Dynamic bifurcation and instability of Dean problem

TL;DR

This paper develops a nonlinear dynamic bifurcation theory for the Dean problem, providing rigorous justification for linear theories and analyzing vortex structures and instabilities.

Contribution

It introduces a nonlinear theory for the Dean problem, rigorously justifies linear approximations, and applies dynamic bifurcation theory to analyze vortex structures and instabilities.

Findings

Rigorous justification of linear theory for Dean problem

Identification of vortex structures during bifurcation

Analysis of instability mechanisms in fluid flow

Abstract

The main objective of this paper is to address the instability and dynamical bifurcation of the Dean problem. A nonlinear theory is obtained for the Dean problem, leading in particular to rigorous justifications of the linear theory used by physicists, and the vortex structure. The main technical tools are the dynamic bifurcation theory [15] developed recently by Ma and Wang.

Equations

$$u_{\theta}^{0}=\frac{1}{2\rho\nu}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}(r\ln r+Ar+\frac{B}{r}),$$

$$\{\begin{array}{rl}& \frac{\partial u_r}{\partial t}+(u\cdot \nabla )u_r-\frac{u_{\theta }^2}{r}=\nu (\Delta u_r-\frac{u_r}{r^2}-\frac{2}{r^2}\frac{\partial u_{\theta }}{\partial \theta })-\frac{1}{\rho }\frac{\partial p}{\partial r},\\ & \frac{\partial u_{\theta }}{\partial t}+(u\cdot \nabla )u_{\theta }+\frac{u_r{u}_{\theta }}{r}=\nu (\Delta u_{\theta }-\frac{u_{\theta }}{r^2}+\frac{2}{r^2}\frac{\partial u_r}{\partial \theta })-\frac{1}{r\rho }\frac{\partial p}{\partial \theta },\\ & \frac{\partial u_z}{\partial t}+(u\cdot \nabla )u_z=\nu \Delta u_z-\frac{1}{\rho }\frac{\partial p}{\partial z},\\ & \frac{\partial (r{u}_r)}{\partial r}+\frac{\partial u_{\theta }}{\partial \theta }+\frac{\partial (r{u}_z)}{\partial z}=0,\end{array}$$

$$\displaystyle\begin{aligned} &(u\cdot\nabla)=u_{r}\frac{\partial}{\partial r}+\frac{u_{\theta}}{r}\frac{\partial}{\partial\theta}+u_{z}\frac{\partial}{\partial z},\\ &\Delta=\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial}{\partial r}\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\partial^{2}}{\partial z^{2}}.\end{aligned}$$

$$R_2-R_1\le \frac{R_1+R_2}{2},$$

$$u=(0,{\overline{u}}_{\theta }(r),0).$$

$$\frac{\partial\overline{p}}{\partial\theta}=\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}=\text{const.}$$

$$\left\{\begin{aligned} &u_{r}=u_{z}=0,\quad u_{\theta}=\overline{u}_{\theta}(r)=\frac{1}{2\rho\nu}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}(r\ln r+Ar+\frac{B}{r}),\\ &\frac{1}{\rho}\frac{\partial\overline{p}}{\partial r}=\frac{u_{\theta}^{2}(r)}{r},\quad\frac{\partial\overline{p}}{\partial\theta}=\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}.\end{aligned}\right.$$

$$\overline{u}_{\theta}(R_{1})=\frac{1}{2\rho\nu}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}R_{2}^{2},\quad\overline{u}_{\theta}(R_{2})=\frac{1}{2\rho\nu}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}R_{1}^{2}.$$

$$\frac{1}{\rho}\frac{\partial p}{\partial r}=\frac{u_{\theta}^{2}(r)}{r},\quad\frac{\partial p}{\partial\theta}=\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}.$$

