Figure 3.5: Mach reflection of a shock wave at a rigid surface [22].
3.2 Spherical bubble dynamics
Figure 3.6 displays schematically a spherical bubble in an infinite liquid. As-suming there is no boundary or shock wave or other sources of perturbation like buoyancy forces, the cavity may oscillate spherically [43]. The density of the liquid ρis supposed to be constant although the compressibility of the liquid at the moment of cavity collapse can have a significant effect. It is also assumed that the bubble contents are homogeneous and temperature TB and pressure PB of the cavity interior are uniform [14]. The fluid motion around the bubble is described by the velocity potential φ, which satisfies the Laplace equation [43]:
∇2φ = 0 , (3.36)
and the fluid velocity is obtained by:
u=∇φ . (3.37)
Considering the conservation of mass, in the absence of mass transfer across the interface, the radial outward velocity of the fluid u(r, t) at distance of r from the bubble center will be [14]:
u(r, t) = R2 r2
dR
dt , (3.38)
where R(t)is the bubble radius at time t. For the velocity potential this yields [43]:
φ=−R2 r
dR
dt . (3.39)
The Navier-Stokes equation of motion for a Newtonian fluid is [14]:
−1
Figure 3.6: Schematic illustration of a spherical bubble in an infinite medium [14].
where νL being the kinematic viscosity of the liquid and the quantity P is the pressure in the medium at a distance of r from the cavity center. By substituting the velocity u(r, t)from equation (3.38) into equation (3.40), one obtains:
with F(t) = R2dRdt. It is worth mentioning that the viscous term has disap-peared in this equation. By integrating equation (3.41) we have:
P −P∞
whereP∞ is the pressure at infinite distance from the bubble center.
Ignoring evaporation or condensation (mass transport across the bubble wall), the net force on an assumed lamina on the bubble surface in the ra-dially outward direction per unit area is zero. Thus:
PB=Pr=R+4µL R
dR dt +2S
R , (3.43)
whereµLand Sare dynamic viscosity and surface tension of the liquid, respec-tively. Substituting the value ofPr=R from equation (3.42) and considering the relationF(t) =R2dRdt, one gets [14]:
This equation is called the generalized Rayleigh-Plesset equation for bubble dynamics. It can be seen that the viscous term is related to the boundary condition at the bubble wall [14].
3.2 Spherical bubble dynamics
By ignoring surface tension and viscosity effects, the equation (3.44) reduces to:
This equation was first derived by Rayleigh (1917) [14].
The pressure inside the bubble (PB) is the sum of gas pressure PG and vapor pressure Pv. The major effect is related to the gas pressure. Neglecting the vapor pressure (PB ≈ PG) and assuming an adiabatic behavior of the gas content of the cavity, the internal pressure becomes [43]:
PB =PB0
where PB0 is the bubble inner gas pressure at initial radius R0 (i.e. at t= 0), and V represents the cavity volume. The parameter V0 is the initial bubble volume (V0 = (4/3)πR30) and γ is the ratio of specific heats of the bubble contents. An analytical solution of equation (3.45) is [14,43]:
3
The effects of viscosity and surface tension have been neglected. This is basi-cally a relation between dR/dt and R. At initial radius of R =R0, the initial velocity dR/dt is zero. At the time when the bubble is at its maximum size (R =Rmax), the velocitydR/dt also drops to zero. Therefore, we have [43]:
Ro3
This equation can be used to calculate R0 numerically, in the case that the other parameters are given.
The Rayleigh-Plesset equation can be solved numerically to derive radius time (R-t) curves for bubble dynamics in an undisturbed medium. In this case, the parameters such as bubble collapse time and maximum cavity radius can be calculated.
The bubble oscillation starts with the radius R0 and pressure PB0. The cavity inner pressure falls below the atmospheric pressure during expansion to the maximum size. In the absence of energy dissipation, the bubble collapses to its original size, and the pressure recovers to its initial value PB0. For explosion bubbles the value of PB0 is between100 and 500 bar [22, 43].
The compressibility of the liquid must be considered for quick cavity collapse.
Acoustic approximation is the simplest step in which a constant sound velocity C is assumed. This assumption holds just in the case where the bubble wall velocity dR/dt is always small relative to the sound speed in the liquid, C.
Then, the loss of energy is taken into account by sound radiation. By neglect-ing surface tension and viscosity, the followneglect-ing equation provides the acoustic
approximation, as Flynn shows [106]: quantity PL is the pressure on the liquid side of the bubble surface.
A more accurate statement regarding energy storage via compression of the liquid and also sound radiation, is given by Herring [106]:
1− 2 ˙R Shock wave formation in the case that the bubble wall speed is near to the sound velocity, is investigated by Gilmore using the Kirkwood-Bethe approximation.
This approximation implies that the propagation speed of the wave is equal to the sound velocity plus the fluid velocity. It gives [106]:
1− R˙ where the parameterC is the sound velocity at the bubble wall. H is the liquid enthalpy difference between the cavity surface and infinity, which is defined as [56]:
In the case that the water is regarded as incompressible, the bubble oscillation is not damped and has a constant period. Nevertheless, by considering the water as slightly compressible, damped oscillations with reducing periods are predicted by Keller and Kolodner [106].