**4.2 Magnetically driven quasi-isentropic compression**

In order to simplify the problem, the one dimensional model of magnetically driven quasiisentropic compression can be described by the model shown in Fig.16. The changes of

Magnetohydrodynamics of Metallic Foil Electrical

original boundary conditions are,

considered.

In equation (20),

the temperature of vaporazation point. For aluminum, is 0.69×10-9 m3/J,

density is up to 0.2 T[34]。

magnetic field diffusion.

MA/cm.

For *t*=0 , 0 : 0, 0

*x BP x BP* 

1: 0, 0

equation (4) can be converted to equations from (17) through (19).

0

Explosion and Magnetically Driven Quasi-Isentropic Compression 365

The controlling equations are one dimensionally magnetohydrodynamic ones, which include mass conservation equation, momentum conservation equation, energy conservation equation and magnetic diffusion equation, as shown in equation (4). The

, and for *t*=*t*n(at some time),

 

0 0 *<sup>D</sup>* <sup>0</sup> *de dV pq e dt dt*

0 () () *VB VB t x Bx* 

> 

0 is the electrical resistivity of conductors at temperatureof 0 ºC, is

The equation of electrical resistivity is also very important for the case of magnetically driven quasi-isentropic compression. In order to simplify the problem, a simple model is

heating factor, *Q* is the heat capacity or increment of internal energy relative to that at

In equation (21), *c*v is specific heat at constant volume, which is close to constant from 0 ºC to

equation (20) is suitable. After that, more complex electrical resisistivity model is needed. In this simulation, the stress wave front is defined when the amplitude of pressure reaches to 0.1 GPa, and thediffusion front of magnetic field is determined when the magnetic flux

Fig.18 gives the distribution of density and temperature of Aluminum sample along Lagrangian coordinates for different times in the condition of loading current density 1.5

The results in Fig.18 show that the density and temperature of aluminum sample vary with the loading time along the direction of sample thickness because of the Joule heating and

 

temperatureof 0 ºC, which is related with temprature at the condensed states.

2

0

2

  0

(18)

<sup>0</sup> 1 *Q* (20)

*Q cT* v (21)

0 is 2.55×10-8 m. Before vaporazation point, the

The calculation coordinate are Lagrangian ones, and for the Lagrangian coordinate, the

*du B p q dt x*

0 0 : 0, 0 1 : ( ), 0

(17)

(19)

.

*x BP x B Jt P* 

electrical parameters caused by the motion of loaded electrode are not considered, and the heat conduction is neglected because it is slow in sub microsecond or one microsecond. A standardly discharging current in short circuit is as input condition presented in Fig.17. The relative magnetic permeability is supposed tobe 1, that is to say , 0.

Fig. 16. Physical model of simulation

Fig. 17. Loading current curves

electrical parameters caused by the motion of loaded electrode are not considered, and the heat conduction is neglected because it is slow in sub microsecond or one microsecond. A standardly discharging current in short circuit is as input condition presented in Fig.17. The

J×B

Vacuum

relative magnetic permeability is supposed tobe 1, that is to say , 0.

Al

*x*=1 *x*=0

Fig. 16. Physical model of simulation

J

B

Fig. 17. Loading current curves

The controlling equations are one dimensionally magnetohydrodynamic ones, which include mass conservation equation, momentum conservation equation, energy conservation equation and magnetic diffusion equation, as shown in equation (4). The original boundary conditions are,

$$\text{For } t = 0, \,\begin{cases} \mathbf{x} = \mathbf{0} : B = \mathbf{0}, P = \mathbf{0} \\ \mathbf{x} = \mathbf{1} : B = \mathbf{0}, P = \mathbf{0} \end{cases}, \text{ and for } t \equiv t\_{\mathbf{n}} \text{ (at some time) } \text{ ) }, \begin{cases} \mathbf{x} = \mathbf{0} : B = \mathbf{0}, P = \mathbf{0} \\ \mathbf{x} = \mathbf{1} : B = \mu\_{0} \mathbf{J}(\mathbf{t}), P = \mathbf{0} \end{cases}$$