$$A=-\frac{R_2^2\ln R_2-R_1^2\ln R_1+R_2^2{R}_1-R_1^2{R}_2}{R_2^2-R_1^2},$$

$$B=\frac{R_1^2{R}_2^2(\ln R_2-\ln R_1)+R_2^4{R}_1-R_1^4{R}_2}{R_2^2-R_1^2}.$$

$$u_r,{\overline{u}}_{\theta }+u_{\theta },u_z,p+\overline{p}.$$

$$\displaystyle\left\{\begin{aligned} &\frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac{u_{\theta}^{2}}{r}=\nu\bigg{(}\Delta u_{r}-\frac{u_{r}}{r^{2}}\bigg{)}-\frac{1}{\rho}\frac{\partial p}{\partial r}\\ &\qquad\qquad\qquad\qquad\qquad+\frac{1}{2\nu\rho}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\bigg{(}2\ln r+2A+\frac{2B}{r^{2}}\bigg{)}u_{\theta},\\ &\frac{\partial u_{\theta}}{\partial t}+(u\cdot\nabla)u_{\theta}+\frac{u_{r}u_{\theta}}{r}=\nu\bigg{(}\Delta u_{\theta}-\frac{u_{\theta}}{r^{2}}+\frac{1}{\nu\rho}\frac{1}{r}\bigg{)}\\ &\qquad\qquad\qquad\qquad\qquad-\frac{1}{2\nu\rho}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}(2\ln r+2A+1)u_{r},\\ &\frac{\partial u_{z}}{\partial t}+(u\cdot\nabla)u_{z}=\nu\Delta u_{z}-\frac{1}{\rho}\frac{\partial p}{\partial z},\\ &\frac{\partial}{\partial r}(ru_{r})+\frac{\partial}{\partial z}(ru_{z})=0,\end{aligned}\right.$$

$$\begin{array}{r}(u\cdot \nabla )=u_r\frac{\partial }{\partial r}+u_z\frac{\partial }{\partial z},\end{array}$$

$$\begin{array}{r}\Delta =\frac{{\partial }^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^2}{\partial z^2}.\end{array}$$

$$\displaystyle\begin{aligned} &(x,t)=\bigg{(}lx^{\prime},\frac{l^{2}}{\nu}t^{\prime}\bigg{)},\quad\quad\quad(x=(r,r\theta,z)),\\ &(u,p)=\bigg{(}\frac{\nu}{l}u^{\prime},\frac{\rho\nu^{2}}{l^{2}}p^{\prime}\bigg{)},\quad\quad(u=(u_{r},u_{\theta},u_{z})).\end{aligned}$$

$$\displaystyle\left\{\begin{aligned} &\frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}-\frac{u_{\theta}^{2}}{r}=\Delta u_{r}-\frac{u_{r}}{r^{2}}-\frac{\partial p}{\partial r}+\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\bigg{(}\ln r+A+\frac{B}{r^{2}}\bigg{)}u_{\theta},\\ &\frac{\partial u_{\theta}}{\partial t}+(u\cdot\nabla)u_{\theta}+\frac{u_{r}u_{\theta}}{r}=\Delta u_{\theta}-\frac{u_{\theta}}{r^{2}}+\frac{2}{r}\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\\ &\qquad\qquad\qquad\qquad\qquad\qquad-\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\bigg{(}\ln r+A+\frac{1}{2}\bigg{)}u_{r},\\ &\frac{\partial u_{z}}{\partial t}+(u\cdot\nabla)u_{z}=\Delta u_{z}-\frac{\partial p}{\partial z},\\ &\frac{\partial}{\partial r}(ru_{r})+\frac{\partial}{\partial z}(ru_{z})=0,\end{aligned}\right.$$

$$\frac{l}{R_1}\ll 1.$$

$$\ln r+A+\frac{1}{2}$$

$$=\ln r-\frac{R_2^2\ln R_2-R_1^2\ln R_1+R_2^2{R}_1-R_1^2{R}_2}{R_2^2-R_1^2}+\frac{1}{2}$$

$$=\ln r-\ln R_2-\frac{R_1^2(\ln R_2-\ln R_1)}{R_2^2-R_1^2}-\frac{R_1{R}_2}{R_1+R_2}+\frac{1}{2}$$

$$=\ln \frac{r}{R_2}-\frac{\ln (1+l/R_1)}{(1+l/R_1{)}^2-1}-\frac{R_1{R}_2}{R_1+R_2}+\frac{1}{2}$$

