## 6 Result and Discussion
### Pressure Field  
This section reports the pressure field calculated with the method proposed in this study for both the light and the heavy inhomogeneity. Analysis of the pressure field, in fact, allows to understand and explain why the inhomogeneity assumes specific shapes during its evolution. In the case of light inhomogeneity  
Figure 8: Shapes comparison at \(\tau=5.07\) (Semi claw shape) and \(\tau=8.90\) (Claw shape) between constant viscosity (**left**) and time-varying viscosity (**right**).  
Figure 9: Shape comparison at \(\tau=3.55\) (Jellyfish shape) between constant viscosity (**left**) and time-varying viscosity (**right**).  
(Figure 10), initially the incident shock wave impacts and reflects on the inhomogeneity (Figure 10a) generating a lower pressure reflected-wave behind the inhomogeneity. At a later stage, a high-pressure region behind the inhomogeneity (Figure 10b) is observed and, consequently, the lighter gas moves away from the high-pressure area giving a crescent moon shape to the inhomogeneity. In the case of heavy inhomogeneity (Figure 11), a reflected-wave (Figure 11a) and a high pressure region (Figure 11b) also forms, but, this time, the reflected wave has a higher pressure than the surroundings and the high-pressure area forms in front of the inhomogeneity. Moreover, the high pressure area is partly outside and partly inside the gas inhomogeneity. As a result of this, the tip of the inhomogeneity stretches, forming the central wedge typical of the jellyfish head shape.  
Figure 10: Pressure field in the driven gas for the light inhomogeneity (\(A=-0.79\)) case at different dimensionless times.  
For clarity, Figures 10 and 11 show the pressure field only in the driven gas, while Figure 12 shows the pressure field both outside and inside the inhomogeneity. Our calculations also capture the twin regular reflection-refraction (TRR) configuration (Figure 12), firstly observed by [PERSON] (1972). The TRR is a four-shock configuration where the refracted shock moves faster than the incident shock, with the reflected shock moving in the opposite direction. The fourth shock is the side shock, which connects the refracted shock with the incident shock (Figure 12b). The pressure field in some of the figures is slightly \"noisy\" at certain locations. This is probably the result of the piecewise constant smoothing length used in this study. However, this occurs far from the gas inhomogeneity and does not affect the inhomogeneity shape evolution analysis carried out in this study.  
light diverged or converged. With respect to the focal point, the shock wave is either diverged or converged with respect to the high pressure region area discussed in the previous section (compare Figures 10b and 11b with Figure 13c,d). Considering, therefore, that the focal point and the high pressure region play a similar role, it is possible to identify the high pressure region as a focal region of the system.  
Another interesting observation concerns the shape of the reflected wave (Figures 10 and 11). Observing the pressure field, it is possible to see how the evolution of the reflected wave mirrors reflects, to a certain degree, that of the inhomogeneity. In the light inhomogeneity, the mirror image (Figure 10c) has a shape similar to the semi-claw standard shape, it is just more elongated in the \(y\)-direction. In the heavy inhomogeneity, the mirror image (Figure 11d) has a shape similar to the jellyfish head standard shape.