Non-modal linear stability analysis of miscible viscous fingering in porous media

The non-modal linear stability of miscible viscous fingering in a two dimensional homogeneous porous medium has been investigated. The linearized perturbed equations for Darcy's law coupled with a convection-diffusion equation is discretized using finite difference method. The resultant initial value problem is solved by fourth order Runge-Kutta method, followed by a singular value decomposition of the propagator matrix. Particular attention is given to the transient behavior rather than the long-time behavior of eigenmodes predicted by the traditional modal analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the $\epsilon-$pseudospectra structures and the largest energy growth function. With the help of non-modal stability analysis we demonstrate that at early times the displacement flow is dominated by diffusion and the perturbations decay. At later times, when convection dominates diffusion, perturbations grow. Furthermore, we show that the dominant perturbation that experiences the maximum amplification within the linear regime lead to the transient growth. These two important features were previously unattainable in the existing linear stability methods for miscible viscous fingering. To explore the relevance of the optimal perturbation obtained from non-modal analysis, we performed direct numerical simulations using a highly accurate pseudo-spectral method. Further, a comparison of the present stability analysis with existing modal and initial value approach is also presented. It is shown that the non-modal stability results are in better agreement, than the other existing stability analyses, with those obtained from direct numerical simulations.