Derivation of KdV-Burgers equation for weakly nonlinear pressure waves in bubbly liquids with a polydispersity

Abstract

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical gas bubbles is theoretically studied. Our especial focus is an initial polydispersity of bubble radius (i.e., an initial nonuniform distribution of bubble size); an initial number density is uniform. By using the method of multiple scales, the derivation of the KdV-Burgers equation including the polydisperse effect from the basic equations is succeeded. Then, an advection term of waves appears as a new term and the coefficient of advection term is varied from the constant coefficient in the previous study to the variable coefficient in the present study. The polydispersity does not affect the nonlinear, dissipation, and dispersion effects. For the case of monodispersity (i.e., without the advection term), the present KdVB equation coincides with the previously derived KdVB equation for the monodisperse case.