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On hydrodynamic instability of chemical oscillations

A study was made of the instability that arises when acoustic and gravity waves propagate in an inhomogeneous medium which is characterized by oscillatory approach of the reaction coordinates to the steady state. It is shown that loss of stability can occur in two different ways: by bifurcation of the periodic solution and by Eckhaus bifurcation. It is considered how an introduced nonlinearity stabilizes the unstable modes when the controlling parameter passes through the bifurcation points.


I. INTRODUCTION
Self-organization effects in nonequilibrium chemical systems with diffusion due to stability loss of the thermodynamic branch have been systematically presented in the well known monograph of Nicolis and Prigogine l and following publications (see, for example, Refs. 2 and 3). It is important to note that for certain values of the system parameters the thermodynamic branch stable with respect to the chemical processes loses stability when diffusion is "switched on." [1][2][3] It turns out that a similar situation occurs when chemical processes are combined with hydrodynamics. It is shown in the present paper that a stable-node-stable-focus transition in a chemical (or thermochemical) system of reactions can be accompanied by the loss of stability of propagating hydrodynamic perturbations. We recall that it was shown in Ref. 4 that if there is departure from equilibrium in the system of only the chemical reactions, then transition from a monotonic to an oscillatory nature of the damping of the reaction coordinates "does not follow from any kind of instability but comes progressively." Using the standard assumption of local equilibrium and also the thermohydrodynamic criterion of stability 4 , we consider in this paper the occurrence of instability in the case of propagation of acoustic and internal gravity waves in a nonequilibrium medium with an oscillatory nature of the approach of the chemical (thermochemical) reaction variables N i (i= 1,2) to the steady state. The controlling parameter of the problem is r=sin β, where 2 /    characterizes the phase shift between the deviations of N i from the current values 0 i N in the wave. The stability region is r - r  r + , (for the considered model of internal gravity waves, r - r  1 ). It is shown that the loss of stability at the boundaries r  occurs in two different ways: through bifurcation of the periodic solution of the nonvanishing critical frequency and through Eckhaus bifurcation. 6 Using the well-known assumption 7 of weak variation of the wave profile due to nonlinearity and dissipation over distances of the order of the characteristic wavelength, we introduce quadratic nonlinearity into the acoustic problem. A model equation of this form was considered for the first time by Malkin and the author of the present paper in Ref. 8. On the basis of the equations obtained in the paper, we shall show how the introduced nonlinearity stabilizes the unstable modes on the passage of the controlling parameter through the bifurcation points.
A fairly large number of nonequilibrium media, 1,9,10 in which behavior in the form of chemical or thermochemical oscillations is realized, is now known. In general, such behavior is not restricted to chemical systems; for example, it was noted in Ref. 11 that nonmonotonic relaxation of a degree of vibrational excitation of molecules can occur in a gas-discharge plasma through a trapping-vibrational instability. It can be seen that the class of nonequilibrium media in which the considered form of hydrodynamic instability can exist is quite large.

ACOUSTIC WAVES
We consider acoustic perturbation of a steady, homogeneous, nonequilibrium state whose stability is maintained by continuously unfolding dissipative processes. There are two interesting cases in which the medium can execute damped chemical or thermochemical oscillations in the absence of hydrodynamic perturbations.

