A general approach to the linear stability analysis of miscible viscous fingering in porous media

We analyze the linear growth of the viscous fingering (VF) instability for miscible, non-reactive, neutral buoyant fluids using the non-modal analysis (NMA). The onset of instability is obscured due to the continually changing base state and the normal mode analysis is not applicable to the non-autonomous linearized perturbed equations. Commonly used techniques such as frozen time method or amplification theory approach with random initial condition using transient amplifications yield substantially different results for the threshold of instability. We present the classical non-modal methods in the short-time limit using singular value decomposition of the propagator matrix. Using the non-modal approach we characterize the existence of a transition region between a domain exhibiting strong convection and a domain where initial perturbations are damped due to diffusion. Further, at the early times the non-modal linear growth is possible, these results suggest that non-modal linear stability analysis could play a fundamental role in the physical phenomenon involving VF.