Abstract

Cavitation is the formation and subsequent dynamics of vapor filled or gas vapor cavities in a liquid when the local pressure becomes sufficiently low. A single cavitation bubble is the simplest experimentally accessible unit of this process. It can be produced by a focused laser pulse, an electric discharge, a rapid pressure reduction, a tensile acoustic pulse, or localized heating. Once the pressure recovers, the surrounding liquid accelerates inward and the bubble collapses. The bubble is called a cavitation bubble rather than an ordinary gas bubble because a substantial part of its volume is normally created by vaporization of the liquid, even though permanent gas and products of the generation process may also be present.

Chapter 3. Single Bubble Collapse Phenomena, Models, and Thermodynamic Effects

Introduction

Cavitation is the formation and subsequent dynamics of vapor filled or gas vapor cavities in a liquid when the local pressure becomes sufficiently low. A single cavitation bubble is the simplest experimentally accessible unit of this process. It can be produced by a focused laser pulse, an electric discharge, a rapid pressure reduction, a tensile acoustic pulse, or localized heating. Once the pressure recovers, the surrounding liquid accelerates inward and the bubble collapses. The bubble is called a cavitation bubble rather than an ordinary gas bubble because a substantial part of its volume is normally created by vaporization of the liquid, even though permanent gas and products of the generation process may also be present.

The classical starting point is Rayleigh's analysis of an empty spherical cavity in an incompressible inviscid liquid [1]. Plesset extended the radial dynamics to include pressure inside the bubble, viscosity, and surface tension [2]. These models establish the characteristic inertial time scale and explain why the final collapse accelerates so rapidly. They also contain an ideal mathematical singularity: if the cavity were truly empty, perfectly spherical, and surrounded by an incompressible liquid, the interface velocity and pressure would diverge as the radius approached zero. Real bubbles do not reach that state. Liquid compressibility, gas compression, vapor condensation, heat transfer, viscosity, and loss of spherical symmetry regularize the collapse. Gilmore's formulation and the analysis of Hickling and Plesset showed how compressibility permits pressure waves and rebound dynamics to be calculated [3,4].

A bubble rarely collapses in a perfectly uniform environment. Gravity, a background pressure gradient, neighboring interfaces, and especially a nearby solid wall make the collapse anisotropic. The side of the bubble exposed to the larger pressure impulse moves faster, and a high speed liquid microjet forms through the bubble. Early boundary studies established the relationship between collapse asymmetry, impulsive pressure, and material damage [5,6]. In parallel, thermal theory showed that the gas response cannot always be represented by a constant polytropic exponent and that heat transfer contributes both stiffness and damping [7]. Compressible radial equations developed by Keller and Miksis, by Fujikawa and Akamatsu, and by Prosperetti and Lezzi improved the treatment of acoustic radiation, nonequilibrium condensation, and finite sound speed [8-11].

High speed optical and acoustic experiments have since resolved the full sequence of growth, collapse, shock emission, jetting, rebound, and fragmentation [12]. At more extreme compression, the bubble interior can reach temperatures and pressures sufficient to produce sonoluminescence and chemical reactions. Models and measurements by Yasui, Pecha and Gompf, Storey and Szeri, Didenko and Suslick, and Flannigan and Suslick established that the peak state depends strongly on water vapor, heat loss, ionization, and the time scale of the final compression [13-18].

Theoretical work has also moved beyond spherical symmetry. Weakly compressible boundary integral formulations describe migration, deformation, jet penetration, and repeated topology changes [19-21]. Energy measurements and scaling laws have connected the shock, rebound, and jet energies to the initial potential energy and to a nondimensional anisotropy parameter [20,22-24]. More recent experiments have directly measured thermal fields around single bubbles and quantified the effects of ambient temperature, vapor pressure, and cryogenic fluids [25-29]. Saturation property correlations supplied by the International Association for the Properties of Water and Steam provide the thermodynamic basis for calculating the large increase in water vapor pressure with temperature [30].

The central physical idea of this report is that bubble collapse is an energy focusing event whose outcome is selected by five coupled mechanisms: inertial liquid motion, compressibility, bubble contents, geometric asymmetry, and thermodynamic exchange.

Figure 3.1. Idealized life cycle of an isolated cavitation bubble.

3.1 Phenomena Occurring at the Collapse of a Single Cavitation Bubble

3.1.1 Inertial focusing, minimum volume, and rebound

At maximum radius, a transient cavitation bubble is nearly at rest. Its internal pressure is usually below the recovered liquid pressure. The pressure imbalance accelerates a large mass of liquid toward the center. Because the same radial flow must pass through progressively smaller spherical surfaces, the interface speed rises sharply as the radius decreases. The kinetic energy density therefore becomes concentrated in a thin region around the bubble.

The ideal Rayleigh solution gives the characteristic collapse velocity scale:

$$U_R=\sqrt{\frac{\mathrm{\Delta }p}{\rho }},$$

(3.1)

where $\mathrm{\Delta }p$ is the pressure difference available to drive collapse. The mean collapse velocity is of order $R_{\max }/t_R$, which is also proportional to $U_R$. The instantaneous wall velocity can become many times larger than this mean value near the end of collapse. The real minimum radius is established by the rapid increase of internal gas and vapor pressure, by liquid compressibility, and by deviations from spherical shape. The pressure work not lost to outgoing waves or vortical motion is stored temporarily in compressed bubble contents and then drives a rebound [3,4,31].

Rebound is therefore an indicator of the state at collapse. A large rebound indicates that appreciable energy remained as internal energy or reversible compression. A small rebound indicates stronger acoustic radiation, jet formation, fragmentation, viscous loss, or thermal loss. Noncondensable gas has a particularly strong effect because it cannot disappear by condensation on the collapse time scale. Even a small amount can arrest the collapse at a larger radius and increase the rebound. Vapor can also cushion collapse if condensation and heat transfer are too slow to remove it.

The collapse and rebound need not be exactly periodic. Acoustic radiation removes energy abruptly near minimum volume, while viscosity and thermal conduction remove energy throughout the cycle. The radius of each subsequent rebound is consequently smaller. Laser produced bubbles may oscillate for many cycles, but the first collapse usually contains the strongest pressure impulse and the largest change in topology [12,31].

