気泡流中における長波と短波の弱非線形相互作用の理論的解明

抄録

This study theoretically investigates one-dimensional weakly nonlinear interaction between short and long waves in a compressible liquid uniformly containing many spherical gas bubbles. Important assumptions are summarized as follows: (i) Two different wave modes coexist, one is the short wave and the other is the long wave; (ii) Bubble does not coalesce, break up, extinct, and appear; (iii) The effect of viscosity in the gas phase, heat conduction in the gas and liquid phases, and phase change across the bubble wall, are neglected for simplicity. The basic equations for bubbly flows are composed of a set of conservation equations of mass and momentum for the gas and liquid phases in a two-fluid model, the Keller equation, and so on. Appropriate choices of scaling relations of some physical parameters, i.e., wavelength, wave frequency, propagation speed, yields the result that amplitudes of the short and long waves are of the same order of magnitude. By the use of a singular perturbation analysis based on the method of multiple scales, we can derive the coupled equations from the set of governing equations for bubbly flows, one is the nonlinear Schrödinger (NLS) type equation for the short wave with a nonlinear interaction term and the other is the Korteweg–de Vries (KdV) type equation for the long wave with a nonlinear term of the short wave.