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Stability of two-dimensional quasiharmonic dissipative structures
Volume 99A, number 4
PHYSICS LETTERS
28 November 1983
STABILITY OF TWO-DIMENSIONAL QUASIHARMONIC DISSIPATIVE STRUCTURES B.A. MALOMED and I.E. STAROSELSKY Landau Institute for Theoretical Physics, Kosygina 2, Moscow, USSR Received 20 July 1983
We discuss here stationary cellular two-dimensional (2D) structures which may form as a result of instabilities of homogeneous states. Examples include gas flames, heat-conductive and hydrodynamic problems. It is proved that the quasi-onedimensional (Q1D) structure is the most stable one in the global sense.
In the preceeding paper [1] we discussed the stability of 2D cellular structures in gaseous flames subject to the action of an external stabilizing factor. In particular, we have demonstrated the absence of stable structures on a plane. This conclusion essentially relies upon the fact that the equation we dealt with in ref. [ 1] had only the quadratic nonlinear term [2]. However, in various physical problems there arises the analogous equation with the cubic nonlinearity [3-5] : ~t + ~ + 2aA~ + A2~ + ~3 = 0.
(1)
The designations are the same as in ref. [1], and we shall again consider only the slightly supercritical case 0 < a - 1 ~ 1. Eq. (1) describes kinetic systems where aperiodic instability arises at a finite value of the perturbation wavenumber. In the same time there are problems [2,6,7] with an aperiodic long-wave instability, i.e. one arising at small wavenumbers. The simplest model equation describing this situation can be written by analogy with (1): ~t + A t + A2~+ ~3 = 0.
(1')
The trivial solution of (1 ') ~ = 0 is unstable against small perturbations o f the form g "~ exp (~2t + ikx) with the wavenumbers k 2 <1, ~2(k) being the instability growth rate: I2(k) = k 2 - k 4. Such a dependence of ~2 on k occurs in different heat-conductive [2,7] and hydrodynamic [6] problems. Note that eq. (1') contrary to (1), does not contain a small parameter. In the present communication we report the results of an investigation of different solutions to (1) and 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(1 ') and their stability. Eqs. (1), (1') possess "pseudohamiltonians", i.e. they can be written in gradient form; ~t =
= f Ma dy,
(2)
and H = I~2 _ ~(V~)2 + ~1 (A~)2 + ~ 4
(3)
or n = ~(A~)2 _ ~ (V~)2 + ~ 4 .
(4)
[we mark by a prime the formulae concerning eq. (1')]. According to (2) d~/dt <~O, i.e. the value of pseudohamiltonian cannot increase with time. Hence the evolution o f the perturbed unstable trivial state (for which ~7 = 0) may result in the formation of the stationary structures ~ (x) for which ~f(~} ~< 0, and among them the "most stable" one is that providing the absolute minimum of the pseudohamiltonian (actually it is convenient to compare the mean densities of the pseudohamiltonian h = ~fff dxdy). The simplest 2D solutions to (1), (1') are Q1D structures = x/~e(k2)cos(~) + a ( k ) c o s ( 3 ~ ) ,
(5)
where 62(k 2) = 2(a - 1) - (k 2 - 1) 2 ,
(6)
or
145
Volume 99A, number 4 e2(k 2) = k 2 - k 4 .
PHYSICS LETTERS (6')
The amplitude of the third harmonic A in the solution (5, 6) is small ~ e 3 and may be directly neglected. As to the solution (5, 6') concerning eq. (1'), for it
A(k) = 2 e3(k2______)(81k 2 _ 9)_ 1 x/Tqk 2 and the solution in the quasiharmonic form (5, 6') is valid in the range of k where X(k 2) = ~
e(k2)A-l(k)
(7)
is numerically small. As we shall see below, the wavenumbers of interest for us meet this condition. Substituting (5) into (2), (4') we obtain h = - g 1 e4(k2)
(8)
i.e. the "most stable" Q I D structure has the wavenumber k 0 corresponding to the maximum of the instability growth rate of the trivial solution [k 2 = 1 for eq. (1) and k 02 = ~l for ( 1 ), ] . Note that l•f k 2 = "1 ~, X from (7) is 0.0053. Proceeding to the properly two-dimensional solutions we should emphasize first of all that searching for these solutions to eq. (1') in quasiharmonic form one should check the validity of the quasiharmonicity condition dealing with the X-parameters defined analogously to (7). Eqs. (1), (1') admit 2D solutions representing the lattice consisting of rectangular ceils:
~(x,y) = ~ {[2 e2(ky2) - e2(k2)l 1/2 cos(kxX) + [2 e2(kx2) - e 2 (ky2)] 1/2 cos(kyy)},
(9)
(10)
Herefrom we find that all the rectangular parkets are stable in comparison with the trivial solution and the most stable ones are those with the maximum aspect ratio. Then, eqs. (1), (1') posess the solutions of hexagonal symmetry describing the parkets o f triangulars: =~
e(k 2) (cos(kx) + cos[~ k(N/~y -- x)]
+ cos[½ k(~r3y + x)] ) .