3.1.2 Shock wave emission

A rapidly collapsing spherical bubble behaves approximately as a monopole acoustic source. For small acoustic amplitudes and at a distance $r$ much greater than $R$, the radiated pressure can be estimated by:

$$p^{\prime }(r,t)\approx \frac{\rho }{r}\frac{d}{dt}{\left(R^2\dot{R}\right)}_{t-r/c}=\frac{\rho }{r}{(R^2\ddot{R}+2R{\dot{R}}^2)}_{t-r/c}.$$

(3.2)

The retarded time $t-r/c$ accounts for propagation at the sound speed. This expression shows that acoustic output is controlled by wall acceleration as well as velocity. During mild oscillation the signal is a smooth pressure pulse. During violent collapse, nonlinear steepening converts the pulse into a shock wave, and a linear far field formula becomes only an estimate [3,4].

Several distinct pressure transients can occur. The generation event itself may launch a shock, particularly for optical breakdown or electric discharge. A second and usually stronger transient is emitted at the first collapse. Further shocks can be emitted by rebound collapse and by the impact of a microjet on the opposite bubble wall or on a nearby solid boundary. Combined high speed optical and acoustic records show that the timing and amplitude of these peaks depend on bubble sphericity and on the location of jet impact [12,24,31].

For a nearly spherical collapse, the source is compact and the shock is emitted approximately in all directions. When the bubble is nonspherical, several separated shock sources can appear along a toroidal bubble or around fragmented lobes. The peak pressure then depends strongly on observation direction and distance. Supponen and coauthors showed that increasing collapse anisotropy redistributes energy away from a single strong spherical shock toward jets and multiple weaker shock sources [24].

Shock pressure is not identical to damaging capability. Material response depends on pulse duration, affected area, incidence angle, repeated loading, and the impedance of the solid. A very high peak acting over a microscopic area can initiate a pit, whereas a lower but repeated pressure can contribute to fatigue. The water hammer estimate:

$$p_{wh}\sim \rho c{U}_j$$

(3.3)

is often used as an upper scale for jet impact pressure, where $U_j$ is the jet velocity. It should not be interpreted as a universal measured pressure because compliance, entrapped gas, finite impact area, and shock propagation reduce or redistribute the load.

3.1.3 Asymmetric collapse and microjet formation

Any pressure anisotropy generates a net Kelvin impulse in the liquid. During growth, the bubble may remain nearly spherical because its interface velocity is modest. During collapse, the same weak asymmetry is amplified as the interface accelerates. The side experiencing the larger pressure impulse indents first, and a reentrant liquid jet crosses the bubble. Near a rigid wall, the jet generally points toward the wall. Near a free surface it points away from the free surface. Buoyancy and imposed pressure gradients similarly select the jet direction [5,6,19–21,23].

Two nondimensional parameters are widely useful. The standoff parameter for a bubble near a flat wall is:

$${\gamma }_s=\frac{h}{R_{\max }},$$

(3.4)

where $h$ is the distance from the bubble center at maximum volume to the wall. Small ${\gamma }_s$ means strong wall influence. A more general measure of weak pressure gradient anisotropy is:

$$\zeta =\frac{|\nabla p|R_{\max }}{\mathrm{\Delta }p}.$$

(3.5)

The parameter $\zeta$ is proportional to the normalized Kelvin impulse for several common forcing fields. It allows bubbles driven by gravity, nearby boundaries, and imposed pressure gradients to be compared on a common basis [20,23]. Supponen and coauthors identified approximate jet regimes: weak jets for $\zeta <{10}^{-3}$, intermediate jets for ${10}^{-3}<\zeta <0.1$, and strong jets for $\zeta >0.1$ [23]. The thresholds are useful scaling guides, not sharp universal boundaries.

When the jet pierces the opposite side of the bubble, the bubble becomes toroidal. Jet impact can launch a local shock and create a vortex ring. The torus may collapse around its circumference, break into daughter bubbles, or rebound while remaining connected to a wall. These topology changes invalidate a purely spherical radius model even if an equivalent radius can still be extracted from images.

Figure 3.2. Schematic sequence for a cavitation bubble near a rigid wall. The pressure field causes a jet toward the wall, followed by toroidal collapse and localized loading.

3.1.4 Material loading and cavitation erosion

Single bubble experiments are important because they isolate elementary erosion mechanisms that are obscured in a cavitating cloud. Benjamin and Ellis linked bubble collapse near a solid boundary to impulsive pressure [5]. Lauterborn and Bolle resolved the deformation and jetting sequence optically [6]. Philipp and Lauterborn then mapped pit formation as a function of bubble size and standoff distance for single laser produced bubbles [14]. These studies established that the most damaging condition is not necessarily the most spherical collapse or the smallest standoff distance. Damage depends on the competition among jet impact, toroidal collapse, shock focusing, and the area over which the load acts.

A typical erosion sequence is as follows. First, the bubble collapses asymmetrically and a jet accelerates toward the wall. Second, the jet penetrates the bubble and impacts either the wall or the liquid layer between the bubble and the wall. Third, the remaining toroidal bubble collapses, producing additional pressure peaks around an annular region. Fourth, rebound and secondary collapse may apply further loads. If the local stress exceeds the yield strength, a plastic pit forms. Repetition produces work hardening, cracking, grain removal, and mass loss.

The scale of the bubble matters. The available potential energy at maximum radius is approximately:

$$E_0=\frac{4\pi }{3}{R}_{\max }^3\mathrm{\Delta }p$$

(3.6)

Thus, at fixed pressure, available energy increases with the cube of radius. However, peak stress and pit geometry also depend on collapse distance and time, which scale with $R_{\max }$. Small bubbles can create intense microscopic loads, while large bubbles affect a larger area. Surface roughness, elastic modulus, yield strength, residual stress, and liquid temperature all modify the observed damage.

3.1.5 Thermal, optical, and chemical events

The final compression of a sufficiently spherical bubble can produce extreme internal conditions. An ideal gas compressed adiabatically would satisfy:

$$\frac{T_g}{T_{g0}}={\left(\frac{R_0}{R}\right)}^{3(\kappa -1)}$$

(3.7)

This relation is only a limiting estimate. In real cavitation bubbles, water vapor dissociates, heat is conducted into the liquid, shocks can form inside the gas, and the composition changes through condensation and chemical reactions. Detailed models therefore predict lower temperatures than a simple adiabatic law and show strong sensitivity to vapor content [13,16].

Sonoluminescence is a short flash emitted near minimum radius. Pecha and Gompf measured picosecond scale light pulses, demonstrating that the luminous state is much shorter than the overall bubble oscillation [15]. Flannigan and Suslick observed plasma related emission from single bubbles under suitable conditions [18]. The existence and spectrum of the flash depend on gas species, dissolved gas concentration, drive amplitude, water vapor, and bubble stability. A highly deformed cavitation bubble generally produces weaker light because the collapse is not focused into one small volume.

