Two Types of Nonlinear Pressure Waves in Bubbly Liquids Incorporating Viscosity and Thermal Conductivity

The present study theoretically elucidates an effect of the viscosity and the thermal conductivity on the propagation process of finite amplitude disturbance in bubbly liquids by deriving two types of weakly nonlinear wave equations. Appropriate choices of a set of scaling relations of physical parameters characterizing waves, that is, the wavelength, incident wave frequency, propagation speed, yield the derivation systematically. From the combination of appropriate scaling relations and the method of multiple scales, we can derive the Korteweg–de Vries–Burgers equation for the low frequency long wave and the nonlinear Schrödinger equation for slowly varying envelope wave of the quasi-monochromatic short carrier wave. As a result, the incorporation of conservation equation of energy affects nonlinear, dispersion, and dissipation terms for both long and short waves. Especially, the viscosity and the thermal conductivity lead to change considerably the form of coefficient of dissipation term.