Theoretical Study on an Effect of Liquid Viscosity and Thermal Conductivity on Weakly Nonlinear Propagation of Long Pressure Waves in Bubbly Liquids

Abstract

This study theoretically clarifies the effect of the liquid viscosity and the thermal conductivity on weakly nonlinear propagation of pressure waves in liquids containing many spherical microbubbles. By the use of the method of multiple scales, a KdV–Burgers equation describing the long-range propagation of a low frequency long wave is derived from the basic equations incorporating the liquid viscosity and the thermal conductivity. As a result, the liquid viscosity and the thermal conductivity affect the dissipation effect. A thermodynamical process of the gas inside bubbles also affects the nonlinear, dissipation, and dispersion effects. From numerical solution of the KdV–Burgers equation, the wave properties appear in the order of the nonlinear and dispersion effects. Comparing the result with the previous study (Kanagawa et al., Trans of the JSME, Ser. B, 76, 1802, 2010) ignoring the liquid viscosity and the thermal conductivity, strong dissipation and dispersion effects appear.