Chemical activity arises because high temperature and pressure can dissociate water vapor and dissolved species, producing radicals and reaction products. Didenko and Suslick compared the energy associated with photons, radicals, and mechanical effects, showing that only a fraction of the bubble energy appears as chemical or optical output [17]. Storey and Szeri demonstrated theoretically that water vapor can substantially reduce peak temperature through heat capacity, dissociation, and nonequilibrium transport [16]. These findings are directly relevant to the thermodynamic effect discussed in Section 3.

3.1.6 Energy partition at collapse

The initial potential energy does not convert into a single output. A useful bookkeeping relation is:

$$E_0\approx E_{shock}+E_{rebound}+E_{jet}+E_{visc}+E_{thermal}$$

(3.8)

The terms are not always separable without a model, and small residual kinetic energy may remain in the surrounding flow. Nevertheless, the relation clarifies why a reduction in shock energy can be accompanied by a larger rebound or stronger jet. Tinguely and coauthors measured shock and rebound energies for nearly spherical collapses and found systematic energy partitioning [22]. Obreschkow and coauthors and Supponen and coauthors extended the picture to jets and anisotropic collapses [20,23,24].

Figure 3.3. Principal energy pathways during first collapse. The diagram is conceptual.

Table 3.1. Main collapse phenomena and their physical consequences.

Phenomenon	Primary mechanism	Main observable	Engineering or scientific consequence
Rapid radial acceleration	Pressure work and geometric focusing	Strong decrease of $R(t)$ and high wall speed	Sets collapse time and velocity scale [1,2]
Shock emission	Liquid compressibility and rapid volume acceleration	Short pressure pulse or shock front	Noise, structural loading, energy loss [3,4,12,24,31]
Microjet	Collapse anisotropy and Kelvin impulse	Reentrant jet and toroidal bubble	Local impact, shear, pumping, erosion [5,6,14,20,23]
Rebound	Compression of permanent gas and retained vapor	Secondary bubble growth	Indicates residual internal energy [4,22,31]
Fragmentation	Jet penetration and interface instability	Daughter bubbles and multiple shocks	Distributed loading and enhanced mixing [12,21,24]
Heating and chemistry	Rapid gas compression and nonequilibrium reactions	Light emission and radical production	Sonochemistry and material modification [13,15–18]
Interfacial cooling or heating	Evaporation, condensation, and latent heat	Local temperature change	Modifies vapor pressure and collapse intensity [25–29]

3.2 Theoretical Models

3.2.1 Spherical kinematics and the pressure field

The simplest derivation assumes a spherical bubble in an incompressible liquid of constant density. Conservation of liquid volume gives radial velocity:

$$u(r,t)=\frac{R^2\dot{R}}{r^2}$$

(3.9)

A corresponding velocity potential is $\varphi =-R^2\dot{R}/r$. Substitution into the unsteady Bernoulli equation gives the liquid pressure field:

$$p(r,t)-p_{\infty }(t)=\rho [\frac{R^2\ddot{R}+2R{\dot{R}}^2}{r}-\frac{R^4{\dot{R}}^2}{2r^4}]$$

(3.10)

At the interface, $r=R$, this reduces to:

$$p_L-p_{\infty }=\rho (R\ddot{R}+\frac{3}{2}{\dot{R}}^2)$$

(3.11)

where $p_L$ is the liquid pressure is immediately outside the bubble. This relation is the inertial core of the Rayleigh and Rayleigh Plesset equations.

The assumptions are restrictive. Incompressibility implies instantaneous pressure communication and cannot represent shock radiation. Spherical symmetry excludes jets and translation. Nevertheless, the equation remains valuable because it exposes the balance between interface acceleration, convective inertia, and pressure.

3.2.2 Rayleigh empty cavity model

Rayleigh considered an empty cavity with constant vapor pressure neglected relative to the external pressure [1]. More generally, define a constant collapse in driving pressure:

$$\mathrm{\Delta }p=p_{\infty }-p_v>0$$

(3.12)

The radial equation is:

$$R\ddot{R}+\frac{3}{2}{\dot{R}}^2=-\frac{\mathrm{\Delta }p}{\rho }$$

(3.13)

Multiplying by $R^2\dot{R}$ and integrating from the maximum radius, where $\dot{R}=0$, gives:

$${\dot{R}}^2=\frac{2\mathrm{\Delta }p}{3\rho }[{\left(\frac{R_{\max }}{R}\right)}^3-1]$$

(3.14)

The wall speed therefore diverges as $R^{-3/2}$ in the ideal model. Integrating again gives the collapse time:

$$t_R=0.914681R_{\max }\sqrt{\frac{\rho }{\mathrm{\Delta }p}}$$

(3.15)

This equation is one of the most useful scaling laws in cavitation. It predicts that collapse time is proportional to bubble size, increases with liquid density, and decreases with the square root of the pressure difference. It is often used to nondimensionalize experimental time histories and to define a Rayleigh factor, the measured collapse time divided by $R_{\max }\sqrt{\rho /\mathrm{\Delta }p}$.

Figure 3.4. Dimensionless Rayleigh collapse trajectory obtained by numerical integration of the empty cavity equation [1]. The ideal model reaches zero radius with a divergent wall speed at.

The model predicts the first order collapse time remarkably well for large vapor cavities when viscosity and surface tension are small and when the bubble remains nearly spherical. It fails near minimum radius, where compressibility and internal pressure dominate. It also cannot predict rebound, shock amplitude, or jetting.

3.2.3 Rayleigh Plesset equation

For a spherical bubble containing gas and vapor, the normal stress condition is:

$$p_L=p_B-\frac{2\gamma }{R}-\frac{4\mu \dot{R}}{R}$$

(3.16)

Combining this relation with the inertial pressure balance gives the Rayleigh Plesset equation [2]:

$$R\ddot{R}+\frac{3}{2}{\dot{R}}^2=\frac{1}{\rho }[p_B(t)-p_{\infty }(t)-\frac{2\gamma }{R}-\frac{4\mu \dot{R}}{R}]$$

(3.17)

An acoustically driven far field may be represented as:

$$p_{\infty }(t)=p_0-p_A\sin (\omega t)$$

(3.18)

where $p_A$ is the acoustic pressure amplitude. For a transient cavitation bubble, $p_{\infty }(t)$ can instead be a step, a measured pressure waveform, or a computed local pressure history.

