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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.

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 [

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 [

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.

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

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

Now,we have the equation of continuity

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

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

Using the boundary conditions that at the two fluid interface

where

Now assuming the potential flow motion, we can write

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

where

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

After some straightforward algebraic calculation and keeping terms up to third order of the ratio

arrive at the following equation

where

However, outer shell fluid satisfy the Laplace’s equation,

Using the boundary condition at the interface

the solution of Laplace’s Equation (11) becomes

Bernoulli’s equation for the fluid with constant

Using

where

neglected

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

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

Equation (16) can be integrated once to obtain a relation between

where

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

where normalized variables are

Now, we solve the set of Equations (18) and (19) using Runge-Kutta-Fehlberg technique and plot interface and its velocity in

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

At this instant the density attains its maximum value

It is shown in

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

varies as

follows the same pattern as the density

(g ® 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 [

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.

LabakantaMandal,SouravRoy,ManoranjanKhan,R.Roychoudhury, (2015) Shock Induced Symmetric Compression in a Spherical Target. Journal of Modern Physics,06,1769-1775. doi: 10.4236/jmp.2015.613178