Shock Induced Symmetric Compression in a Spherical Target

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Journal of Modern Physics, 2015, 6, 1769-1775
[Published Online October 2015 in SciRes. http://www.scirp.org/journal/jmp](http://www.scirp.org/journal/jmp)
http://dx.doi.org/10.4236/jmp.2015.613178

# Shock Induced Symmetric Compression in a Spherical Target

Labakanta Mandal 1 , Sourav Roy 2 , Manoranjan Khan 2 , R. Roychoudhury 2,3

1Department of Physics, Budge Budge Institute of Technology, Kolkata, India
2Department of Instrumentation Science, Jadavpur University, Kolkata, India
3Advanced Centre for Nonlinear and Complex Phenomena, Kolkata, India
[Email: labakanta@gmail.com](mailto:labakanta@gmail.com)

Received 22 June 2015; accepted 11 October 2015; published 15 October 2015

Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
[http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)

Abstract

Shock induced symmetric compression has been studied in a spherical target. The shock induced
interfacial radius will shrink and would reach a minimum point during implosion situation. However, after implosion the plasma tries to expand in blow off/explosion situation and as a result the
interfacial radius will increase. Effects of plasma parameters like density and temperature have
been studied numerically. It is seen that the density increases many times due to the mass conservation in imploding situation of a compressible shell like ICF. However, temperature will
change rapidly due to change of inner density and so would be the pressure of compressible fluid
following adiabatic law. Our analytical results agree qualitatively with those of simulation results
in spherical geometry and also experimental observations conducted in cylindrical container.

Keywords

Two Fluid Interface Instability, Shock Impingement, Adiabatic Symmetric Compression, Implosion

1. Introduction

Dynamics of interfacial radius of an imploded spherical shell like ICF is an interesting task to the fusion
community. The motion of the interface of two fluids and the effects of the plasma parameters like density and
temperature on the motion have been studied both analytically and numerically. This study can explain the
underwater explosion [1] or find out the time taken in filling up the cavity suddenly formed inside an
incompressible fluid [2]. Stability conditions of symmetric compression for incompressible and immiscible
viscous fluids were discussed earlier by Plesset [1] considering a source and sink at the origin of the spherical
shells. Linear perturbation method shows that the thermal electron conduction may be symmetric in laser driven

How to cite this paper: Mandal, L., Roy, S., Khan, M. and Roychoudhury, R. (2015) Shock Induced Symmetric Compression
in a Spherical Target. _Journal of Modern Physics_, 6 [, 1769-1775. http://dx.doi.org/10.4236/jmp.2015.613178](http://dx.doi.org/10.4236/jmp.2015.613178)

L. Mandal _et al_ .

fusion [3] [4]. In both cases, analytical results predict that spherical shell will shrink depending on the pressure
difference at the fluid interface [5]. However, implosion dynamics has been studied in spherical targets considering the solid compressibility and the mass loss by ablation [6]. Later, he described the two layer spherical
shell target implosion considering a single pressure [7]. Mikaelian [8] considered only radial incompressible
flow to explain RT and RM growth in a cylindrical target. Later, he pointed out that smooth density gradient at
the interface could stabilize the fluid instabilities after considering the constant steep density gradient at the
multiple-shell interface [9] in ICF related problems. Again, the problem of implosion and explosion has been
studied in spherical geometry [10] [11]. In ICF, deuterium-tritium fuel is kept in a micro balloon surrounded by
a foam like target. The laser shock immediately produces the target into plasma which tries to expand whereas
shock velocity impinges inside the lighter DT fluid. In this stage hydrodynamic instabilities arise, which oppose
to efficient compression. There are several answers to consider the potential flow model to describe the motion
of the interface. Hwang _et al_ . considered potential flow model for incompressible fluid of a hydromagnetic
situation to study the linear behavior of the dynamic effects of plasma in the cylindrical geometry [12]. The
problem of implosion in cylindrical geometry under a variety of physical conditions has recently been investigated both experimentally and numerically [13]-[15]. The results of these studies show that during a laser
induced implosion (deceleration phase), the interface of cylinder/spherical target is compressed and the radius of
the cylinder/sphere decreases and consequently, the density of the internal plasma increases [16] [17] and
reaches a maximum value for a critical value of the two fluid interface radius. This is followed by a blow off
stage when the interfacial radius expands with associated lowering of density. These features characterize the
compression in ICF process. The interfacial radius will shrink due to shock impingement at the interface and
reaches a minimum during implosion situation. However, after implosion the plasma expands due to blow off/
explosion situation and consequently the interfacial radius will increase.

