  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.
