## 4 Conclusions  
In this study, we conducted Hele-Shaw cell displacement experiments using skim milk and aqueous citric acid solution to investigate the effects of gel film formation on the fingering instability of miscible fluids and develop a mathematical model for the sequential growth of gel film formation at the fingertip. We found that mixing skim milk and an aqueous citric acid solution induces gel film formation, resulting in interface instability. A distinctive feature of the fingering pattern, such as tentacles, appears when miscibility is suppressed, and the growth of the finger is localized at the fingertip. The gel film formation thickens with time owing to the diffusion of citric acid through the existing gel film formation.  
The finger width remains constant with increasing flow rate, whereas the number of fingers increases linearly. However, at a certain point, where the injection rate is too high to be compensated by the increase in the number of fingers, the fingers merge and the number of fingers decreases. The finger patterns that appear at each injection rate are similar, regardless of the gap width. However, the finger width increases with the gap width. In addition, the growing front of the displacement pattern has a flower-like appearance of wide fingers when the displaced and displacing fluids are exchanged.  
The growth of the bubble-like fingertip comprises a series of expansions, stagnations, and explosions that repeats over time. A mathematical model of sequential film thickness growth for a bubble-like fingertip structure was developed. The model is based upon the interplay between the diffusion of citric acid through the existing gel film formation and  
Figure 9: Sequential change in the interfacial gel film thickness based on Equation (12) considering periodic rupture at the fingertip. The parameters are set as follows: \(D=9.3\times 10^{-11}\) m\({}^{2}\)/s, \(b=0.4\) mm, \(Q=1.0\) mL/h, \(r_{0}=0.4\) mm, and \(C=3.404\times 10^{-5}\) m\(\sqrt{s}\). The film thickness changes from decreasing to monotonically increasing as a one-cycle bubble formation. Periodic rupture occurs approximately every 9 s.  
elongation of the fingertip, which provides an understanding of the fundamental mechanism for the growth of the bubble-like fingertip, owing to gel film formation. This model is potentially relevant to various processes, including the growth of bacterial biofilms [19] and fingering instability in confined chemical gardens [24; 25]. The results of this investigation can also be a reference for engineering applications, such as remediation of potential CO\({}_{2}\) leakage from storage reservoirs [27; 28] and enhanced heavy oil recovery methods based on alkali flooding [29; 30]. We believe that future research must be directed toward further analysis based on pore-scale studies using a porous medium structure that resembles the structure of geological formation. Pore-scale studies offer a detailed observation for the underlying mechanisms, which are macroscopically unobserved [43].  
Conceptualization, T.S. and R.Y.; methodology, M.N., R.Y. and Y.S.; validation, A.P., M.A.M. and Z.L.; formal analysis, M.N., R.Y. and W.W; investigation, R.Y., Y.S. and W.W.; writing--original draft preparation, M.N. and R.Y.; writing--review and editing, M.N., A.P. and M.A.M.; visualization, M.N. and Z.L.; supervision, S.M., A.P. and T.S.; project administration, S.M.; funding acquisition, A.P. and T.S. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.