Wave dynamics in the drift-flux two-phase flow model

The present study provides a comprehensive symmetry analysis for a simplified two-phase flow model with the logarithmic equation of state. Under a one-parameter Lie group of transformations, we generate the local symmetry of the model, preserving the invariance of the system. Subsequently, we classify one-dimensional optimal subalgebras, which is a systematic framework for computing invariant solutions efficiently. With the characteristic method, we developed explicit solutions for the model utilizing the optimal subalgebras. Further, we prove that nonlocal symmetries exist for the considered model, and then some new exact solutions were developed where local symmetries cannot provide. Furthermore, the existence of the nonlinear self-adjointness property of the model is demonstrated with the construction of conservation laws. This study concludes by examining the complex hyperbolic nature, such as $C^1$-wave, characteristic shock, and their interaction with one of the solutions derived from nonlocal symmetry, highlighting the critical wave dynamics of the model.