A common bubble pressure closure is:

$$p_B=p_{sat}\left(T_i\right)+p_{g0}{\left(\frac{R_0}{R}\right)}^{3\kappa }$$

(3.19)

with the equilibrium gas pressure:

$$p_{g0}=p_0+\frac{2\gamma }{R_0}-p_{sat}\left(T_{\infty }\right)$$

(3.20)

This closure treats vapor pressure as spatially uniform and permanent gas as polytropic. It is adequate for many moderate oscillations. It becomes unreliable during violent collapse because $T_i$ and vapor mass change rapidly, the gas may be spatially nonuniform, and chemical reactions can occur [7,9,13,16].

The surface tension term is most important for small bubbles, while viscosity is important for small radii and lower Reynolds number. For a millimeter scale transient cavity in water, both terms are usually secondary through most of the first collapse but can become significant near the minimum radius.

3.2.4 Interior thermodynamic closures

The main uncertainty in a spherical bubble equation is often not liquid inertia but the expression for $p_B$. Several levels of closure are used.

Constant vapor pressure and polytropic gas. This is the expression above. The exponent $\kappa =1$ represents isothermal gas behavior and $\kappa$ equal to the heat capacity ratio represents ideal adiabatic behavior. Real oscillations generally lie between these limits, and the effective value depends on frequency and radius [7].

Uniform temperature energy model. The bubble contents are treated as well mixed. An energy equation determines a time dependent gas temperature, and vapor mass changes through an interfacial flux. This approach captures latent heat and variable pressure at moderate computational cost.

Radially resolved gas vapor model. Conservation equations for species, momentum or low Mach flow, and energy are solved inside the bubble. This permits thermal boundary layers, vapor diffusion, nonequilibrium condensation, and chemical reactions [9,13,16]. It is needed for accurate peak temperature and sonochemical predictions.

Equation of state closure. At very high pressure, ideal gas behavior may fail. A real gas equation of state and variable transport properties may be required. For cryogenic fluids or mixtures, phase equilibrium and fluid properties must be evaluated at the local composition and temperature [27,28].

The choice of closure can change the predicted minimum radius, peak pressure, rebound energy, and radiated shock. A radius curve that matches experiment does not guarantee that the internal temperature or composition is correct.

3.2.5 Keller Miksis equation

The Rayleigh Plesset equation neglects the finite speed at which pressure disturbances propagate. The Keller -Miksis equation retains first order terms in the bubble wall Mach number $\dot{R}/c$ [8]. The compact form is:

$$(1-\frac{\dot{R}}{c})R\ddot{R}+\frac{3}{2}(1-\frac{\dot{R}}{3c}){\dot{R}}^2=\frac{1}{\rho }(1+\frac{\dot{R}}{c}+\frac{R}{c}\frac{d}{dt})[p_L-p_{\infty }(t)]$$

(3.21)

Here:

$$p_L=p_B-\frac{2\gamma }{R}-\frac{4\mu \dot{R}}{R}$$

(3.22)

The additional terms represent acoustic radiation and the delay associated with compressibility. The equation is widely used because it is only modestly more expensive than Rayleigh Plesset while providing realistic damping and collapse pressure for wall Mach numbers that remain moderate. It is not fully reliable when $\left|\dot{R}\right|/c$ approaches unity or when the liquid equation of state becomes strongly nonlinear.

A practical advantage is that measured or prescribed acoustic pressure waveforms can be inserted directly. A practical limitation is that numerical integration becomes stiff near collapse. The internal pressure model and its time derivative must also be implemented consistently. Discontinuous pressure forcing or a crude vapor pressure closure can create unphysical spikes.

3.2.6 Gilmore equation

Gilmore formulated spherical bubble dynamics in terms of the enthalpy and local sound speed of a compressible liquid [3]. The equation is:

$$(1-\frac{\dot{R}}{C})R\ddot{R}+\frac{3}{2}(1-\frac{\dot{R}}{3C}){\dot{R}}^2=(1+\frac{\dot{R}}{C})H+(1-\frac{\dot{R}}{C})\frac{R}{C}\dot{H}$$

(3.23)

Where:

$$H={\int }_{p_{\infty }}^{p_L}\frac{dp}{\rho (p)}$$

(3.24)

is the enthalpy difference and $C$ is the sound speed evaluated at the liquid pressure adjacent to the wall. A liquid equation of state, commonly a Tait type relation for water, supplies $\rho (p)$ and $C(p)$.

The Gilmore equation is preferred when the pressure near the wall is high enough that the sound speed and density differ substantially from ambient values. It can represent strong compression and shock emission more robustly than a constant sound speed equation. It remains a spherical model and cannot resolve a propagating shock profile after it leaves the bubble. The predicted far field pressure must be obtained through an acoustic propagation relation or a separate compressible flow calculation.

3.2.7 Prosperetti Lezzi and related compressible equations

Prosperetti and Lezzi derived a systematic family of compressible bubble equations through expansion in a small Mach number parameter [10,11]. The first order theory contains a free parameter that recovers several earlier equations for particular choices, while the second order theory retains additional compressibility terms. The analysis is important because it distinguishes approximations that are formally equivalent to first order from differences that matter at the next order.

In practical computations, the selection among Keller Miksis, Gilmore, and Prosperetti Lezzi formulations should be based on the pressure range, the expected wall Mach number, and the required output. For radius histories and moderate acoustic radiation, first order equations are often sufficient. For very high pressures, a variable property formulation or direct compressible simulation is more appropriate. No radial equation should be expected to predict a wall directed jet.

3.2.8 Nonequilibrium condensation and phase change models

Fujikawa and Akamatsu showed that nonequilibrium condensation can alter the pressure wave produced by bubble collapse [9]. In a pure vapor cavity, an assumption of instantaneous equilibrium would force $p_v=p_{sat}\left(T_i\right)$ at every instant. In reality, condensation is limited by molecular accommodation, diffusion, and heat removal. Vapor can therefore remain supersaturated or be compressed before it condenses.

A simplified kinetic relation for vapor mass flux is the Hertz-Knudsen form:

$$j={\alpha }_m\frac{p_{sat}\left(T_i\right)-p_v}{\sqrt{2\pi R_v{T}_i}}$$

(3.25)

where ${\alpha }_m$ is a mass accommodation coefficient and $R_v$ is the specific gas constant of the vapor. Positive $j$ denotes net evaporation into the bubble. The formula is a useful closure, but ${\alpha }_m$ is often treated as an effective parameter because interface kinetics, contamination, and temperature nonuniformity are not fully resolved.

Phase change models become especially important for high vapor pressure liquids, hot water, cryogenic propellants, and mixtures. In those conditions, vapor mass is not passive pressure offset. It is a dynamic energy and momentum reservoir.

