Stability Criteria to the Incompressible Inviscid Linear Fluid between Two Rotation Coaxial Cylinders

The stability and instability phenomenon coupled with the rotation effect and the thermal convection between two concentric cylinders was studied. By means of the Normal-Modes method, the stability or instability criteria for the linearized system in terms of the oscillation frequency, the axial wave-length and the background thermal gradient are proved. Besides, some numerical simulation for the axisymmetric perturbations is presented.

axial thermal stratification has received little attention [3] [11] [12], which combines horizontal shear and thermal convection, is of great interest in astrophysics, for example, the stratified Taylor-Couette flow serves as a model for instabilities in equatorial oceans [13] and [14]. The purpose of this paper is to study the Taylor-Couette flow with thermal convetion.
The study of the hydrodynamic stability for rotation flow (see Figure 1) begins with Rayleigh [1] (see also the overview in [15]). In the work of Rayleigh [1], a necessary and sufficient condition for the stability of linearized system is stated as ( ) is the angular velocity of the fluid at the distance r ( 1 2 r r r ≤ ≤ ) from the axis. More precisely, when the cylinders rotate in the same direction, only one mode of instability is present, which corresponds to the convection mode. When the cylinders rotate in opposite directions, two types or instability are presented. The second instability mode is of an oscillatory type. These results suggest that the "exchange of stability" may be valid when the cylinders rotate in the same direction, while it may not be valid when the cylinders rotate in opposite direction [16]. When the thermal varies in this area, the convection triggered by the thermal variation, also named as Bernard convection, will cause instability. There were many efforts about the stability or not on this case, see more details in [3] [17]. As to the coupled system with both rotation and the thermal convection, in [2] [18], the case has been studied in a rotation coordinate. However, in the work of [2] [18], in terms of the rotation and thermal convection, which one is the dominant role of the instability is not very obvious.
In this paper, we shall study the stability or instability criteria of the linearized incompressible inviscid fluid with both rotation and vertical background thermal variation. The vertical thermal variation is commonly observed in the atmosphere and in the oceans. Certainly, there also exists a horizontal variation of thermal across the latitudes due to differential heat radiation by the sun [19], however, in a small scale region, it is reasonable to consider only vertical thermal variation case. Namely, we shall study the following system: where ( ) is the velocity. The scalers ρ and P be the thermal and the pressure respectively. The system (2) also named as Boussinesq system, which is widely used to model the dynamics of the ocean or the atmosphere, see [20]. This system includes the weak nonlinear and dispersive effects, it can effectively interpret the dispersion wave in atmospheric dynamics [19] [21]. The goal of present work is to understand the stability problem of system (2) connected with the rotation and the thermal variation. The main theorems state as following: Theorem 1.1 The necessary and sufficient condition to be stable of the linearized system of (2) with axisymmetric perturbation is ( )  (3) is the well-known Rayleigh's criteria [1]. In this theorem, we conclude that when d dz ρ ≠ +∞ , the variation of thermal shall transfer the unstable modes. In other words, when d 0 dz ρ < , which implies hot fluid under the cold fluid, for the static fluid, the buoyancy tends to overturn the fluid. In terms of the rotation fluid, the thermal convection only affects the low-wave numbers modes, whether on the case ( ) 0 Theorem 1.2 The necessary and sufficient condition to be stable of the linearized system of (2) with non-axisymmetric perturbation is 1) For the special situation: To be more precisly, our results state: • The necessary condition for the linear stable: The paper is organized as follows. In Section 3, we shall prove Theorem 1.1, and then theorem 1.2 shall be proved in Section 4.

The Perturbation Equations and the Basic State
We're dealing with the system (2) in a coaxial cylinders area (see Figure 1), it's more convenient to study the system (2) , we investigate the following system: where n denotes the normal exterior vector. r r = respectively. In this paper, we shall study the stability of the following stationary Couette flow where ( ) V r is an arbitrary smooth function of r.
In the experiments the basic temperature state is, a priori, a time-dependent state because the initial temperature gradient is not maintained by the boundary conditions. However, if we consider the time scale smaller than the typical diffusion time over the height of the cylinders, this gradient can be considered as constant in time. Therefore, we choose as time scale small enough, which allows us to write the basic state for the temperature as time-independent. In this paper, we consider the variation of background thermal as the following case: According to the (11) and (12), the basic pressure distribution is determined by the system (9). It takes the form as Consider an infinitesimal perturbation of the system (11)-(13), we write the perturbed state as , , , , .
Substitute (14) into (9), we get the system for the perturbations as The linearized equations governing these perturbations system (15) is We analyze the disturbance by using the Normal-modes method. It is natural to write that the various quantities describing the perturbation have a ( )  (16) and (17), then we get , , then there holds Due to the fluid is confined between two coaxial cylinders of radii 1 r and 2 r , we must require that the radial components of the velocity vanish for these values of r. Thus, we consider the Equations (33) and (35) with the boundary con-

Linear Stability for Axisymmetric Perturbations
In this section, we shall prove Theorem 1.1, the results for the axisymmetric perturbation case. Namely in (17) Eliminating P between these equations, we obtain We shall first show that the principle of the exchange of stabilities is invalid.

Exchange of Stability Is Invalid
Proof. Testing Equation (40) Similarly, from the right hand side of the Equation (41) Recalling that Now we prove the exchange of stability is invalid by using the proof of contradiction. Let Since the Equation (46) holds for k ∈  , we conclude that 0 a ≡ / , which implies the principle of exchange the stability is invalid.

The Critical Value of z d d ρ at the Marginal State
The marginal state illustrates that the transition from stability to instability. In the following, we present the critical value for the d dz ρ at the marginal state.
The equations governing the marginal state as (by setting 0 Eliminating P between these equations, and denoting  In (51), the characteristic value of min R will be a minimumin terms of the characteristic functions of a variation problem. To verify this fact, we denote R δ be the variation in R when r ξ is subjected to a small variation r δξ which is also compatible with the boundary conditions on r ξ .
According to the Equation (51), we obtain ( ) which comes from the Equation (48).

The Linear Stability for Axsymmetric Perturbation
Proof. Recalling (45), for simplicity we write From (57), we get To keep the linearized system stable, the necessary and sufficient condition is that ω be real. That is to say, It is equivalent to study Recalling (58), the inequality (61) is equivalent to the following one: ( )

Non-Axisymmetric Linear Stability Analysis
In this section, we study Theorem 1.2, the non-axisymmetric perturbation situation.
Proof. We consider non-axisymmetric perturbations, namely, In this situation, both the rotation and the thermal convection play the stabilization or destabilize role simutanously.

Conclusion
In this paper, we analyze the stability and instability criteria for the coupled thermal effects of fluids between coaxial rotating cylinders. The perturbation equation is analyzed by normal-modes method. We extend Rayleigh stability criterion by the analysis of axisymmetric perturbation in some cases, and we also analyze the case of non-axisymmetric perturbation, the results are presented in Theorem 1 and Theorem 2. Finally, through numerical simulation experiments, the results obtained by our experiments are consistent with the results obtained by our analysis on the specific cases of axisymmetric perturbations under certain given special values. For the fluid instability near the cylinder boundary, we also found some new problems waiting for us to further deal with.