 = b2 = 0. The corresponding The first order solutions will reproduce the lincritical frequency, ωc is zero for this case.
ear wave solutions for the critical case and the
Introducing ² as a small parameter, we as- solutions of (2.3) subject to boundary conditions
yield
sume the following expansion of the variables:
η=
3
X
n
4
² ηn (θ, z, t0 , t1 , t2 ) + O(² ),
(2.21)
n=1
φ(j) =
3
X
4
²n φ(j)
n (r, θ, z, t0 , t1 , t2 )+O(² ), (j = 1, 2)
n=0
(2.22)
where tn = ²n t(n = 0, 1, 2).0 The quantities
appearing in the field equations (2.3) and the
boundary conditions (2.6), (2.11), and (2.13) can
now be expressed in Maclaurin series expansion
around r = R. Then, we use (2.21), and (2.22)
and equate the coefficients of equal power series
in ² to obtain the linear and the successive nonlinear partial differential equations of various orders.
To solve these equations in the neighborhood
of the linear critical wave number kc , because of
the nonlinear effect, we assume that the critical
wave number is shifted to
k = kc + ²2 µ.
η1 = A(t1 , t2 )eiϑ + Ā(t1 , t2 )e−iϑ ,
(3.1)
¶
µ
α
(1)
(1)
φ1 =
+ ikU1 A(t1 , t2 )Em
(k, r)eiϑ + c.c.,
ρ(1)
(3.2)
µ
¶
α
(2)
(2)
φ1 =
+ ikU2 A(t1 , t2 )Em
(k, r)eiϑ + c.c.,
ρ(2)
(3.3)
where
0 (ka) − I 0 (ka)K (kr)
Im (kr)Km
m
m
(1)
Em
(k, r) = 0
,
0 (ka) − I 0 (ka)K 0 (kR)
Im (kR)Km
m
m
(3.4)
0 (kb) − I 0 (kb)K (kr)
I
(kr)K
m
m
m
m
(2)
Em
(k, r) = 0
,
0 (kb) − I 0 (kb)K 0 (kR)
Im (kR)Km
m
m
(3.5)
¯
∂
0
Im (kr)¯r=a , etc.
ϑ = kz + mθ, Im (ka) =
∂r
with Im and Km are the modified Bessel functions of the first and second kinds, respectively.
4. Second order solutions.
With the use of the first order solutions , we
obtained the equations for the second order problem
3. First Order Solutions.
(j)
We take
(j)
φ0
= Uj z.
(j = 1, 2)
∇2 φ2 = 0,
(j = 1, 2)
and the boundary conditions at r = R.
(4.1)
——————————————————————————————
½
(j)
ρ
·
½
¾½
µ
¶
¾
¸
α
1
m2
(j)
2
(j)
− αη2 = ρ
+ iUj
− 2 k + 2 Em + αα2
R
R
ρ(j)
µ
¶
∂A iϑ
1
×A2 e2iϑ + ρ(j)
e + c.c. + 2α
+ α2 |A|2 ,
(j = 1, 2)
(4.2 − 4.3)
∂t1
R
(j)
∂φ2
∂η2
−
Uj
∂r
∂z
¾
µ 2
¶
(1)
(2)
∂φ2
∂ η2
1 ∂ 2 η2
η2
∂φ2
(1)
− ρ U1
+σ
+ 2
+ 2
ρ U2
∂z
∂z
∂z 2
R ∂θ2
R
½·· µ
¶2 ½
µ 2
¶ ¾
¸¸
1
α
m
2
2
2 2
=−
ρ
+ ikU
−1 −
+ k Em + 3αU ki − 2ρU k
2
ρ
R2
(2)
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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT (IJTSRD)ISSN:2456-6470
¾
·· µ
¶ ¸¸
σ
ρ α
∂A iϑ
2 2
2
2 2iϑ
+ 3 (R k + 2 − 7m ) A e +
+ ikU Em
e + c.c.
R
k ρ
∂t1
½·· µ 2
¶½
µ 2
¶¾¸¸
¾
m
σ
α
2 2
2
2
2 2
2
−1 + Em
+k
+ 3 (R k + m − 2) |A|2 .
