Effect of Pe on miscible viscous fingering with an effective interfacial tension

When the displacing fluid is less viscous, the interface of the two fluids gives a finger-like pattern. This phenomenon is called viscous fingering (VF). Fluid-fluid miscibility plays an important role in the VF dynamics. For example, diffusion affects the VF dynamics in fully miscible systems and thus, the fully miscible VF is characterized by the Péclet number, Pe, defined as the ratio of convective transport rate and diffusive transport rate. The viscous fingers generally become thinner with increasing Pe. However, when the two non-premixed miscible fluids are brought in contact with each other, there are the initial concentration gradient between the two fluids. Therefore, on time scales shorter than that of interface relaxation, there exists a capillary force because there are necessarily differences in intermolecular interactions. Such a capillary force is called effective interfacial tension (EIT). In this study, we have elucidated the effect of Pe on the miscible VF by numerical simulation. The results show that the fingers are thinner with increase in Pe without EIT but wider with increase in Pe with EIT. This is because EIT, which acts to widen the finger, is proportional to square of the concentration gradient and the gradient becomes larger as Pe increases.