Study of viscous fingering of a finite slice using time-dependent strategies

Abstract

This study provides a theoretical analysis of how time-dependent injection strategies affect the development of viscous fingering in a miscible finite slice. We are specifically examining a sinusoidal injection, which is characterized by its amplitude ($\Gamma$) and periodicity (T). This study presents two case studies: injection-extraction ($\Gamma >0$) and extraction-injection ($\Gamma <0$), which have been examined for different values of T. The extraction-injection process demonstrates that diffusion plays a prominent role, leading to a notable delay in the interaction between stable and unstable interfaces. In the injection-extraction case, the onset of fingers does not change with an increase in T, whereas the onset gets delayed in the case of extraction-injection in a nearly linear fashion as the time period increases. Additionally, by quantifying the impact of convective forces on the time-dependent injection strategy, we can gain a deeper understanding of the underlying physical mechanism responsible for the observed delay in onset at higher T values. The delayed onset of instability will have a positive impact on the separation of chemicals in chromatographic applications.