spatially developing gas flow
Agneta M. BALINT
1, Stefan BALINT*
,2*Corresponding author
1
West University of Timisoara, Department of Physics
Blvd. V. Parvan 4, 300223 Timisoara, Romania
[email protected]
*
,2West University of Timisoara, Department of Computer Science
Blvd. V. Parvan 4, 300223 Timisoara, Romania
[email protected]
DOI: 10.13111/2066-8201.2015.7.4.5Received: 05 September 2015 / Accepted: 20 October 2015
Copyright©2015 Published by INCAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
The 36th“Caius Iacob” Conference on Fluid Mechanics and its Technical Applications
29 - 30 October, 2015, Bucharest, Romania, (held at INCAS, B-dul Iuliu Maniu 220, sector 6) Section 3. Equations of Mathematical Physics
Abstract: In this paper different types of stabilities (global, local) with respect to instantaneous perturbations and permanent source produced time harmonic perturbations are presented in case of a spatially developing gas flow. Some types of instabilities (global absolute, local convective) are also presented. For this purpose the Euler equations linearized at the constant gas flow are used. It is shown for instance, that the constant gas flow is global absolutely unstable with respect to some instantaneous and some permanent source produced time harmonic perturbations. The locally convective instability is also proven with respect to some instantaneous and permanent source produced time harmonic perturbations.
Key Words: global/ local stability/ instability; spatially developing gas flow.
1. INTRODUCTION
The nonlinear Euler equations governing the spatially developing flow of an inviscid, compressible, non heat conducting, isentropic, perfect gas, are:
0 1
x p z
v v y v v x v v t
v x
z x y x x x
0
1
y p z
v v y v v x v v t
v y
z y y y x y
0
1
z p z
v v y v v x v v t
v z
z z y z x z
0
z v y v x v z
v y v x v t
z y x z
y
x
Here:
t
- time, vx, vy, vz- velocity components along the Ox,Oy,Oz axis respectively; p- pressure, - density. The pressure p, the density and the absolute temperature T satisfy the state equation of the perfect gasT R
p (2)
where the gas constant Rcpcv; cpand cvbeing the specific heat capacities at constant pressure and constant volume, respectively. If the spatially developing gas flow is constant and vxU0const0; vy0; vz 0; 0const0; pp0const0, then
according to (1), the associated isentropic sound speed c0 verifies 0 0
0 2
0 R T
p
c
.
Linearizing (1) at vx U0;vy vz 0;p p0;0 and using that the flow is isentropic, i.e.:
' 02 '
00
c p x U
t (3)
for the perturbations v'x,v'y,v'z,p' the following system of homogeneous linear partial differential equations is obtained:
0 1
0
0
x p x
v U t
vx x
0 1
0
0
y p x
v U t
vy y
0 1
0
0
z p x
v U t
vz z
0 0
2 0
0
x p U z v y v x v c t
p x y z
(4)
For the null solution of (4) two types of perturbations will be considered: instantaneous perturbations and permanent source produced time harmonic perturbations.
Definition 1.1. An instantaneous perturbation, denoted by I
F,G,H,P
is defined by a system of four real valued functions F,G,H,P:R3R1 representing the perturbation of the three components of the velocity field (v'x0;v'y0;v'z0) and that of the pressure field (p'=0) at a certain moment of time, let say t0. It is commonly accepted that the propagation of the instantaneous perturbation I
F,G,H,P
is given by that solution of (4) which is equal to zero for t0 and at t 0 verifies); , , ( ) 0 , , , (
' x y z F x y z
vx v'(x,y,z,0) G(x,y,z);
y vz'(x,y,z,0)H(x,y,z); )
, , ( ) 0 , , , (
' x y z P x y z
p (5)
Definition 1.2. A permanent source produced time harmonic perturbation of amplitude )
, , ,
(F G H P
t z
y x F t
h( ) ( , , )sinf
;
h(t)G(x,y,z)sinft;
h(t)H(x,y,z)sinft;
tz y x P t
h() ( , , )sinf
or
tz y x F t
h() ( , , )cosf
;
h(t)G(x,y,z)cosft;
h(t)H(x,y,z)cosft;
tz y x P t
h() ( , , )cosf
(6)
with (x,y,z)R3, t0 and h(t) the Heaviside function.
