Nonlinear bubble dynamics

Abstract

The standard approach to the analysis of the pulsations of a driven gas bubble is to assume that the pressure within the bubble follows a polytropic relation of the form p=p0(R0/R)3κ, where p is the pressure within the bubble, R is the radius, κ is the polytropic exponent, and the subscript zero indicates equilibrium values. For nonlinear oscillations of the gas bubble, however, this approximation has several limitations and needs to be reconsidered. A new formulation of the dynamics of bubble oscillations is presented in which the internal pressure is obtained numerically and the polytropic approximation is no longer required. Several comparisons are given of the two formulations, which describe in some detail the limitations of the polytropic approximation.

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

“…Therefore, we incorporate the nonlinear model based on solving the continuity and the energy conservation equations for a gas (as in Ref. [30]) into the formulation.…”

“…The explicit expressions for the damping coefficient ܾ ୲୭୲ and the bubble resonance frequency ߱ can be found in [28,30].…”

A reduced-order model of diffusive effects on the dynamics of bubbles

“…The following assumptions are made in the remainder of this paper: ͑i͒ The internal bubble pressure is spatially uniform, 1,11 ͑ii͒ the liquid temperature is uniform, 12 ͑iii͒ there is no diffusion of noncondensable gas in the liquid, 12 and ͑iv͒ the vapor pressure is in equilibrium at the gas-liquid interface. 12 Each of these additional assumptions has been validated for all the cases considered by comparisons with the full computations of the preceding section.…”

“…[1][2][3][4] The alternative is to solve the full set of coupled radial transport equations for each bubble, but this is too computationally intensive to implement in continuum models of complex bubbly flows. Consequently, reduced-order models that accurately capture diffusive effects are needed.…”

Behaviour of a cloud of bubbles filled with vapour and a small amount of noncondensable gas (A theoretical and numerical study)

“…So even a small amount of noncondensable gas is of great importance! Sorne non-essential simplifications are made: spherical symmetric bubbles ; condensation only occurs at the surface of a bubble ; pressure is uniform within each bubble (PROSPERETTI [2]); damping is assumed to be caused mainly by thermal conduction and radiation of sound. If the radius of the bubbles is not too smaIl and aIl velocities are small compared to the velocity of sound in the vapour, surface tension, viscosity and compressibility can be neglected.…”