The calculation coordinate are Lagrangian ones, and for the Lagrangian coordinate, the equation (4) can be converted to equations from (17) through (19).

$$
\rho \rho\_0 \frac{d\mu}{dt} + \frac{\partial}{\partial \mathbf{x}} \left( p + q + \frac{B^2}{2\mu\_0} \right) = 0 \tag{17}
$$

$$
\rho\_0 \frac{de}{dt} + (p+q)\frac{dV}{dt} - \rho\_0 \dot{e}\_D = 0\tag{18}
$$

$$\frac{\partial \langle VB \rangle}{\partial t} = \frac{\partial}{\partial \mathbf{x}} \left[ \frac{\eta}{\mu\_0 B} \frac{\partial \langle VB \rangle}{\partial \mathbf{x}} \right] \tag{19}$$

The equation of electrical resistivity is also very important for the case of magnetically driven quasi-isentropic compression. In order to simplify the problem, a simple model is considered.

$$
\eta = \eta\_0 \left( 1 + \beta Q \right) \tag{20}
$$

In equation (20),0 is the electrical resistivity of conductors at temperatureof 0 ºC, is heating factor, *Q* is the heat capacity or increment of internal energy relative to that at temperatureof 0 ºC, which is related with temprature at the condensed states.

$$Q = \mathcal{c}\_\mathbf{V} T \tag{21}$$

In equation (21), *c*v is specific heat at constant volume, which is close to constant from 0 ºC to the temperature of vaporazation point.

For aluminum, is 0.69×10-9 m3/J, 0 is 2.55×10-8 m. Before vaporazation point, the equation (20) is suitable. After that, more complex electrical resisistivity model is needed.

In this simulation, the stress wave front is defined when the amplitude of pressure reaches to 0.1 GPa, and thediffusion front of magnetic field is determined when the magnetic flux density is up to 0.2 T[34]。

Fig.18 gives the distribution of density and temperature of Aluminum sample along Lagrangian coordinates for different times in the condition of loading current density 1.5 MA/cm.

The results in Fig.18 show that the density and temperature of aluminum sample vary with the loading time along the direction of sample thickness because of the Joule heating and magnetic field diffusion.

Magnetohydrodynamics of Metallic Foil Electrical

1.5MA/cm.

Explosion and Magnetically Driven Quasi-Isentropic Compression 367

Fig.19 gives the calculated results of distribution of magnetic induction strength along Lagrangian coordinates for different times in the condition of loading current density

Fig. 19. Distribution of magnetic induction strength along Lagrangian coordinates for

(a) current density of 1MA/cm (b) current density of 3MA/cm

front under the Lagrangian coordinates

Fig. 20. Physical characteristics of hydrodynamic stress wave front and magnetic diffusion

And Fig.20 gives the physical characteristics of hydrodynamic stress wave front and magnetic diffusion front under the Lagrangian coordinates. The velocity of stress wave front is far more than that of the magnetic diffusion front, which is the prerequisite of magnetically driven quasi-isentropic compression. And the velocity of magnetic diffusion

different times in the condition of loading current density 1.5MA/cm

front increases gradually with the increasing of loading current density.

Fig. 18. Distribution of density and temperature of Aluminum sample along Lagrangian coordinates for different times under the condition of loading current density 1.5 MA/cm at time of 0.09 s (a), 0.18 s (b), 0.27 s (c), 0.36 s (d) and 0.54 s (e)

(a) (b)

(c) (d)

time of 0.09 s (a), 0.18 s (b), 0.27 s (c), 0.36 s (d) and 0.54 s (e)

(e) Fig. 18. Distribution of density and temperature of Aluminum sample along Lagrangian coordinates for different times under the condition of loading current density 1.5 MA/cm at Fig.19 gives the calculated results of distribution of magnetic induction strength along Lagrangian coordinates for different times in the condition of loading current density 1.5MA/cm.

