**3. Similarity Transformation of Fundamental Equations**

As the process is self-similar, *x* = *R*(*t*) and *U* = *dR* ˜ *dt* . It gives us the characteristic scale. We, therefore, introduce the following new independent non-dimensional variables in place of *x* and *S* :

$$\mathbf{x}/\bar{\mathbf{R}} = \mathbf{r}, \quad (\mathbb{C}/\mathcal{U})^2 = \mathbf{s}. \tag{15}$$

*x*/*R* ˜ = *r* is known as similarity variable. Non-dimensional quantities *<sup>u</sup>*/*U*, *p*/*p*0, *ρ*/*ρ*0 and *<sup>h</sup>*/(*p*0(*U*/*C*)<sup>2</sup>) are assumed to be equal to *F*, *G*, Π, and *H*, respectively which are consistent with Equations (1)–(4). We, therefore, can take the similarity transformations as:

$$u = \mathcal{U}F(r, s),\tag{16}$$

$$p = p\_0 (\mathcal{U}/\mathcal{C})^2 G(r, s), \tag{17}$$

$$
\rho = \rho\_0 \Pi(r, s),
\tag{18}
$$

$$h = p\_0 (\mathcal{U}/\mathbb{C})^2 H(r, \mathbf{s}) = p\_0 H(r, \mathbf{s}) / \text{s.t.} \tag{19}$$

Using the system of Equations (16)–(19), we obtain

$$\frac{\partial}{\partial x} = \frac{1}{\overline{\mathbb{R}}} \frac{\partial}{\partial r} \tag{20}$$

$$\frac{D}{Dt} = \frac{U}{\mathcal{R}} \left[ (F - r) \frac{\partial}{\partial r} + \lambda s \frac{\partial}{\partial s} \right],\tag{21}$$

where *λ* = *<sup>λ</sup>*(*s*) = *<sup>R</sup>*˜(*ds*/*dR*˜)/*<sup>s</sup>*. Substituting (16)–(19) into the fundamental Equations (1)–(4), we obtain 

$$\ln\left(F - r\right)\Pi\_r + \lambda \operatorname{s\,II}\_s + \Pi \left(F\_r + \frac{mF}{r}\right) = 0,\tag{22}$$

$$
\Pi \left( -\frac{\lambda}{2} F + (F - r) F\_r + \lambda s F\_\delta \right) + \frac{1}{\gamma} (G\_r + H\_\delta) = 0,\tag{23}
$$

$$-\lambda \mathcal{G} + (F - r)\mathcal{G}\_r + \lambda s \mathcal{G}\_s + \frac{\gamma \mathcal{G}}{(1 - \tilde{b}\bar{\Pi})} \left(F\_r + \frac{mF}{r}\right) = 0,\tag{24}$$

$$-\lambda H + (F - r)H\_r + \lambda s \, H\_s + 2H \left( F\_r + \frac{mF}{r} \right) = 0,\tag{25}$$

where ¯ *b* = *bρ*0. Using Equation (14), we obtain

$$s\left(\frac{\mathbb{R}\_0}{\overline{R}}\right)^{m+1} = \int\_0^1 \left(\frac{1}{2}\gamma I\Gamma F^2 + \frac{(1-\overline{b}\Gamma\Gamma)G}{(\gamma - 1)} + H\right)r^m dr - \frac{(1-\overline{b}\Gamma\Gamma)s}{(\gamma - 1)(m+1)} - \frac{\gamma C\_0}{2(m+1)},\tag{26}$$

where

$$\mathcal{R}\_0 = \left(\frac{E}{p\_0}\right)^{1/(m+1)}.\tag{27}$$

Using (16)–(19), RH conditions become

$$\Pi(1,s) = \frac{(\gamma+1)}{(\gamma-1)} \left[ 1 - \frac{2a}{\gamma-1} - \frac{2}{\gamma-1}s \right],\tag{28}$$

$$F(1,s) = \frac{2}{\gamma + 1}(1 - \alpha - s),\tag{29}$$

$$G(1,s) = \frac{2\gamma}{\gamma+1} \left[1 - a - \frac{(\gamma-1)s}{2\gamma}\right] - \frac{1}{2} \left(\frac{\gamma+1}{\gamma-1}\right)^2 \mathbb{C}\_0 \gamma \left[1 - \frac{4a}{\gamma-1} - \frac{4s}{\gamma-1}\right],\tag{30}$$

$$H(1,s) = \frac{\gamma}{2} \mathbb{C}\_0 \left(\frac{\gamma+1}{\gamma-1}\right)^2 \left[1 - \frac{4\alpha}{\gamma-1} - \frac{4s}{\gamma-1}\right].\tag{31}$$

