Coupled effects of viscosity contrast and phase separation in hydrodynamically stable radial Hele-Shaw flow

We investigate a nonlinear interaction between viscous fingering (VF) and phase separation (PS) in a binary fluid system within a radial Hele-Shaw cell. Through nonlinear simulations, we analyse displacement under favourable viscosity contrast, in which a more viscous fluid displaces a less viscous one, and PS may induce VF. The system undergoes PS when the concentration lies within the spinodal region, where the second derivative of free energy is negative. In the absence of viscosity contrast, the flow exhibits distinct morphologies, rings under the strongest PS conditions and droplets otherwise. A distinct composition of separated droplets results from uphill and downhill diffusion imbalances. The prominence of PS is characterised by the interfacial tension within the spinodal region, while the extent of pattern rupture is quantified by the interfacial length in the fully separated region. We identify interfacial tension as a reliable and experimentally accessible indicator of PS. It offers a practical alternative to the second derivative of free energy, which is challenging to quantify directly. We find that higher miscibility stabilises the overall pattern, as evidenced by reductions in both interfacial tension and interfacial length. In contrast, viscosity contrast plays a complex role: while a favourable viscosity contrast generally stabilises the flow by reducing interfacial length, there are specific flow conditions under which the interfacial length increases despite a weaker PS condition. Our results reveal instability patterns consistent with experimental observations, reinforcing the reliability of our findings.