− ρ 2 +k U
ρ
R2
R
The non secularity condition for the existence of the uniformly valid solution is0
∂A
= 0.
∂t1
Equations (4.1) to (4.4) furnish the second order solutions:
¶
µ
1
η2 = −2
+ α2 |A|2 + A2 A2 e2iϑ + Ā2 Ā2 e−2iϑ ,
R
(j)
(j)
(j)
φ2 = B2 A2 e2iϑ E2m (2k, r) + c.c. + b(j) (t0 , t1 , t2 ),
where
(j = 1, 2)
½··
½ µ 2
¶
¾µ
¶2
ρ
m
α
2
2
−ρi2kU E2m β +
Em
+k +1
+ ikU
2
R2
ρ
¸¸
¾
σ
2
2 2
2
+2ρ(kU ) − i3αkU + 3 (2 + R k − 7m ) ,
2R
½
¾
α
(j)
(j)
B2 = β +
+ 2ikUj A2 ,
ρ(j)
½
¾½
µ 2
¶¾
α
1
αα2
(j)
(j) m
2
β =
+ ikUj
− 2Em
+k
+ (j) ,
2
(j)
R
R
ρ
ρ
½·· µ 2
¶½
µ 2
¶¾¸¸
(2)
(1)
∂b
∂b
α
m
2
2
ρ(2)
− ρ(1)
=
ρ 2 + k2 U 2
1 − Em
(k, R)
+
k
∂t0
∂t0
ρ
R2
µ
¶¾
σ
|A|2 ,
− 3 k 2 R2 + m2 − 4 − 2Rα2
R
(4.4)
(4.5)
(4.6)
(4.7)
1
A2 =
D(0, 2k, 2m)
(j)
(4.8)
(4.9)
(4.10)
(4.11)
(j)
whereE2m = E2m (2k, R).
5. Third order solutions
We examine now the third order problem:
(i)
∇20 φ3 = 0.
(i)
(i = 1, 2)
(5.1)
(i)
On substituting the values of η1 , φ1 from (3.1)-(3.3) and η2 , φ2 from (4.6)-(4.7) into (A.7), we
obtain
∂A iϑ
(j)
(j) (j)
φ3 = C3 E2m (k, r)A2 Āeiϑ + E (j) (k, r)
e + c.c.,
(5.2)
∂t2
where
·½
µ 2
¶
¾
½
µ 2
¶
¾µ
¶
m
1
1
α
(j)
(j)
(j)
2
(j) m
2
C3 = − E2m 2
+k −
B2 − 2 Em
+k −
+ ikUj
R2
R
R2
R
ρ(j)
µ
¶
½
¶¾µ
¶
(j) µ
1
1 2 2 + m2 Em 3m2
3α
2
×
+ α2 +
k +
−
+k
+ ikUj
R
2
R2
R
R2
ρ(j)
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µ
¶µ 2
¶
½
µ
¶
¾
α
2m (j) m2
α
1
2
+ (j) − ikUj
+ α2 − 3α3
E − 2 − k + (j) 4α2
R3 m
R
R
ρ
ρ
½µ
µ 2
¶½
¶
¾
¾ ¸
α
1
2αα2
(j) m
2
(j = 1, 2)
−
Em
− ikUj
+k +
+ (j) A2 .
R2
R
ρ(j)
ρ
(5.3)
We substitute the first- and second-order solutions into the third order equation. In order to
avoid nonuniformity of the expansion, we again impose the condition that secular terms vanish.
