Dynamics of bubble-bubble interactions experiencing viscoelastic drag

The subject of the present theoretical study is the dynamics of bubble-bubble interactions in a viscoelastic medium. First, new equations for calculating the viscoelastic drag exerted on bubbles during their translational motion in a viscoelastic medium are derived. The drag equations are incorporated in the bubble-bubble interaction model in which, thereby, both the translational and radial motions of the bubbles are affected by the viscoelastic features of the medium. Second, the derived equations are applied to investigate how the viscoelastic properties of the medium can affect the dynamics of multiple bubbles, as well as how the bubbles can affect each other. It was discovered that the bubble-bubble interaction can significantly influence the dynamics of a single bubble. As the distance between the bubbles increases, their effect on each other decreases, and at a distance of several millimeters, this effect can be neglected. Moreover, it was concluded that with increasing elasticity and viscosity of the medium, as well with decreasing relaxation time, the effects of other bubbles on the current bubble's radial motion can become negligible. The translational motion of the bubbles was investigated for different viscoelastic models. The elasticity resists the motion of bubbles in space, resulting in a dynamical steady state of the distance between the bubbles at high elasticity values. The relaxation time of the medium was also found to be important in terms of the bubbles' translational movement.

Figure 1

Schematic of the bubble-bubble interaction model. Two bubbles with radii $R_1\left(t\right)$ and $R_2\left(t\right)$ and center positions $x_1\left(t\right)$ and $x_2\left(t\right)$ are attracted or repelled from each other in a biological viscoelastic fluid. The biological fluid is exposed to an ultrasound driving pulse $p\left(t\right)$. The bubbles are experiencing bubble-bubble interaction forces and viscoelastic drag $D_{\text{ve}}^{\left(i\right)}$, $i=\{1,2\}$.

Figure 2

Validation and verification of the current model by comparing our results with the available experimental and numerical data. $R\left(t\right)$ is a bubble radius. (a) Comparison with the experimental data presented in Ref. [70]. $f=2.9\times {10}^6\mathrm{Hz}$, $R_0=0.975\mu \mathrm{m}$, $c_{\infty }=1480\mathrm{m}/\mathrm{s}$, ${\rho }_t=1000\mathrm{kg}/{\text{m}}^3$, $\sigma =\sigma \left(R\right)$ [70], $\mu =0.001\mathrm{Pa}$ s, $G=0\mathrm{Pa}$, ${\lambda }_1=0$ s. (b) Comparison with the simulation results presented in Ref. [23]. $A=1.21\times {10}^5\mathrm{Pa}$, $f=20\times {10}^3\mathrm{Hz}$, $R_0^{\left(1\right)}=6\mu \mathrm{m}$, $R_0^{\left(2\right)}=4\mu \mathrm{m}$, $x_0^{\left(1\right)}=0\mu \mathrm{m}$, $x_0^{\left(2\right)}=200\mu \mathrm{m}$, $c_{\infty }=1500\mathrm{m}/\mathrm{s}$, ${\rho }_t=998\mathrm{kg}/{\text{m}}^3$, $\sigma =0.0725\mathrm{kg}/{\text{s}}^2$, $\mu =0.001\mathrm{Pa}\times$ s, $G=0\mathrm{Pa}$, ${\lambda }_1=0$ s. (c) Comparison with the experimental data presented in Ref. [71]. $p\left(t\right)=0$, $R_0=355\mu \mathrm{m}$ ($R_{\text{equilibrium}}=55\mu \mathrm{m}$), $c_{\infty }=1430\mathrm{m}/\mathrm{s}$, ${\rho }_t=1060\mathrm{kg}/{\text{m}}^3$, $\sigma =0.056\mathrm{kg}/{\text{s}}^2$, $\mu =0.089\mathrm{Pa}$ s, $G={10}^4\mathrm{Pa}$, ${\lambda }_1=0$ s, ${\lambda }_2=0$ s.

Figure 3

The effect of the presence of the surrounding bubbles and viscoelastic nature of the drag on the single-bubble dynamics. Three cases were considered (on the basis of the Gilmore-Akulichev-Zener model): single isolated bubble, bubble-bubble interaction model with Levich viscous drag, and bubble-bubble interaction model with viscoelastic drag in a Zener form. (a) The first bubble dynamics. (b) The second bubble dynamics. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$, $x_0^{\left(1\right)}=0\mu \mathrm{m}$, $x_0^{\left(2\right)}=170\mu \mathrm{m}$. $\mu =0.015\mathrm{Pa}\mathrm{s}$, $G={10}^5\mathrm{Pa}$, $\lambda =3\times {10}^{-9}\mathrm{s}$.

Figure 4

The effect of the presence of the surrounding bubbles on the single-bubble dynamics for different initial distances $d_0$ between the bubbles. The solid black line denotes the dynamics of the isolated bubble. When $\zeta \le 0.1$, the effect can be neglected. When $\zeta >0.1$, the effect is noticeable (red font). ${\overline{R}}_0=\left[R_0^{\left(1\right)}+R_0^{\left(2\right)}\right]/2$. (a) $R_0^{\left(1\right)}=0.1\mu \mathrm{m}$, $R_0^{\left(2\right)}=0.1\mu \mathrm{m}$. (b) $R_0^{\left(1\right)}=0.6\mu \mathrm{m}$, $R_0^{\left(2\right)}=0.2\mu \mathrm{m}$. (c) $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$. (d) $R_0^{\left(1\right)}=7\mu \mathrm{m}$, $R_0^{\left(2\right)}=6\mu \mathrm{m}$. The viscoelastic parameters are $\mu =0.015\mathrm{Pa}\mathrm{s}$, $G={10}^5\mathrm{Pa}$, $\lambda =3\times {10}^{-9}\mathrm{s}$.

