Viscous effects on the oscillations of two equal and deformable bubbles under a step change in pressure

Abstract

According to linear theory and assuming the liquids to be inviscid and the bubbles to remain spherical, bubbles set in oscillation attract or repel each other with a force that is proportional to the product of their amplitude of volume pulsations and inversely proportional to the square of their distance apart. This force is attractive, if the forcing frequency lies outside the range of eigenfrequencies for volume oscillation of the two bubbles. Here we study the nonlinear interaction of two deformable bubbles set in oscillation in water by a step change in the ambient pressure, by solving the Navier–Stokes equations numerically. As in typical experiments, the bubble radii are in the range 1–1000 μm. We find that the smaller bubbles (~5 μm) deform only slightly, especially when they are close to each other initially. Increasing the bubble size decreases the capillary force and increases bubble acceleration towards each other, leading to oblate or spherical cap or even globally deformed shapes. These deformations may develop primarily in the rear side of the bubbles because of a combination of their translation and harmonic or subharmonic resonance between the breathing mode and the surface harmonics. Bubble deformation is also promoted when they are further apart or when the disturbance amplitude decreases. The attractive force depends on the Ohnesorge number and the ambient pressure to capillary forces ratio, linearly on the radius of each bubble and inversely on the square of their separation. Additional damping either because of liquid compressibility or heat transfer in the bubble is also examined.

Numerical modeling of the dynamics of bubble oscillations subjected to fast variations in the ambient pressure with a coupled level set and volume of fluid method

“…Regarding the verifications, the numerical predictions of Figs. 13(b)-13(d) show qualitatively similar behavior to the results of Chatzidai, Dimakopoulos, and Tsamopoulos [24] and Oguz and Prosperetti [53]. Figure 14 reports the snapshots of the shape of two volume oscillating bubbles with pressure contour and velocity field at four different instants of dimensionless time.…”

“…As with the previously examined numerical study of Chatzidai et al [23] and Chatzidai, Dimakopoulos, and Tsamopoulos [24], the interaction of two coaxial bubbles subjected to pressure changes in the ambient liquid is revisited to assess the ability of the present proposed method. For this purpose, the simulations are performed within computational domain size 32R 10 × 32R 10 , shown in Fig.…”

“…However, a comprehensive theoretical explanation for understanding the highly nonlinear behavior of multiple bubbles associated with nonspherical pulsations in an unbounded liquid still has unexplored aspects. On the other hand, to the best of the author's knowledge, very few numerical studies [17,23,24] have been conducted revealing the mutual interaction between two oscillating bubbles in liquids. However, the methods proposed by the authors of the studies [17,23,24] can be used only for potential flows [17] or Stokes flows [23,24].…”

“…where e z is the unit vector in the vertical direction. In addition, the combined center of volume Z c of the two oscillating bubbles [24] can be determined as…”

“…However, these techniques are restricted to capture weak deformations of bubble. An alternative numerical approach for simulating flows with moving interfaces is the so-called arbitrary-Lagrangian-Eulerian (ALE) method proposed by Chatzidai et al [23] and Chatzidai, Dimakopoulos, and Tsamopoulos [24]. In the ALE method, the nodes of the moving, unstructured, and finite element mesh conform to the moving boundary or interface.…”