$$\sim -\frac{l/R_1}{l/R_1(1+l/R_1)}-\frac{\ln (1+l/R_1)}{(1+l/R_1{)}^2-1}-\frac{R_1{R}_2}{R_1+R_2}+\frac{1}{2}$$

$$\sim -\frac{R_1{R}_2}{R_1+R_2},$$

$$\ln r+A+\frac{B}{r^2}$$

$$=\ln r+A+\frac{1}{2}+\frac{B}{r^2}-\frac{1}{2}$$

$$\sim -\frac{R_1{R}_2}{R_1+R_2}+\frac{1}{r^2}\frac{R_1^2{R}_2^2(\ln R_2-\ln R_1)+R_2^4{R}_1-R_1^4{R}_2}{R_2^2-R_1^2}-\frac{1}{2}$$

$$=-\frac{R_1{R}_2}{R_1+R_2}+\frac{R_2^2}{r^2}\frac{R_1^2(\ln R_2-\ln R_1)}{R_2^2-R_1^2}+\frac{R_1{R}_2}{r^2}\frac{R_2^3-R_1^3}{R_2^2-R_1^2}-\frac{1}{2}$$

$$\sim -\frac{R_1{R}_2}{R_1+R_2}+\frac{1}{2}+\frac{R_2^2+R_1{R}_2+R_1^2}{R_2^2-R_1^2}-\frac{1}{2}$$

$$\sim \frac{2R_1{R}_2}{R_1+R_2}.$$

$$\displaystyle\left\{\begin{aligned} &\frac{\partial u_{r}}{\partial t}+(u\cdot\nabla)u_{r}=\Delta u_{r}-\frac{\partial p}{\partial r}+\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\frac{2R_{1}R_{2}}{R_{1}+R_{2}}u_{\theta}\\ &\frac{\partial u_{\theta}}{\partial t}+(u\cdot\nabla)u_{\theta}=\Delta u_{\theta}+\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\frac{R_{1}R_{2}}{R_{1}+R_{2}}u_{r},\\ &\frac{\partial u_{z}}{\partial t}+(u\cdot\nabla)u_{z}=\Delta u_{z}-\frac{\partial p}{\partial z},\\ &\frac{\partial u_{r}}{\partial r}+\frac{\partial u_{z}}{\partial z}=0,\end{aligned}\right.$$

$$\Delta =\frac{{\partial }^2}{\partial r^2}+\frac{{\partial }^2}{\partial z^2}.$$

$$\displaystyle\left\{\begin{aligned} &\frac{\partial u_{r}}{\partial t}=\Delta u_{r}-\frac{\partial p}{\partial r}+\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\frac{\sqrt{2}R_{1}R_{2}}{R_{1}+R_{2}}u_{\theta}-(u\cdot\nabla)u_{r}\\ &\frac{\partial u_{\theta}}{\partial t}=\Delta u_{\theta}+\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\frac{\sqrt{2}R_{1}R_{2}}{R_{1}+R_{2}}u_{r}-(u\cdot\nabla)u_{\theta}\\ &\frac{\partial u_{z}}{\partial t}=\Delta u_{z}-\frac{\partial p}{\partial z}-(u\cdot\nabla)u_{z}\\ &\frac{\partial u_{r}}{\partial r}+\frac{\partial u_{z}}{\partial z}=0,\end{aligned}\right.$$

$$\{\begin{array}{rl}& u_r=0,u_{\theta }=0,\frac{\partial u_z}{\partial r}=0,\mathrm{when}r=r_1,r=r_2,\\ & \frac{\partial u_r}{\partial r}=\frac{\partial u_{\theta }}{\partial z}=0,u_z=0,\mathrm{when}z=0,L.\end{array}$$