Then from (A.8), we find0
½
·· µ
¶ ¸¸
¾
α
∂D(0, k, m) ∂A
+ 2σkc µ + ρU
i
+ ikU Em kc iµ A + qA2 Ā = 0,
(5.4)
∂ω
∂t2
ρ
where
·· µ
µ
¶
α
2 2
q = ρ ikU C3 Em + A2 i 3kU − k U
ρ
µ
¶½
µ 2
¶
¾
µ
¶ µ
¶
α
m
5
α
2
+B2
− iU k
2Em E2m
+k −1 −i
+ 2α2 kU
+ ikU
ρ
R2
2R
ρ
µ 2
µ
¶
µ
¶
¶¸¸
m
α
m2 2 α2
α
3 α2
2 kU
2 2
− Em
+k
i 7 + 5kU i + 2 Em 2 + 3k U − i2 kU
+
R ρ2
R2
2
ρ
R
ρ
ρ
ª
3
1
σ ©
− 4 (2A2 R−4−4Rα2 )(1−m2 )−2A2 R(m2 +k 2 R2 )− (m2 +k 2 R2 )2 + (9m2 +k 2 R2 −6) . (5.5)
R
2
2
——————————————————————————————
(iii) a1r > 0, and |A0 |2 < −a1r /a2r : stable
and |A|2 → 0 as t2 → ∞. Thus, a sufficient con(5.6) dition for stability is a2r > 0, which is due to the
finite amplitude effect. The cylindrical system
is nonlinearly stabile if a1r > 0 and the initial
|A(t2 )|2 = a1r |A0 |2 exp(−2a1r t)
amplitude is sufficiently small.
2
2
−1
×[a1r + a2r |A0 | − a2r |A0 | exp(−2a1r t)] ,
(5.7)
where A0 is the initial amplitude and ajr = 6.Viscous asymmetric linear cylindrical
<ãj , (j = 1, 2) .
flow
With a finite initial value |A0 |, |A| may beIn this section we consider the viscous potencome infinite when the denominator in (5.7) van- tial flow. For the viscous fluid, (2.12) is now
ishes. Otherwise, |A| will be asymptotically replaced by
bounded. The situation can be summarized as
follows:
µ
¶
∂F
(1)
(1)
(1)
(1) a2r > 0; stable.
ρ (∇ϕ · ∇F )
+ ∇ϕ · ∇F
∂t
2
(i) a1r > 0; |A| → 0, as t2 → ∞
¶
µ
∂F
(2)
(2)
(2)
(ii) a1r < 0; |A|2 → −a1r /a2r , as t2 → ∞
+ ∇ϕ · ∇F
= ρ (∇ϕ · ∇F )
∂t
(2) If a2r < 0,
+(p2 − p1 − 2µ2 n · ∇ ⊗ ∇ϕ(2) · n
(i) a1r < 0 ; unstable.
We rewrite (5.4) as
∂A
+ (ã1 + ã2 |A|2 )A = 0,
∂t2
which can be easily integrated as
(ii) a1r > 0, and |A0 |2 > −a1r /a2r : unstable.
+2µ1 n · ∇ ⊗ ∇ϕ(1) · n + σ∇ · n)|∇F |2 ,
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(6.1)
8
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where µ1 , µ2 are viscosities of fluid ’1’ and ’2’, for the viscous fluid is same as (2.14), however
respectively and we modify (2.13) accordingly.
(1)
(2)
a0 = ρ(1) Em
− ρ(2) Em
,
The nonlinear analysis for the viscous fluid is
(2) (2)
(1) (1)
too onerous when the perturbation is asymmet- a1 = 2k{ρ Em U2 − ρ Em U1 },
ric , we are content here with the linear analysis. b1 = α{E (1) − E (2) } + 2(µ1 E (1) − µ2 E (2) ),
m
m
t
t
Then linearizing (2.6), (2.11) and (6.1) we have
2 (1) (1) 2
(2) (2) 2
a2 = k {ρ Em U1 − ρ Em U2 }
σ
− 2 (R2 k 2 + m2 − 1)
·· µ
¶
¶¸¸
Rµ
∂φ ∂η
∂η1
¶
ρ
−
−
U
= 0,
(6.2)
µ1 (1)
µ2 (2)
∂r
∂t
∂z
,
−2α (1) Et − (2) Et
ρ
ρ
0
µ
ρ(1)
∂φ(1)
(2)
(1)
b2 = αk{Em
U2 − Em
U1 }
¶
∂η ∂η
−
U = αη,
∂r
∂t
∂z
·· µ
¶
¸¸
∂φ ∂φ
∂2φ
ρ
+
U + 2µ 2
∂t
∂z
∂r
µ 2
¶
∂ η
η
1 ∂2η
= −σ
+
+
.
