384 research outputs found
On the Stability of Fluid Flows with Spherical Symmetry
The conditions for the stability or instability of the interface between two immiscible incompressible fluids in radial motion are deduced. The stability conditions derived by Taylor for the interface of two fluids in plane motion do not apply to spherical flows without significant modifications
A Nonsteady Heat Diffusion Problem with Spherical Symmetry
A solution in successive approximations is presented for the heat diffusion across a spherical boundary with radial motion. The approximation procedure converges rapidly provided the temperature variations are appreciable only in a thin layer adjacent to the spherical boundary. An explicit solution for the temperature field is given in the zero order when the temperature at infinity and the temperature gradient at the spherical boundary are specified. The first-order correction for the temperature field may also be found. It may be noted that the requirements for rapid convergence of the approximate solution are satisfied for the particular problem of the growth or collapse of a spherical vapor bubble in a liquid when the translational motion of the bubble is neglected
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On the Classical Model of Nuclear Fission
The first experiments on neutron bombardment
of various elements carried out by
Fermi and his collaborators included the study
of the group of activities observed in uranium
which were at that time ascribed to transuranic
elements. The great number of studies following
this first work led finally to the results of Hahn
and Strassmann which showed clearly that many
of the activities ascribed to transuranic elements
came, instead, from nuclei of approximately
half the mass of uranium. The startling conclusion
that these activities must arise from the
splitting of the uranium nucleus under neutron
bombardment into two fragment nuclei was
pointed out by Meitner and Frisch, and was
quickly confirmed by subsequent experiments. In
the first theoretical discussion of this new type of
nuclear reaction, Meitner and Frisch proposed
the name fission for the process, and compared it
with the splitting that may take place in a liquid
drop in oscillation. This model was supported by
Bohr who correlated it with other nuclear
properties and, at the same time, emphasized
how far the phenomenon of nuclear fission may
be described classically. A very complete theoretical
discussion of both the classical and
quantum aspects of fission was given by Bohr and
Wheeler, and it is proposed here to describe
some of the classical theory of fission developed
by these authors
The Dynamics of Cavitation Bubbles
Three regimes of liquid flow over a body are defined,
namely: (a) noncavitating flow; (b) cavitating flow with a
relatively small number of cavitation bubbles in the field
of flow; and (c) cavitating flow with a single large cavity
about the body. The assumption is made that, for the
second regime of flow, the pressure coefficient in the flow
field is no different from that in the noncavitating flow.
On this basis, the equation of motion for the growth and
collapse of a cavitation bubble containing vapor is derived
and applied to experimental observations on such bubbles.
The limitations of this equation of motion are pointed
out, and include the effect of the finite rate of evaporation
and condensation, and compressibility of vapor and
liquid. A brief discussion of the role of "nuclei" in the
liquid in the rate of formation of cavitation bubbles is
also given
Phase Diagrams for Sonoluminescing Bubbles
Sound driven gas bubbles in water can emit light pulses. This phenomenon is
called sonoluminescence (SL). Two different phases of single bubble SL have
been proposed: diffusively stable and diffusively unstable SL. We present phase
diagrams in the gas concentration vs forcing pressure state space and also in
the ambient radius vs gas concentration and vs forcing pressure state spaces.
These phase diagrams are based on the thresholds for energy focusing in the
bubble and two kinds of instabilities, namely (i) shape instabilities and (ii)
diffusive instabilities. Stable SL only occurs in a tiny parameter window of
large forcing pressure amplitude atm and low gas
concentration of less than of the saturation. The upper concentration
threshold becomes smaller with increasing forcing. Our results quantitatively
agree with experimental results of Putterman's UCLA group on argon, but not on
air. However, air bubbles and other gas mixtures can also successfully be
treated in this approach if in addition (iii) chemical instabilities are
considered. -- All statements are based on the Rayleigh-Plesset ODE
approximation of the bubble dynamics, extended in an adiabatic approximation to
include mass diffusion effects. This approximation is the only way to explore
considerable portions of parameter space, as solving the full PDEs is
numerically too expensive. Therefore, we checked the adiabatic approximation by
comparison with the full numerical solution of the advection diffusion PDE and
find good agreement.Comment: Phys. Fluids, in press; latex; 46 pages, 16 eps-figures, small
figures tarred and gzipped and uuencoded; large ones replaced by dummies;
full version can by obtained from: http://staff-www.uni-marburg.de/~lohse
Analytical Approximations for the Collapse of an Empty Spherical Bubble
The Rayleigh equation 3/2 R'+RR"+p/rho=0 with initial conditions R(0)=Rmax,
R'(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an
ideal, infinite liquid with far-field pressure p and density rho. The solution
for r=R/Rmax as a function of time t=T/Tcollapse, where R(Tcollapse)=0, is
independent of Rmax, p, and rho. While no closed-form expression for r(t) is
known we find that s(t)=(1-t^2)^(2/5) approximates r(t) with an error below 1%.
