## 3 Results and Discussion
### Effects of Injection Rate and Concentration of the Citric Acid on Displacement Pattern  
We compared the fingering patterns that appeared under each condition when the injection rate and concentration of the citric acid were varied. Figure 3 shows the fingering patterns when citric acid is injected into skim milk under different injection rates and citric acid concentrations. Instability does not occur when pure water is used because no gel forms. However, the displacement becomes unstable when the citric acid concentration increases; thus, fingers emerge. The fingering pattern occurs because of casein precipitation at the interface, as the pH value decreases with increasing citric acid concentration. However, the displacement pattern becomes more stable with an increase in the injection rate. Figure 2: **(a)** A typical finger structure consists of bubble-like patterns with gel formation (red) generated at a 15 wt% concentration of the citric acid, an injection rate \(Q\) of 1.0 mL/h, and a gap width \(b\) of 0.4 mm. The skim milk was doped with a methyl red pH indicator. (**b**) The sequential thickening of the gel film at the interface of the skim milk and aqueous citric acid solution. The blue dot represents the experimental result, whereas the red line is the fitting curve based on Equation (1). The mixing between the skim milk and the aqueous citric acid solution occurs when the concentrations of the citric acid are 0.0, 0.50, 1.5 (all injection rates), and 5.0 wt% (injection rates of 15 and 50 mL/h). The mixing part is represented by the color gradient between white and pink at the interface of the fluid pair. At a low concentration of citric acid, the miscibility is not suppressed because of insufficient gel formation at the interface of the fluid pair; thus, mixing occurs. Moreover, the small amount of gel formation enables branching of the finger to occur at the sidewall of the fingers, mimicking the displacement pattern of viscous fingering. However, at a high citric acid concentration of 15 wt% (all injection rates) and 5 wt% (injection rates of 150 and 500 mL/h), the mixing between the skim milk and the aqueous citric acid solution is suppressed. A high concentration of citric acid results in high casein precipitation, which turns into a large amount of gel formation and becomes a continuous wall at the interface between skim milk and aqueous citric acid solution. This gel formation hinders the miscibility of the two fluids. Moreover, unlike the finger pattern that emerges when the aqueous citric acid solution is mixed with skim milk, remarkable fingering patterns, identical to tentacle-like structures, appear in the state where miscibility is suppressed (at a citric acid concentration of 15 wt% and injection rates of 15 and 50 mL/h). The number of branches on the finger is extremely small. The gel formation prevents the finger from branching, as previously mentioned;  
Figure 3: Displacement patterns when citric acid (white) is injected into skim milk (pink) upon injection rates of 15, 50, 150, and 500 mL/h and citric acid concentrations of 0.0 (pure water), 0.50, 1.5, 5.0, and 15 wt%. Images are taken when the injection volume of the aqueous citric acid solution is 2 mL and the gap width b is 0.4 mm. The red dash line represents the conditions at which the miscibility of citric acid and skim milk is suppressed. hence, the growth of the finger is localized at the tip. When the finger reaches a certain length, the growth stops, and a new finger is generated from the base of the finger. Figure 4 shows how the number of finger changes upon different injection rates \(Q\), at a citric acid concentration of 15 wt%. Increasing the injection rate up to an injection rate of 200 mL/h leads to a linear increase in finger numbers. However, although the number of fingers increases, the finger width is constant. As a result, more fingers emerge to compensate for a larger injected volume of aqueous citric acid solution.