A non-modal linear stability analysis (NMA) of the miscible viscous fingering in a porous medium is studied for a toy model of non-monotonic viscosity variation. The onset of instability and its physical mechanism are captured in terms of the singular values of the propagator matrix corresponding to the non-autonomous linear equations. We discuss two types of non-monotonic viscosity profiles, namely, with unfavorable (when a less viscous fluid displaces a high viscous fluid) and with favorable (when a more viscous fluid displaces a less viscous fluid) end-point viscosities. A linear stability analysis yields instabilities for such viscosity variations. In addition, we also show that to understand the spatiotemporal evolution of the perturbations it is necessary to analyse the viscosity gradient with respect to the concentration and the location of the maximum concentration $c_m$. For the favorable end-point viscosities, a weak transient instability is observed when the viscosity maximum moves close to the pure invading or defending fluid. This instability is attributed to an interplay between the sharp viscosity gradient and the favorable end-point viscosity contrast. We demonstrate the dissimilarity between the quasi-steady-state approach and NMA in finding the correct perturbation structure and the onset, for both the favorable and unfavorable viscosity profiles. The correctness of the linear perturbation structure obtained from the non-modal stability analysis is validated through nonlinear simulations. We have found that the nonlinear simulations and NMA results are in good agreement. In summary, a non-monotonic variation of the viscosity of a miscible fluid pair is seen to have a larger influence on the onset of fingering instabilities, than the corresponding Arrhenius type relationship.
A spatial analog of the Ruelle-Takens-Newhouse scenario developing in reactive miscible fluids
本文对恒重力下垂直Hele-Shaw池中混溶流体通过二级反应$A + B \to S$进行斑图形成的理论研究。我们最近报道，浓度依赖性扩散的物种加上正面中和反应可以产生一个多层系统，其中低密度耗尽区可以嵌入之间的致密层。这导致激发的化学对流模式空间上彼此分离的静止流体。在本文中，我们表明，层可以通过扩散机制相互作用。由于扩散耦合不稳定性最初具有不同的波长，这导致一种模式被另一种模式的长波调制。我们已经开发了一个数学模型，其中包括一个系统的反应扩散对流方程。通过计算每个不稳定层的李雅普诺夫指数的增长率，研究了瞬态基态的线性稳定性。数值模拟支持的相图重建和傅立叶谱计算表明，非线性动力学始终通过（i）一个完美的空间周期系统的化学对流细胞;（ii）准周期系统的相同的细胞;（iii）无序的指状结构。我们发现，在这个系统中，坐标共同指向的反应前矛盾地发挥着时间的作用，时间本身作为一个分岔参数，并观察到一个完整的空间模拟的Ruelle-Takens-Newhouse方案的混沌发病。
