Maximum bubble radius

The maximum bubble radius occurs when dRIdt = 0 at time t = tm, where... 

Relative Potential Bubble Energy (RPBE) The cube of the ratio of the maximum bubble radius constants (J s) ... 

Maximum bubble radius Initial bubble radius Bubble radius at the commencement of asymptotic growth... 

Time at which bubble collapses to zero radius Time at which maximum bubble radius is attained Waiting period between bubbles... 

An analysis of the equation of motion for a single transient bubble under a constant driving sound pressure can be performed by solving the Kirkwood-Bethe-Gilmore model. The best combination of sound pressure and initial bubble radius can be found by demanding a maximum bubble radius prior to collapse, a very small final radius and a collapse that is timed to be finished at the maximum positive pressure. 

Assuming isothermal expansion to maximum bubble radius R ax and subsequent adiabatic collapse, the pressures and temperatures generated in a gas-filled transient cavity can be derived as... 

Figure 8.1.7 Maximum bubble radius for a transient cavity with an initial radius of 1 pm in a sound field with a sound pressure of 4 bar.

Fig. 8. a Jet formation upon collapse of a spherical cavity in the neighbourhood of a plane solid boundary in water. Maximum bubble radius 1.1 mm distance of bubble centre from the wall 4.5 mm framing rate 75000 frames/s. [The first frame (upper left) is doubly exposed with another cavity]. 

As Pritchett and Cole have described, the simple incompressible model predicts that the maximum bubble radius is inversely proportional to the hydrostatic pressure to the 1 /3 power, and the period is inversely proportional to the hydrostatic pressure to the 5/6 power. 

From the theoretical considerations of Noltin and Neppiras 64, 65) many conclusions can be drawn. The most important result is that the maximum radius of the cavitation bubble is inversely proportional to the frequency of the ultrasound within a large range. Once the maximum bubble size has been reached, collapse follows in a maimer determined mainly by the value of the maximum bubble radius. Therefore, cavitation effects diminish with increasing frequency. Noltii and Neppiras also derived an expression for the (nressure distribution in the liquid surrounding a cavity filled with gas and which collapses adiabatically. The results are depicted in... 

F. 6. Calculated pressure distribution in liquid sunoundii gas-filled collapring cavitation bubble. Pg. static pressure exerted on liquid Rfff. maximum bubble radius. Parameter of the curves is Rfn/R). Curves show that for relatively small changes in ratio R, maximum pressure developed can be enormous [Ref. (d5)J... 

The maximum bubble pressure method is good to a few tenths percent accuracy, does not depend on contact angle (except insofar as to whether the inner or outer radius of the tube is to be used), and requires only an approximate knowledge of the density of the liquid (if twin tubes are used), and the measurements can be made rapidly. The method is also amenable to remote operation and can be used to measure surface tensions of not easily accessible liquids such as molten metals [29]. 

According to the simple formula, the maximum bubble pressure is given by f max = 27/r where r is the radius of the circular cross-section tube, and P has been corrected for the hydrostatic head due to the depth of immersion of the tube. Using the appropriate table, show what maximum radius tube may be used if 7 computed by the simple formula is not to be more than 5% in error. Assume a liquid of 7 = 25 dyn/cm and density 0.98 g/cm. ... 

A number of experimental studies have supplied numerical values for these, using either the classical maximum bubble pressure method, in which tire maximum pressure requhed to form a bubble which just detaches from a cylinder of radius r, immersed in tire liquid to a depth jc, is given by... 

Fig. 31. Maximum pressures produced throughout the bursting process plotted against bubble radius [118]...

Fig. 32. Maximum energy dissipation rates produced throughout the bursting process, plotted against bubble radius. The logarithmic scale indicates an exponential dependence of maximum stress on bubble radius for large bubbles. The slight drop in the data point for the smallest bubble as compared to the next smallest may be because of the difficulty in locating the exact place and time of the peak, due to large spatial and temporal gradients beneath the forming jet [113]...

Figure 9. Bubble radius and pressure transients of the water vapor inside the bubbles. The first maximum in pressure at 650 ps marks the collapse of the bubbles. The following modulations are only expected for oscillatory bubble motion.

The maximum ambient radius for an active bubble is smaller than or comparable to the linear resonance radius (Rres) given by (1.18) [64]. 

Fig. 4.8 Schematic illustration of the working principle of the dynamic bubble pressure method. If the bubble radius equals the capillary radius, maximum pressure is detected. The pressure minimum occurs on bubble detachment.