Now differentiating Equation (26) regarding *s*, we obtain the expression for *λ* as

$$\lambda = \frac{J(m+1) - \frac{(1-\tilde{b}\Pi)s}{\gamma - 1} - \frac{\gamma \mathbb{C}\_0}{2}}{J - s\frac{dI}{ds} - \frac{\gamma \mathbb{C}\_0}{2(m+1)}}. \tag{32}$$

Here

$$J = \int\_0^1 \left(\frac{1}{2}\gamma\Pi F^2 + \frac{(1-\bar{b}\Pi)G}{(\gamma -1)} + H\right)r^m dr.\tag{33}$$

The self-similar process of the explosion has clearly explained the power rule particularly in Equation (27).

## **4. Construction of Solutions in Power Series of** *s*

As we are seeking solution for strong shocks, the velocity of leading shock front *S* is very large than the velocity of sound waves *V* in ideal gas. The quantity *s* = (*V*/*S*)<sup>2</sup> is very small for strong shocks. We, therefore, expand the non-dimensional quantities *F*, *G*, Π and *H* in power series *s* = (*V*/*S*)<sup>2</sup> as follows

$$\begin{aligned} F &= F^{(0)} + sF^{(1)} + s^2 F^{(2)} + s^3 F^{(3)} + \dots \\ G &= G^{(0)} + sG^{(1)} + s^2 G^{(2)} + s^3 G^{(3)} + \dots \\ \Pi &= \Pi^{(0)} + s\Pi^{(1)} + s^2 \Pi^{(2)} + s^3 \Pi^{(3)} + \dots \\ H &= H^{(0)} + sH^{(1)} + s^2 H^{(2)} + s^3 H^{(3)} + \dots \end{aligned} \tag{34}$$

Here *<sup>F</sup>*(*i*), *<sup>G</sup>*(*i*), Π(*i*) and *H*(*i*) (*i* = 0, 1, 2, ...) are either constants or functions of *r*. We use the power series expansions from Equation (34) in Equation (33) to obtain the value of *J* as

$$J = J\_0(1 + \sigma\_1 s + \sigma\_2 s^2 + \dots),\tag{35}$$

where

$$\begin{split} l\_{0} &= \quad \int\_{0}^{1} \left[ \frac{\gamma}{2} \Pi^{(0)}(F^{(0)})^{2} + \frac{(1 - b\Pi^{(0)})}{\gamma - 1} G^{(0)} + H^{(0)} \right] r^{\pi} dr, \\ \varphi\_{1} l\_{0} &= \quad \int\_{0}^{1} \left[ \frac{\gamma}{2} \Pi^{(1)}(F^{(0)})^{2} + \gamma \Pi^{(0)} F^{(0)} F^{(1)} + \frac{1}{\gamma - 1} (G^{(1)} - b\Pi^{(0)} G^{(1)} - b\Pi^{(1)} G^{(0)}) + H^{(1)} \right] r^{\pi} dr, \\ \varphi\_{2} l\_{0} &= \quad \int\_{0}^{1} \left[ \frac{\gamma}{2} \Pi^{(2)}(F^{(0)})^{2} + \gamma \Pi^{(0)} F^{(0)} F^{(2)} + \frac{\gamma}{2} \Pi^{(0)} (F^{(1)})^{2} + 2\Pi^{(1)} F^{(1)} F^{(0)} + H^{(2)} \right] r^{\pi} dr \\ &\quad + \quad \int\_{0}^{1} \frac{1}{\gamma - 1} [G^{(2)} - b\Pi^{(2)} G^{(0)} - b\Pi^{(1)} G^{(1)} - b\Pi^{(0)} G^{(2)}] r^{\pi} dr, \\ \dots \end{split} \tag{36}$$