∂z 2 R2 r2 ∂θ2
−
(1)
(6.3)
−2k(µ1 U1 Et
(2)
− µ2 U2 Et ),
with
µ
¶
m2
1
2
=
k + 2 − ,
k
R
and necessary and sufficient stability conditions
are
b1 > 0,
(6.5)
and
a0 b22 − a1 b1 b2 + a2 b21 < 0,
(6.6)
since a0 is always positive.
(i)
Et
(6.4)
When the interface is perturbed to η =
A exp[i(kz + mθ − ωt), we recover the first order
solutions (3.1)-(3.3), and the dispersion relation
(i)
Em
7. Numerical examples
In this section we do numerical works using the expressions presented in previous sections for the
film boiling conditions. The vapor and liquid are identified with phase 1 and phase 2, respectively,
FIGURE 1. The critical wave number for m=1.
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9
so that T1 > T0 > T2 .
In the film boiling, the liquid-vapor interface is of saturation condition and the temperature T0
is set equal to the saturation temperature. The properties of both phases are determined from this
condition. First, in figure 1 we display critical wave number kc , i.e., the value for which ω = 0 in
(2.14) Here we chose ρ1 = 0.001gm/cm3 , ρ2 = 1gm/cm3 , σ = 72.3dyne/cm, b = 2cm, a = 1cm, R =
1.2cm, α = 0.1gm/cm3 s 0
FIGURE 2. The stability diagram for the flow when m=1. The system is stable in the region
between the two upper and lower curves.
Fig.3.Viscous cylindrical flow for m=0.The region above the curve is stable region.
From this figure we can notice that critical wave number increases as the velocity of fluid increases,
the increment rate of the inviscid fluid being sharper at higher fluid velocities. In figure 2 we display
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the region of stability of fluid in the nonlinear analysis as the velocity of one fluid increases while
that of the other fluid remains unchanged. In these figures , u1 remains constant as 1 cm/sec while
u2 varies from 1 cm/sec to 10cm/sec. The region between the two curves is the region of stability,
while in the region above the upper curve, the fluid is unstable.0
In Fig.3 and Fig.4 we present the results for viscous cylindrical linear flow.Here we chose
ρ1 = 0.0001gm/cm3 , ρ2 = 1gm/cm3 , σ = 72.3dyne/cm, b = 2cm, a = 1cm, R = 1.2cm, α =
.1gm/cm3 s, µ1 = 0.00001poise, µ2 = 0.01poise
Fig.4.Viscous cylindrical flow for m=1.The region above the curve is stable region.
8. Conclusions.
The stability of liquids in a cylindrical flow when there is mass and heat transfer across the
interface which depicts the film boiling is studied. Using the method of multiple time scales, a first
order nonlinear differential equation describing the evolution of nonlinear waves is obtained.With
the linear theory the region of stability is the whole plane above a curve like in Fig.3,4, however
with the nonlinear theory it is in the form of a band as shown in Fig.2. Unlike linear theory,
with nonlinear theory, it is evident that the mass and heat transfer plays an important role in the
stability of fluid, in a situation like film boiling.
Appendix
The interfacial conditions are given on r = R as
Order O(²)
¶
¶¸¸
·· µ
∂η1
∂η1 ∂φ0
∂φ1
−
−
= 0,
ρ
∂r
∂T0
∂z ∂z
µ
(1)
ρ
(1)
∂η1
∂η1 ∂φ0
∂φ1
−
−
∂r
∂T0
∂z ∂z
(A.1)
¶
= αη1 ,
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(A.2)
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11
·· µ
¶¸¸
µ 2
¶
∂φ1 ∂φ1 ∂φ0
∂ η1
η1
1 ∂ 2 η1
ρ
+
= −σ
+ 2+ 2
.