The system of four functions represents the perturbation of the three components of the velocity field (v'x0;v'y0;v'z0) and that of the pressure field (p'0) produced by a source which begins to act at the moment of time t0. The modules of the real valued functions F,G,H,P represent the amplitude of the perturbations.
It is commonly accepted that the propagation of a permanent source produced time harmonic perturbation of amplitude A(F,G,H,P) and angular frequency
f is given by that solution of the system of partial differential equations:
t z
y x F t h x p x
v U t v
f x
x
sin ) , , ( ) ( 1
0 0
t z
y x G t h y p x
v U t v
f y
y
sin ) , , ( ) ( 1
0 0
t z
y x H t h z p x
v U t v
f z
z
sin ) , , ( ) ( 1
0 0
t z
y x P t h x p U z v y v x v c t
p
f z
y
x
sin ) , , ( ) ( 0
2 0 0
(7)
which is equal to zero for t0.
In the right hand side of (7) instead of sinft we can have alsocosft.
Definition 1.3. An instantaneous perturbation I
F,G,H,P
and a permanent source produced time harmonic perturbation of amplitude A(F,G,H,P) is called localizedperturbation, if the supports (carriers) of the functions F,G,H,P are subsets ofR3. Otherwise, the perturbation is called global perturbation.
2. STABILITY AND INSTABILITY WITH RESPECT TO INSTANTANEOUS
PERTURBATION
Definition 2.1. The null solution of (4) is globally stable with respect to the instantaneous perturbation I
F,G,H,P
if the magnitude of the solution of the initial value problem (4), (5) (which is equal to zero fort0) at any point (x,y,z)R3 and at every moment of time t0 is less than a prior given value0. The term “global” is used here to underline that the whole velocity and pressure field is concerned [5].Example 2.2. If the instantaneous perturbation I
F,G,H,P
is given by:) sin(
3 )
, , (
; ) sin(
) , , ( ) , , ( ) , , (
0
0 x y z
c K z y x P
z y x K z y x H z y x G z y x F
(8)
where K is a real number, then the solution of (4), (5) (equal to zero fort0) is given by:
x y z U c t
c K t h t z y x p
t c U z y x K t h t z y x v t z y x v t z y x
vx y z
3 sin
3 )
( ) , , , ( '
3 sin
) ( ) , , , ( ' ) , , , ( ' ) , , , ( '
0 0 0
0
0 0
(9)
where h(t) the Heaviside function.
The magnitude of the solution at any point and at every moment verifies:
K t z y x v t z y x v t z y x
v'x( , , , ) 'y( , , , ) 'z( , , , ) ; p'(x,y,z,t) K c00 3 (10)
and at any point for some moments of time tn, tending to , verifies:
K t z y x v t z y x v t z y x
v'x( , , , n) 'y( , , , n) 'z( , , , n) ; p'(x,y,z,tn) Kc00 3 (11)
For a prior given 0 if K verifies:
1, 3
max 00
c
K (12)
then the magnitude of the solution at any point and at every moment is less than . If K verifies:
1, 3
max 00
c
K (13)
then the magnitude of the solution at any point for some moments of time tn is greater than . In other words, if (12) holds, then the null solution is globally stable and if (13) holds, then the null solution is globally unstable. For U080m/s; 0 1.20kg/m3;
s m
when the magnitude of the solution of the initial value problem (4), (5) at any points increases and tends to infinity for t tending to .