Using (35) in (26), we obtain

$$s\left(\frac{\tilde{R}\_0}{\mathcal{R}}\right)^{m+1} = J\_0\left[\left(1 - \frac{\gamma \mathbb{C}\_0}{2(m+1)l\_0}\right) + \left(\sigma\_1 - \frac{1}{(m+1)(\gamma - 1)l\_0}\right)s + \sigma\_2 s^2 + \dots\right].\tag{37}$$

In view of (15), Equation (37) becomes

$$\left(\frac{V}{S}\right)^2 \left(\frac{\mathcal{R}\_0}{\overline{R}}\right)^{m+1} = I\_0 \left[ \left(1 - \frac{\gamma \mathcal{C}\_0}{2(m+1)I\_0}\right) + \left(\sigma\_1 - \frac{1}{(m+1)(\gamma - 1)I\_0}\right) \left(\frac{V}{S}\right)^2 + \sigma\_2 \left(\frac{V}{S}\right)^4 + \dots \right]. \tag{38}$$

This equation provides a relation between shock velocity and its position at time *t*. We expand *λ* by using Equations (32) and (35) as follows:

$$
\lambda = (m+1)[1 + \sigma\_1's + 2\sigma\_2's^2 + \dots]\_\prime \tag{39}
$$

where

$$\begin{aligned} \sigma\_1' &= \frac{\sigma\_1 - \frac{1}{(m+1)(\gamma - 1)f\_0}}{1 - \frac{\gamma \mathbb{C}\_0}{2(m+1)(\gamma - 1)f\_0}}, \\ \sigma\_2' &= \frac{\sigma\_2}{1 - \frac{\gamma \mathbb{C}\_0}{2(m+1)f\_0}}, \\ \sigma\_3' &= \frac{\sigma\_3}{1 - \frac{\gamma \mathbb{C}\_0}{2(m+1)f\_0}}. \end{aligned}$$

For simplification, let us consider *λ*1 = *σ*1, *λ*2 = <sup>2</sup>*σ*2, *λ*3 = <sup>3</sup>*σ*3,..., then the Equation (39) becomes

$$
\lambda = (m+1)[1 + \lambda\_1 s + \lambda\_2 s^2 + \dots]. \tag{40}
$$

We use the relations from Equations (34) and (40) in Equations (22)–(25). Comparing the likewise powers of s on both sides of an equation, we ge<sup>t</sup> relations in terms of Ordinary Differential Equations (ODEs). Comparing the terms free from *s*, we ge<sup>t</sup>

$$\left( (F^{(0)} - r) \Pi\_r^{(0)} + \Pi^{(0)} \left( F\_r^{(0)} + \frac{m F^{(0)}}{r} \right) = 0,\tag{41}$$

$$(F^{(0)} - r)\Pi^{(0)}F\_r^{(0)} + \frac{1}{\gamma}(G\_x^{(0)} + H\_x^{(0)}) = \frac{(m+1)F^{(0)}\Pi^{(0)}}{2},\tag{42}$$

$$-(m+1)G^{(0)} + (F^{(0)} - r)G\_r^{(0)} + \frac{\gamma G^{(0)}}{(1 - b\Pi^{(0)})} \left( F\_r^{(0)} + \frac{mF^{(0)}}{r} \right) = 0,\tag{43}$$

$$-(m+1)H^{(0)} + (F^{(0)} - r)H\_r^{(0)} + 2H^{(0)}\left(F\_r^{(0)} + \frac{mF^{(0)}}{r}\right) = 0.\tag{44}$$