In this paper, we have shown that the interfacial radius will shrink due to the shock impingement producing
the interfacial velocity and the density will increase many times due to the mass concentrated in imploding
situation of a compressible shell like ICF. However, temperature also changes the inner shell obeying adiabatic
compression for very fast process. Our analytical results are found to agree qualitatively with recent simulation
and experimental results obtained in cylindrical geometry. Also in the spherical geometry, during compression
the density of the DT fuel increases many times more than the density of the DT gas [18] [19] in the compressed
shell within the core of the spherical target.

We have described the paper as follows: In section 2, we describe the geometry and physical situation of the
problem with analytical results. Numerical results and its discussion has been given in section 3.

2. Formulation of the Problem: Spherical Target

In this paper, we assume a model which focuses on the post shock phenomena of interface motion during
implosion-explosion situation and thereby changes density, temperature. Consider two concentric spherical
shells containing two different fluids of different densities: fluid having density ρ 1 = ρ 1 ( _r_, _t_ ) in the inner shell
having radius _R_ ( _t_ ) and another fluid of constant density of ρ 2 in the outer shell with radius _R_ 2 where we
have taken _R_ 2  _R_ ( _t_ ) . The inner shell with radius _R_ ( _t_ ) is set in motion by impingement of a shock. We are
interested in the dynamics of the interface leading to consequent compression of the contained fluid and the
effects of plasma parameters like density and temperature on the dynamics ( Figure 1 ).

Assuming mass conservation relation for inner fluid, we can write for fluid density ρ 1 :

4 π 3 ρ 1 = _M_ = Constant; ρ 1 ( _r_, _t_ ) ∼ 31
3 _r_ _r_ (

1

( _r_, _t_ ) ∼ _r_ 3 ( _t_ )

1

ρ 1 = _M_ = Constant; ρ 1 ( _r_, _t_ ) ∼ _r_ 3 ( _t_ ) (1)

Now,we have the equation of continuity

∂∂ρ _t_ 1 + ∇ ⋅(ρ 1 _v_ ) = 0 (2)

where _v_ is interface velocity after passage of shock.

Assuming that the impingement of shock gives rise to uniform compression from all directions so that the
above equation may be considered to be independent of angular variable. In spherical coordinates, Equation (2)
can be written as

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L. Mandal _et al_ .

Figure 1. Geometry of the problem.

∂∂ρ _t_ 1 + _r_ 12 ∂∂ _r_  _vr_  = 0

(3)

Solving it one can get the expression of velocity in the following form

_v_ = _f_ ( _t_ ) _r_ 4 (4)

Using the boundary conditions that at the two fluid interface

_r_ = _R_ ( _t_ ) ; -  ∂φ 1  = _R_  (5)
 ∂ _r_ 

where _R_  is the shock generated interfacial velocity. Then velocity of the interface in Equation (4) leads to

 _r_  4

_v_ = _R_   _R_  (6)

Now assuming the potential flow motion, we can write

 _r_ 5  _R_  
_R_  ; so that φ 1 ( _r_, _t_ ) = − 4   (7)

∂∂φ _r_ 1 = −  _Rr_  4 _R_  ; so that φ 1 ( _r_, _t_ ) = − _Rr_ 54  _R_ 5  

( _r_, _t_ )

Since, there is no source/sink of velocity at the origin, the integration constant can be neglected. Now, we
employ Bernoulli’s equation for compressible fluid in inner shell

_r_ = _R_ ( _t_ ) d _Pl_

= _R_

( _t_ )

_r_ = _R_ ( _t_ )

∂φ _t_ 1 + 12 ( ∇ φ 1 ) 2  _r_ = _R_ 1 = −∫ _rr_ == _RR_ 1 ( _t_ ) d ρ _P_ 1 _l_

_r_ = _R_ ( _t_ )