Definition 2.3. The null solution of (4) is globally absolutely unstable with respect to the instantaneous perturbation I
F,G,H,P
if for any point (x,y,z)R3 any real numbers M 0,N0 (M and N big) there exists a moment of time tN such that the magnitude of the solution of the initial value problem at (x,y,z) and moment of time t is greater than M .a. evolution of v'x b.evolution of p'
Fig. 1 - Global stability with respect to an instantaneous perturbation at an arbitrary point of the plane m
z y
x 100 during the first 600s.
a. evolution of v'x b. evolution of p'
Fig. 2 -Global instability with respect to an instantaneous perturbation at an arbitrary point of the plane
m z y
x 100 during the first 600 . s
Example 2.4. If the instantaneous perturbation I
F,G,H,P
is given by:) sin(
3 )
, , (
; ) sin(
) , , ( ) , , ( ) , , (
) ( 0 0
) (
z y x e
c K z y x P
z y x e
K z y x H z y x G z y x F
z y x
z y x
where K is a real number, then the solution of (4), (5) (equal to zero for t0) is given by:
x y z
U c
t
e c K t h t z y x p t c U z y x e K t h t z y x v t z y x v t z y x v t c U z y x t c U z y x z y x 3 sin 3 ) ( ) , , , ( ' 3 sin ) ( ) , , , ( ' ) , , , ( ' ) , , , ( ' 0 0 3 0 0 0 0 3 0 0 0 0 (15)where h(t) the Heaviside function. If U0c0 30 and K 0, then for an arbitrary point 3
) , ,
(x y z R at the moments of time
2
3 1 2 0 0
x y z
c U n
tn nN (16)
the magnitude of the solution at (x,y,z) and ttn is given by:
n c U n z n y n
x x y zt v x y zt v x y z t K e
v' ( , , , ) ' ( , , , ) ' ( , , , ) 2( 00 3)2
n c U
n K c e
t z y x
p 2( 3)2 0
0 3 0 0
) , , , ( ' (17)
Therefore, for a prior given M 0 (M big) and nN with N given by
/20 0 0 0 1 , 1 max ln 3 2 1 e K c M U c N (18)
the magnitude of the solution is greater than M. In other words, if K 0 and 0
3 0 0c
U , then the null solution of (4) is globally absolutely unstable with respect to the considered instantaneous perturbation. For U0 80m/s; 0 1.20kg/m3;
s m
c0345 / and K1 the v' component of the velocity field and the pressure field x evolutions at an arbitrary point situated in the plane defined by the equation xyz0 during the first 0.0123sare represented in Fig. 3a and Fig. 3b, respectively.
a. evolution of v'x b. evolution of p'
Fig. 3 - Global absolute instability with respect to an instantaneous perturbation at an arbitrary point of the plane 0
y z
Another type of stability of the null solution of (4) is the local stability.
Definition 2.5. The null solution of (4) is locally stable on R3, with respect to the instantaneous perturbation I
F,G,H,P
if the magnitude of the solution of the initial value problem (4), (5) (which is equal to zero for t0) at any point (x,y,z) and moment of time t0is less than a prior given value 0.Example 2.6. If the instantaneous perturbation I
F,G,H,P
is given by2
2 ( )
0 0 )
( ; ( , , ) 3
) , , ( ) , , ( ) , ,
(x y z G x y z H x y z e x y z P x y z c e x y z
F (19)
then the solution of the initial value problem (4), (5) is given by:
20 0
2 0 0
3 0
0
3
3 )
( ) , , , ( '
) ( ) , , , ( ' ) , , , ( ' ) , , , ( '
t c U z y x
t c U z y x z
y x
e c
t h t z y x p
e t h t z y x v t z y x v t z y x v
(20)
For a prior given 0 if (x,y,z)and t satisfy
2 0 0
0
0 ,ln
1 ln max
3 t c
c U z y
x , (21)
then the magnitude of the solution of (4), (5) at the point (x,y,z) and the moment of time t
is less than . For U0c0 30 at every point of the set defined by:
(x,y,z) R3 x y z max ln1,lnc0 0 , (22)
the magnitude of the solution of (4), (5) is less than at any moment of time t0. For U0c0 30 at every point of the set defined by:
(x,y,z) R3 x y z max ln1,lnc0 0 ,
(23)
the magnitude of the solution of (4), (5) is less than at any moment of time t0.
