Numerical study of the interaction between a pulsating coated microbubble and a rigid wall. II. Trapped pulsation

The dynamic response of an encapsulated bubble subject to an acoustic disturbance in a wall restricted flow is investigated, when the viscous forces of the surrounding liquid are accounted for. The Galerkin finite element methodology is employed and the elliptic mesh generation technique is used for updating the mesh. As the bubble accelerates towards the wall, the dominant force balance is between Bjerknes forces and the viscous drag that develops. In this process a prolate shape is acquired by the bubble, due to excessive compression at the equator region. When the bubble reaches the wall lubrication pressure develops in the near wall region that resists further approach. As long as the sound amplitude remains below a threshold value determined by the onset of parametric shape mode excitation saturated, or “trapped,” pulsations are performed around a certain small distance from the wall. The balance between Bjerknes attraction and the lubrication pressure that arises due to shell bending along the flattened shell portion that faces the wall generates an oblate shape. Elongation is now observed along the equatorial plane where a local liquid overpressure is established generating large tensile strain. The time-averaged deflection of the microbubble at the pole that lies close to the wall is determined by the bending and stretching resistances of the shell in the manner described by Reissner's linear law for a static compressive load on an elastic shell, corrected for the effect of surface tension. The oscillatory part of the bubble motion in that same region, the contact region, follows the forcing frequency and consists of a pressure driven and a shear flow in the form of a Stokes layer where a significant amount of instantaneous wall shear is generated. The thickness of the film that occupies the Stokes layer is on the order of a few tenths of nm and is determined by the balance between liquid and shell tangential viscous stresses. The steady streaming effect on the wall shear is absent owing to the negligible phase difference between the volume and center of mass pulsations.

Figure 1

A contrast agent in a wall-restricted flow.

Figure 2

Schematic representation of the physical and the computational domain for the case of a bubble in wall restricted flow.

Figure 3

Temporal evolution of (a) bubble shape and (b) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$, and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=6$; an acoustic disturbance of forcing frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 4

(a) Zoom-in on the shape mode evolution during trapped oscillations; interfacial distribution of (b), (d) pressure, and (c),(e) viscoelastic stresses, at maximum compression and expansion respectively during trapped pulsations; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=6$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 5

Temporal evolution of (a) bubble shape, (b) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition, and (c) zoom-in on the shape mode evolution during trapped oscillations; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=4$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 6

Interfacial distribution of (a),(b) pressure, (c),(d) shell viscoelastic stresses, and (e),(f) liquid viscous tangential stresses when the volume pulsation crosses from compression to expansion and vice versa, respectively, during trapped pulsations; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=4$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 7

(a),(b) Interfacial distribution of tangential and normal velocities when the volume pulsation crosses from compression to expansion and vice versa, respectively, (c)–(f) velocity profile during maximum expansion, maximum compression, and when the volume pulsation crosses from compression to expansion and vice versa, respectively, during trapped pulsations; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=4$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 8

Temporal evolution of (a),(c) bubble shape and (b),(d) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition, when the sound amplitude ε is set to 1.and 1.7, respectively; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$ and the acoustic disturbance is at a forcing frequency of $f=1.7$ MHz.

Figure 9

Temporal evolution of (a),(c) bubble shape and (b),(d) breathing ${\mathrm{P}}_0$, translational $P_1$ modes, and shape mode decomposition, when the sound amplitude ε is set to 3 and 4, respectively; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are $\chi =0.24\mathrm{N}/\mathrm{m}, {\mu }_s=120\times {10}^{-9}\mathrm{kg}/\mathrm{s}$, and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$ and the acoustic disturbance is at a forcing frequency of $f=3.4$ MHz.

Figure 10

(a) Schematic diagram of the time averaged force balance in the near wall region of the microbubble, (b) values obtained for $\frac{F_B}{\sqrt{{\chi }^{k_B}}}$ numerically and comparison with the function $16\mathrm{\Delta }/R_0$, and (c) numerical obtained static calculations and fitting of the results [31].

Figure 11

Temporal evolution of (a),(c),(e) bubble shape and (b),(d),(f) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition when the area dilatation modulus χ is set to 0.12, 0.24, and 0.48 N/m, respectively; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=120\times {10}^{-9}\mathrm{kg}/\mathrm{s}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$; an acoustic disturbance of frequency $f=1.7$ MHz and amplitude $\varepsilon =2$ is imposed.

Figure 12

Temporal evolution of (a),(b) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition when the bending resistance is set to $k_B=6\times {10}^{-14}\mathrm{N}\mathrm{m}$ and $k_B=12\times {10}^{-14}\mathrm{N}\mathrm{m}$, respectively, (c) bubble shape at maximum compression $t=285.2$ for the different $k_B$ values, (d) zoom-in of the shape close to the wall at $t=285.2$, (e),(f) indicative radial velocity profiles below the contact region at the time instants corresponding to transition from expansion to compression and vice versa when $k_B=12\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}$ and χ = 0.24 Ν/m; the initial distance from the wall is set to $z_{c0}=4$; an acoustic disturbance of frequency $f=1.7$ MHz and amplitude $\varepsilon =2$ is imposed.

Figure 13

Temporal evolution of (a),(c),(e) bubble shape and (b),(d),(f) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition when ${\mu }_s=120\times {10}^{-9}\mathrm{kg}/\mathrm{s}$ and ε = 2, 2.2, and 3.0, respectively. Temporal evolution of (g) bubble shape and (h) breathing $P_0$, translational $P_1$ modes, and shape mode decomposition when ${\mu }_s=300\times {10}^{-9}\mathrm{kg}/\mathrm{s}$ and $\varepsilon =3$; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties are $\chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$; an acoustic disturbance of frequency $f=1.7$ MHz is imposed.

Figure 14

Interfacial distribution of (a) pressure, (b) viscoelastic stresses, and (c) radial velocity profile during maximum expansion during trapped pulsations; shell properties are ${\mu }_s=120\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.

Figure 15

Interfacial distribution during trapped pulsations of (a),(b) normal and tangential interfacial velocity, (c),(d) radial velocity profiles, (e),(f) interfacial distribution of pressure, (g),(h) interfacial distribution of viscoelastic stresses, and (i),(j) interfacial distribution of liquid tangential stresses, when the volume pulsation crosses from compression to expansion and from expansion to compression, respectively; the initial bubble radius is $R_0=3.6\mu \mathrm{m}$; shell properties ${\mu }_s=120\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$ and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$; the initial distance from the wall is set to $z_{c0}=2$; an acoustic disturbance of frequency $f=1.7$ MHz and sound amplitude $\varepsilon =2$ is imposed.