∂T0
∂z ∂z
∂z 2
R
r ∂θ2
(A.3)
Order O(²2 )0
·· µ
¶¸¸
∂φ2 ∂ 2 φ1
∂η2
∂η1
∂η1 ∂φ1
1 ∂η1 ∂φ1 ∂η2 ∂φ0
ρ
+
η1 −
−
−
− 2
−
= 0,
(A.4)
∂r
∂r2
∂T0 ∂T1
∂z ∂z
r ∂θ ∂θ
∂z ∂z
µ (1)
¶
(1)
(1)
∂ 2 φ1
∂η2
∂η1 ∂η1 ∂φ1
1 ∂η1 ∂φ1 ∂η2 ∂φ0
(1) ∂φ2
ρ
+
η1 −
−
−
− 2
−
= α(η2 + α2 η12 ), (A.5)
∂r
∂r2
∂T0 ∂T1
∂z ∂z
r ∂θ ∂θ
∂z ∂z
µ
·· ½
·µ
¶
¶
µ
¶ ¸
µ
¶
∂ 2 φ1
∂φ2 ∂φ1
1 ∂φ1 2 1 ∂φ1 2
∂φ1 2
∂φ2 ∂φ0 ∂φ1 ∂η1 ∂φ1
+
+
−
ρ
η1 +
+ 2
+
+
+
∂T0 ∂T1 ∂T0 ∂r
2
∂r
r
∂θ
∂z
∂z ∂z
∂r ∂T0
∂r
µ
µ
¶2
¶¾¸
¸
∂φ0
∂η1 ∂η1
∂η1 ∂φ1
∂η1 ∂φ0
∂ 2 φ1
+
−
+2
−
+
η1
∂z
∂z ∂t0
∂z ∂r
∂z
∂z
∂z∂r
µ
¾
½ 2
µ
¶
¶
∂ η2
1 ∂η1 2
η2
η12
3 1 ∂η1 2
1 ∂ 2 η2
2 ∂ 2 η1
= −σ
+
+ 2− 3−
+ 2
− 3 η1 2 .
(A.6)
∂z 2
2R ∂z
R
R
2 R3 ∂θ
r ∂θ2
r
∂θ
Order O(²3 )
½
(j)
(j)
(j)
(j)
∂φ3
∂ 2 φ2
∂ 2 φ1
1 ∂ 3 φ1 2 ∂η3
∂η2
∂η1
+
η
+
η
+
η1 −
−
−
1
2
2
2
3
∂r
∂r
∂r
2 ∂r
∂T0 ∂T1 ∂T2
µ (j)
¶
(j)
(j)
(j)
(j)
(j)
∂η1 ∂φ2
∂ 2 φ1
∂η2 ∂φ1
∂η3 ∂φ0
1 ∂η2 ∂φ1
1 ∂η1 ∂φ2
−
+
η1 −
−
− 2
− 2
∂z
∂z
∂z∂r
∂z ∂z
∂z ∂z
R ∂θ ∂θ
R ∂θ ∂θ
¾
(j)
(j)
2 ∂η1 ∂φ1
1 ∂η1 ∂ 2 φ1
+ 3 η1
= α(η3 + 2α2 η1 η2 + α3 η13 ), (j = 1, 2),
(A.7)
− 2 η1
R
∂θ ∂θ∂r
R
∂θ ∂θ
·· ½
µ 2
¶
∂φ3 ∂φ2 ∂φ1
∂ 2 φ1
∂ φ1
∂ 2 φ2
ρ
+
+
+
η2 +
+
η1
∂T0 ∂T1 ∂T2 ∂T0 ∂r
∂T1 ∂r ∂T0 ∂r
µ
µ
¶
¶
1 ∂ 3 φ1 2 ∂φ1 ∂φ2 ∂ 2 φ1
∂ 2 φ1
∂φ1 ∂φ2
+
η +
+
η1 +
+
η1
2 ∂T0 ∂r2 1
∂r
∂r
∂r2
∂z
∂z
∂r∂z
µ
¶
1 ∂φ1
1 ∂φ1 ∂φ2
1 ∂φ1 ∂ 2 φ1
−
η1 + 2
+ 2
R ∂θ ∂r∂θ R ∂θ
R ∂θ ∂θ
µ
¶µ
¶
µ
¶
∂φ1 ∂η1
∂φ2 ∂φ1 ∂η1
1 ∂φ1 ∂η1
∂φ1 ∂η2 ∂η1 ∂φ2 ∂φ1 ∂η1
1 ∂φ1 ∂η1
−
−
−
−
+
+
−
+
+
∂r
∂t0
∂r
∂z ∂z R2 ∂θ ∂θ
∂r ∂t0 ∂t1
∂r
∂z ∂z R2 ∂θ ∂θ
µ
¶
µ
∂ 2 φ1 ∂η1
∂φ1
∂φ0 ∂φ3
∂ 2 φ1
∂ 2 φ2
∂ 3 φ1 η12
+η1 2
−2
+
+
η2 +
η1 + 2
∂r
∂T0
∂r
∂z
∂z
∂r∂z
∂r∂z
∂r ∂z 2
2
∂φ2 ∂η1
∂φ1 ∂η2
∂ φ1 ∂η1
∂η2 ∂η1 ∂η1 ∂η2
+2
+2
+2 2
η1 −
−
∂r ∂z
∂r ∂z
∂r ∂z
∂t0 ∂z
