Simulation of light scattering on a deformed gas bubble

The paper describes simulating light scattering of an electromagnetic field on a deformed gas bubble suspended in a liquid. The field intensity distributions around the deformed bubble were obtained. It turned out that field scattering increases with bubble deformation up to a certain limit of the deformation coefficient A (relative deformation amplitude). At A = 0.7 and higher, the scattering remains independent of the magnitude of the deformation.


Introduction
Purifying water from impurities is an important and urgent task of modern science and technology [1], [2], [3], [4].One of the methods of water purification is to pass gas bubbles through a layer of liquid, at the interface of which impurities settle and are carried away along with the bubbles (or the gas enters into a chemical reaction with impurities in the liquid).Moreover, the smaller the bubbles, the larger the interfacial surface that cleans the liquid from impurities [5], [6], [7], [8].To study the factors influencing the size of gas bubbles suspended in a liquid, a system for measuring this size is needed, for example, based on the principle of light scattering in a two-phase liquid [9], [10], [11], [12].Optical examination of a two-phase liquid is the least invasive, since light makes minimal changes to the medium being examined.Since gas and liquid have different optical properties, when light hits the bubble, the electromagnetic field is disturbed: part of the light is scattered, part is refracted, and the intensity of the passing beam decreases.In optical examination of a two-phase liquid, a measuring device (photosensitive sensor) receives light that is either scattered by a bubble or passed through an area of the liquid and reduced by scattering.
One of the ways to "grind" gas bubbles in a liquid is exposure to ultrasonic radiation (US).In this case, a gas bubble introduced from the outside using mechanical means (bubbling) is broken into several small bubbles.A similar effect was obtained experimentally, but only for small volumes, since ultrasonic vibrations decay with distance quite quickly.When a large bubble breaks into smaller ones, deformations occur on its surface.For non-invasive measurement of deformations, it is necessary to modify existing methods for measuring the parameters of the dispersed phase in a continuous medium [10,11].To modify measurement methods, theoretical studies of the scattering of an electromagnetic field on a deformed dispersed inclusion (in this case, a bubble) are necessary, which are described below.

Problem statement
To develop a theoretical model, the following simplifications of the research object were proposed, i.e. deformed bubble suspended in liquid: 1) first, research is carried out in a two-dimensional domain, i.e. the bubble is a cylinder.This is possible, for example, in a flat Hele-Shaw cell, the thickness of which is comparable to the diameter of the bubble.The cross section of the cylinder is described by the parametric equation ( 1): where R is the distance from the center of the bubble to a point on the surface, m; R0 -radius of an undeformed bubble, m; A is the relative amplitude of disturbances; θ is the angle between the direction of propagation of the electromagnetic wave and the radius vector of the point of the interphase surface.
2) the bubble density is constant, i.e. we are considering an incompressible gas.This can happen when the capillary jump on the surface of the bubble is small compared to atmospheric pressure (with a bubble radius of more than 0.5 mm) and also at bubble deformation rates not exceeding 10 m/s.The consequence of this assumption is that the cross-sectional area of the bubble is constant.Therefore, the radius R0 in equation ( 1) must be consistent with the deformation coefficient A as follows: After performing the transformations and taking the integral, we get: Thus, the cross-sectional radius of the perturbed bubble is determined by the equation: where R00 is unperturbed bubble radius.Magnetic and dielectric permeabilities inside the body and outside (in the environment) are different.Based on the complex geometry of the problem, we consider the system as consisting of one region, but filled with an inhomogeneous dielectric, i.e. μ = μ(r ⃗) (magnetic permeability), n = n(r) (refractive index).In this case, the bubble-liquid system is considered as a single medium in which the interphase surface is modeled by a thin transition layer with a smooth change in magnetic permeability and refractive index.From Maxwell's equations we obtain the wave equation for an inhomogeneous dielectric.Let us represent the electric and magnetic field strength vectors in complex form A ⃗ ⃗⃗ ̃= A ⃗ ⃗⃗ e −iωt , where A ⃗ ⃗⃗ is a spatial vector independent of time, and write Maxwell's equations in the frequency domain: where k = ω c From the last two equations we find: Considering that μ depends on the coordinate, we write the double rotor from the equation ( 3): Multiplying the resulting equation by μ, we get: To implement calculations using the finite element method, we multiply equation ( 5) scalarly by the test function v ⃗⃗: Integrating equation ( 8) over volume, we obtain: Applying the divergence theorem: The following are the results of calculations based on the resulting equation.

Results and discussion
Using the equations presented above, the modulus of the scattered electromagnetic field was calculated using the FreeFEM++ package.The dependences presented make it possible to calculate the relative amplitude (coefficient) of bubble deformation depending on the intensity of the scattered field if the radius of the unperturbed bubble is known.

Conclusion
The work involved numerical modeling of the scattering of an electromagnetic field on a deformed air bubble suspended in water.The study showed that the difference in intensity of the initial and scattered light can be used to judge the deformation of the illuminated bubble: with increasing deformation, the difference in intensity increases, i.e. the light is scattered more.This conclusion can be used in further experimental studies of light scattering by air bubbles suspended in water.
Figures 1, 2 and 3 present visualizations of the electromagnetic field under the condition of an incident wave with circular polarization at different values of the strain coefficient A. Scattering was calculated as the difference between the root mean square values of the electric field for the original and scattered fields, which were calculated as integrals along the upper boundary (initial field) and along a straight line located at a distance of 2R0, from the center of the bubble, as shown in Figure 1a.

Figure 1 .
Figure 1.Distribution of the electromagnetic field around a deformed bubble with an initial radius of a round cylinder R00 = 1 μm and various values of the coefficient A: a) A = 0 (undeformed bubble); b) A = 0.3; c) A = 0.6; d) A = 0.9.

Figure 2 .
Figure 2. Distribution of the electromagnetic field around a deformed bubble with an initial radius of a circular cylinder R00 = 2 μm and various values of the coefficient A: a) A = 0 (undeformed bubble); b) A = 0.3; c) A = 0.6; d) A = 0.9.

Figure 3 .Figure 4 .
Figure 3. Distribution of the electromagnetic field around a deformed bubble with an initial radius of a circular cylinder R00 = 3 μm and different values of the coefficient A: a) A = 0 (undeformed bubble); b) A = 0.3; c) A = 0.6; d) A = 0.9.