3.2.9 Nonspherical potential flow and boundary integral models

When viscosity and vorticity are confined to thin regions, the external liquid can be approximated as potential flow:

$${\nabla }^2\varphi =0$$

(3.26)

The bubble surface moves with the liquid,

$$\frac{Dx}{Dt}=\nabla \varphi$$

(3.27)

and the potential on the interface satisfies:

$$\frac{D\varphi }{Dt}=\frac{1}{2}|\nabla \varphi {|}^2+\frac{p_{\infty }-p_L}{\rho }$$

(3.28)

A boundary integral method converts the Laplace problem into an integral over the bubble and solid boundaries. Only the surfaces need to be discretized, so the method is efficient for large interface deformation. Image systems or explicit boundary panels enforce rigid wall or free surface conditions. Weak compressibility corrections can be included, as in the formulations of Wang and Blake [19,21].

Boundary integral methods predict migration, jet formation, toroidal transition, and the dependence of jet direction on the forcing field. Their principal limitations are the assumptions of irrotational flow and a single valued interface representation. Jet impact creates a topological change that requires special reconnection algorithms. After impact, vortex rings, splashing, and fragmentation become increasingly important. Strong shock propagation also lies outside incompressible potential flow.

3.2.10 Compressible multiphase computational fluid dynamics

Direct numerical simulation solves conservation equations in both phases and in a mixture formulation. Interface capturing methods such as volume of fluid, level set, and diffuse interface schemes can handle jet impact, toroidal bubbles, breakup, and shock interaction. A general compressible two phase formulation includes:

$$\frac{\partial 𝛒}{\partial 𝐭}+\nabla \cdot \left(𝛒𝐮\right)=\mathrm{𝟎}$$

(3.29)

$$\frac{\partial (\rho u)}{\partial t}+\nabla \cdot (\rho uu+pI-\tau )=\rho g+f_{\gamma }$$

(3.30)

and

$$\frac{\partial (\rho E)}{\partial t}+\nabla \cdot [(\rho E+p)u-\tau \cdot u+q]={\dot{Q}}_{phase}$$

(3.31)

The equation of state, interface method, surface tension treatment, and phase change source terms determine the physical fidelity. The computational challenge is severe because the bubble radius, thermal boundary layer, jet tip, and shock thickness occupy very different length scales. Numerical diffusion can smear the interface and reduce peak pressure, while insufficient resolution can delay jet impact or alter fragmentation.

Compressible computational fluid dynamics is most appropriate when the research question concerns shock structure, topology change, interaction with complex geometry, or local wall loading. It should be validated against radius histories, jet speeds, shock arrival times, and energy conservation, not against a single output alone.

Figure 3.5. Hierarchy of common single bubble models. Increasing physical completeness generally requires increasing computational cost.

3.2.11 Dimensionless groups and model selection

Choose the characteristic velocity $U_R=\sqrt{\mathrm{\Delta }p/\rho }$ and length $R_{\max }$. The following groups indicate the relative importance of physical mechanisms.

Table 3.2. Dimensionless groups for single bubble dynamics.

Group	Definition	Physical interpretation
Mach number	$Ma=U_R/c$ or $|\dot{R}|/c$	Importance of liquid compressibility and shock formation
Reynolds number	$Re=\rho R_{\max }{U}_R/\mu$	Ratio of inertia to viscosity
Weber number	$We=\rho R_{\max }{U}_R^2/\gamma$	Ratio of inertia to surface tension
Thermal Peclet number	$P{e}_T=R_{\max }{U}_R/{\alpha }_l$	Ratio of advection time to thermal diffusion time
Fourier number	$Fo={\alpha }_{l}t/R_{\max }^2$	Fraction of the radius reached by thermal diffusion
Jakob number	$Ja={\rho }_l{c}_{p,l}|T_{\infty }-T_i|/({\rho }_{v}L)$	Sensible heat available relative to latent heat demand
Standoff parameter	${\gamma }_s=h/R_{\max }$	Strength of a nearby wall influence
Anisotropy parameter	$\zeta =|\nabla p|R_{\max }/\mathrm{\Delta }p$	Normalized pressure gradient or Kelvin impulse scale

A practical model selection rule follows.

• Use the Rayleigh model for a first collapse time and velocity scale.

• Use Rayleigh Plesset for spherical motion when compressibility is weak and shock amplitude is not required.

• Use Keller Miksis or a first order Prosperetti Lezzi equation when acoustic radiation and moderate wall Mach number matter.

• Use Gilmore when liquid properties vary strongly with pressure or collapse is very violent.

• Use a boundary integral method for nonspherical shape and jetting when shocks and viscosity are secondary.

• Use compressible multiphase computational fluid dynamics for simultaneous shock, jet, topology change, and complex boundary interaction.

• Add coupled heat and mass transfer whenever vapor pressure, condensation rate, or peak temperature is central to the question.

Table 3.3. Comparison of principal theoretical models.

Model	Included physics	Best use	Main limitation
Rayleigh [1]	Spherical inertia and constant pressure difference	Collapse time and scaling	Empty cavity, no rebound, no shock physics
Rayleigh Plesset [2]	Gas pressure, vapor pressure, surface tension, viscosity	Moderate spherical oscillations	Incompressible liquid
Keller Miksis [8]	First order liquid compressibility	Acoustic driving and radiative damping	Constant reference sound speed, spherical shape
Gilmore [3]	Variable density, sound speed, and enthalpy	Strong spherical collapse	Requires liquid equation of state, no jets
Prosperetti Lezzi [10,11]	Systematic first and second order compressibility	Error controlled radial modeling	More complex implementation
Boundary integral [19,21]	Nonspherical potential flow and boundaries	Migration, jetting, topology studies	Weak shock and viscous capability
Compressible multiphase CFD	Shocks, viscosity, complex geometry, topology change	Local loads and coupled flow structure	High computational cost and resolution sensitivity
Thermal phase change model [9,13,16]	Heat transfer, vapor transport, reactions	Vapor rich, hot, cryogenic, or luminous collapse	Property data and closure uncertainty

3.3 The Thermodynamic Effect

3.3.1 Meaning of the thermodynamic effect

In cavitation research, the thermodynamic effect is the feedback of evaporation, condensation, latent heat, and temperature dependent material properties on bubble dynamics. It is sometimes described only as a suppression of cavitation in cryogenic liquids, but the concept is broader. It operates during both growth and collapse.