Fig. 19. Distribution of magnetic induction strength along Lagrangian coordinates for different times in the condition of loading current density 1.5MA/cm

And Fig.20 gives the physical characteristics of hydrodynamic stress wave front and magnetic diffusion front under the Lagrangian coordinates. The velocity of stress wave front is far more than that of the magnetic diffusion front, which is the prerequisite of magnetically driven quasi-isentropic compression. And the velocity of magnetic diffusion front increases gradually with the increasing of loading current density.

Fig. 20. Physical characteristics of hydrodynamic stress wave front and magnetic diffusion front under the Lagrangian coordinates

Magnetohydrodynamics of Metallic Foil Electrical

loading current density of 3 MA/cm.

around between experiments.

**5.1.1 Short-pulse shock initiation of explosive** 

data to test the capability of improved models.

Explosion and Magnetically Driven Quasi-Isentropic Compression 369

Fig. 22. The particle velocities of copper sample at different thickness in the condition of

**5. Applications of metallic foil electrically exploding driving highvelocity** 

The apparatus of metallic foil electrically exploding driving high velocity flyer offers an attractive means of performing shock initiation experiments. And the impact of an electrically exploding driven flyer produces a well-defined stimulus whose intensity and duration can be independently varied. Experiments are low-cost and there is fast turn-

Short-pulse shock initiation experiments will be very useful in developing more realistic theoretical shock initiation models. For the present, the models predicting shock initiation thresholds is short of, where very short pulses are employed . The technique can provide

Based on our experimental apparatus, the shock initiation characteristics of TATB and TATB-based explosives are studied[35,36]. Fig.23 and Fig.24 show the experimental results of shock initiation thresholds and run distance to detonation of a TATB-based explosive.

**flyers and magnetically driven quasi-isentropic compression** 

**5.1 Metallic foil electrically exploding driving highvelocity flyers** 

Fig.21 presents the relationships between the velocity of magnetic diffusion front and loading current density. The results show that an inflection poin occurs at the loading current density of 1 MA/cm, and that the results can be expressed with two linear equations (22)

$$\begin{cases} D = 0.008 + 0.46 \,\text{J}, & 1.0 < \text{J} \le 3 \quad \text{MA/cm} \\\\ D = 0.36 + 0.06 \,\text{J}, & 0.5 \le \text{J} \le 1.0 \quad \text{MA/cm} \end{cases} \tag{22}$$

In equation (22), *D* is the velocity of magnetic diffusion, and *J* is loading current density.

Fig. 21. The relationship of magnetic diffusion velocity varying with loading current densities.

Fig.22 is the case of copper samples under magnetically driven quasi-isentropic compression. The calculated results show that the particle velocity curves become steeper with the increasing of sample thickness, and that the shock is formed when the thickness is more than 2.5 mm for this simulating condition.

Fig.21 presents the relationships between the velocity of magnetic diffusion front and loading current density. The results show that an inflection poin occurs at the loading current density of 1 MA/cm, and that the results can be expressed with two linear equations

0.008 0.46 , 1.0 3 MA/cm

In equation (22), *D* is the velocity of magnetic diffusion, and *J* is loading current density.

Fig. 21. The relationship of magnetic diffusion velocity varying with loading current

more than 2.5 mm for this simulating condition.

Fig.22 is the case of copper samples under magnetically driven quasi-isentropic compression. The calculated results show that the particle velocity curves become steeper with the increasing of sample thickness, and that the shock is formed when the thickness is

*D JJ*

*DJJ*

0.36 0.06 , 0.5 1.0 MA/cm

(22)

(22)

densities.

Fig. 22. The particle velocities of copper sample at different thickness in the condition of loading current density of 3 MA/cm.