In other words, if U0c0 30, then the null solution of (14) is locally stable on or on
. For U080m/s; 01.20kg/m3; c0345m/s the component v' of the velocity x field and the pressure field evolutions given by (20) are represented in Fig. 4a and Fig. 4b, respectively for xyzd, d in the range
260,...,50
m and t in the range
0,...,0.5
s. For U0697m/s; 0 1.20kg/m3; c0345m/s the component v' of the velocity field x and the pressure field evolutions given by (20) are represented in Fig. 5a and Fig. 5b, respectively for xyzd, d in the range
50,...,260
m andt
in the range
0,...,50
s. In the definition 2.5. there is no requirement concerning the magnitude of the solution at the points which belong to the complement of . In Fig. 4a, Fig. 4b, Fig. 5a, Fig. 5b can be seen that the perturbation swept away from the set of points where its magnitude is significant at t0.solution of (4), (5) is less than and there exists a sequence of points (xn,yn,zn) in , which tends to (i.e. xn2yn2zn2 ) and a sequence of moments of times tn, which tends also to such that the magnitude of the solution of (4), (5) at the points
n n n y z x , ,
( and moments of times tn is greater than 0.
a. evolution of
v
'
x b. evolution of p'Fig. 4 - Local stability on of an instantaneous perturbation at an arbitrary point of the plane xyzd,
d in the range
260,...,50
m, during the first 0.5s.a. evolution of v'x b. evolution of p'
Fig. 5 - Local instability on of an instantaneous perturbation at an arbitrary point of the plane xyzd,
d in the range
50,...,260
m, during the first 50 . sExample 2.8. The null solution of (4) is locally convectively unstable with respect to the instantaneous perturbation given in Example 2.6 on 1CR3 if U0c0 30 and on
3. STABILITY AND INSTABILITY OF THE NULL SOLUTION WITH
RESPECT TO PERMANENT SOURCE PRODUCED HARMONIC
PERTURBATION
In this section we present different types of stabilities and instabilities of the null solution of (1.4) with respect to the permanent source produced harmonic perturbation. Stabilities and instabilities discussed here are related to the local, global, absolute and convective instabilities analyzed in [3]-[10].
Definition 3.1. The null solution of (4) is globally stable with respect to the permanent source produced time harmonic perturbation of amplitude A(F,G,H,P) and angular frequency f if the magnitude of the solution of (1.6) (which is equal to zero fort0) at any point (x,y,z)R3 and moment of time t0 is less than a prior given value0. In this definition the term "global" is used to underline that the whole velocity and pressure field are concerned [5].
Example 3.2. If the permanent source produced time harmonic perturbation of amplitude A(F,G,H,P)is given by:
) sin( 3 ) , , ( ; ) sin( ) , , ( ) , , ( ) , , ( 0
0 x y z
c K z y x P z y x K z y x H z y x G z y x F
, (24)
where K is a real number, then the solution of (1.6) (which is equal to zero fort0) for
U0 c0 3,c0 3 U0
f
is given by:
3 2 sin 3 2 sin 3 3 3 sin ) ( ) , , , ( ' ) , , , ( ' ) , , , ( ' 0 0 0 0 0 0 0 0 0 0 c U t z y x c U t z y x c U c U t c U z y x K t h t z y x v t z y x v t z y x v f f f f f f f z y x ) , , , ( ' 3 ) ( ) , , , (' x y z t ht c0 0 v x y zt
p x ,
(25)
where h(t) the Heaviside function.
The magnitude of the solution at any point and at every moment verifies:
) , , , ( ' 3 ) , , , ( ' 3 2 1 3 2 1 3 3 ) , , , ( ' ) , , , ( ' ) , , , ( ' 0 0 0 0 0 0 0 0 0 0 t z y x v c t z y x p c U c U c U c U K t z y x v t z y x v t z y x v x f f f f f z y x (26)
For a prior given 0 if K satisfies:
1, 3
max 00
c M K (27)
3 2
1 3
2
1 3
3 0 0 0 0 0 0
0
0 c U c U c U c
U M
f f
f f
f
(28)
This means that if f U0c0 3,c0 3U0 and K verifies (27), then the null solution of (4) is globally stable with respect to the considered permanent source produced time harmonic perturbation.
