Dynamics of a Spherical Bubble in Non-Newtonian Liquids

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.