  The effect of a linear adsorption isotherm on the onset of fingering
instability in a miscible displacement in the application of liquid
chromatography, pollutant contamination in aquifers etc. is investigated. Such
fingering instability on the solute dynamics arise due to the miscible viscus
fingering (VF) between the displacing fluid and sample solvent. We use a
Fourier pseudo-spectral method to solve the initial value problem appeared in
the linear stability analysis. The present linear stability analysis is of
generic type and it captures the early time diffusion dominated region which
was never expressible through the quasi-steady state analysis (QSSA). In
addition, it measures the onset of instability more accurately than the QSSA
methods. It is shown that the onset time depends non-monotonically on the
retention parameter of the solute adsorption. This qualitative influence of the
retention parameter on the onset of instability resemblances with the results
obtained from direct numerical simulations of the nonlinear equations.
Moreover, the present linear stability method helps for an appropriate
characterisation of the linear and the nonlinear regimes of miscible VF
instability and also can be useful for the fluid flow problems with the
unsteady base-state.
