Abstract—

A series of the problems of dynamics of a spherical gas cavity with the uniformly distributed pressure inside and, in particular, without pressure are considered within the framework of hydrodynamic theory of incompressible power-law non-Newtonian liquids. A special attention is given to investigation of the behavior of solutions as functions of the exponent (index) in the power-law non-Newtonian model and determination of the extreme properties of the solutions. The problems of calculation of the necessary external pressure that leads to conservation of the kinetic energy of liquid or the dissipation rate in the process of compression are solved. Other solutions are constructed in a particular case of the Newtonian model. They represent the exact implementation of the linear-resonance behavior of the cavity radius within the framework of the nonlinear formulation of problem and, on the contrary, the law of cavity dynamics under a given harmonic external pressure at the linear-resonance frequency is corrected using numerical methods. The law of dependence of the concentration of the kinetic energy of liquid on the index in the non-Newtonian model and the generalized Reynolds number is established analytically and numerically under the piecewise constant external pressure in the case of the vacuum cavity. It is shown that for a certain indices there is no energy concentration at all. The critical values of the generalized Reynolds number at which the energy concentration also disappears are calculated for the remaining indices. The total energy dissipation is minimized in the case of the cavity occupied by a gas.

中文翻译：

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非牛顿液体中球形气泡的动力学

摘要-

在不可压缩幂律非牛顿液体流体力学理论框架内，研究了内部压力均匀分布，特别是无压力的球形气腔的一系列动力学问题。特别注意研究作为幂律非牛顿模型中指数（指数）函数的解的行为以及解的极端性质的确定。解决了导致液体动能守恒或压缩过程耗散率守恒的必要外压计算问题。其他解决方案是在牛顿模型的特定情况下构建的。它们代表了问题的非线性公式框架内腔半径的线性谐振行为的精确实现，相反，线性谐振频率下给定谐波外部压力下的腔动力学定律使用以下方法进行校正数值方法。在真空腔情况下，在分段恒定外压下，分析和数值建立了非牛顿模型中液体动能浓度对指数和广义雷诺数的依赖规律。结果表明，对于某些指标，根本没有能量集中。计算其余指数的广义雷诺数的临界值，在该临界值处能量集中也消失。