∂t0 ∂z
µ
¶2
¶¾¸¸
· 2
µ
¶
∂η1 ∂η2 ∂φ0
∂η1 ∂φ1
2 ∂η1 ∂η1 ∂φ1
∂ η3 3 ∂ 2 η1 ∂η1 2
−2
−2
− 2
= −σ
−
∂z ∂z ∂z
∂z
∂z
R ∂z ∂θ ∂θ
∂z 2
2 ∂z 2 ∂z
(j)
ρ
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INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT (IJTSRD)ISSN:2456-6470
¶
µ
¶
∂η1 2
1 ∂η1 ∂η2
η3
2η1 η2
η13
3 ∂η1 ∂η2
1 ∂ 2 η1 ∂η1 2
+
+ 2−
+
−
−
∂z
R ∂z ∂z
R
R3
R4 R3 ∂θ ∂θ
2R2 ∂z 2 ∂θ
µ
¶2
½ µ
¶2
µ
¶ ¾
2
2
2
9
∂η1
1 ∂ η3
1 ∂ η1
1 ∂η1
2η2 3η1
3
∂η1 2
+ 4 η1
+ 2
+ 2
−
−
+ 2 −
2R
∂θ
R ∂θ2
R ∂θ2
2 ∂z
R
R
2R2 ∂θ
¸
2 ∂ 2 η2
2 ∂η1 ∂ 2 η1 ∂η1
− 3 η1 2 − 2
.
(A.8)
R
∂θ
R ∂θ ∂θ∂z ∂z
−
1 η1
2 R2
µ
REFERENCES
0
4.
Hsieh, D.Y. Effect of heat and mass
transfer on Rayleigh-Taylor instability. Trans
ASME,1972; 94D: 156-162.
1. Hsieh, D.Y. Interfacial stability with mass
and heat transfer, Phys. Fluids, 1978; 21(5): 5. Lee, D.-S. Nonlinear stability of a cylindrical
interface with mass and heat transfer, Z. natur745-748
forsch. 2000; 55a: 837-842
2. Nayak, A.R. and Chakraborty, B.B. KelvinHelmholtz stability with mass and heat transfer, 6. Lee, D.-S., Nonlinear instability of cylindrical
interface with mass and heat transfer in magPhys. Fluids, 1984; 27(8): 1937-1941
netic fluids. Z. Angew. Math. Mech. 2002; 82
3. Elhefnawy, A. R. F., Stability properties of a 8: 567-575
cylindrical flow in magnetic fluids:effect of mass
and heat transfer and periodic radial field. Int. 7. Awashi, M.K. , Asthana, R., and Uddin, Z.
Nonlinear study of Kelvin-Helmholtz instability
J. Engng Sci., 1994;32(5): 805-815
of cylindrical flow with mass and heat transfer,
Inter.Comm. Heat and Mass Trans. 2016;71:
216-224
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