Equating the power *s*, we ge<sup>t</sup>

(*F*(0) − *r*)Π(1) *r* + *F*(1)Π(0) *r* + (*m* + 1)Π(1) + Π(0)*F*(1) *r* + *mF*(1) *r* + Π(1)*F*(0) *r* + *mF*(0) *r* = 0, (45) − Π(0) 2 (*m* + 1)[*F*(1) + *<sup>λ</sup>*1*F*(0)] − (*m* + 1)Π(1)*F*(0) 2 + Π(0)[(*F*(0) − *r*)*F*(1) *r* + *F*(1)*F*(0) *r* ] + Π(1)(*F*(0) − *r*)*F*(0) *r* + (*m* + 1)Π(0)*F*(1) + 1*γ*[*G*(1) *r* + *H*(1) *r* ] = 0, (46) − *λG*(1) + *F*(1)*G*(0) *r* + *<sup>λ</sup>*1(*G*(0) *r* + *G*(1)) + *F*(0)*G*(1) *r* + [1 + *b*Π(1)*G*(0) + {*b*Π(0) + (*b*Π(0))<sup>2</sup> + ...}*G*(1)][*F*(0) + *mF*(0) *r* ] = 0, (47) − (*m* + <sup>1</sup>)*<sup>λ</sup>*1*H*(0) + (*F*(0) − *r*)*H*(1) *r* + *F*(1)*H*(0) *r* + 2*H*(0)*F*(1) *r* + *mF*(1) *r* + 2*H*(1)*F*(0) *r*+ *mF*(0) = 0, (48)

From Equations (28)–(31) and (34), we have

*r*

$$\begin{split} F^{(0)}(1) &= \frac{2}{\gamma+1} (1-a), \qquad G^{(0)}(1) = \frac{2\gamma}{\gamma+1} (1-a) - \left(\frac{\gamma+1}{\gamma-1}\right)^2 \frac{\gamma \mathbb{C}\_0}{2} \left(1 - \frac{4a}{\gamma-1}\right), \\ \Pi^{(0)}(1) &= \frac{\gamma+1}{\gamma-1} \left(1 - \frac{2a}{\gamma-1}\right), \quad H^{(0)}(1) = \left(\frac{\gamma+1}{\gamma-1}\right)^2 \frac{\gamma \mathbb{C}\_0}{2} \left(1 - \frac{4a}{\gamma-1}\right). \end{split} \tag{49}$$

$$\begin{split} F^{(1)}(1) &= -\frac{2}{\gamma+1}, \qquad G^{(1)}(1) = -\frac{\gamma-1}{\gamma+1} + 2\gamma \mathbb{C}\_0 \frac{(\gamma+1)^2}{(\gamma-1)^3}, \\ \Pi^{(1)}(1) &= -\frac{2(\gamma+1)}{(\gamma-1)}, \qquad H^{(1)}(1) = -2\gamma \mathbb{C}\_0 \frac{(\gamma+1)^2}{(\gamma-1)^3}. \end{split} \tag{50}$$

To ge<sup>t</sup> the first approximate solution, we determine *<sup>F</sup>*(0), *<sup>G</sup>*(0), Π(0) and *H*(0) from the system of nonlinear ODEs (41)–(44) with the boundary conditions given in Equation (49). Finally, these values are used in Equation (36)1 to ge<sup>t</sup> approximate solution as follows

$$\begin{aligned} \mu &= SF^{(0)}(r), \quad p = p\_0 (\mathcal{S}/V)^2 G^{(0)}(r), \\ \rho &= \rho\_0 \Pi^{(0)}(r), \quad h = p\_0 (\mathcal{S}/V)^2 H^{(0)}(r). \end{aligned} \tag{51}$$

To ge<sup>t</sup> the second approximation, we need to determine the values of *<sup>F</sup>*(1), *<sup>G</sup>*(1), Π(1) and *H*(1) in Equation (45). *<sup>F</sup>*(0), *<sup>G</sup>*(0), Π(0) and *H*(0) are used from the first approximation. *<sup>F</sup>*(1), *<sup>G</sup>*(1), Π(1) and *H*(1) contain *λ*1. We use these values in terms of *λ*1 in Equation (36)2 to finally obtain the value of *λ*1. The obtained value of *λ*1 is used to ge<sup>t</sup> the second approximate solution. The other steps involve the repetition of the above process to ge<sup>t</sup> higher order approximate solutions of the problem.