The case of chemical oscillations
We consider a chemically active (reacting) medium with two degrees of freedom-the mass concentrations N 1 , and N 2 , which describe a rapid (compared with the consumption of the original materials) stage of a complex reaction 1 and are characterized by an oscillatory nature of the relaxation of N i to the steady state.
We take the equations of the chemical kinetics in the form of an expansion around the steady state, retaining the first term of the expansion: As usual, we assume that the dissipation has a weak effect on the change in the wave amplitude over times of the order of a period, and therefore in (1) where  (>0) is the frequency of the concentration oscillations, and  -1 is the characteristic time of their damping. We write down the equation of state, retaining only the terms linear in the increments: where h is the enthalpy per unit mass, C p and  are the "frozen" specific heat and "frozen" coefficient of expansion at constant pressure; x is the spatial coordinate, is the "frozen" compressibility and  is the coefficient of thermal conductivity. We define and K S 0 is the compressibility corresponding to a very slow process, so that the medium is at all times in a steady state. In accordance with the adopted approximation of local equilibrium, we assume that K S  0 (5) Inequality of the type (5) are well known in the acoustics of linearly nonequilibrium media. 12 The contribution of the chemical reactions to the change in the volume during propagation of the wave is taken into account by the second term on the right-hand side of Eq. (3). From (1) and (2), we can obtain a relation for the deviation of  i S i N V  from its steady value: In (6), it is convenient to parametrize the constant factors multiplying J C and J S as follows, taking into account (5): with constants >0,   /2. The introduced parameter  characterizes the phase shift between the deviations of we write the dependence (6) with allowance for (7) in the differential form To investigate the stability of pressure waves in the dissipative medium, we shall proceed from the local thermodynamic and hydrodynamic criterion of stability for the Lyapunov function    In the case of one spatial variable, the balance equation for  2 z has the form Here L is the flux of the function  2 z ,  i is the chemical potential, and  and  are the coefficients of first and second viscosity. On the right-hand side of Eq. (10), we have retained terms of order not higher than the second in the increments. Smallness of the dissipation presupposes smallness of the dimensionless dissipative coefficients, and also smallness of the change in the volume and enthalpy during one step of each reaction. Therefore, in order to retain on the right-hand side of (10) terms of not higher than third order (we assume that the increments of the dimensionless pressure and the velocity are quantities of first order of smallness), it is with respect to P and N j at S= S 0 . Thus, bearing in mind that in the acoustic wave in the first approximation  , we obtain the following expression from (10) after transformations (with the indicated accuracy): Here We consider the evolution of a periodic wave, . Going over to complex variables 5 by a substitution of the where the asterisk denotes the complex conjugate, and . sin cos (12) is written within the same order of accuracy as Eq. (11). A similar expression [without the coefficient 2 on the right-hand side of (12)] is obtained if the right-hand side of (11) is averaged with respect to  over the period 2/.

The case of thermochemical oscillations
We consider a thermochemical system that includes photochemical stages with one concentration degree of freedom. Here we also assume an oscillatory nature of the relaxation to the steady state. Examples of such systems can be found in Refs. 1 and 9. For acoustic problems, the most interesting case is one in which the nonequilibrium state is maintained solely by exchange of energy with the ambient medium.
In the simplest case, the balance equation for the part of the entropy S e , which is responsible for the exchange of energy with the ambient medium, has the form ) ( where N is the mass concentration,  and  are the coefficients of absorption of external radiation and heat transfer, and T ex is the temperature of the external medium. The temperature increment in an acoustic wave is The equation of state (3) and the stability conditions (11) and (12)  , and also the dimensionless variations of the volume and enthalpy during one stage of the reaction must be regarded to be quantities of order unity. As before, the dimensionless dissipative coefficients, and also the increments of the dimensionless temperature, pressure, and velocity are assumed to be first-order quantities. Such a scale of orders of smallness makes the thermal and chemical effects of the same order, while the dissipation, as before, has only a slight effect on the wave amplitude over times of the order of the period.

Stability conditions
The requirement of positive definiteness of the right hand side of the expression (12) for all  gives the stability conditions (  and >0) The conditions (15) determine the region of stability in the space of the parameters     / ˆ , , and r  sin  (the controlling parameter): The critical frequency corresponds to the state r r For r r   , the Eckhaus instability 6 arises. As will be seen from what follows, the complex and real parts of the dispersion relation vanish when (17)  At the point r r   , bifurcation of the periodic solution occurs with critical frequency ris the smaller root of the equation (for 0      ): For r r   , a band of finite width of modes near  c becomes unstable.
To consider the propagation of acoustic waves in a medium with monotonic approach of N j to the stable state N j 00 (that is a singular point of the type of a stable node), it is necessary to make the formal substitutions . In this case, the stability condition (12) is satisfied identically for all . In fact, become positive at all   .
The stability of the acoustic waves in the medium with monotonic damping of the reaction coordinates near equilibrium is, as is well known, 13 a consequence of Onsager's principle. It is clear from the above discussion that far from equilibrium the stable-node-stable-focus transition in a chemical (or thermochemical) system of reactions can be accompanied by the loss of stability of propagating acoustic perturbations.