During growth, liquid evaporates into the bubble. Vaporization consumes latent heat from the interface and adjacent liquid. The interfacial temperature falls, lowering the local saturation pressure. This reduction in $p_{sat}\left(T_i\right)$ opposes further evaporation and can limit bubble growth.

During collapse, vapor condenses and releases latent heat. The interface can warm relative to the surrounding liquid. If condensation and heat removal are fast, vapor mass decreases and the collapse approaches an inertial cavity collapse. If they are slow, vapor is compressed and maintains a higher internal pressure. This vapor cushioning increases the minimum radius, reduces wall velocity, weakens the collapse shock, and can increase the symmetry of the bubble by reducing the time available for an anisotropic jet to develop [9,25–29].

The thermodynamic effect therefore has two linked aspects: a local temperature effect on phase equilibrium, and a finite rate effect on mass transfer. Treating vapor pressure as a constant equal to the ambient saturation pressure includes neither aspect fully.

Figure 3.6. Thermodynamic feedback during growth and collapse.

3.3.2 Heat and mass balances

The temperature field in the liquid surrounding a spherical bubble satisfies, in a simplified constant property form:

$$\frac{\partial T_l}{\partial t}+u\frac{\partial T_l}{\partial r}={\alpha }_l\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial T_l}{\partial r}\right)$$

(3.32)

The radial velocity $u=R^2\dot{R}/r^2$ advects the thermal boundary layer. A characteristic thermal penetration thickness is:

$${\delta }_T\sim \sqrt{{\alpha }_{l}t}$$

(3.33)

If ${\delta }_T\ll R$, only a thin layer of liquid participates in rapid heat exchange. The bulk liquid can remain close to $T_{\infty }$ while the interface is several kelvin colder or hotter.

For a spatially uniform bubble interior, a useful energy balance is:

$$\frac{d{U}_B}{dt}=-p_B\frac{dV}{dt}+\dot{Q}+{\dot{m}}_v{h}_v$$

(3.34)

where $\dot{Q}$ is heat transferred into the bubble and ${\dot{m}}_v$ is positive for vapor entering the bubble. The last term carries the enthalpy of phase changing mass. In a detailed model, separate gas and vapor species equations are solved and the work term is coupled to a real gas or mixture equation of state.

At the interface, heat flux and latent heat must balance. Independent of sign convention, the magnitude of the latent heat demand is of order:

$$|j|L\sim |q_l-q_v|q=-k\nabla T$$

(3.35)

This relation explains why a large latent heat and a limited conductive flux produce a substantial temperature departure. It also explains why thermodynamic effects can be strong even when the measured bulk temperature is uniform.

3.3.3 Saturation pressure feedback

The Clausius Clapeyron relation connects phase equilibrium to temperature:

$$\frac{d{p}_{sat}}{dT}=\frac{L}{T(v_v-v_l)}\approx \frac{L{p}_{sat}}{R_v{T}^2}$$

(3.36)

where the approximation treats the vapor as ideal and neglects the liquid specific volume. Because the derivative increases with saturation pressure, the same interfacial temperature change has a much larger pressure consequence at high temperature or in a volatile liquid.

For water, the IAPWS saturation correlation can be written in the form:

$$\ln \left(\frac{p_{sat}}{p_c}\right)=\frac{T_c}{T}\sum _{i=1}^6{a}_i{\tau }^{b_i}\tau =1-\frac{T}{T_c}$$

(3.37)

with coefficients specified in the IAPWS release [30]. Between 20 and 90 degrees Celsius, water saturation pressure rises from approximately 2.34 kPa to 70.2 kPa. At one atmosphere, the nominal Rayleigh pressure difference therefore falls from about 99.0 kPa to 31.1 kPa.

Figure 3.7. Saturation pressure of water calculated from the IAPWS correlation [30]. The dashed line is standard atmospheric pressure.

The collapse time contribution from the pressure difference alone follows:

$$t_R(T)\propto {[p_{\infty }-p_{sat}(T)]}^{-1/2}$$

(3.38)

Relative to 20 degrees Celsius and at constant density,

$$\frac{t_R(T)}{t_R\left({20}^{\circ }C\right)}={\left[\frac{p_{\infty }-p_{sat}\left({20}^{\circ }C\right)}{p_{\infty }-p_{sat}(T)}\right]}^{1/2}$$

(3.39)

The calculated ratio reaches about 1.78 at 90 degrees Celsius. This curve isolates only the reduced pressure drive. Real bubble dynamics also involve temperature dependent density, viscosity, surface tension, gas content, heat transfer, and nonequilibrium phase change.

Figure 3.8. Normalized Rayleigh collapse time for water at 101.325 kPa, calculated using the IAPWS saturation pressure [30] and constant liquid density. The curve shows only the pressure drive contribution to the thermodynamic effect.

3.3.4 Effective gas behavior and thermal damping

The polytropic relation $p_g{V}^{\kappa }=C_g$, where $C_g$ is constant for a prescribed polytropic process, is an economical representation of gas thermodynamics. The exponent $\kappa$ is not universal for an oscillating bubble. Prosperetti showed that thermal conduction makes the effective exponent and damping frequency dependent [7]. A slowly oscillating gas bubble tends toward isothermal behavior because heat has time to cross the gas. At sufficiently rapid compression, the core tends toward adiabatic behavior, while a thermal boundary layer persists near the interface.

A thermal Peclet number:

$$P{e}_T=\frac{R\left|\dot{R}\right|}{{\alpha }_l}$$

(3.40)

compares interface motion with liquid thermal diffusion. Large $P{e}_T$ means that the interface moves faster than heat can diffuse across a radius, so a thin thermal layer develops. The Fourier number $Fo={\alpha }_{l}t/R^2$ gives the complementary diffusion measure. These groups vary by orders of magnitude during one collapse because both $R$ and $\left|\dot{R}\right|$ change rapidly.

Vapor adds another relaxation process. If the condensation time is short compared with collapse time, vapor pressure remains near equilibrium and vapor mass falls rapidly. If the condensation time is long, vapor behaves more like a compressible gas. The transition changes both the effective stiffness and the energy dissipation. This is why the same radial equation can predict different shock amplitudes depending on the chosen phase change model [9,16].

3.3.5 Direct measurements of interfacial temperature change

Dular and Coutier Delgosha performed an experimental study of temperature variations around a single cavitation bubble [25]. They reported cooling of approximately 3 K in the liquid near the bubble during growth and heating of up to approximately 4 K during collapse. The measurements provided direct evidence that latent heat creates a transient thermal boundary layer even in room temperature water.