$$\displaystyle\begin{aligned} &H=\bigg{\{}u=((u_{r},u_{z}),u_{\theta})\in H^{2}(\Omega,\mathbb{R}^{2})\times H^{2}(\Omega)~{}|~{}u~{}\mathrm{satifies}~{}(\ref{211})\\ &\qquad\qquad\qquad\qquad\qquad\qquad\mathrm{and}~{}\frac{\partial u_{r}}{\partial r}+\frac{\partial u_{z}}{\partial z}=0\bigg{\}},\\ &H_{1}=\bigg{\{}u=((u_{r},u_{z}),u_{\theta})\in L^{2}(\Omega,\mathbb{R}^{2})\times L^{2}(\Omega)~{}|~{}u~{}\mathrm{satifies}~{}(\ref{211})\\ &\qquad\qquad\qquad\qquad\qquad\qquad\mathrm{and}~{}\frac{\partial u_{r}}{\partial r}+\frac{\partial u_{z}}{\partial z}=0\bigg{\}}.\end{aligned}$$

$$\displaystyle\begin{aligned} &Au=-\big{(}P(\Delta u_{r},\Delta u_{z}),\Delta u_{\theta}\big{)},\\ &Bu=\big{(}P(u_{\theta},0),u_{r}\big{)},\\ &Gu=\big{(}P((u\cdot\nabla)u_{r},(u\cdot\nabla)u_{z}),(u\cdot\nabla)u_{\theta}\big{)},\end{aligned}$$

Full text

keywords

Dean problem; instability; dynamic bifurcation.

1 Introduction

The instability of rotating flows is an important issue in fluid dynamics. The problem go back to the pioneering work of Rayleigh[16]. He also considered a basic swirling flow of an inviscid fluid which moves with angular velocity $\Omega(r)$, an arbitrary function of the distance $r$ from the axis of rotation, which is determined by the radial rate of change of pressure, then Rayleigh’s criterion for stability is “the square of the circulation, $r^{2}\Omega^{2}$, is an increasing function of.” Synge has pointed out that Rayleigh’s argument is inconclusive, and that a rigorous demonstration of stability can only be determined by considering the stability of the basic flow to arbitrary perturbations. Further, Synge showed that Rayleigh’s criterion is valid for arbitrary perturbations which are symmetric with respect to the axis of rotation[2]. However, Rayleigh’s criterion can predicts instability for an inviscid rotating fluid. The stability of an inviscid fluid with respect to non-axisymmetric disturbance has also been studied by Bisshopp[4] and Krueger and Diprima[9] in the narrow-gap approximation.

Note that the instability of viscous rotation flows also have been considered by several authors including Taylor[17] and Dean[4]. In 1923, Taylor [17] conducted a famous experiment, and observed and studied the stability of an incompressible viscous flow between two rotating coaxial cylinders. He found that when the Taylor number $T$ is smaller than a critical value $T_{c}>0$, called the critical Taylor number, the basic flow, called the Couette flow, is stable. While the Taylor number crosses the critical number, the Couette flow breaks into a cellular pattern that is radially symmetric. Such instability called Taylor instability. More research about Taylor instability see [1, 2, 5, 8, 19]. In 1928, Dean[4] found that a similar type of instability can also occur when a viscous fluid in a curved channel owing to a pressure gradient acting round the channel. Such instability is called Dean instability, which has been considered again by H$\ddot{a}$mmerl[7] and Walowit, Tsao and DiPrima [18]. For Couette flow, especially, in the case that the cylinders rotate in the same direction, a simple formula for predicting the critical speed is derived in [18]. The effect of a radial temperature gradient on the stability of Couette flow is also considered.

Recently, Ma and Wang [10, 15] have developed a bifurcation theory [9] for nonlinear partial differential equations, which has been used to develop a nonlinear analysis for the Rayleigh-Bénard convections [11, 12] and Taylor stability and bifurcation [13, 14]. This bifurcation theory is centered on a new notion of bifurcation, called attractor bifurcation, for nonlinear evolution equations. In particular, based on this bifurcation theory, for the Rayleigh-Bénard convection problem, Ma and Wang have shown [11, 12] that the problem bifurcates from the trivial solution to an attractor $\mathcal{A}_{R}$ when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_{c}$ , for all physically sound boundary conditions and regardless of the multiplicity of the eigenvalue $R_{c}$ for the linear problem. It is worth pointing out that the instability of viscous flows is the nonlinear phenomena.

