Derivation of Weakly Nonlinear Wave Equations for Pressure Waves in Bubbly Flows with Different Types of Nonuniform Distribution of Initial Flow Velocities of Gas and Liquid Phases

Abstract

In this study, the weakly nonlinear propagation of pressure waves in bubbly flows is theoretically investigated, with a particular focus on different types of initial flow patterns. At the initial state, the gas and liquid phases exhibit a high velocity with uniform distribution and a low velocity with nonuniform distribution, respectively. Using basic conservation equations based on a two-fluid model and the method of multiple scales with perturbation expansions, we can derive three cases of nonlinear wave equations: (i) Korteweg–de Vries Burgers equation for long waves; (ii) Nonlinear Schrödinger (NLS)-I equation for short waves in slow nonuniform flows; (iii) NLS-II equation for short waves in fast nonuniform flows. Consequently, the initial velocities contribute to an advection effect of the waves, the initial nonuniform flow distribution induces a variable coefficient of advection, and the prominent role of relative velocity is elucidated.