This section of the programme determines the maximum and minimum acoustic pressure and bubble radius and ensures optimum use of the screen. 

A more quantitative gas bubble history than that of Fig 1 is shown in Fig 4 (from Ref 1). The dashed horizontal line is the bubble radius at which the bubble pressure equals the hydrostatic pressure of the water. Note that, over most of the first bubble cycle in this example, the gas pressure in the bubble is below the surrounding hydrostatic pressure. The maximum velocity of the bubble surface is about 200ft/sec... 

In the Maximum-bubble-pressure method the surface tension is determined from the value of the pressure which is necessary to push a bubble out of a capillary against the Laplace pressure. Therefore a capillary tube, with inner radius rc, is immersed into the liquid (Fig. 2.9). A gas is pressed through the tube, so that a bubble is formed at its end. If the pressure in the bubble increases, the bubble is pushed out of the capillary more and more. In that way, the curvature of the gas-liquid interface increases according to the Young-Laplace equation. The maximum pressure is reached when the bubble forms a half-sphere with a radius r/s V(j. This maximum pressure is related to the surface tension by 7 = rcAP/2. If the volume of the bubble is further increased, the radius of the bubble would also have to become larger. A larger radius corresponds to a smaller pressure. The bubble would thus become unstable and detach from the capillary tube. 

FIGURE 16.2 Three possible modes through which inertial cavitation may enhance SC permeability, (a) Spherical collapse near the SC surface emits shock waves, which can potentially disrupt the SC lipid bilayers, (b) Impact of an acoustic microjet on the SC surface. The microjet possessing a radius about one tenth of the maximum bubble diameter impacts the SC surface without penetrating into it. The impact pressure of the microjet may enhance SC permeability by disrupting SC lipid bilayers, (c) Microjets may physically penetrate into the SC and enhance the SC permeability. (From Mitragotri, S., and Kost J., Adv. Drug Deliv. Rev., 56, 589, 2004. With permission.)... 

A better method uses two capillaries of differing radius, at the same immersion depth, and involves measuring the differential maximum bubble pressure between the two capillaries. In this case the two (Apgt) terms cancel out and the differential pressure is the difference between the two (2y/b) terms where b is the radius of curvature at the apex of the bubble. 

The maximum bubble pressure method. If a bubble is blown at the bottom of a tube dipping vertically into a liquid, the pressure in the bubble increases at first, as the bubble grows and the radius of curvature diminishes. It was shown in Chap. I, 13, that when the bubble is small enough to be taken as spherical, the smallest radius of curvature and the maximum pressure occurs when the bubble is a hemisphere further growth causes diminution of pressure, so that air rushes in and bursts the bubble. At this point the pressure in the bubble is... 

The average bubble radius (as well as the biggest and smallest size), the maximum distance between the opposite walls of the bubble (relative diameter) [8] and the specific liquid/air interface are involved in the estimation of dispersity. However, the distribution of bubbles by size, for example by radius of equivalent spheres, reveals completely the foam dispersity. Additional information about dispersity which takes into account the difference in polyhedral shapes is gathered from the number and shape of polyhedron faces (see Section 1.2). 

The principle is sketched in fig. 1.18. Consider a vertical capillary in the liquid to be studied. If left to itself, the liquid would rise as in fig. 1.4a. If now a gradually increasing pressure p is applied inside the capillary the liquid level can be pushed down. Upon arriving at the bottom of the capillary the meniscus passes successively through the stages 1, 2 and 3. In this order the bubble radius first diminishes, then rises, so p vrill pass through a maximum p(max). This maximum can be recorded as the pressure where the bubble starts to expand spontaneously, generally within lO s. At that point... 

A variant is the micro-pipette method, which is also similar to the maximum bubble pressure technique. A drop of the liquid to be studied is drawn by suction into the tip of a micropipette. The inner diameter of the pipette must be smaller than the radius of the drop the minimum suction pressure needed to force the droplet into the capillary can be related to the surface tension of the liquid, using the Young-Laplace equation [1.1.212). This technique can also be used to obtain interfacial tensions, say of individual emulsion droplets. Experimental problems include accounting for the extent of wetting of the inner lumen of the capillary, rate problems because of the time-dependence of surfactant (if any) adsorption on the capillary and, for narrow capillaries accounting for the work needed to bend the interface. Indeed, this method has also been used to measure bending moduli (sec. 1.15). 

At t = 0 (initial state), the pressure is low (note that the pressure is equal to 2y/r since r of the bubble is large, p will be small). At t = r (the smallest bubble radius that is equal to the tube radius), p reaches a maximum, whilst at... 

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