Thermal effect inside bubbles for weakly nonlinear pressure waves in bubbly liquids: Theory on short waves

Abstract

In this study, weakly nonlinear pressure waves in quiescent compressible liquids comprising several uniformly-distributed spherical microbubbles, at moderately high-frequency and short-wavelength, are theoretically investigated. The energy equation at the bubble–liquid interface and the effective polytropic exponent are utilized to clarify thermal effects inside bubbles on wave dissipation. In addition, thermal conduction is investigated in detail using four temperature-gradient models. The following results are drawn: (i) Nonlinear Schrödinger equation is derived as an effective equation, wherein three types of dissipation factors, i.e., liquid viscosity, liquid compressibility, and thermal conduction, are unified into a linear combination as the dissipation coefficient. This is different from our previous result treating the low-frequency and long-wavelength case [Kamei et al., Phys. Fluids 33, 053302 (2021)], i.e., two types of dissipation terms appeared and did not unify into a linear combination. (ii) Dissipation due to thermal conduction is more than four times larger than that due to other dissipation factors. (iii) Dissipation due to thermal conduction at the bubble–liquid interface is considerably larger than that due to thermal conduction through the bubbly liquid. (iv) It is found that the dissipation effect in the short-wave case is smaller than that in the long-wave case.