Combustion, Explosion, and Shock Waves, Vol. 37, No. 4, pp. 470 474, 2001
Simulation of the Behavior of Mixtures of Heavy Particles
Behind a Shock-Wave Front
D. V. Voronin1 and V. F. Anisichkin1 UDC 532.529
Translated from Fizika Goreniya i Vzryva, Vol. 37, No. 4, pp. 116 121, July August, 2001.
Original article submitted May 19, 2000.
The two-dimensional inviscid nonstationary flow behind a shock wave passing through
solid uranium dioxide or carbide particles suspended in liquid iron was simulated nu-
merically. Such layers can appear inside planets in the vicinity of the planet s solid
core. Shock waves passing in the interior of a planet (resulting from a possible asteroid
impact on the planet) can change parameters of the layer. The calculations demon-
strated that the local particle mass concentration behind the incident and reflected
shock waves increases considerably, which can cause a transition of the layer into a
supercritical state and a nuclear explosion inside the planet. The problem was solved
taking into account possible particle collisions and their deformation and fission as
well as changes in the fields of major thermodynamic parameters inside and outside
each particle.
In considering the detonation of mixtures of par- Thus, at the initial stages of the process, disturbances
ticles with different densities, suspensions, and shock- from one particle to another can pass only via liquid iron
wave compaction of multicomponent mixtures located (carrier phase). In what follows, however, the situation
near a rigid wall, one needs to know the behavior of can change. The initial diameter of the particles equals
suspended heavy particles behind a shock-wave (SW) 10 cm. On the right, the layer is bounded by a rigid
front. In the present paper, we study the passage of impermeable piston (planet s solid core). We assume
shock-wave disturbance through a layer of uranium ox- that initially the medium is in thermodynamic equilib-
ide particles suspended in iron nickel melt. Such a layer rium. The piston starts moving leftward with constant
can occur in the interior of a planet in the vicinity of its velocity U0 at the time t = 0 and initiates a SW in front
solid core. A SW from, for example, an asteroid impact of itself. This case corresponds to the passage of a SW
on the planet can change the relative concentration of from the planet s solid core into the outer liquid core,
active particles. If the local concentration exceeds the where the two-phase mixture considered is located.
critical value, an explosion can occur inside the planet This study was carried out in a hydrodynamic
as a result of a nuclear chain reaction [1]. The possi- approximation. The equations describing the two-
bility of particle compaction is analyzed in the present dimensional nonstationary movement of an inviscid
paper. compressible continuous medium at t > 0 are based on
the laws of conservation of mass and momentum and
are written in dimensionless variables:
MATHEMATICAL FORMULATION
dÁ "u "v v
OF THE PROBLEM
+ Á + + S = 0,
dt "z "r r
Let us consider a mixture of liquid iron (initial den-
du "p
sity 12.14 g/cm3) and solid crystals of uranium diox-
Á + = 0, (1)
dt "z
ide (initial density 20.05 g/cm3) in a planetary core.
Uranium dioxide particles are located chaotically but
dv "p
Á + = 0.
are not initially in contact with one another (Fig. 1a).
dt "r
1
Here u and v are the longitudinal and transverse compo-
Lavrent ev Institute of Hydrodynamics, Siberian Division,
Russian Academy of Sciences, Novosibirsk 630090. nents of the velocity of the medium, respectively, p and
470 0010-5082/01/3704-0470 $25.00 © 2001 Plenum Publishing Corporation
Behavior of Mixtures of Heavy Particles Behind a Shock-Wave Front 471
Fig. 1. Field of longitudinal velocities u during movement of the piston (closed right boundary) for t = 40 µsec.