Figure 5

Viscoelastic effects on the bubbles' radial motion. (a) Elasticity effect. Sum of bubble radii $R_1\left(t\right)+R_2\left(t\right)$ plotted for the case of $x_0^{\left(1\right)}=0$ m, $x_0^{\left(2\right)}=200\mu \mathrm{m}$, ${\lambda }_1=3·{10}^{-10}\mathrm{s}$. (b) Relaxation time effect. Sum of bubble radii $R_1\left(t\right)+R_2\left(t\right)$ plotted for the case of $x_0^{\left(1\right)}=0$ m, $x_0^{\left(2\right)}=200\mu \mathrm{m}$, $G=3·{10}^3\mathrm{Pa}$. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$. $\mu =0.015\mathrm{Pa}\mathrm{s}$.

Figure 6

The distance $d_0$ between bubbles at which their effect on each other is negligible ($\zeta \le 0.1$), plotted for different viscoelastic properties of the medium. (a) Variation of the elasticity and viscosity of the medium, ${\lambda }_1=3\times {10}^{-9}\mathrm{s}$. (b) Variation of the relaxation time and viscosity of the medium, $G={10}^5\mathrm{Pa}$. (c) Variation of the elasticity and relaxation time of the medium, $\mu =0.015\mathrm{Pa}\mathrm{s}$. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$.

Figure 7

Distance between the bubbles for the Zener, Maxwell, Newton, and Kelvin-Voigt viscoelastic models with the corresponding viscoelastic drags. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$, $x_0^{\left(1\right)}=0\mu \mathrm{m}$, $x_0^{\left(2\right)}=200\mu \mathrm{m}$.

Figure 8

Viscoelastic drag effect. (a) Motion of the bubble centers exerted by Zener viscoelastic drag. (b) Motion of the bubble centers exerted by Levich viscous drag. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$, $x_0^{\left(1\right)}=0\mu \mathrm{m}$, $x_0^{\left(2\right)}=200\mu \mathrm{m}$.

Figure 9

Repulsion of the bubbles at lower ultrasound frequencies (and amplitudes, $A=1.3\times {10}^5\mathrm{Pa}$, $f=20\times {10}^3\mathrm{Hz}$). The black line is the distance between bubble centers experiencing the Levich viscous drag. The red line is the distance between bubble centers experiencing the Kelvin-Voigt viscoelastic drag. $R_0^{\left(1\right)}=3\mu \mathrm{m}$, $R_0^{\left(2\right)}=2\mu \mathrm{m}$, $x_0^{\left(1\right)}=0\mu \mathrm{m}$, $x_0^{\left(2\right)}=150\mu \mathrm{m}$, $G=2\times {10}^3\mathrm{Pa}$.

Figure 10

Viscoelastic effects on the translational bubbles' motion at high ultrasound frequency and amplitude. Zones of the relaxation time effect are shown by Latin letters. The area $ABCD$: relaxation time enlarges the oscillations of the bubbles' distance. The area $CDEF$: large relaxation times accelerate bubbles' attraction. The area $FGHB$: relaxation time effect vanishes.

Figure 11

Elasticity effect on the distance between the bubbles $d\left(t\right)=x_2\left(t\right)-x_1\left(t\right)$. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$, $x_0^{\left(1\right)}=0$ m, $x_0^{\left(2\right)}=200\mu \mathrm{m}$, ${\lambda }_1=3·{10}^{-10}\mathrm{s}$, $\mu =0.015\mathrm{Pa}\mathrm{s}$.

Figure 12

Relaxation time effect on the distance between the bubbles $d\left(t\right)=x_2\left(t\right)-x_1\left(t\right)$. (a) $G={10}^4\mathrm{Pa}$. (b) $G=10\mathrm{Pa}$. $R_0^{\left(1\right)}=5\mu \mathrm{m}$, $R_0^{\left(2\right)}=3\mu \mathrm{m}$, $x_0^{\left(1\right)}=0$ m, $x_0^{\left(2\right)}=200\mu \mathrm{m}$, $\mu =0.015\mathrm{Pa}\mathrm{s}$.

Figure 13

Schematic of the cell model used for the drag derivation. A bubble of radius $R_B$ with surface $S_1$ is surrounded by fluid of volume $V$ with surface $S_2$ and radius $R_S$. ${\mathbf{n}}_1$ is a normal to $S_1$, ${\mathbf{n}}_2$ is a normal to $S_2$, ${\mathbf{t}}_1^{\left(\alpha \right)}$ are unit vectors on $S_1$, and ${\mathbf{t}}_2^{\left(\alpha \right)}$ are unit vectors on $S_2$. $\{{\mathbf{n}}_1,{\mathbf{t}}_1^{\left(1\right)},{\mathbf{t}}_1^{\left(2\right)}\}$ and $\{{\mathbf{n}}_2,{\mathbf{t}}_2^{\left(1\right)},{\mathbf{t}}_2^{\left(2\right)}\}$ are orthogonal. The bubble with the changing radius $R_B$ is moving with velocity $U$ along ${\mathbf{e}}_x$. $\varepsilon$ is the distance between the bubble center and the center of the outer fluid sphere.