The collapse of a spherical cavity in a compressible liquid

This paper presents numerical solutions for the flow in the vicinity of a collapsing spherical bubble in water. The bubble is assumed to contain a small amount of gas and the solutions are taken beyond the point where the bubble reaches its minimum radius up to the stage where a pressure wave forms and propagates outwards into the liquid. The motion up to the point where the minimum radius is attained, is found by solving the equations of motion both in the Lagrangian and in the characteristic forms. These are in good agreement with each other and also with the approximate theory of Gilmore which is demonstrated to be accurate over a wide range of Mach number. The liquid flow after the minimum radius has been attained is determined from a solution of the Lagrangian equations. It is shown that an acoustic approximation is quite valid for fairly high pressures and this fact is used to determine the peak intensity of the pressure wave at a distance from the center of collapse. It is estimated in the case of typical cavitation bubbles that such intensities are sufficient to cause cavitation damage

Flow of vapour in a liquid enclosure

A solution is developed for the flow of a vapour in a liquid enclosure in which different portions of the liquid wall have different temperatures. It is shown that the vapour pressure is very nearly uniform in the enclosure, and an expression for the net vapour flux is deduced. This pressure and the net vapour flux are readily expressed in terms of the temperatures on the liquid boundary. Explicit results are given for simple liquid boundaries: two plane parallel walls at different temperatures and concentric spheres and cylinders at different temperatures. Some comments are also made regarding the effects of unsteady liquid temperatures and of motions of the boundaries. The hemispherical vapour cavity is also discussed because of its applicability to the nucleate boiling problem

Reply to comments on "General analysis of the stability of superposed fluids"

Previous results by Plesset and Hsieh on the effects of compressibility for Rayleigh–Taylor instability are shown to be valid, and an alternative brief deduction is given

Viscous effects in Rayleigh-Taylor instability

A simple, physical approximation is developed for the effect of viscosity for stable interfacial waves and for the unstable interfacial waves which correspond to Rayleigh‐Taylor instability. The approximate picture is rigorously justified for the interface between a heavy fluid (e.g., water) and a light fluid (e.g., air) with negligible dynamic effect. The approximate picture may also be rigorously justified for the case of two fluids for which the differences in density and viscosity are small. The treatment of the interfacial waves may easily be extended to the case where one of the fluids has a small thickness; that is, the case in which one of the fluids is bounded by a free surface or by a rigid wall. The theory is used to give an explanation of the bioconvective patterns which have been observed with cultures of microorganisms which have negative geotaxis. Since such organisms tend to collect at the surface of a culture and since they are heavier than water, the conditions for Rayleigh‐Taylor instability are met. It is shown that the observed patterns are quite accurately explained by the theory. Similar observations with a viscous liquid loaded with small glass spheres are described. A behavior similar to the bioconvective patterns with microorganisms is found and the results are also explained quantitatively by Rayleigh‐Taylor instability theory for a continuous medium with viscosity

Reply to Comments of P. W. Smith, Jr.

In his comments on this subject, Smith has put emphasis on the special nature of the plane-wave solution in acoustic problems. It is perhaps unnecessary to defend the importance of the plane-wave solution in a linear theory

Tensile Strength of Liquids

The theory of the tensile strength of a pure liquid is developed and it is shown that it predicts much larger tensile strengths than are observed. This theory is modified and extended under the supposition that liquids usually contain nuclei which are here taken to be solid particles. It is shown that the theory leads to more moderate predictions of tensile strength provided the solid particles are not wetted by the liquid. It is also shown that Brownian motion will serve as the mechanism whereby solid particles can remain in suspension in liquids

Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary

Vapour bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)^½ where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For Δp/ρ=10^6cm^2/sec^2 ≈ 1 atm/density of water the jet had a speed of about 130m/sec in the first case and 170m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapour are not important

Collapse of an initially spherical vapor cavity in the neighborhood of a solid boundary

Vapor bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. The method uses finite time steps and features an iterative technique for applying the boundary conditions at infinity directly to the liquid at a finite distance from the free surface. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)^1/2 where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For Δp/ρ = 10^6 (cm/sec)^2 ~ 1 atm./density of water the jet had a speed of about 130 m/sec in the first case and 170 m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapor are not important

Cavitating Flows

There are very few differences between the fluid dynamics of liquids and gases. The viscosity of water, for example, is 10^-2 poise at 20° C while the viscosity of air at this temperature is about 2 X 10^-4 poise. The kinematic viscosity of water is 10^-2 cm^2/sec compared with 0.15 cm^2/sec for air. As one would expect from simple kinetic theory, the viscosity of gases increases with increasing temperature; the viscosity of liquids on the other hand decreases rather rapidly as the temperature rises. While the speed of sound in water is about four times that in air, there is a more interesting consequence of the equation of state. A pressure pulse with an intensity of several hundred psi, which propagates as a strong shock in air, will propagate acoustically in water with a negligible production of entropy