Á are the pressure and density, z and r are the longitu- Here p1 and p2 are pressure values and Un,1 and Un,2 are
dinal and transverse spatial coordinates, and S = 0 for values of the component of the medium s velocity vector
the case of plane symmetry and S = 1 for axial symme- U = (u, v) normal to the discontinuity surface on differ-
try. ent sides of the contact surface. We consider the right
To close system (1), we used the following shock- boundary of the region (piston) closed; here, the non-
adiabat relations: penetration condition (the medium s velocity is equal to
the piston s velocity) is specified; the upper and lower
D = C + LU, Á(D - U) = Á0D,
boundaries are also closed (v = 0); the left boundary is
open and the boundary conditions there correspond to
p = Á0DU + p0, (2)
the conditions at the free boundary.
p + p0 1 1
E = - + E0.
2 Á0 Á
METHOD OF SOLUTION
Here D is the SW velocity, U is the particle velocity
behind the SW front, L and C are constants of the
System (1), (2) with the specified boundary con-
material, and E is the internal energy of the material.
ditions was solved by the Harlow method of particles
Hereinafter, the subscript  zero indicates the initial
in cells, which was improved in [2]. The dimensions of
state before the SW.
the calculation domain were 950 mm along the z axis
We assume that the boundaries between the ura- and 350 mm along the r axis. The lower boundary of
nium dioxide particles and liquid iron are discontinuous.
the calculation domain is the symmetry axis. The num-
The following boundary conditions are specified for each
ber of particles in cells varies. The numerical algorithm
discontinuity:
takes into account the possibility of fusion and fission
Un,1 = Un,2, p1 = p2. of individual particles determined by Harlow s method
472 Voronin and Anisichkin
depending on the parameters of the medium. The max- the front. Filtration of iron occurs in this zone: ura-
imum number of particles in a cell equals seven. The nium dioxide particles squeeze iron from the layer. The
cell size is 0.4 mm. velocity of uranium dioxide particles in the cross section
We note that the problem of a SW passing through considered is about 11.6 m/sec, and the velocity of iron
two-phase layers located near a rigid wall has been in the space between the particles is about -38 m/sec.
solved previously [3, 4] in a one-dimensional formula- A similar situation occurs in other transverse cross sec-
tion within the framework of the mechanics of inter- tions located closer to the piston. The uranium diox-
penetrating continua, where two phases with particular ide velocity is about the same, and the velocity of the
values of thermodynamic parameters are present simul- carrier phase approaches zero. Thus, the particles are
taneously at any point of the domain. This approach intensely compacted in the relaxation zone.
imposes certain limitations on the parameters of the The flow pattern at the time t = 1800 µsec is shown
mixture and the nature of interfacial interactions. For in Fig. 2. It is evident that uranium dioxide particles are
example, the size of a condensed-phase particle should deforming and settling on the piston. In certain calcula-
be much smaller than that of the calculation cell, and tions (with variation in the intensity of the incident SW)
the particles can interact only via the carrier phase. fission of the particles is possible. When the SW passes
In the present work, the problem is solved in a two- from the less dense carrier phase into a uranium diox-
dimensional formulation. Only one phase is present at ide particle, the interface loses stability. Surface waves
each point. The velocity, density, and pressure fields appear whose amplitude is comparable to the particle
are calculated not only in the carrier phase (liquid iron) size. This can be seen in Fig. 2. Sometimes, this pro-
but also inside each of the uranium dioxide particles. cess results in breakup of the particle. Thus, fission of
With time, an individual uranium dioxide particle can particles in this case follows the mechanism of bound-
deform, split, collide, or adhere to other particles. No ary instability. We note that in this process any type of
limitations on the particle size are imposed. instability  Rayleigh Taylor or Kelvin Helmholtz 
can be the determining one. The algorithm used allows
both types of instability to be calculated. In addition,
CALCULATION RESULTS
in the case of small distance between particles, uranium
dioxide particles can stick together.
