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Woods-Burnett方程是Boltzmann方程的二阶近似,是Burnett方程的一种修正,能够描述轻微偏离热力学平衡时的稀薄气体流动。但是Woods-Burnett方程在小扰动下不稳定,这是限制Woods-Burnett方程广泛应用的一个重要原因。本文在一维稳定性分析的基础上,通过线性小扰动理论,首次得到了二维Woods-Burnett方程的稳定性特征方程,并把稳定性方程的解表示在复平面上,得到了二维稳定性特征曲线。通过扰动增长系数和扰动波数的关系,得到二维Woods-Burnett方程的临界努森数为0.130,而一维Woods-Burnett方程的临界努森数为0.184。说明在二维情况下Woods-Burnett方程更加不稳定。
The Woods-Burnett equation, which is a second-order approximation of the Boltzmann equation, is a modification of the Burnett equation that describes the lean gas flow that deviates slightly from the thermodynamic equilibrium. However, the Woods-Burnett equation is unstable under small perturbations, which is an important reason for limiting the widely used Woods-Burnett equation. Based on the one-dimensional stability analysis, the stability characteristic equation of the two-dimensional Woods-Burnett equation is obtained for the first time by the linear perturbation theory. The solution of the stability equation is presented on the complex plane and the two-dimensional stability is obtained Sex characteristic curve. By the relationship between the growth coefficient of disturbance and the number of disturbance wave, the critical Knudsen number of two-dimensional Woods-Burnett equation is 0.130, while that of one-dimensional Woods-Burnett equation is 0.184. Which shows that the Woods-Burnett equation is more unstable in two-dimensional case.