1 2  _r_ = _R_ ( _t_ ) d
2 ( ∇ φ 1 )  _r_ = _R_ 1 = −∫ _r_ = _R_ 1

_r_ = _R_ ( _t_ )

− ∂∂φ _t_ 1 + 12 ( ∇ φ 1 ) 2  _r_ = _R_ 1 = −∫ _rr_ == _RR_ 1 ( _t_ ) d ρ _P_ 1 _l_ (8)

( _t_ )

( ∇ φ 1 )

_r_ = _R_ ( _t_ )

_r_ = _R_ 1 _r_ = _R_ 1

_r_ =

= _R_ 1

_r_ = _R_ 1

_t_ 1 + 12 ( ∇ φ 1 ) 2  _r_ = _R_ 1 = −∫ _rr_ == _RR_ 1 ( _t_ ) d ρ _P_ 1 _l_

where _R_ 1 is the minimum value of _R_ ( _t_ ) can reach. _P_ _l_ is internal pressure of lighter fluid.
To calculate the r.h.s term of the above equation, we assume adiabatic compression _i.e_ ., for very fast process
as in ICF situation, which leads to

d _Pl_ =  _k_ γ ρ 1 γ

ρ 1 γ− 1 

d _Pl_ =  _k_ γ ρ 1 γ− 1 /

ρ 1 γ− 1 

## ∫ rr == RR 1 ( t ) d ρ P 1 l = γ k −γ 1 ρ 1 γ− 1 / rr == RR 1 ( t )                (9)

_R_ ( _t_ ) d _Pl_  _k_ γ ργ− 1 / _r_ = _R_ ( _t_ )

_r_ = _R_ ( _t_ ) d _Pl_ =  _k_ γ ργ− 1 / _r_ = _R_ ( _t_ )

== _RR_ 1 ( _t_ ) d ρ _Pl_ = γ _k_ −γ 1 ρ 1 γ− 1 / _rr_ == _RR_

_l_ =  _k_ γ ρ 1 γ− 1

1 γ− 1 

_r_ = _R_ 1 ρ = γ− 1  1 _r_ = _R_ 1

1 ρ γ− 1  1 _r_ = _R_ 1

After some straightforward algebraic calculation and keeping terms up to third order of the ratio  _R_ 1 

 _R_ 

, we

arrive at the following equation

 _R_  _R_ - 3 _R_   =  γ _R_ 3   _R_ 1  3 _P_ 1 - _p_ 1 ( _R_ ( _t_ ))
 5 10  γ− 1  _R_  

_R_  _R_ - 3 _R_   =  γ _R_ 3   _R_ 1  3 _P_ 1 - _p_ 1 ( _R_ ( _t_ )) (10)

_R_  _R_ - 3 _R_   =  γ _R_ 3   _R_ 1  3 _P_ 1 - _p_ 1 ( _R_ ( _t_
5 10  γ− 1  _R_ 

( _R_ ( _t_ ))

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L. Mandal _et al_ .

where _P_ 1, _p_ 1 are the pressure at _r_ = _R_ 1 and _r_ = _R_, respectively.
However, outer shell fluid satisfy the Laplace’s equation,

∇ 2 φ 2 = 0 (11)

Using the boundary condition at the interface

 - ∂φ 2  =
 ∂ _r_  _r_ = _R_ ( _t_ )

φ 2  = _R_  ( _t_

∂ _r_  _r_ = _R_ ( _t_ )

2  = _R_  ( _t_ )

_R_  ( _t_ ) (12)

_r_ = _R_ ( _t_ )

_R_ ( _t_ )

the solution of Laplace’s Equation (11) becomes

_R_ 2
φ 2 ( _r_, _t_ ) = _r_ _R_  ; for _R_ ( _t_ ) < _r_ < _R_ 2 ( _R_ ( _t_ )) (13)

_R_ 2

( _r_, _t_ ) = _R_  ; for _R_ ( _t_ ) < _r_ < _R_ 2 ( _R_ ( _t_ ))

Bernoulli’s equation for the fluid with constant ρ 2, gives

_r_ = _R_ 2

− ∂∂φ _t_ 2 + 12 ( ∇ φ 2 ) 2  _rr_ == _RR_ 2 ( _t_ ) = −∫ _rr_ == _RR_ ( 2 _t_ ) ρ d _p_ 2 (14)

