标题： 关于具有线性粘性分布的增强采油三层Hele-Shaw模型 显示英文标题

标题： On a three-layer Hele-Shaw model of enhanced oil recovery with a linear viscous profile

摘要： 我们提出一个非标准特征值问题，该问题出现在增强油回收的三层Hele-Shaw模型的线性稳定性中。 引入一种非线性变换，使得当中间层的粘性剖面为线性时，可以将非标准特征值问题重新表述为Kummer方程的边界值问题。 利用现有的关于Kummer方程的研究成果，我们构建了特征值问题的精确解，并通过非平凡解的存在准则隐式地给出了色散关系。 我们还讨论了级数解的收敛性。 结果表明，该解在两个渐近极限中退化为物理上有意义的解：(i) 当线性粘性剖面趋近于常数粘性剖面时；或(ii) 当中间层的长度趋近于零时。 显示英文摘要

摘要： We present a non-standard eigenvalue problem that arises in the linear stability of a three-layer Hele-Shaw model of enhanced oil recovery. A nonlinear transformation is introduced which allows reformulation of the non-standard eigenvalue problem as a boundary value problem for Kummer's equation when the viscous profile of the middle layer is linear. Using the existing body of works on Kummer's equation, we construct an exact solution of the eigenvalue problem and provide the dispersion relation implicitly through the existence criterion for the non-trivial solution. We also discuss the convergence of the series solution. It is shown that this solution reduces to the physically relevant solutions in two asymptotic limits: (i) when the linear viscous profile approaches a constant viscous profile; or (ii) when the length of the middle layer approaches zero.