Bubble dynamics: Long‐time behavior and interaction in viscous liquids

Abstract

A mathematical two-phase model is used to numerically investigate physical and rheological effects on small, individual bubbles in high-viscosity liquids under pressure impact. It is found out that bubbles remain stable over time at high viscosity and surface tension. The steady case is considered and connected to the stability behavior of the bubble. An upper bound for the bubble radius is derived and the new equilibrium state of the bubble can be predicted by means of stability theorems of differential equations. Finally, the interaction of a limited number of well separated bubbles in an Hele-Shaw flow is mathematically analyzed to visualize and physically interpret their trajectories. Model and parameter studiesThe two-phase system consists of a small, gas-filled bubble and a highly viscous, non-Newtonian and incompressible liquid phase. Thus, gravity effects can be neglected. Let R(t), R 0 := R(0) be the bubble radius at time t and at rest. The flow is assumed to be spherically symmetric. Furthermore, the pressure p is isotropic under isothermal conditions. By the balances for mass and momentum the governing equation in spherical coordinates is derived [1]. Hence, from the continuity equation ∇ · (r 2 v(r, t)) = 0 and the kinematic boundary condition v(R(t), t) =Ṙ(t) the velocity field is obtained as v(r, t) = R 2 (t)Ṙ(t)/r 2 for r ≥ R(t). Due to the high viscosity of the dispersion medium, a constitutive law for slowly varying deformation processes is applied for the stress tensor σ, i. e.where A 1 and A 2 denote the Rivlin-Ericksen tensors [2], η the dynamic viscosity of the liquid, α and β non-Newtonian material coefficients. Representation (1) is an approximation of second order for a simple fluid [2], in which the η-term is the contribution of a Newtonian liquid, while the last two summands reflect the non-Newtonian character of the liquid including viscoelastic properties. Moreover, assuming a polytropic gas p g (t)R 3 (t) = p g0 R 3 0 , where p g with p g0 := p g (0) stands for the gas pressure, and the dynamic boundary condition on the bubble surface p(R, t) − σ rr = p g − 2σ/R, one can derive the dimensionless initial value problem (IVP) describing the bubble dynamics bywhere p l0 := p l (0) represents the liquid pressure at rest,p ∞ =p ∞ (t) the far-field pressure, R 0 the undisturbed bubble radius, l the liquid density, σ the surface tension and τ = R 0 /ν a characteristic time with kinematic viscosity ν. The IVP (2)- (4) is solved numerically by the classical fourth order Runge-Kutta method with adaptive step size control. Defining as standard case A * = B * = M * = σ * = η * = 1 and takingp ∞ (t) = 2 − exp(−t) we perform parameter studies by increasing all dimensionless constants. The inertia term M * causes more radial oscillations, larger amplitudes and wavelengths. These phenomena are intensified by B * which includes the memory effects of the liquid having a resonant effect. Consequently, the bubble response is that of an elastic body. Furthermore, a control of the characteristic time is po...