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

Non-Newtonian coefficient condition for a stable long-time behavior of a single bubble: existence and characteristics of stable solutions

“…The bubble is assumed to be spherically symmetric and the origin of the coordinate system is placed in the center of the bubble. Combination of the corresponding continuity and momentum equation connected with the condition of incompressibility, i.e., ∂ ∂t (r 2 v(r, t)) = 0, and the kinematic boundary condition on the bubble surface, i.e., v(R, t) =Ṙ, yields a reduced equation of motion for the bubble radius R = R(t); see [5] for more details. The derived dimensionless ODE is then given byR…”

“…where the convergence speed for large times t approximatively behaves as ce − λ 2 t +g(t). Table 1 The experimental data for the non-Newtonian coefficients α and β for thickened whole milk are taken from [5], those for polyisobutylene and osteoarthritic fluid come from [20] and for silicone oil from [21] Material The corresponding dimensionless parameters A * and B * are given as well…”

“…In [5], a two-phase system consisting of a small, spherical gas bubble in a highly viscous, incompressible liquid was theoretically examined by the conservation equations for mass and momentum at changing pressure conditions. The important point to note here is that the material law of a second-order liquid was applied for the stress tensor to take into account slow and slowly varying bubble motions [6][7][8][9].…”

“…The one connected to the nonlinear character of the fluid causes at increasing values a damping of bubble surface vibrations, while that one connected to the memory effects leads to an increment in surface oscillations almost similar to the effects of inertia. Deriving an analytical solution by linearization of the named IVP from [5] yields an explicit expression for the bubble radius taking into account only one of the non-Newtonian material constants. For stationary solutions of the original ordinary differential equation (ODE), an upper bound could be derived only depending on surface tension.…”

“…A phenomenological description of the fluid-mechanical transport coefficients for the compressible case is thoroughly performed in [10]. In the incompressible case, the author obtains a special variant of the Rayleigh-Plesset equation (RPE) (comparable to that in [5]), but only the linearization including just one of the non-Newtonian coefficients is taken into consideration. In [12], the second-order constitutive relation is used for incompressible fluids in plane creeping flows.…”