The sign change is physically consistent. Evaporation during growth removes energy from the nearby liquid, while condensation and compression during collapse return energy. The spatial and temporal resolution of such measurements is challenging because the thermal layer is thin, the interface moves rapidly, and optical access can be distorted by the curved bubble surface. Consequently, measured temperature excursions should be understood as resolved values over a finite sensing volume rather than exact molecular interface temperatures.

The Dular measurements also show why a bulk temperature measurement is insufficient. A liquid bath can be isothermal to within a small fraction of a kelvin, while the bubble interface experiences a much larger transient excursion. Models that use $p_{sat}\left(T_{\infty }\right)$ at all times miss this local feedback.

3.3.6 Ambient temperature effects in water

Phan and coauthors investigated single bubble dynamics in water over a broad ambient temperature range [26]. They found that maximum radius, first minimum radius, and collapse time increased with temperature, while predicted peak internal pressure and temperature decreased. The trend is consistent with a lower collapse in driving pressure and stronger vapor cushioning at higher saturation pressure.

Pei and coauthors extended experiments to 23 to 90 degrees Celsius in free field and near wall configurations [29]. They reported a monotonic weakening of collapse with temperature, evidenced by changes in the Rayleigh factor, minimum bubble volume, maximum collapse velocity, jet velocity, and bubble migration. Above approximately 70 degrees Celsius, secondary cavitation nuclei appeared near the primary bubble surface around maximum expansion and later contributed to surface wrinkles, instability, and fission.

Several mechanisms operate simultaneously as water is heated:

• $p_{sat}$ increases sharply, reducing $\mathrm{\Delta }p$ at fixed ambient pressure.

• More vapor can be present in the bubble at maximum radius.

• Condensation must remove a larger vapor mass during collapse.

• Latent heat exchange produces stronger pressure temperature feedback.

• Viscosity and surface tension decrease, which can promote deformation even while the radial collapse becomes weaker.

Dissolved gas solubility and nucleation conditions change.

The resulting collapse is generally slower and less intense, but not necessarily simpler. High temperature can reduce peak shock and jet velocity while increasing instability, secondary cavitation, and fragmentation. A model that changes only $p_{sat}$ captures the leading pressure effect but not the complete behavior.

3.3.7 Cryogenic liquids

Cryogenic cavitation is often strongly thermodynamic because the vapor density is relatively high and latent heat consumption can cool the liquid appreciably. Chen and coauthors compared single bubbles in liquid nitrogen with bubbles in room temperature water at similar ambient pressure [28]. Their experiments and model indicated a weaker collapse in liquid nitrogen. The minimum bubble radius was more than three times that in water in their comparison, and the maximum wall Mach number was approximately one fifth of the water value. The authors attributed the difference to the smaller effective pressure difference and to strong heat and mass transfer.

The result illustrates an important distinction. A cryogenic liquid can have a low absolute temperature but a large thermodynamic effect. What matters is not temperature alone, but the combination of saturation pressure slope, vapor density, latent heat, thermal conductivity, heat capacity, and the available thermal subcooling. Cryogenic pump and rocket propellant cavitation therefore cannot be extrapolated directly from room temperature water tests by matching only a conventional cavitation number.

A useful qualitative parameter is the Jakob number:

$$Ja=\frac{{\rho }_l{c}_{p,l}(T_{\infty }-T_i)}{{\rho }_{v}L}$$

(3.41)

Small $Ja$ indicates that limited sensible heat is available relative to the latent heat required to create vapor, so local cooling can strongly suppress evaporation. Definitions vary in the cavitation literature, and care is required when comparing values based on different reference temperatures or vapor densities.

3.3.8 High vapor pressure mixtures

Preso and coauthors used aqueous ammonia to vary condensable vapor pressure while retaining a water based liquid system [27]. Increasing ammonia content increased vapor pressure and altered the collapse. The higher vapor pressure bubbles resisted collapse, dissipated less total energy, and remained more spherical. The study also showed that vapor can be compressed far from equilibrium during collapse.

Mixtures introduce additional physics. The vapor composition may differ from the liquid composition, each species has its own volatility, diffusion rates may differ, and condensation can enrich one component. A single saturation pressure function is then insufficient. Vapor liquid equilibrium, species diffusion, and mixture thermodynamics must be coupled to the bubble equation. The aqueous ammonia experiments are valuable because they separate vapor pressure effects from a simple change of bulk temperature.

3.3.9 Consequences for shock, jet, rebound, and erosion

Thermodynamic effects alter every major collapse output.

Shock wave. Higher retained vapor pressure limits the maximum inward velocity and softens the pressure rise at minimum radius. The principal collapse shock is therefore generally weaker. Nonequilibrium condensation can still create a sharp pressure transient if vapor is compressed and then condenses rapidly [9].

Microjet. A lower pressure difference reduces the characteristic jet velocity. Pei and coauthors observed lower near wall jet speed and migration as temperature increased [29]. However, a longer collapse time can allow some perturbations to grow, and reduced surface tension can increase interface distortion. Jet strength and shape must therefore be evaluated together.

Rebound. Retained vapor and permanent gas increase the minimum radius and can supply energy to the rebound. At the same time, weaker collapse can reduce shock energy loss. A relatively large rebound does not necessarily mean a more violent collapse; it may indicate stronger cushioning.

Erosion. Reduced shock and jet velocities usually decrease the severity of individual impacts. This is one reason cavitation erosion can decrease at elevated temperature or in cryogenic liquids. Material properties also vary with temperature, and cloud scale cavitation can behave differently from a single bubble. Single bubble results isolate the hydrodynamic contribution but do not by themselves predict component life.

Optical and chemical output. Additional vapor increases heat capacity and endothermic dissociation, which lowers peak temperature and can suppress light emission or alter radical yields [13,16–18]. A more spherical but more vapor rich bubble can therefore have weaker luminescence than a drier bubble with the same maximum radius.

3.3.10 Modeling recommendations for thermodynamic regimes

A thermodynamically consistent single bubble calculation should use the following minimum structure:

• A compressible radial equation, normally Keller Miksis or Gilmore, if the collapse shock or rebound is required.

• Separate permanent gas and vapor partial pressures.

• An energy equation for the bubble contents, with temperature dependent heat capacities where needed.

• A liquid thermal diffusion model or a validated boundary layer approximation.

• A phase change of law driven by $p_{sat}\left(T_i\right)-p_v$.

• Temperature dependent on liquid density, viscosity, surface tension, sound speed, latent heat, and saturation pressure.

• A real fluid equation of state for cryogenic liquids or high pressure collapse.

• Validation against both $R(t)$ and at least one energy related quantity such as rebound radius, shock pressure, or interfacial temperature.