Motivated by the above papers [11, 12, 13, 14], our purpose of this article is to address the dynamic transition of the Dean problem under narrow gap approximation. The main technical tool are the dynamical bifurcation theory [10, 15]. The basis of our research is a basic solution of Dean problem as follows

[TABLE]

where $A$ and $B$ are determined by boundary velocity. In this paper, we shall get the expressions of $A$ and $B$ by the physical assumption, which are different from expressions in [4, 7, 18] where the boundary velocity vanishes. Based on the basic solution determined by boundary condition and narrow gap approximation, we get a new simplified governing model for the Dean problem. In addition, the expressions of $A$ and $B$ guarantee that the linear part of dynamical equation is symmetrical.

The article is organized as follows. In Section 2, we will give one type of boundary conditions for the steady state solution to guarantee that the corresponding linear part is symmetrical. The simplified model and the set-up are given in Section 2, and the principle of exchange of stability(PES) is given in Section 3, and the main theorems are proved in Section 4.

2 Simplified Governing equations

2.1 the basic flow

The governing equations of a viscous incompressible fluid in the space between two concentric cylinders are the following equations in the cylindrical coordinates, which are given by

[TABLE]

where $\nu$ is the kinematic viscosity, $\rho$ is the density, $u=(u_{r},u_{\theta},u_{z})$ is the velocity field, $p$ is the pressure function, and

[TABLE]

To simplify (2.1), we make the following physical assumptions:

narrow gap approximation,

[TABLE]

where $R_{1}$ and $R_{2}$ are the radius of of the two concentric cylinders. 2. 2.

the steady state solution of (2.1) only related to the $\theta$-direction and only depend on the variable $r$, i.e.,

[TABLE] 3. 3.

the partial derivatives of pressure function $\widetilde{p}$ with respect to $\theta$ is a constant, i.e.,

[TABLE]

From [5], the basic flow for (2.1) is a steady state solution, defined by

[TABLE]

Remark 1**.**

In this paper, the basic flow (2.4) we considered is different from the Couette flow for the Taylor problem.

By the boundary conditions

[TABLE]

We also assume that the pressure $p$ is a function depending on $\theta$ and $r$, satisfying that

[TABLE]

Basing on the above assumptions, together (2.5) with (2.6), we get

[TABLE]

Remark 2**.**

Note that (2.8), (2.9) and the narrow gap approximation assumption (2.2) can guarantee the linear part of the perturbation equation by (2.5) is symmetrical.

2.2 perturbed non-dimensionless equation

In the following, in order to investigate the stability of the flow described by (2.5), we need to consider the perturbed state

[TABLE]

Assume that the perturbations are axi-symmetric and independent of $\theta$, we derive from (2.1) that

[TABLE]

where

[TABLE]

To derive the non-dimensionless form of equations (2.10), let

[TABLE]

Omitting the primes, we obtain the non-dimensionless form of (2.10) as follows

[TABLE]

Taking the length scale $l=R_{2}-R_{1}$, then the narrow gap condition is given by

[TABLE]

Furthermore, by (2.14) and (2.8), we have

[TABLE]

Moreover,

[TABLE]

Under the assumption (2.14), we can neglect the terms containing $r^{-n}(n\geq 1)$ in (2.13). Furthermore, by (2.15) and (2.16), we have

[TABLE]

where $\Delta$ is defined as follows

[TABLE]

Let $u_{r}=u^{\prime}_{r}$, $u_{\theta}=\frac{1}{\sqrt{2}}u^{\prime}_{\theta}$, $u_{z}=u^{\prime}_{z}$, $p=p^{\prime}$. Omitting the primes, we can rewrite the equations (2.17) as follows

[TABLE]

Remark 3**.**

It should be pointed out that the existence of global weak solution, the regularity of global weak solution and the existence of global attractor for the simplified model (2.18) under various boundary conditions has been obtained in our other submitted papers.

2.3 Abstract operator form

For convenience, we denote \lambda=\bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}\frac{\sqrt{2}R_{1}R_{2}}{R_{1}+R_{2}}. In this paper, we only consider the following free boundary conditions:

[TABLE]

where $r_{1}=\frac{R_{1}}{l}$ and $r_{2}=\frac{R_{2}}{l}$ are non-dimensionless constants.

