Experimental and analytical approaches toward controlling viscous fingering at the interface of two immiscible fluids using the optimal constant piecewise injection rate

Abstract

Viscous fingering (VF) at the interface of two immiscible fluids can have disruptive effects in many technological applications. Therefore finding methods for reducing it with an effective and simple approach plays a significant role. A proposed solution that restrains the expansion of such complex morphologies, is the time-dependent injection rate. This proposed solution is often done either analytically or by numerical simulations which need to be experimentally proven. In this study, we perform multiple experiments of low viscosity fluid injection into high viscosity fluid in order to study the formation of VF and find injection rates or protocols that reduce VF at the onset of the nonlinear regime and in the fully nonlinear regime by applying several constant piecewise injection rates. Moreover, analytically and numerically studies are done, in which we derive the optimal parameters in piecewise injection profiles to limit the VF and the tip-splitting effect. This optimal constant piecewise injection rate includes specified number of stages as well as the injection time and its rate in each stage. The obtained results revealed that depending on being at the beginning of the nonlinear regime or in the fully nonlinear regime, the optimal constant piecewise injection rate can considerably reduce the length of fingers and the interfacial fingering perturbations. Empirical results show similar trend in terms of instability reduction by applying the same optimal number of piecewise injection rate stages. In addition, evaluating the difference between this piecewise results and the results of linear injection rates, which we have done before, reveals that employing linear injection rates is more effective in controlling the instability in lower injection rates, while for higher injection rates, it is required to utilize a piecewise injection rate.