In the calculations, we used the following values of
The concentration of uranium dioxide particles (²)
the constants: for iron, C = 10.26 km/sec and L = 1.55,
at the time corresponding to Fig. 2 reached 0.39. Never-
for uranium dioxide, C = 8.06 km/sec and L = 1.39,
theless, the compaction process continues (as indicated
p0 = 330 GPa according to [5, 6], U0 = -0.1 km/sec,
by the velocity profiles in Fig. 2b and c). The carrier
and the initial volume concentration of uranium dioxide
phase has a considerable negative velocity in the cross
particles was ²0 = 0.35.
section r = 0 (see Fig. 2c), which corresponds to the
The field of longitudinal velocities u at the time
motion in the clearance between uranium particles. In
t = 40 µsec is shown in Fig. 1a. Hereinafter, the ve-
the cross section r = 12 cm, the longitudinal phase ve-
locity u is given relative to the piston. Light areas in
locities are close to one another. Here, iron flows around
the figure correspond to large values of the velocity u.
uranium dioxide particles predominantly in the trans-
It can be seen that the SW propagates leftward. At
verse direction. It turned out that the possibility of
the moment the SW leaves the layer of uranium dioxide
compaction is independent of the arrangement of par-
particles suspended in liquid iron, it has an intensity of
ticles, their volume concentration or size, the type of
H"13 GPa. The longitudinal profiles of the velocity u
symmetry, velocity of piston, and fission or sticking of
in the center of the channel and in the cross section
the particles. This was confirmed by calculations with
corresponding to the value of r = 12 cm are shown in
variation in the corresponding parameters.
Fig. 1c. Figure 1b shows the transverse profile of u in
the vicinity of the SW front at z = 61.5 cm (it cor-
responds to the vertical dashed line in Fig. 1a). Due
SW REFLECTION FROM A RIGID WALL
to dispersion of the velocity of sound in the material,
the SW front leaving the layer is not flat. The veloc- We also solved the problem for the case where a
ity of sound in the carrier phase is higher than that in SW passes from the left through a layer of liquid iron
uranium dioxide particles, and, hence, the SW velocity with uranium dioxide particles, is reflected from a rigid
in iron is higher. Therefore, by that time, the SW has wall (closed right boundary), and passes through the
not passed from the particles into the undisturbed, less layer for the second time. This case corresponds to
dense medium, as is evident from Fig. 1. The particle the case where a SW first passes through the planet s
velocity and pressure relaxation zone is located behind outer core (where the layer considered is located) and
Behavior of Mixtures of Heavy Particles Behind a Shock-Wave Front 473
Fig. 2. Field of longitudinal velocities u during movement of the piston for t = 1800 µsec.
then enters the solid inner core and is reflected from it. crease in mass concentration) is determined by the time
The main parameters of the mixture components are between the passages of the incident SW and the SW
the same as in the previous case, and ²0 = 0.23. The reflected through a fixed point in space. It is consider-
incident SW has an intensity of "p H" 10 GPa, and for ably smaller that the time of velocity relaxation of the
the reflected SW, "p H" 23 GPa. The results of cal- medium. Therefore, ultimately, the compaction process
culations for t = 40 µsec (from the moment it passes also takes place here.
through the center of the calculated field) are shown in In what follows, the uranium dioxide particles start
Fig. 3. Behind the incident wave, the medium starts coming closer to each other and squeezing iron from the
moving rightward. Particles of uranium dioxide have a layer. The particles located directly near the rigid wall
velocity of u H" 63.6 m/sec and the iron layer moves with first decelerate and, then, acquire a small negative ve-
a velocity of u H" 102 m/sec (cross section z = 64.5 cm; locity. In this process, the leftmost layer of the particles
curve 2 in Fig. 3b). The velocities are given in the lab- continues moving rightward. Thus, the layers are mov-
oratory coordinates. It is evident that iron flows into ing toward one another. Then, the leftmost layer also
the layer, and the uranium dioxide particles are still less acquires a negative velocity but the squeezing of iron
mobile. That is, the particle mass concentration in the continues. This is evident from Fig. 4, which shows lon-
layer decreases over this period of time. However, this gitudinal velocity profiles at t = 100 µsec (² = 0.24).