If f U0c0 3,c0 3U0 and K verifies
1, 3
max 00
c M
K , (29)
then the magnitude of the solution given by (25) is uniformly bounded with respect to 3
) , ,
(x y z R and t0, but the null solution of (4) is unstable in general. More precisely, if (29) holds, then the magnitude of the solution at any point for some moments of time
n
t is greater than .
In other words, if (27) holds, then the null solution is globally stable and if (29) holds, then the null solution is globally unstable.
For U0 80m/s; 01.20kg/m3; c0345m/s; 0.1; f 100 and K 0.0001 satisfying (27) the
v
'
xcomponent of the velocity field and the pressure field evolutions at an arbitrary point situated in the plane defined by the equation xyz0 during the firsts
600 are represented in Fig. 6a and Fig. 6b, respectively. For K 0.001, satisfying (29), the evolutions of the same quantities at the same points during the same period of time are represented on Fig. 7a and Fig. 7b, respectively.
a. evolution of v'x b. evolution of p'
a. evolution of v'x b. evolution of p'
Fig. 7 - Global instability with respect to a permanent source produces time harmonic perturbation at any point of the plane xyz0 during the first600s.
Figs. 6a, 6b illustrate global stability and Figs. 7a, 7b illustrate global instability. A special type of global instability is that when the magnitude of the solution of (6) (which is zero fort0) at any points increases and tends to infinity for t tending to.
Definition 3.3. The null solution of (4) is globally absolutely unstable with respect to the permanent source produced time harmonic perturbation of amplitude A(F,G,H,P)
and angular frequency f if for any point (x,y,z)R3 any real numbers M 0,N0 (M and N big) there exists a moment of time tN such that the magnitude of the solution of problem (6) (which is equal to zero for t0) at (x,y,z) and moment of time t is greater than M.
Example 3.4. If the permanent source produced time harmonic perturbation amplitude is given by (24) and the angular frequency isf U0c0 3, then the solution of (6) is given by:
) , , , ( ' 3 )
( ) , , , ( '
cos 2
sin 4
1 sin
1 2 cos 4
1
cos 2 ) ( ) , , , ( ' ) , , , ( ' ) , , , ( '
0
0 v x y z t
c K t h t z y x p
t z y x t t
z y x t
t z y x t K t h t z y x v t z y x v t z y x v
x
f f
f f
f f
f z
y x
(30)
For a given (x,y,z)R3there exists a sequence of moments( ) n
t , tending to such that
1cos xyzf tn . (31)
The situation is similar forf U0c0 3. This means that if the angular frequency
f
belongs to the set
U0c0 3,U0c0 3
, then the null solution of (4) is globally absolutely unstable with respect to the considered permanent source produced time harmonic perturbation. For U080m/s; 301.20kg/m
; c0 345m/s;K1; f U0c0 3 the
x
v' component of the velocity field and the pressure field evolutions at an arbitrary point situated in the plane xyz0 during the first 600 are represented in Fig. 8a and Fig. s 8b, respectively. These figures illustrate global absolute instability.
Definition 3.5. The null solution of (4) is locally stable onR3, with respect to the permanent source produced time harmonic perturbation of amplitude A(F,G,H,P) and angular frequency fif the magnitude of the solution of the problem (6) (which is equal to zero fort0) at any point (x,y,z) and moment of time t0is less than a prior given value0. The term “local” is used to underline that just a part of the velocity and pressure field is concerned [5].
a. evolution of v'x b. evolution of p'
Fig. 8 - Global absolute instability with respect to a permanent source produces time harmonic perturbation at any point of the plane xyz0 during the first600s.
