<p>The nonlinear oscillations of a micron-sized air bubble in water driven by a spatially uniform, temporally periodic, single-frequency electric field are considered. Using a model that accounts for shape mode interactions to second order, thermal damping of the interior gas, viscous damping of the liquid and weak compressibility, the resultant volume mode oscillations and shape deformation are studied in detail. For a range of driving frequencies and electric field strengths, after an initial transition phase, the bubble is shown to assume a sustained, finite amplitude, oscillating ellipsoidal shape dominated by the prolate/oblate mode (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> shape mode) which oscillates at twice the driving frequency. Both the volume mode and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> shape mode are directly excited by the electric field while higher-order even shape modes are excited through nonlinear shape mode interactions consistent with previous work. The dynamical behaviour of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> shape mode is shown to depend on the difference between the shape mode’s natural frequency and twice the driving frequency and on whether the bubble is driven below, at, or above the resonance of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> shape mode. In all considered cases, the volume mode oscillations are shown to be prohibitively small, even at volume resonance, to induce parametric instability growth, leaving the directly excited <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> shape mode to dominate the resultant bubble dynamics.</p>

Shape deformation of an air bubble in water under an oscillatory electric field

期刊论文
发表日期：

2026年7月22日

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Stephen J. Shaw

摘要

The nonlinear oscillations of a micron-sized air bubble in water driven by a spatially uniform, temporally periodic, single-frequency electric field are considered. Using a model that accounts for shape mode interactions to second order, thermal damping of the interior gas, viscous damping of the liquid and weak compressibility, the resultant volume mode oscillations and shape deformation are studied in detail. For a range of driving frequencies and electric field strengths, after an initial transition phase, the bubble is shown to assume a sustained, finite amplitude, oscillating ellipsoidal shape dominated by the prolate/oblate mode ( $k=2$ $k = 2$ shape mode) which oscillates at twice the driving frequency. Both the volume mode and $k=2$ $k = 2$ shape mode are directly excited by the electric field while higher-order even shape modes are excited through nonlinear shape mode interactions consistent with previous work. The dynamical behaviour of the $k=2$ $k = 2$ shape mode is shown to depend on the difference between the shape mode’s natural frequency and twice the driving frequency and on whether the bubble is driven below, at, or above the resonance of the $k=2$ $k = 2$ shape mode. In all considered cases, the volume mode oscillations are shown to be prohibitively small, even at volume resonance, to induce parametric instability growth, leaving the directly excited $k=2$ $k = 2$ shape mode to dominate the resultant bubble dynamics.