Let

[TABLE]

Define the mappings $A,B,G:H\rightarrow H_{1}$ as follows

[TABLE]

where $P$ is the Leray projection.

Setting $L_{\lambda}=-A+\lambda B$, then the equation (2.18) is equivalent to the following operator equation

[TABLE]

where $u_{0}$ is the initial value of (2.18).

3 Principle of exchange of stability

Let the eigenvalue of operator $L_{\lambda}$ and the corresponding eigenvector be $\beta$ and $v=(v_{r},v_{\theta},v_{z})$ respectively such that

[TABLE]

Obviously, (3.1) is equivalent to

[TABLE]

Clearly, $v_{r},~{}v_{\theta}\in W_{1}$ and $v_{z}\in W_{2}$ respectively, where

[TABLE]

We know that

[TABLE]

and

[TABLE]

are isomorphic, where

[TABLE]

Noticing that the eigenvectors $e_{mn}$ and $\overline{e}_{mn}$ of $\Delta$ in $W_{1}$ and $W_{2}$ are the orthogonal basis of $W_{1}$ and $W_{2}$ respectively. The eigenvalues $\lambda_{mn}$ and eigenvectors $e_{mn}$ of $\Delta$ in $W_{1}$ are as follows

[TABLE]

The eigenvalues $\overline{\lambda}_{mn}$ and eigenvectors $\overline{e}_{mn}$ of $\Delta$ in $W_{2}$ are as follows

[TABLE]

where $m,n=1,2,\cdots$.

Hence, $v_{r},v_{\theta}$ are linear combinations of the eigenvectors $e_{mn}$ and $v_{z}$ is linear combination of the eigenvectors $\overline{e}_{mn}$, that is

[TABLE]

Substituting (3.3) into (3.2) to get

[TABLE]

Eliminating $p$ of (3.4) to get

[TABLE]

Furthermore, we obtain

[TABLE]

Then, we can deduce from (3.6) that

[TABLE]

Let

[TABLE]

we rewrite (3.7) as follows

[TABLE]

Obviously, the eigenvalue of (3.2) is also the eigenvalue of (3.8). It follows from (3.7) that

[TABLE]

and the eigenvectors corresponding $\beta_{mn}^{i}(i=1,2)$ are given by

[TABLE]

From (3.11), it is easy to get the following properties of $f_{mn}^{i}$.

Proposition 1**.**

Let $f_{mn}^{i}$ satisfy (3.11), then

$\{f_{mn}^{i}\}$* is an orthogonal basis of $H$ and $H_{1}$, and*

[TABLE] 2. 2.

[TABLE]

where $G(f_{mn}^{i},f_{sk}^{j})=(f_{mn}^{i}\cdot\nabla)f_{sk}^{j}$, 3. 3.

[TABLE]

3.1 First eigenvalue and critical number

Obviously, it is easy to check that

[TABLE]

Hence, the first eigenvalue of $L_{\lambda}$ is determined by the following formulation:

[TABLE]

(3.12) implies

[TABLE]

Note that $\lambda_{m}$ is the function of $m$, which can obtain the minimum $\lambda_{0}$ when $m=m_{0}\in N$.

Let \lambda_{0}=\pi^{2}\sqrt{1+\frac{L^{2}}{m^{2}_{0}}}\big{(}1+\frac{m^{2}_{0}}{L^{2}}\big{)}. Then there exists a positive $\delta>0$ such that $\lambda_{0}-\delta<\lambda<\lambda_{0}+\delta$, and $\beta_{1}=\beta_{m_{0}1}^{1}$,

[TABLE]

Let

[TABLE]

which means that $\text{dim}E_{1}=1$.

4 Main conclusions

Based on Theorem 2.3.1[15], we can get the following conclusions.

Theorem 4.1**.**

For system (2.21), the following conclusion hold.

if $0\leq\lambda<\lambda_{0}$, it does not have bifurcation, and the only equilibrium point $u=0$ is globally asymptotically stable; 2. 2.

there exists a constant $\varepsilon>0$ such that $\lambda_{0}<\lambda<\lambda_{0}+\varepsilon$, system (2.21) bifurcates from $(0,\lambda_{0})$ to an attractor $\Omega_{\lambda}$, and attractor $\Omega_{\lambda}$ attracts $H_{1}/\Gamma$, where $\Gamma$ is the stable manifold $(0,0)$, which has codimension $1$ in $H_{1}$.

