## 3 Results and Discussion
### Mathematical Model of Sequential Growth of Gel Film Thickness at the Fingertip  
Figure 8a shows the sequential growth of bubble-like fingers that emerges at a high citric acid concentration. The growth of this finger comprises a series of expansions, stagnations, and explosions that repeat over time. The pressure of the neighboring fingers does not influence the emerging bubble-like fingers. As shown in Figure 2a, even a single finger may possess this feature. Figure 7: Displacement pattern when the skim milk doped with methyl red is injected into the aqueous citric acid solution. The dark red part is the gel resulting from precipitated casein micelles, whereas the pink part is the water separated from the gel. The image was captured at the initial injection stage when the skim milk was injected into the aqueous citric acid solution, and the skim milk injection volume was 0.4 mL. The citric acid concentration \(c\) was 15 wt% at an injection rate \(Q\) of 15 mL/h and a gap width \(b\) of 0.4 mm. The reason why this structure appeared is because of the change in the gel film thickness over time. The film thickness decreases when the fingertip is elongated. However, the diffusion of citric acid through the existing gel formation increases the gel film thickness. Figure 8b illustrates the growth of the fingertip. Initially, the fingertip expands, resulting in a reduction in gel film thickness. However, at the same time, diffusion overcomes the elongation effect, resulting in a thicker gel film over time, while the fingertip continues to expand. At a certain point, the expansion stops because the gel film formation is sufficiently thick to act against the injection pressure. When finger growth stops, the pressure continues to increase. The increased pressure causes the gel film to burst. After the rupture, the finger grows again, and the pressure returns to its original state, marking the cycle repetition. An attempt was made to mathematically model the change in the film thickness at the tip of the finger when this phenomenon occurred. First, the increase in the film thickness per unit time resulting from diffusion was derived. The time change in the film thickness owing to diffusion is expressed by Equation (1), based on Fick's law. Differentiating Equation (1) for time, the increase in gel film thickness per unit time can be expressed as:  
\[\frac{d\sigma}{dt}=\sqrt{\frac{\pi D}{t}} \tag{2}\]  
Next, the decrease in gel film thickness per unit time caused by extension was derived. When the number of fingers is one, the shape of the fingertip of a newly emerged bubble-like tip is modeled as a circle, the radius is \(r\), the injection velocity is \(q\), and the cell gap width is \(b\). Then, the following equation holds from the preservation of the volume increase in the injected fluid:  
\[2\pi brdr=Qdt \tag{3}\]  
Integrating both sides of the above equation, one obtains  
\[\pi br^{2}=Qt+C \tag{4}\]  
Figure 8: (a) Sequential growth of bubble-like finger structure that appears at a citric acid concentration \(c\) of 15 wt%, injection speed \(Q\) of 15 mL/h, and gap width \(b\) of 0.4 mm. (b) Illustration of the growth of fingertip comprising expansion, stagnation, and explosion.