Numerical study of the interaction between a pulsating coated microbubble and a rigid wall. I. Translational motion

The dynamic response of an encapsulated bubble to an acoustic disturbance in a wall restricted flow is investigated in the context of axial symmetry, 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. The bubble is accelerated towards the wall as a result of the secondary Bjerknes forces and consequently the translational velocity gradually increases in a nearly quadratic fashion as the bubble approaches the wall. Proximity to the wall affects the resonance frequency that is seen to be reduced as the initial distance between the bubble and the wall decreases, as long as the sound amplitude remains below a threshold value that is determined by the onset of parametric shape mode excitation. While the microbubble remains far from the wall an overpressure develops in the upstream region that causes flattening and bending of the shell. However, shell elasticity coupled with viscous shell stresses prevents jet formation. Thus the bubble remains spherical during the expansion phase of the pulsation and deforms mainly in the compressive phase, during which most of the translation takes place due to the reduced added mass effect. As it approaches the wall the maximum overpressure is moved to the downstream pole region and this generates an excess of viscous shell stresses during compression that balance compressive elastic stresses. As a result the latter are attenuated in the downstream region of the shell, in comparison with the bulk of the shell where they are balanced solely by the cross membrane pressure drop, leading to a gradually more pronounced prolate bubble shape. Viscous drag due to the surrounding liquid develops mainly in the bulk of the shell where it is balanced by viscous shell stresses in the tangential stress balance. Over a period of the pulsation it counteracts the Bjerknes force that accelerates the bubble, via a force balance that is almost instantaneously established due to the relatively large shell viscosity. This is in marked difference with the case of rising gas bubbles that acquire oblate shapes as a result of the balance between buoyancy and pressure drag. In the case of coated microbubbles the drag coefficient is seen to obey a law previously obtained for no-slip interfaces for large radial and relatively small translational Reynolds numbers.

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

(a) Construction of the grid in one domain and (b) construction of the grid in two separate subdomains.

Figure 4

Phase diagram for a contrast agent with ${\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$, and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$ subject to an acoustic disturbance with forcing frequency $f=1.7\mathrm{MHz}$.

Figure 5

Temporal evolution of (a), (b) bubble shape and (c), (d) breathing, $P_0$, and shape mode, $P_4, P_6, P_8$, decomposition for a contrast agent with $R_0=3.6\mu \mathrm{m}, {\mu }_s=60\times {10}^{-9}\mathrm{kg}/\mathrm{s}, \chi =0.24\mathrm{N}/\mathrm{m}$, and $k_B=3\times {10}^{-14}\mathrm{N}\mathrm{m}$ that freely pulsates away from nearby surfaces, subject to an acoustic disturbance with forcing frequency $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2.5$ and 3, respectively.

Figure 6

Temporal evolution of (a), (c) bubble shape and (b), (d) breathing, $P_0$, translational $P_1$ modes, and shape mode decomposition for grids $G_1$ and $G_2$, 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 $z_{c0}=6$; an acoustic disturbance of frequency $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.

Figure 7

(a) Zoom-in on the shape mode evolution during the acceleration phase of the pulsation; distribution of (b,d) pressure and (c,e) viscoelastic stresses, at maximum compression during the acceleration phase when $t=52.6$ and 410.8, respectively; (f) tangential $u_t$ and normal $u_n$ velocities of the liquid when $t=410.8$; 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 $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.

Figure 8

Comparison of the evolution of the dimensionless translational bubble velocity as a function of distance from the wall calculated numerically and theoretically; 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 $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.

Figure 9

(a) Prolate shapes during maximum compression for different ${\mathrm{Re}}_T$; (b), (c) viscoelastic stresses when ${\mathrm{Re}}_T=3.06$ and ${\mathrm{Re}}_T=8.18$, respectively; (d) liquid tangential stresses when ${\mathrm{Re}}_T=3.06$ and ${\mathrm{Re}}_T=8.18$, respectively; (e) temporal evolution of the viscous drag force and the absolute value of the secondary Bjerknes force, both obtained numerically; and (f) interfacial distribution of pressure 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 $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.

Figure 10

Temporal evolution of (a) bubble shape; (b) breathing, $P_0$; translational $P_1$ modes; and shape mode decomposition; (c) comparison of the evolution of the dimensionless translational bubble velocity as a function of distance from the wall calculated numerically and theoretically: 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 $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.

Figure 11

Temporal evolution of (a,c) bubble shape; (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 $z_{c0}=2$; an acoustic disturbance of forcing frequency $f=1.7\mathrm{MHz}$ is imposed.

Figure 12

Prolate shapes during compression for gradually increasing ${\mathrm{Re}}_T$ and ${\mathrm{We}}_T$ numbers of the translating microbubble.

Figure 13

Comparison between the drag coefficient for the microbubble translational motion as a function of the translational Reynolds number, ${\mathrm{Re}}_T$, based on the average translational velocity of its center of mass, as obtained via application of Eq. (30) on the numerically obtained bubble average radial and translational velocity, and via the correlation in Eq. (31) provided in [17].

Figure 14

(a), (b) Temporal evolution of energy dissipation due to liquid viscosity, $E_{\mathrm{liq}}$, and shell damping, $E_{\mathrm{shell}}$, in an unbounded flow and in the vicinity of a wall, respectively; (c), (d) comparison of the evolution of $E_{\mathrm{liq}}$ and $E_{\mathrm{shell}}$, respectively, in an unbounded flow and in the vicinity of a wall; 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 in wall bounded simulations is set to $z_{c0}=4$; an acoustic disturbance of forcing frequency $f=1.7\mathrm{MHz}$ and sound amplitude $\varepsilon =2$ are imposed.