A constant vapor pressure Rayleigh Plesset model remains useful for preliminary estimates. It should be labeled as such, particularly above about 60 to 70 degrees Celsius in water at atmospheric pressure, where saturation pressure becomes a large fraction of ambient pressure. In cryogenic liquids and volatile mixtures, a phase change model is not an optional refinement but part of the primary force balance.

Table 3.4. Representative experimental observations of thermodynamic effects in single bubbles.

Study	Liquid and condition	Main observation	Modeling implication
Dular and Coutier Delgosha [25]	Water near ambient conditions	About 3 K cooling during growth and up to about 4 K heating during collapse	Resolve a transient liquid thermal layer and local $T_i$
Phan et al. [26]	Water, 20 to 80 degrees Celsius	Longer collapse and larger minimum radius at higher temperature; lower predicted peak pressure and temperature	Include temperature dependent saturation pressure and heat transfer
Preso et al. [27]	Water and aqueous ammonia	Higher vapor pressure resisted collapse and reduced total energy dissipation	Model mixture vapor composition and nonequilibrium compression
Chen et al. [28]	Liquid nitrogen compared with water	Larger minimum radius and much lower maximum Mach number in liquid nitrogen	Use cryogenic properties, phase change, and variable compressibility
Pei et al. [29]	Water, 23 to 90 degrees Celsius	Monotonic collapse weakening, lower jet speed, secondary cavitation above about 70 degrees Celsius	Couple thermal attenuation with instability and nucleation behavior

3.4 Conclusions

A single cavitation bubble provides a controlled setting for studying one of the strongest forms of hydrodynamic energy focusing. The bubble grows when local pressure falls or energy is deposited, reaches a maximum volume, and collapses when the surrounding pressure recovers. During collapse, liquid inertia drives rapid radial acceleration. The finite sound speed, bubble contents, and geometry then determine how the energy is partitioned.

The principal collapse phenomena are shock wave emission, rebound, microjet formation, toroidal transition, fragmentation, localized wall loading, heating, light emission, and chemical activity. Spherical collapse concentrates energy into a compact shock and internal compression. Nonspherical collapse redirects part of that energy into a jet, translation, vortex motion, and multiple shock sources. A nearby rigid wall generally attracts the bubble and directs the jet toward the wall, but erosion is governed by the complete jet and torus sequence rather than by one pressure peak alone.

The Rayleigh model remains the correct first step because it provides the collapse time $t_R=0.914681R_{\max }\sqrt{\rho /\mathrm{\Delta }p}$ and the fundamental inertial scale. The Rayleigh Plesset equation adds gas pressure, vapor pressure, viscosity, and surface tension. Keller Miksis and Prosperetti Lezzi equations add first order compressibility, while the Gilmore equation accounts for pressure dependent density and sound speed. Boundary integral methods are needed for deformation and jetting. Compressible multiphase computational fluid dynamics is needed when shocks, viscosity, topology change, and complex wall geometry must be resolved simultaneously.

The thermodynamic effect is dynamic feedback, not merely a correction to vapor pressure. Evaporation cools the interface and suppresses further growth. Condensation releases latent heat, while finite condensation rates can leave vapor inside the bubble and cushion collapse. As water temperature rises, saturation pressure increases sharply and the collapse driving pressure decreases. Experiments in hot water, liquid nitrogen, and aqueous ammonia confirm larger minimum radii, lower collapse velocity, weaker shocks or jets, and altered rebound and stability. Direct measurements also confirm several kelvin of transient cooling and heating near the interface.

For engineering prediction, the model must be matched to the output of interest. Collapse time can often be estimated from Rayleigh scaling. Shock amplitude requires compressibility. Jet loading requires nonspherical dynamics. Peak temperature, vapor content, and cryogenic behavior require coupled heat and mass transfer. Reliable simulations should be validated against multiple observables, because agreement with the radius history alone does not establish that energy partition, internal pressure, or thermal behavior is correct.

Nomenclature

The following key defines the symbols and abbreviations used in this chapter. Unless otherwise stated, quantities are expressed in SI units.

Symbol	Definition	SI unit or status
c	Small signal sound speed in the liquid	m s⁻¹
C	Local sound speed at the bubble wall in the Gilmore model	m s⁻¹
$E_0$	Initial bubble potential energy	J
$E_{\mathrm{jet}}$	Energy in the liquid jet and directed flow	J
$E_{\mathrm{rebound}}$	Rebound bubble energy	J
$E_{\mathrm{shock}}$	Radiated shock wave energy	J
$E_{\mathrm{thermal}}$	Energy transferred by heat and phase change	J
$E_{\mathrm{visc}}$	Viscous and vortical energy loss	J
h	Distance from the bubble center to a nearby wall	m
H	Liquid enthalpy difference	m² s⁻²
j	Interfacial vapor mass flux, positive into the bubble	kg m⁻² s⁻¹
Ja	Jakob number	dimensionless
$L_v$	Latent heat of vaporization	J kg⁻¹
Ma	Mach number	dimensionless
$p_B$	Total pressure inside the bubble	Pa
$p_g$	Noncondensable gas partial pressure	Pa
$p_{sat}$	Saturation vapor pressure	Pa
$p_{\infty }$	Far field liquid pressure	Pa
Pe	Thermal Peclet number	dimensionless
R	Equivalent bubble radius	m
$R_0$	Reference or equilibrium radius	m
$R_{\max }$	Maximum bubble radius	m
$R_{\min }$	Minimum bubble radius	m
r	Radial coordinate from the bubble center	m
$t_R$	Rayleigh collapse time	s
$T_i$	Interfacial temperature	K
$T_{\infty }$	Ambient liquid temperature	K
$U_j$	Microjet velocity	m s⁻¹
$U_R$	Rayleigh velocity scale	m s⁻¹
$V_b$	Bubble volume	m³

Greek Letters

Symbol	Letter name	Meaning in this chapter
${\alpha }_l$	Alpha	Liquid thermal diffusivity
$\gamma$	Gamma	Surface tension
${\gamma }_s$	Gamma	Bubble stand off parameter relative to a wall
$\kappa$	Kappa	Polytropic exponent of the bubble gas
$\mu$	Mu	Dynamic viscosity
$\rho$	Rho	Liquid density
$\zeta$	Zeta	Collapse anisotropy parameter
$\varphi$	Phi	Velocity potential in potential flow models
$\tau$	Tau	Viscous stress tensor in multiphase flow equations
$\mathrm{\Delta }$	Delta	Collapse driving pressure difference

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