Abstract:
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. Using the optimal perturbation
structure, we are able to show that an initially unconditional stable state becomes unstable corresponding to the most unstable initial disturbance. 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 cm. 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. Further, the usefulness
of the non-modal analysis demonstrating the physical mechanism of the quadruple structure of the
perturbations from the optimal concentration disturbances is discussed. 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.
