Nonlinear interactions of phase separation and viscous fingering in radial Hele–Shaw flow

We investigate the coupled dynamics of viscous fingering (VF) and phase separation (PS) in a radial Hele–Shaw flow. Using a combination of linear stability analysis (LSA) and nonlinear simulations (NLS), we analyze how the initial concentration of injected fluid (ci) and viscosity contrast (R) determine the dominant instability mechanism. VF arises when a less viscous fluid displaces a more viscous one, i.e., R>0. In comparison, PS is quantified by parameter Θ, which is defined as the negative of the second derivative of free energy and depends on both miscibility and ci. A positive Θ indicates thermodynamic instability within the spinodal region, driving PS; otherwise, it is a metastable region. For R>0, PS dominates at low R, producing droplet or ring-like morphologies with limited influence from viscosity contrast. However, at higher R, ci determines the dominant mechanism: VF dominates for ci>0.5, while PS remains dominant for ci≤0.5. We construct a pattern diagram between effective viscosity contrast and Θ that classifies the different regimes and identifies regions where both mechanisms interact. These nonlinear patterns are consistent with the LSA predictions. The LSA reveal that viscosity contrast controls the onset and nature of instability: for R<0, the onset is delayed, whereas R>0 accelerates onset through the combined action of PS and VF. The strong agreement among NLS, LSA, and experimental observations confirms the robustness of our framework. This also validates the novel LSA approach developed here for partially miscible displacements, with direct relevance to transport in porous media.