Proof.

It is obvious that the eigenvalue $\beta^{i}_{mn}<0$ when $0<\lambda<\lambda_{0}$. Hence, the conclusion 1 naturally holds.

Note that for any $u\in H$,

[TABLE]

Based on the results in [6], system (2.21) has a global attractor in $H$, that is, all invariant sets are uniformly bounded. Then we only need to prove that [math] is the unique invariant set as $\lambda=\lambda_{0}$. We infer from (3.14) that any invariant set $\Sigma$ of system (2.21 ) is only in $E_{1}$. Suppose $\Sigma$ has a non-zero element, for $u=(u_{r},u_{\theta},u_{z})\in\Sigma$, we know $u$ satisfies the following equations

[TABLE]

Clearly, $bu(bt)$ satisfies the above equations as well, $bu(bt)\in b\Sigma\subset E_{1}$, where $b$ is any real number. This contradicts to the boundaries of $\Sigma$. Then the invariant set of the system (2.21) can only be [math], and [math] is a globally asymptotically stable attractor. ∎

Theorem 4.2**.**

The set $\Omega_{\lambda}$ given in theorem 4.1, which contains two equilibrium point, that is $\Omega_{\lambda}=(\Omega_{1}^{\lambda},~{}\Omega_{2}^{\lambda})$. And there are following conclusion:

\Omega_{1}^{\lambda}=a\sqrt{\beta_{m_{0}1}^{1}}f_{m_{0}1}^{1}+o\big{(}\sqrt{\beta_{m_{0}1}^{1}}\big{)}*,  \Omega_{2}^{\lambda}=-a\sqrt{\beta_{m_{0}1}^{1}}f_{m_{0}1}^{1}+o\big{(}\sqrt{\beta_{m_{0}1}^{1}}\big{)}, *

where $a$ is a positive real number; 2. 2.

$H_{1}$* is decomposed into two open sets $U_{1}$ and $U_{2}$:*

[TABLE]

such that $\Omega_{i}^{\lambda}\in U_{i}$(i=1,2), and

[TABLE]

Proof.

We only need to prove the conclusion 1.

It is clear that the steady state equation of (2.21) is

[TABLE]

Let the first feature space of $L_{\lambda}$ be $E$, and the orthogonal complement of $E$ be $E_{2}$, then

[TABLE]

Then (4.1) can be rewritten as

[TABLE]

where $P_{1}:H\rightarrow E_{1}$ and $P_{2}:H\rightarrow E_{2}$ are projections.

Thus, (4.2)-(4.3) are equivalent to the equations

[TABLE]

where $f_{m_{0}1}^{1}$ is the first eigenvector of $L_{\lambda}$.(4.4) is the finite-dimensional bifurcation equations.

For any $u\in H$, $u$ can be expressed as

[TABLE]

Then (4.4) and (4.5) are equivalent to the following two equations:

[TABLE]

By the proposition 1 and the implicit function theorem [10], we can get

[TABLE]

Taking (4.7) into (4.6), we have

[TABLE]

where

[TABLE]

From (3.14), we can infer that (4.8) bifurcates two singular points from $(0,\lambda_{0})$ when $\lambda>\lambda_{0}$:

[TABLE]

Then

[TABLE]

where $a=\frac{1}{\sqrt{\gamma}}>0$. ∎

Physical explanation of theorem 4.2. The fact is that $R_{1},R_{2},\rho$ and $\nu$ are fixed, and we can only adjust the value of \bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0}. Then (3.14) and the expression of $\lambda$ imply that the basic flow becomes unstable, and the movement of the flow in $r$ and $z$ directions will occur if \bigg{(}\frac{\partial p}{\partial\theta}\bigg{)}_{0} exceeds a certain critical value. The structure of flow in $r$ and $z$ directions is shown in Figure 1, and the number of vortexes is determined by $L$ and $m_{0}$.