process is of short duration: it continues only until the The velocity profiles in Fig. 4b correspond to the cross
wave reflected from the rigid wall passes through the sections z = 64.5 and 86 cm. Here the compaction of
layer. In this process, the difference between the phase the layer becomes more intense than during the passage
velocities near the rigid wall is not so large (cross section of the SW initiated by the piston through the two-phase
z = 86 cm; curve 1 in Fig. 3b). layer. For large values of t, no principal changes in the
The velocity of the carrier phase behind the re- nature of the process occur. Pressure waves that change
flected SW is virtually zero. The particles of uranium the situation drastically are absent, velocity relaxation
dioxide continue moving rightward with a velocity of takes place in the medium, and the velocities of the car-
u 10 m/sec, and compaction of the particle layer rier phase and the particles tend to zero asymptotically.
commences. The time of particle decompaction (de- A comparison of Figs. 2 and 4 demonstrates that in
474 Voronin and Anisichkin
Fig. 3. Field of longitudinal velocities u during SW reflection from the rigid wall for t = 40 µsec and z = 86 (curve 1)
and 64.5 cm (curve 2).
Fig. 4. Field of longitudinal velocities u during SW reflection from the rigid wall for t = 100 µsec.
the first case, uranium dioxide particles settle directly This work was carried out under the Integration
on the piston, whereas in the second case, maximum project No. 24 of the Presidium of the Siberian Di-
compaction takes place at a certain distance from the vision of the Russian Academy of Sciences  Study of
rigid wall. This means that if a SW passes from a the Possibility of Explosive Liberation of Energy in the
dense medium (planet s solid core) into a less dense Interior of Planets.
medium, where the two-phase layer considered is lo-
REFERENCES
cated, then particle compaction occurs directly near the
core surface. If a SW initiated by an asteroid first passes
1. V. F. Anisichkin,  Do planets burst? Fiz. Goreniya
through the two-phase layer and only then reaches the
Vzryva, 33, No. 1, 138 142 (1997).
solid core and is reflected from it, then particle com-
2. V. A. Agureikin and B. P. Kryukov,  The method of
paction and nuclear explosion are also possible at a dis-
individual particles for calculating fluxes of multicom-
tance from the inner core. Variation of the incident SW
ponent media with large deformations, in: Numerical
parameters and particle sizes and arrangement confirms
Methods of Continuum Mechanics [in Russian], Inst. of
the established regularity.
Theor. and Appl. Mech., Sib. Div., Acad. of Sci. of the
We also note that the times corresponding to
USSR, Vol. 17, No. 1 (1986), pp. 17 31.
Figs. 3 and 4 are still small for development of inter- 3. A. I. Ivandaev and A. G. Kutushev,  Effect of dispersed
face instability; therefore, the shape of uranium dioxide particles on attenuation of blast waves in gas mixtures
particles changed over these times insignificantly. and their interaction with barriers, in: Non-Stationary
Flows of Multiphase Systems with Physicochemical Con-
versions [in Russian], Izd. Mosk. Univ., Moscow (1983),
CONCLUSIONS
pp. 60 79.
4. A. A. Zhilin and A. V. Fedotov,  Reflection of shock
During passage of the SW reflected from the rigid
waves from a rigid boundary in a mixture of condensed
wall, as in the case of a moving impermeable piston, the
materials. I. Equilibrium approximation, Prikl. Mekh.
local mass concentration of heavy particles increases. In
Tekh. Fiz., 40, No. 5, 73 78 (1999).
the case of active particles, the concentration can exceed 5. V. N. Zharkov, Internal Structure of the Earth and Plan-
the critical level and, ultimately, lead to an explosion. ets [in Russian], Nauka, Moscow (1983).
6. V. F. Anisichkin,  Shock-wave data as evidence of the
The concentration increases over a wide range of initial
presence of carbon in the Earth s core and lower man-
parameters of the mixture.
tle, Fiz. Goreniya Vzryva, 36, No. 4, 108 114 (2000).