Motion of the bubble wall

In any cavitation field most of the visible bubbles will be oscillating in a stable manner and it is perhaps pertinent that we concentrate our discussions first on the fate of such bubbles in the acoustic field. If we assume that we have a bubble with an equilibrium radius, R, existing in a liquid at atmospheric pressure Pjj, then the oscillation of the bubble and in particular the motion of the bubble wall, under the influence of the applied sinusoidal acoustic pressure (P ) is a simple dynamical problem, akin to simple harmonic motion for a spring. 

Although there exist many sophisticated mathematical treatise which derive the motion of the bubble wall, all yield equations similar in form to Eq. 2.28. 

It has been argued (Appendix 3, Eq. A.21) that the collapse time for a bubble, initially of radius R, is considerably shorter than the time period of the compression cyde. Thus the external pressure Pj (= P + Pjj), in the presence of an acoustic field, maybe assumed to remain effectively constant (Pj ) during the collapse period. Neglecting surface tension, assuming adiabatic compression (i. e. very short compression time), and replacing R, by R, the size of the bubble at the start of collapse, the motion of the bubble wall becomes... 

U nlike Rayleigh s original example of a collapsing empty cavity, this bubble will reduce to a minimum size, on compression, after which it will expand to Rj and subsequently it will oscillate between the two extremes R and Rf in. Obviously at the two extremes of radii, motion of the bubble wall is zero - i. e. R = 0. To determine these radii it is necessary to integrate Eq. A.25. With Z = (R /R), the integration yields ... 

By integrating the stress equation for the radial motion of an incompressible Newtonian fluid from the bubble wall to infinity, the equation of motion for the bubble wall was obtained by Scriven (S3), upon using Eq. (3), in the form... 

The RPNNP equation, or more pertinently the nature of the link between the radial response and the pressure field described by this equation, explains the particular character of the radial pulsation of a bubble because of the presence of a term in which the square of the bubble-wall speed appears. When the acoustic pressure is very small, the bubble radius evolves proportionally to this pressure constraint. However, non-linear behavior occurs as soon as the acoustic pressure increases. Figure 16 shows the different responses of a bubble for which all the physical parameters are equal except the initial radius. Again, the analogy with a pendulum can be used, the motion of which is described by Eq. 25 ... 

Piezoelecttic impulse ink-jet printers ate especially sensitive to bubbles in the ink. A bubble in the firing chamber absorbs some of the comptessional force from the flexing of the chamber wall and reduces drop volume and drop velocity, thereby affecting print quaHty. Because of the limited range of motion of the crystal, bubbles ate not readily ejected, and the loss of print quaHty owing to their presence is persistent. 

The velocity u0 and the laminar path length x0 can be related to measurable physical quantities by using dimensional analysis. Indeed, the circulatory motion is induced by the buoyancy force gAp, where Ap is the difference between the density near the wall (assumed to be equal to that of the liquid) and the density of the bubble bed... 

Slug flow-chum flow transition. As the gas flow is increased even more, a transition to chum flow occurs. The subjective discrimination between slug flow and chum flow makes it difficult to identify the transition exactly. Taitel et al. [3] use the definition that is based on the behavior of the liquid film between the Taylor bubble and the wall. In this case the chum flow is characterized as the condition where oscillatory motion of the liquid is observed. 

Various mathematical models have been put forth to describe the rate of bubble growth and the threshold pressure for rectified diffusion.f ° The most widely used model quantifies the extent of rectified diffusion (i.e., the convection effect and bubble wall motion) by separately solving the equation of motion, the equation of state for the gas, and the diffusion equation. To further simplify the derivation, Crum and others made two assumptions 1) the amplitude of the pressure oscillation is small, i.e., the solution is restricted to small sinusoidal oscillations, and 2) the gas in the bubble remains isothermal throughout the oscillations.Given these assumptions, the wall motion of a bubble in an ultrasonic field with an angular frequency of co = 2nf can be described by the Rayleigh-Plesset equation ... 

Another similar problem is the problem of extrusion by air of a viscous liquid from a capillary (Fig. 17.7) [19]. The flow is analogous to the flow in the problem of a lubricating layer between rotating cylinders. To convince ourselves in this similarity, it is necessary to attach the coordinate system to a bubble moving with constant velocity U. Then, in its proper system, the bubble is motionless, and the walls of the capillary move with the velocity U to the left. Near the front of the bubble, the surface of the liquid forms a meniscus-like interface, which then transfigures into a film flow at the surface of the capillary. For small values of Re and Ca, the equation of motion reduces to a balance of viscous and capillary... 

Fig. 17.7 depicted the motion of a semi-infinite bubble inside a capillary in the absence of a surfactant. Consider now the case when a surfactant is present in the liquid. Let s switch to a coordinate system where the bubble is at rest and the capillary wall moves from the right to the left with the velocity U (Fig. 17.10). We are mostly interested in two parameters that can be calculated and determined experimentally the thickness Sj of the liquid film between the bubble and the capillary wall, and the additional pressure drop Ap inside the liquid, caused by the presence of the bubble.

Experiments have shown that adsorption of a surfactant on the interface creates additional stresses that block the motion of long bubbles inside capillaries. Therefore the presence of these stresses necessitates the creation of a larger pressure drop to push bubbles through the capillary. It also leads to an increase in the thickness of the moistening film at the capillary wall. 

Let us estimate the additional pressure drop caused by the presence of surfactant in the liquid. Although in reality the front and rear parts of the bubble s surface have slightly different shapes (the curvature radius of the back cap is greater than that of the front cap ), for simplicity s sake, we consider them as identical hemispheres with radii equal to the radius of the capillary. The length of the liquid film confined between the wall and the bubble is 1. The presence of a surfactant in the liquid flowing around the motionless bubble, leads to a transfer of the surfactant to the bubble surface via convective diffusion. This gives rise to a non-uniform distribution of the surfactant at the bubble surface. The surfactant is pushed to the back of the bubble and accumulates there. An increased surfactant concentration caused a reduction of S. Therefore, 2 decreases from the front to the rear of the bubble. As a result, the pressure in the rear becomes higher than in the front the bubble, and the difference p2 — pi should increase the velocity of the bubble s motion. 

So, when a surfactant is present, the velocity of the bubble s motion relative to the capillary wall will be much greater than in the absence of the surfactant. 

The formation and subsequent growth of a vapor bubble on a heated wall covered with a liquid is controlled by the forces arising from the excess pressure inside the bubble, the surface tension forces at the liquid-vapor interface and at the contact line formed by the interface at the heater surface, and the inertia forces resulting from the motion of the flow as well as the interface. The resistance to the phase change process at the liquid-vapor interface is quite small in comparison to the... 

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