DEVELOPMENT OF INSTABILITY AS A RESULT OF PROPAGATION OF GRAVITY WAVES IN A LAYER
We consider a horizontal layer of liquid in a constant gravity field between two infinitely extended parallel planes. A constant temperature gradient =(T 1 -T 0 )/z 0 > 0, is maintained. Here T 1 and T 0 are the temperatures on the upper and lower boundaries, respectively. Such a temperature gradient prevents convection but creates a density gradient that facilitates the generation of internal gravity waves. We assume that the layer is chemically active and has one concentration and one temperature degree of freedom. The liquid is assumed to be incompressible in the Boussinesq approximation. 14 We investigate the stability of the steady state determined by the distributions where  > 0 is the isobaric coefficient of expansion,  is the bulk reaction effect at constant . In addition, we assume that in (19) the bulk reaction effect does not change the direction of the density gradient due to the heating, i.e., The mass and energy balances are written in the form , and then in (21) we can make a change of variables of the form (w, N, T) = (w, N, T) sin(nz), where n = m/z 0 , m is an integer, z 0 is the thickness of the layer, and the variables with the prime do not depend on z. We shall take into account the diffusion and heat conduction only in the z direction (across the layer). In accordance with (21), the nature of the relaxation of N and T will then be determined by the set f ij : , , ; where we assume that for some n * the f ij satisfy the conditions (2); i.e., N and T have an oscillatory nature of the damping to the steady state. Further, we consider the mode n * , and values  > 0 and  > 0 corresponding to it.
As in the previous section, we introduce the constant  and variable R : where g is the acceleration of free fall. Taking into account the assumptions that have been made, we obtain from (21) Here and in what follows, the prime on w is omitted; sin and cos (-/2    /2) will be determined by (7) if we set ; , , , it also follows from (20) that >0. Note that Eqs. (8) and (23) can be transformed into each other by the substitution After transformations, we write the equation of momentum conservation in the form , , and x is the horizontal coordinate;  is the coefficient of kinematic viscosity, and for brevity the asterisk on n is omitted here and below. In deriving (24), we have used the standard transformations employed to treat the Benard problem 14 (with allowance for diffusion 15 ).
We seek the solution of (23) and (24) in the form of plane waves and real k. Assuming that the dissipation is small ( In contrast to the acoustic case, there is no instability here in the long-wavelength limit ), but for negative sin  the value of I  can become positive, i.e., loss of stability can occur. The condition for neutral stability ( I  = 0) gives the critical wave number k c and the minimum value of C  tan at which stability loss occurs: (26) The formal substitutions ) plane into a stable node. In this case, the stability condition I  < 0 is satisfied for all values of the parameter ': It can be seen that here, as in the previous section, the stable-node-stable-focus transition in the thermochemical system of reactions can be accompanied by a loss of stability of propagating hydrodynamic perturbations.
The considered form of instability must also occur in the case of the propagation of gravity waves on the surface of a nonequilibrium active liquid.

NONLINEAR EQUATIONS
, >0 is determined from (7). And R' is determined by the dimensionless equation (8): By means of the substitution Equation (29)

Behavior near r 
In this case, it is convenient to proceed from the system (27). At the bifurcation point r  , the solution of the system (27) linearized near the homogeneous state h =0, R = 0 has the form of a traveling wave with nonvanishing frequency  c (18): Near the point r  , we can set r r  -r 2  2 , <<1. It is assumed that the nonlinear dynamics is described by a weakly varying amplitude W: where  Assuming that the amplitude of this periodic solution is a function of the "slow" constant X 5 , and then using the standard procedure for eliminating secular terms, we can show that in the given case the Eckhaus instability has a supercritical nature.

CONCLUSIONS
In this paper, we have considered the appearance of instability when acoustic and internal gravity waves propagate in a nonequilibrium medium characterized by oscillatory approach of the reaction coordinates to the steady state. Monotonic approach of the reaction coordinates to the steady state cannot lead to a loss of stability of propagating waves. For linear nonequilibrium systems, this last fact is a direct consequence of Onsager's principle. 13 For acoustic waves, the stability problem has been considered by means of the thermohydrodynamic criterion. 5 We have shown that loss of stability can occur in two ways: through bifurcation of the periodic solution of the nonvanishing critical frequency and through Eckhaus bifurcation. 6 We have considered the propagation of nonlinear acoustic waves and shown that the introduced nonlinearity stabilizes the unstable modes on the passage of the controlling parameter through the bifurcation points.
We have constructed a model of the propagation of internal gravity waves in a stratified layer of a nonequilibrium medium with damped thermochernical oscillations. We have shown that in this case loss of stability can occur only through bifurcation of the periodic solution. The considered form of instability must also occur in the case of the propagation of gravity waves on the surface of a nonequilibrium liquid.