A systematic development in orders of t^2 further yields the
0.001%-approximation r*(t)=s(t)[1-a Li(2.21,t^2)], where a=-0.01832099 is a
constant and Li is the polylogarithm. The usefulness of these approximations is
demonstrated by comparison to high-precision cavitation data obtained in
microgravity.Comment: 5 pages, 2 figure
On the Dynamics of Small Vapor Bubbles in Liquids
When a vapor bubble in a liquid changes size, evaporation
or condensation of the vapor takes place at the surface of the bubble. Because of the latent heat requirement of evaporation, a change in bubble size must
therefore be accompanied by a heat transfer across the bubble wall, such as to cool the surrounding liquid when the bubble grows (or heat it when the bubble
becomes smaller). Since the vapor pressure at the bubble wall is determined by the temperature there, the result of a cooling of the liquid is a decrease of the
vapor pressure, and this causes a decrease in the rate of bubble growth. A similar effect occurs during the collapse of a bubble which tends to slow down the collapse.
In order to obtain a satisfactory theory of the behavior of a vapor bubble in a liquid, these heat transfer effects must be taken into account.
In this paper, the equations of motion for a spherical vapor bubble will be derived and applied to the case of a bubble expanding in superheated liquid and a bubble collapsing in liquid below its boiling point. Because of the inclusion of the heat transfer effects, the equations are nonlinear, integro-differential
equations. In the case of the collapsing bubble, large temperature variations occur; therefore, tabulated vapor pressure data were used, and the equations of
motion were integrated numerically. Analytic solutions are obtainable for the case of the expanding bubble if the period of growth is subdivided into several
regimes and the simplifications possible in each regime are utilized. The growth is considered here only during the time that the bubble is small. An asymptotic
solution of the equations of motion, valid when the bubble becomes large (i.e. observable), has been presented previously, together with experimental verification.
We shall be specifically concerned in the following discussion with the dynamics of vapor bubbles in water. This restriction was made for convenience
only, since the theory is applicable without modification to many other liquids
The Growth of Vapor Bubbles in Superheated Liquids
The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius-time behavior is made with experimental observations in superheated water, and very good agreement is found
On the Mechanism of Cavitation Damage
A new method for producing cavitation damage in the
laboratory is described in which the test specimen has no
mechanical accelerations applied to it in contrast with the
conventional magnetostriction device. Alternating pressures
are generated in the water over the specimen by exciting
a resonance in the "water cavity." By this means
the effects of cavitation have been studied for a variety of
materials. Photomicrographs have been taken of several
ordinary (polycrystalline) specimens and also of zinc
monocrystals. The zinc monocrystal has been exposed to
cavitation damage on its basal plane and also on its
twinning plane. X-ray analyses have been made of polycrystalline specimens with various exposures to cavitation. The results show that plastic deformation occurs in the specimens so that the damage results from cold-work of the material which leads to fatigue and failure. A variety of materials has been exposed to intense cavitation for extended periods to get a relative determination of their resistance to cavitation damage. It is found that, roughly speaking, hard materials of high tensile strengths are the most resistant to damage. While this survey is not complete, it has been found that titanium 150-A and tungsten are the most resistant to damage of the materials tested. Cavitation-damage studies, which have been carried out in liquid toluene and in a helium atmosphere, show that chemical effects can be, at most, of secondary significance
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