∂φ _t_ 2 + 12 ( ∇ φ 2 ) 2  _rr_ == _RR_ 2 ( _t_ ) = −∫ _rr_ == _RR_ ( 2 _t_ ) ρ d _p_ 2

1 2  _r_ = _R_ 2 _r_ = _R_ 2 d
2 ( ∇ φ 2 )  _r_ = _R_ ( _t_ ) = −∫ _r_ = _R_ ( _t_ ) ρ

2 + 12 ( ∇ φ 2 ) 2  _rr_ == _RR_ 2 ( _t_ ) = −∫ _rr_ == _RR_ ( 2 _t_ ) ρ d _p_ 2

( ∇ φ 2 )

_r_ = _R_

_r_ = _R_ ( _t_
_r_ = _R_ ( _t_ )

_r_ =

= _R_ ( _t_ )

( _t_ )

Using φ 2 as given by Equation (13) we get,

_RR_  + 3 _R_  2 = 1  _p_ 2 ( _R_ ( _t_ )) - _P_ 2
2 ρ 2 ( _t_ )

2 = 1  _p_ 2 ( _R_ ( _t_ )) - _P_ 2

ρ 2 ( _t_ )

 + 3 _R_  2 = 1  _p_ 2 ( _R_ ( _t_ )) - _P_ 2  (15)

2 ρ ( _t_ )

where _P_ 2, _p_ 2 are the pressure at very large distance from the interface and _r_ = _R_, respectively. Also, we have

for _R_ ( _t_ )  _R_ 2 . Using boundary condition _p_ 1 ( _R_ ( _t_ )) = _p_ 2 ( _R_ ( _t_ )) and combining

 _R_ ( _t_ )

neglected 

_R_ ( _t_

 _R_ ( _t_ ) 
 
 _R_ 2 

_R_ 2

Equations (10) and (15) we arrive at the following equation for _R_ ( _t_ ),

_RR_  ρ 2 +  5 γγ− _R_ 13  + 32 ρ 2 -  5 γγ− _R_ 13  _R_  2 =  _P_ 1  _RR_ 1  3 - _P_ 2  (16)

which will describe the evolution of the interface and other parameters.

Equation (16) can be integrated once to obtain a relation between _V_ 2 ( _R_ ( _t_ )) and _R_ ( _t_ ) ( Figure 2 ).

Figure 2. Graph of _V_ 2  _R_ ( _t_ ) vs _R_ , from analytical result in (Equation

(17)). Numerical calculation com- pletely agrees with the analytical result.

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L. Mandal _et al_ .

 3 3 
###  R ( t ) = ρ 2 ( R 2 k 3 2 + Rb ρ 2 ) ( k 1 k 2 − b a ) log e R − k 2 k baR 1 3 − R 3  + a ( 1 + 2 kb 2 a ) 2  kk b 21 a + 13  + 1 

 3 3
### 2  R ( t ) = ρ 2 ( R 2 k 3 2 + Rb ρ 2 ) ( k 1 k 2 − b a ) log e R − k 2 k baR 1 3 − R 3  + a ( 1 + 2 kb 2 a ) 2  kk b 21

 3 3
### V 2  R ( t ) = ρ 2 ( R 2 k 3 2 + Rb ρ 2 ) ( k 1 k 2 − b a ) log e R − k 2 k baR 1 3 − R 3  + a ( 1 + 2 kb 2 a ) 2  kk b 21 a + 13  + 1

### ( t ) = ρ 2 ( R 2 k 3 2 + Rb ρ 2 )

( _k_ 1 _k_ 2 - _b_ _a_ )

( 1 + _b_ _a_ )

(17)

where _k_ 1 = _PR_ 1 13, _k_ 2 = _P_ 2, _a_ = ρ 2, _b_ = γ 5 −γ 1 .

