This study investigates the linear stability of viscosity-stratified miscible two-layer flows in a channel, addressing the inherent complexity introduced by the time-dependent base state resulting from diffusion of solute concentration at the interface. Unlike previous studies that rely on the quasi-steady-state approximation (QSSA), which neglects the transient nature of the base state by assuming it remains frozen, we employ an initial value problem (IVP) approach to incorporate the unsteady base state into the stability analysis. Employing the IVP approach, we solve the linearized incompressible Navier-Stokes and convection-diffusion equations. We demonstrate that, unlike QSSA, the IVP method effectively provides critical insights into early-time perturbation growth, primarily driven by initial diffusion, and accurately predicts the onset time of instability. Through energy amplification analysis, we compute perturbation growth rates over time, revealing their nonmonotonic dependence on the log-mobility ratio ($R$) and mean interface location ($h^{\prime }$). We show that the growth rates increase monotonically with increasing Reynolds ($\text{Re}$) and Péclet ($\text{Pe}$) numbers, and decrease with increasing interface thickness ($\delta$). Systematic analyses further unveil a nonmonotonic dependence of onset time of instability ($t_{\text{on}}$) on $R$, identifying critical values of the log-mobility ratio for instability onset. These critical values, $R_l^c$ and $R_u^c$, are influenced by $\text{Re}, \text{Pe}, h^{\prime }$, and $\delta$. Through an energy budget analysis of concentration perturbations, we highlighted the interplay between convection and diffusion in determining the overall perturbation growth rates, thereby explaining the observed nonmonotonic variation with the log-mobility ratio $R$.

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• Channel flow
• Flow instability
• Kelvin-Helmholtz instability
• Multiphase flows