Dynamics and long-time behavior of a small bubble in viscous liquids with applications to food rheology

In this work, the physical behavior of a small, spherical bubble in a highly viscous, incompressible liquid phase is analyzed under specific pressure impacts. This represents an attractive topic in the food industry, since it is of interest to know under which conditions this two-phase dispersion exhibits a stable state. The specific material law of a second-order liquid, which includes Newtonian and non-Newtonian material constants, provides a nonlinear initial value problem for the radius of the bubble. This system is solved numerically by an efficient version of the classical Runge–Kutta method. By parameter variation, the impact of the dimensionless quantities associated with inertia, non-Newtonian material coefficients, pressure, surface tension and viscosity on the two-phase system is investigated. This particularly yields insights into the stability behavior of the bubble surface. The solution curves show various characteristics such as asymptotic oscillations or monotonically decreasing profiles. These results are transferred to a specific non-Newtonian and Newtonian substance. Finally, by studying stationary solutions, it becomes obvious that only the excitation pressure and the surface tension determine the new equilibrium state of the bubble, which in particular represents its long-time behavior. Furthermore, a sinusoidal driving pressure is used to investigate unstable solutions. The aim of the paper was to bring together these mathematical stability results to practice-oriented considerations.