Banach Zbigniew, Larecki Wieslaw
Institute of Fundamental Technological Research, Department of Theory of Continuous Media, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 1):061137. doi: 10.1103/PhysRevE.78.061137. Epub 2008 Dec 31.
A detailed treatment of the Chapman-Enskog method for a phonon gas is given within the framework of an infinite system of moment equations obtained from Callaway's model of the Boltzmann-Peierls equation. Introducing no limitations on the magnitudes of the individual components of the drift velocity or the heat flux, this method is used to derive various systems of hydrodynamic equations for the energy density and the drift velocity. For one-dimensional flow problems, assuming that normal processes dominate over resistive ones, it is found that the first three levels of the expansion (i.e., the zeroth-, first-, and second-order approximations) yield the equations of hydrodynamics which are linearly stable at all wavelengths. This result can be achieved either by examining the dispersion relations for linear plane waves or by constructing the explicit quadratic Lyapunov entropy functionals for the linear perturbation equations. The next order in the Chapman-Enskog expansion leads to equations which are unstable to some perturbations. Precisely speaking, the linearized equations of motion that describe the propagation of small disturbances in the flow have unstable plane-wave solutions in the short-wavelength limit of the dispersion relations. This poses no problem if the equations are used in their proper range of validity.
在由卡拉韦的玻尔兹曼 - 派尔斯方程模型得到的无穷矩方程组框架内，给出了声子气体的查普曼 - 恩斯科格方法的详细处理。该方法对漂移速度或热流各分量的大小不设限制，用于推导能量密度和漂移速度的各种流体动力学方程组。对于一维流动问题，假设正常过程主导电阻过程，发现展开式的前三个层次（即零阶、一阶和二阶近似）给出了在所有波长下线性稳定的流体动力学方程。这个结果可以通过研究线性平面波的色散关系或者通过为线性微扰方程构建显式二次李雅普诺夫熵泛函来实现。查普曼 -恩斯科格展开的下一阶导致方程对某些微扰不稳定。确切地说，描述流动中小扰动传播的线性化运动方程在色散关系的短波长极限下有不稳定的平面波解。如果在其适当的有效范围内使用这些方程，这不会造成问题。
