We perform a linear stability analysis of three-layer radial porous media and Hele-Shaw flows with variable viscosity in the middle layer. A nonlinear change of variables results in an eigenvalue problem that has time-dependent coefficients and eigenvalue-dependent boundary conditions. We study this eigenvalue problem and find upper bounds on the spectrum. We also give a characterization of the eigenvalues and prescribe a measure for which the eigenfunctions form an orthonormal basis of the corresponding $L^2$ space. This allows for any initial perturbation of the interfaces and viscosity profile to be easily expanded in terms of the eigenfunctions by using the inner product of the $L^2$ space, thus providing an efficient method for simulating the growth of the perturbations via the linear theory. The limit as the viscosity gradient goes to zero is compared with previous results on multi-layer radial flows. We then numerically compute the eigenvalues and obtain, among other results, optimal profiles within certain classes of functions.

中文翻译：

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径向多孔介质和两个移动界面之间可变粘度的 Hele-Shaw 流的稳定性结果

我们对三层径向多孔介质和中间层具有可变粘度的 Hele-Shaw 流进行线性稳定性分析。变量的非线性变化会导致特征值问题，该问题具有与时间相关的系数和与特征值相关的边界条件。我们研究这个特征值问题并找到频谱的上限。我们还给出了特征值的表征，并规定了特征函数形成相应 $L^2$ 空间的标准正交基的度量。这允许通过使用 $L^2$ 空间的内积，根据特征函数轻松扩展界面和粘度分布的任何初始扰动，从而提供一种通过线性模拟扰动增长的有效方法理论。将粘度梯度变为零时的极限与多层径向流动的先前结果进行了比较。然后，我们对特征值进行数值计算，并在其他结果中获得某些函数类别中的最佳配置文件。