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

Abstract

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. © 2021 American Physical Society.