Nonlinear and dissipation effects of pressure waves in water flows containing translational bubbles with a drag force

Abstract

Weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows uniformly containing many spherical bubbles is theoretically studied. Drag force acting bubbles and translation of bubbles are newly considered by introducing in momentum conservation equations in a two fluid model and the bubble dynamics equation for volumetric oscillations, respectively. Although these assumptions are the same as our previous paper, in this study, the energy conservation equation for each bubble describing a thermal conduction inside bubble is introduced. By using the method of multiple scales, the Korteweg–de Vries–Burgers equation for low-frequency long wave was derived from the set of basic equations in the two-fluid model. As a result, the dissipation effect was described by two types of terms, i.e., one was the second-order partial derivative owing to the liquid compressibility and the other was the term without differentiation owing to the drag force and the thermal conduction. Finally, we clarified that the dissipation owing to the drag force was smaller than that owing to the thermal conduction.