146
Besides that, eq. (1) admits the solution corresponding to hexagon parkets which can be obtained from (11) on replacing sin by cos. For all these solutions sax ea(k2) h = - s-a-6~
(12)
Comparing (8), (9) and (12) we find that they are the Q1D structures (6, 6') which are the most stable ones in the global sense. So, it is interesting to exam. ine their local stability. The "dangerous" eigenmode of small perturbations on the background o f ( 6 ) may be chosen in the form = exp(~2t)[~cos(p + k)x] + ¢cos[(p - k)x I , ~, ~ = const.
(13)
Using (13) it can be readily demonstrated that in the 2D case the local stability criterion for the Q1D solution (5, 6) to (1) coincides with that for purely 1D case [6]: ~2(p) < 0 if 4 (a - 1) ~< e2(k 2) ~< 2(~ - 1). Contrarily, in the case of eq. (1') the stability criterion for the Q1D solution (5, 6') to (1'): i ~< k2 <~ 0.805 appears to be more stringent than the purely one-dimensional: 0.266 ~< k 2 ~< 0.805. We f'md it to be very surprizing that the band of the wavenumbers corresponding to stable structures is restricted in the 2D case so as to choose the more quasiharmonic structures. (It is clear that X(k) drops with the increase of k; for instanse X(0.266) = 0.019, X(0.805) = 0.0015). We are deeply grateful to Professor S.I. Anisimov.
References
for which h = 27 [e4(kx2) - 4 eZ(k2x) e2(k 2) + e4(ky2)].
28 November 1983
(11)
[ 1 ] B.A. Malomed and I.E. Staroselsky, Phys. Lett. 99A (1983) 143. [2] G.I. Sivashinsky, SIAM J. Appl. Math. 39 (1980) 67. [3] A.P. Aldushin, B.A. Malomed and Ya.B. Zeldovich, Comb. Flame 42 (1981) 1. [4] S.I. Anisimov, S.M. Goldberg, B.A. Malomed and M.I. Tribelsky, Dokl. Akad. Nauk SSSR 262 (1982) 1117. [5] A.B. Konstantinov, B.A. Malomed and S.M. Goldberg, Dokl. Akad. Nauk SSSR 262 (1983) 270. [6] M.A. Nepomnyashchy, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, No. 3. (1974) 28. [7 ] G.I. Sivashinsky, Acta Astronautica 4 (1977) 1117.
PHYSICS LETTERS
28 November 1983
STABILITY OF TWO-DIMENSIONAL QUASIHARMONIC DISSIPATIVE STRUCTURES B.A. MALOMED and I.E. STAROSELSKY Landau Institute for Theoretical Physics, Kosygina 2, Moscow, USSR Received 20 July 1983
We discuss here stationary cellular two-dimensional (2D) structures which may form as a result of instabilities of homogeneous states. Examples include gas flames, heat-conductive and hydrodynamic problems. It is proved that the quasi-onedimensional (Q1D) structure is the most stable one in the global sense.
In the preceeding paper [1] we discussed the stability of 2D cellular structures in gaseous flames subject to the action of an external stabilizing factor. In particular, we have demonstrated the absence of stable structures on a plane. This conclusion essentially relies upon the fact that the equation we dealt with in ref. [ 1] had only the quadratic nonlinear term [2]. However, in various physical problems there arises the analogous equation with the cubic nonlinearity [3-5] : ~t + ~ + 2aA~ + A2~ + ~3 = 0.
(1)
The designations are the same as in ref. [1], and we shall again consider only the slightly supercritical case 0 < a - 1 ~ 1. Eq. (1) describes kinetic systems where aperiodic instability arises at a finite value of the perturbation wavenumber. In the same time there are problems [2,6,7] with an aperiodic long-wave instability, i.e. one arising at small wavenumbers. The simplest model equation describing this situation can be written by analogy with (1): ~t + A t + A2~+ ~3 = 0.
(1')
The trivial solution of (1 ') ~ = 0 is unstable against small perturbations o f the form g "~ exp (~2t + ikx) with the wavenumbers k 2 <1, ~2(k) being the instability growth rate: I2(k) = k 2 - k 4. Such a dependence of ~2 on k occurs in different heat-conductive [2,7] and hydrodynamic [6] problems. Note that eq. (1') contrary to (1), does not contain a small parameter. In the present communication we report the results of an investigation of different solutions to (1) and 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(1 ') and their stability. Eqs. (1), (1') possess "pseudohamiltonians", i.e. they can be written in gradient form; ~t =
= f Ma dy,
(2)
and H = I~2 _ ~(V~)2 + ~1 (A~)2 + ~ 4
(3)
or n = ~(A~)2 _ ~ (V~)2 + ~ 4 .
