**2. Fundamental Equations**

The fundamental equations which govern unsteady planar ( *m* = 0) or cylindrically ( *m* = 1) symmetric flow in a non-ideal gas in the presence of transverse magnetic field can be expressed as [12,21]

$$
\rho\_t \rho\_t + \rho u\_x + u \rho\_x + \frac{m \rho u}{x} = 0,\tag{1}
$$

$$
\mu\_t + \mu u\_x + \frac{1}{\rho}(p\_x + h\_x) = 0,\tag{2}
$$

$$
\mu p\_t + \mu p\_x + a^2 \rho \left( u\_x + \frac{m\mu}{\varkappa} \right) = 0,\tag{3}
$$

$$
\mu h\_t + uh\_x + 2h\left(u\_x + \frac{mu}{x}\right) = 0.\tag{4}
$$

Here *ρ*, *u*, *p* and *h* = *μH*<sup>2</sup> 2 denote the density, velocity, pressure, and magnetic pressure, respectively. *μ* denotes the magnetic permeability and *H* being used for transverse magnetic field. *a*2 = *γp ρ*(<sup>1</sup>−*bρ*) is speed of sound in real gases. *γ* is the ratio the specific heats at constant pressure and volume of the gas. *t* and *x* represents the time and space variables, respectively. *m* = 0 denotes to planar symmetry while *m* = 1 denotes cylindrical symmetry of the flow. It is assumed that electrical resistivity of the medium is zero and magnetic field is orthogonal to the radial motion of gas particles.

This system of Equations (1)–(4) is supplemented with the equation of state for real gas as follows

$$p = \frac{\rho RT}{(1 - b\rho)}.\tag{5}$$

*R*, *T* represent the gas constant and temperature, respectively.

The position of leading shock at any time *t* is given by *R* ˜ = *R* ˜(*t*). Therefore, the velocity of shock front is ( *dR* ˜ *dt*= *C*). The conditions for flow variables ahead of shock are characterized by

$$
\rho = \rho\_0(\mathbf{x}), \quad \mathbf{u} = \mathbf{0}, \quad \mathbf{p} = p\_{0\prime} \quad h = h\_0(\mathbf{x}). \tag{6}
$$

RH conditions at the leading shock (*x* = *R*˜(*t*)) are obtained by the conservation of laws that can be expressed in simplified form as follows [22]

$$\rho\_0(\rho)\_{\mathbf{x}=R} = \frac{(\gamma+1)}{(\gamma-1)}\rho\_0 \left[1 - \frac{2a}{\gamma-1} - \frac{2}{\gamma-1} \left(\frac{C}{\overline{\mathcal{U}}}\right)^2\right],\tag{7}$$

$$S\_{\lambda}(u)\_{\mathbf{x}=R} = \frac{2}{(\gamma - 1)} S \left[ 1 - a - \left(\frac{V}{S}\right)^2 \right],\tag{8}$$

$$\mathbf{r}(p)\_{\mathbf{x}=R} = \frac{2\rho\eta lI^2}{\gamma+1} \left[1 - a - \frac{\gamma - 1}{2\gamma} \left(\frac{C}{U}\right)^2\right] - \frac{1}{2} \left(\frac{\gamma + 1}{\gamma - 1}\right)^2 \mathbb{C}\_0 \rho\_0 lI^2 \left[1 - \frac{4a}{\gamma - 1} - \frac{4}{\gamma - 1} \left(\frac{C}{U}\right)^2\right],\tag{9}$$

$$\zeta(h)\_{\mathbf{x}=R} = \frac{1}{2} \left( \frac{\gamma + 1}{\gamma - 1} \right)^2 \mathbb{C} \rho \rho\_0 \mathcal{U}^2 \left[ 1 - \frac{4a}{\gamma - 1} - \frac{4}{\gamma - 1} \left( \frac{\mathbb{C}}{\mathcal{U}} \right)^2 \right],\tag{10}$$

where *C*0 = 2*h*0 *<sup>ρ</sup>*0*U*<sup>2</sup> is the shock cowling number, *C*<sup>2</sup> = *γp*0 *ρ*0 is the velocity of shock and *α* = (*γ* − <sup>1</sup>)*bρ*0. The initial density *ρ*0 is assumed to follow the power law given as

$$
\rho\_0 = \rho\_\varepsilon \mathbf{x}^{-\delta},
\tag{11}
$$

where *x* is the perpendicular distance of the point on leading shock from the point of explosion. *ρc* is the constant density and *δ* is an exponent. Using the conservation of total energy, we ge<sup>t</sup>

$$E = \int\_0^{\mathbb{R}} \left[ \frac{1}{2} \mu^2 + \frac{(1 - b\rho)}{(\gamma - 1)} \left( \frac{p}{\rho} - \frac{p\_0}{\rho\_0} \right) + \left( \frac{h}{\rho} - \frac{h\_0}{\rho\_0} \right) \right] \rho \mathbf{x}^{\mathbf{m}} d\mathbf{x}, \quad m = 0, 1, \tag{12}$$

where *E* denotes the surface energy carried by blast waves per unit of area. We obtain the following relation by the Lagrangian equation of continuity:

$$\int\_0^{\mathbb{R}} \frac{\rho}{\rho\_0} \mathbf{x}^m d\mathbf{x} = \frac{\bar{\mathbb{R}}^{m+1}}{m+1}.\tag{13}$$

We use the value of *R* ˜ 0 *ρ ρ*0 *xmdx* from Equation (13) in Equation (12) to ge<sup>t</sup> the following expression

$$E = \int\_0^{\mathbb{R}} \left[ \frac{1}{2} \rho u^2 + \frac{(1 - b\rho)}{(\gamma - 1)} p + h \right] x^m dx - \left\{ \frac{(1 - b\rho) p\_0}{\gamma - 1} + h\_0 \right\} \frac{\mathbb{R}^{m + 1}}{m + 1}.\tag{14}$$