3. Numerical Results and Discussions

To solve Equation (16), we first make the equation dimensionless and write a set of first order equations,

d _X_

d τ = _Y_ (18)

 _XP_ 3   _RR_ 10  3 -  5 γ _X_ γ 3 −ρ 12 _R_ 03  - 52 
=

γ− 1
γ _X_ 3 ρ 2

_Y_  _XP_ 3   _RR_ 10  3 -  5 γ _X_ γ 3 −ρ 12 _R_ 03  - 52  _Y_

d _Y_  _XP_ 3   _RR_ 10  3 -  5 γ _X_ γ 3 −ρ 12 _R_ 03  - 52

d τ   γ− 1 

_X_  1 +  5 γ _X_ 3 ρ 2 _R_ 03 

3   _RR_ 10  -  5 γ _X_ γ 3 −ρ 12 _R_ 03  - 52  _Y_ 2

(19)

τ   γ− 1

_X_  1 +  5 γ _X_ 3 ρ 2

  γ− 1 
 1 +  5 γ _X_ 3 ρ 2 _R_ 03 

 1

_X_  1 +  5 γ _X_ 3 ρ 2 _R_

3 ρ 2 _R_ 03

where normalized variables are _X_ = _R_, _Y_ = _R_  ρ 2, τ= _t_ ρ 2, _P_ = _P_ 1 .

_R_ 0 _P_ 2 _R_ 0 _P_ 2 _P_ 2

Now, we solve the set of Equations (18) and (19) using Runge-Kutta-Fehlberg technique and plot interface
and its velocity in Figure 3(a) and Figure 3(b) . It is seen that the induced shock compresses the interface which

Figure 3. (a) and (b) represent nondimensional interfacial radius _X_ (τ) and

nondimensional velocity of the interface _Y_ (τ), respectively. Initial condition:

_X_ (τ) = 1.0 and _Y_ (τ) = − 1.0 . Taking _P_ = 20, ρ 2 _R_ 03 = 15, γ= 5 3 .

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L. Mandal _et al_ .

reaches a minimum point near the origin of the target during deceleration phase. This is the implosion situation.
Mass conservation suggests that since the volume is reduced, the density increases and reaches a maximum
corresponding to the minimum of interfacial radius. Then plasma tries to expand in acceleration phase. Thus
interfacial radius increases with time. This is the explosion situation. The interfacial radius reaches minimum

_R_ (τ) = _Rs_ at stagnation time

d _X_

τ=τ _s_ when d τ τ=τ _s_ = _V_ = 0 (20)

_s_

d _X_
when = _V_ = 0
d τ τ=τ _s_

_X_

= _V_

At this instant the density attains its maximum value

_M_
ρ 1 (τ=τ _s_ ) = ρ max = _R_ 3 (21)

It is shown in Figure 4(a) . Again, the velocity changes like a shock during changing of implosion state to
explosion state. It has been shown in Figure 3(b) . It is to be noted that in our uniform compression model predicts a good qualitative agreement with the corresponding experimental and simulation results for both cylindrical [13]-[15] and spherical geo- metries [18] [19]. However, there may be more complicated physics behind
the compression during stagnation time.

Finally, if we assume that the compression of the inner spherical shell fluid occurs adiabatically and that there
occurs little or no conduction of heat across the interfacial radius _R_ (τ), the shell fluid temperature _T_ 1 ( τ )

ρ 1 (τ)γ− 1, where γ is the adiabatic constant. Thus the time evolution of the temperature _T_ 1 ( τ )

ρ 1 (τ)γ

varies as ρ 1 (τ)

(γ

τ) ∼ρ 1 (τ)

(γ− 1 )

∼ρ 1 (τ) against τ

follows the same pattern as the density ρ 1 (τ) . In Figure 4(b) we have plotted _T_ 1 (τ) ∼ρ 1 (τ)

(γ− 1 )

_T_ 1 (τ) ∼ρ 1

( γ→ adiabatic index). It is to be noted that the time dependence of temperature thus exhibited also agrees
qualitatively with experimental and simulation results in cylindrical geometry [15].

Figure 4. (a) and (b) represent density and temperature variations, respectively. Initial
conditions are the same as those in Figure 1 .

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Acknowledgements

L. Mandal _et al_ .

This work is supported by the Department of Science & Technology, Government of India under grant no. SR/S2/
HEP-007/2008. Author L. K. Mandal is thankful to Prof. S. Ghosh and Dr. N. Chakraborty for helpful discussion.

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