(4)
[we mark by a prime the formulae concerning eq. (1')]. According to (2) d~/dt <~O, i.e. the value of pseudohamiltonian cannot increase with time. Hence the evolution o f the perturbed unstable trivial state (for which ~7 = 0) may result in the formation of the stationary structures ~ (x) for which ~f(~} ~< 0, and among them the "most stable" one is that providing the absolute minimum of the pseudohamiltonian (actually it is convenient to compare the mean densities of the pseudohamiltonian h = ~fff dxdy). The simplest 2D solutions to (1), (1') are Q1D structures = x/~e(k2)cos(~) + a ( k ) c o s ( 3 ~ ) ,
(5)
where 62(k 2) = 2(a - 1) - (k 2 - 1) 2 ,
(6)
or
145
Volume 99A, number 4 e2(k 2) = k 2 - k 4 .
PHYSICS LETTERS (6')
The amplitude of the third harmonic A in the solution (5, 6) is small ~ e 3 and may be directly neglected. As to the solution (5, 6') concerning eq. (1'), for it
A(k) = 2 e3(k2______)(81k 2 _ 9)_ 1 x/Tqk 2 and the solution in the quasiharmonic form (5, 6') is valid in the range of k where X(k 2) = ~
e(k2)A-l(k)
(7)
is numerically small. As we shall see below, the wavenumbers of interest for us meet this condition. Substituting (5) into (2), (4') we obtain h = - g 1 e4(k2)
(8)
i.e. the "most stable" Q I D structure has the wavenumber k 0 corresponding to the maximum of the instability growth rate of the trivial solution [k 2 = 1 for eq. (1) and k 02 = ~l for ( 1 ), ] . Note that l•f k 2 = "1 ~, X from (7) is 0.0053. Proceeding to the properly two-dimensional solutions we should emphasize first of all that searching for these solutions to eq. (1') in quasiharmonic form one should check the validity of the quasiharmonicity condition dealing with the X-parameters defined analogously to (7). Eqs. (1), (1') admit 2D solutions representing the lattice consisting of rectangular ceils:
~(x,y) = ~ {[2 e2(ky2) - e2(k2)l 1/2 cos(kxX) + [2 e2(kx2) - e 2 (ky2)] 1/2 cos(kyy)},
(9)
(10)
Herefrom we find that all the rectangular parkets are stable in comparison with the trivial solution and the most stable ones are those with the maximum aspect ratio. Then, eqs. (1), (1') posess the solutions of hexagonal symmetry describing the parkets o f triangulars: =~
e(k 2) (cos(kx) + cos[~ k(N/~y -- x)]
+ cos[½ k(~r3y + x)] ) .
146
Besides that, eq. (1) admits the solution corresponding to hexagon parkets which can be obtained from (11) on replacing sin by cos. For all these solutions sax ea(k2) h = - s-a-6~
(12)
Comparing (8), (9) and (12) we find that they are the Q1D structures (6, 6') which are the most stable ones in the global sense. So, it is interesting to exam. ine their local stability. The "dangerous" eigenmode of small perturbations on the background o f ( 6 ) may be chosen in the form = exp(~2t)[~cos(p + k)x] + ¢cos[(p - k)x I , ~, ~ = const.
(13)
Using (13) it can be readily demonstrated that in the 2D case the local stability criterion for the Q1D solution (5, 6) to (1) coincides with that for purely 1D case [6]: ~2(p) < 0 if 4 (a - 1) ~< e2(k 2) ~< 2(~ - 1). Contrarily, in the case of eq. (1') the stability criterion for the Q1D solution (5, 6') to (1'): i ~< k2 <~ 0.805 appears to be more stringent than the purely one-dimensional: 0.266 ~< k 2 ~< 0.805. We f'md it to be very surprizing that the band of the wavenumbers corresponding to stable structures is restricted in the 2D case so as to choose the more quasiharmonic structures. (It is clear that X(k) drops with the increase of k; for instanse X(0.266) = 0.019, X(0.805) = 0.0015). We are deeply grateful to Professor S.I. Anisimov.
References
for which h = 27 [e4(kx2) - 4 eZ(k2x) e2(k 2) + e4(ky2)].
28 November 1983
(11)
[ 1 ] B.A. Malomed and I.E. Staroselsky, Phys. Lett. 99A (1983) 143. [2] G.I. Sivashinsky, SIAM J. Appl. Math. 39 (1980) 67. [3] A.P. Aldushin, B.A. Malomed and Ya.B. Zeldovich, Comb. Flame 42 (1981) 1. [4] S.I. Anisimov, S.M. Goldberg, B.A. Malomed and M.I. Tribelsky, Dokl. Akad. Nauk SSSR 262 (1982) 1117. [5] A.B. Konstantinov, B.A. Malomed and S.M. Goldberg, Dokl. Akad. Nauk SSSR 262 (1983) 270. [6] M.A. Nepomnyashchy, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, No. 3. (1974) 28. [7 ] G.I. Sivashinsky, Acta Astronautica 4 (1977) 1117.