Example 3.6. If the permanent source produced time harmonic perturbation amplitude )
, , ,
(F G H P
A is given by
) ( 0 0 )
( ; ( , , ) 3
) , , ( ) , , ( ) , ,
(x y z G x y z H x y z e x y z P x y z c e x y z
F (32)
then the solution of (6) (equal to zero fort0) is given by:
) , , , ( ' 3 )
( ) , , , ( '
sin )
( ) , , , ( ' ) , , , ( ' ) , , , ( '
0 0
0
) ( 3
0 0
t z y x v c
t h t z y x p
d e
t h t z y x v t z y x v t z y x v
x
f t
t c U z y x z
y x
(33)
0 31 ln
0 0
U c
z y
x , (34)
then the magnitudes of the velocity components are less than . It follows that at every point of the set defined by:
0 0
0 0
0 0
3
3 3 ln
, 3 1 ln max )
, , (
U c
c U
c z
y x R z y
x (35)
the magnitude of the solution of the problem (6) (which is zero fort0) is less than 0 at any moment of timet0. In other words, the null solution is locally stable on with respect to the permanent source produced time harmonic perturbation of which amplitude is given by (32) and the angular frequency f is arbitrary. For U080m/s;
3 01.20kg/m
; c0345m/s; f 100; 0.1 the v' component of the velocity field x and the pressure field evolutions given by (33) are represented in Fig. 9a and Fig. 9b, respectively, for xyzd, d in the range
2.6099,...,500
m and t in the range
0,...,600
s.
a. evolution of v'x b. evolution of p'
Fig. 9 - Local stability on of a permanent source produced time harmonic perturbation at an arbitrary point of
the planexyzd, d in the range
2.6099,...,500
m, during the first600s.The evolutions of the v' component of the velocity field and that of the pressure field are x represented in Fig.10a and Fig.10b, respectively at an arbitrary point of the plane defined by
0
y z
x during the first 600 seconds. Figures 9a and 9.b illustrate that the null solution is locally stable in, while figures 10.a and 10.b illustrate that in the points of the plane defined by xyz0 stability conditions are not satisfied.
Definition 3.7. The null solution of (4) is locally convectively unstable on R3 with respect to the permanent source produced time harmonic perturbation of amplitude
) , , ,
(F G H P
A and angular frequency fif for the prior given 0 there exists a
sequence of points (xn,yn,zn) in , which satisfies 2 2 2
n n
n y z
of moments of times tn, which tends to , such that the magnitude of the solution of the problem (6) (which is equal to zero for t0) at the points (xn,yn,zn), at the moments of times tn is greater than .
a. evolution of
v
'
x b. evolution of p'Fig. 10 - Local stability conditions with respect to the permanent source produced time harmonic perturbation are not satisfied in the points of the planexyz0. Illustration in a point during the first600s.
Example 3.8. The null solution of (4) is locally convectively unstable with respect to the permanent source produced time harmonic perturbation given in Example 3.6 on, given by:
( , , ) 3 3
x y z R x y z . (36)
The evolutions of the v' component of the velocity field and that of the pressure field, given x by (3.10), are represented in Fig. 11a and Fig. 11b respectively, at an arbitrary point of the plane defined byxyzd, d in the range[500,...,3]m, during the first 600 seconds.
a. evolution of v'x b. evolution of p'
4. COMMENTS AND CONCLUSIONS
The explicit instantaneous perturbations as well the permanent source produced time harmonic perturbations and their propagations used in this paper were constructed using the results presented in [11].
Example 2.4. shows the existence of instantaneous perturbation which leads to absolute global instability; example 2.6. shows the existence of instantaneous perturbation which leads to local stability and example 2.8. shows the existence of instantaneous perturbation which leads to locally convective instability.
Example 3.4. shows the existence of permanent source produced time harmonic perturbation which leads to absolute global instability; example 3.6. shows the existence of permanent source produced time harmonic perturbation which leads to local stability and example 3.9 shows the existence of permanent source produced time harmonic perturbation which leads to locally convective instability.
When the stability and instability are discussed in a given function space it can happen that due to the choice of the function space the instantaneous perturbations appearing in the examples 2.4, 2.6 and 2.8 are not in the function space (which was chosen) and their effect is not reflected in the analysis. This can be a problem, because there are neither set rules, nor understanding of the “right way” to choose the function space. Similar facts appear in the case of permanent source produced time harmonic perturbation appearing in the examples 3.4, 3.6 and 3.8.
ACKNOWLEDGEMENT
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project number PN-II-ID-PCE-2011-3-0171.
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