Impact of dust in the decay of blast waves produced by a nuclear explosion

Abstract

In this paper, we have studied the impact created by the introduction of up to 5% dust particles in enhancing the decay of blast waves produced by a nuclear explosion. A mathematical model is designed and modified using appropriate assumptions, the most important being treating a nuclear explosion as a point source of energy. A system of partial differential equations describing the one-dimensional, adiabatic, unsteady flow of a relaxing gas with dust particles and radiation effects is considered. The symmetric nature of an explosion is captured using the Lie group invariance and self-similar solutions obtained for the gas undergoing strong shocks. The enhancements in decay caused by varying the quantity of dust are studied. The energy released and the damage radius are found to decrease with time with an increase in the dust parameters.

1. Introduction

Science has created deadly nuclear bombs that can destroy the world. And only science can find their opposite counterpart to mitigate or reduce the effects of explosion. Our present work is one such step in that direction. Studies have shown that atmospheric dust absorbs the effects of shock waves [1–3]. We concentrate on studying the effects of dust particles in a pseudo-fluid (dusty gas) and focus on the absorption of shock waves by the dust particles. Mathematically, such problems are altogether inaccessible unless we make some assumptions. The general problem of the decay of the blast wave can be treated by either approximate analytical methods or numerically, or by a combination of these.

Point blast theory was formulated to describe the phenomena produced in continuous media by exploding charges of small size and weight but with high specific heat [4]. It deals with strong explosions in the atmosphere, explosive processes in outer space, etc. In a nuclear explosion, the temperatures reached are around 50 million kelvin inside the active material compared with about 5000 kelvin inside an ordinary explosion [5]. This brings about considerable differences in the treatment of these two explosions, the most important being considering the nuclear explosion as a point source of energy. The point source solution determines the development of the shock wave as its radius increases and pressure decreases.

The physical situation of the dynamics of a fluid–particle system is very complicated. The fluidized bed and burning of well-packed gun powder are some examples of a fluid–particle system. There are different phases in the flow of a mixture of fluid and solid particles which are dependent on the rate of the fluid flow, the details of which are given in [6]. The pseudo-fluid that we refer to is the dilute phase of the two-phase flow of a mixture of solid particles and a fluid, in which the solid particles occupy less than 5% of the total volume of the mixture and mix well with the fluid in the flow field. The existence of the solid particles changes the properties of the mixture significantly; as a result, the dilute phase of solid–gas flow is a good approximation of the actual condition.

The similarity solutions for blast waves produced by nuclear explosion were first found by Taylor [7,8], Sedov [9] and von Neumann [10] independently. They used dimensional analysis to find the flow variables and the energy released. Thereafter, blast waves have been a subject of interest to many. Sharma & Radha [11] have investigated the self-similar solutions in an ideal gas with relaxing effects. Singh & Vishwakarma [3] found non-similar spherical solutions for isothermal shocks in a dusty gas with radiation heat flux and the initial density obeying the exponential law. Strong shocks in dusty gases with planar symmetry have been investigated by Pai et al. [1]. The similarity solution was sought when the surrounding medium was of constant density and negligible counter pressure. The shock-free flow field emanating from a pulsating sphere in a vibrationally relaxing gas without viscosity and heat conduction was studied by Scott & Johannesen [12]. Nath [13] investigated the propagation of a cylindrical shock wave in a non-ideal dusty gas in the presence of conductive and radiative heat fluxes with increasing energy and with variable azimuthal and axial fluid velocities.

Our present work is an extension of our previous work [2], wherein we studied the self-similar solutions in a non-ideal dusty gas. We now study the classical point blast problem in a one-dimensional, adiabatic, unsteady flow of a relaxing gas consisting of up to 5% dust particles considering the radiation effects. Chapman–Jouguet conditions relate the flow immediately ahead of and behind the blast wave. The decay of a blast wave is studied using the self-similarity property of a point blast explosion. An expression for the energy released during the blast, which is a function of time, is derived using the reduced flow variables. The impact of adding dust particles to enhance the decay of blast waves is studied. To take into account all factors that affect blast wave propagation will make the problem very complex, hence we take into account factors that predominate in the process [4]. The decay of blast waves depends upon many factors: yield of explosive material, density of the air, humidity, altitude of detonation, vibrational energy of the molecules which are ionized as a result of high temperatures near the vicinity of the blast, thermal inversion of the atmosphere, etc. Temperature inversion layers, which are also called thermal inversion layers, are areas where the normal decrease in air temperature with increasing altitude is reversed and the air above is warmer than the air below. Inversion layers are significant as they limit the diffusion of air pollutants and cause stagnation in weather [14].

Consider a small explosive mass concentrated in a volume much smaller than the ambient medium. An explosion produced at some instant of time releases energy of density (per unit volume) much higher than in the ambient medium. The pressure and energy of the medium surrounding the explosion will then increase almost instantaneously. The difference in the ambient pressure and the high pressure generated inside the explosion causes the high-pressure waves to move forward spherically into the undisturbed medium and cause destruction. A schematic showing the different temperature regions as given in [5] is depicted in figure 1, and the notations used in this article are given in table 1.

Figure 1.

A schematic showing the top view of the self-similar temperature regions. (Online version in colour.)

Table 1.

Notations used.

ρ	density of mixture	u	velocity of mixture
p	pressure of mixture	Γ	ratio of specific heat of mixture
Z	volume fraction	σ	vibrational energy
F	radiative flux	Q	change of vibrational energy
m	geometry (planar, cylindrical, spherical)	x	spatial coordinate
t	time coordinate	kp	mass fraction
cp	specific heat of gas at constant pressure	cv	specific heat of gas at constant volume
ρsp	density of solid particles	csp	specific heat of solid particles
β	ratio of the specific heat of solid particles to the specific	Vg	total volume of gas
	heat of the gas at constant pressure	γ	ratio of specific heat of the perfect gas
G	ratio of the density of the solid particles to the species	$\overline{\sigma }$	equilibrium value of σ
	density of the gas	H	radiation parameter
τ	relaxation time	ϕ	ratio of vibrational specific heat to specific
T	temperature of mixture		gas constant
c	speed of light	μR	Rossland’s absorption coefficient
a	Stefan–Boltzmann radiation constant	e	internal energy
V	shock velocity	$Q_{\ast }$	blast energy
E	total energy	ξ	similarity variable
mg	total mass of the mixture	R	gas constant
C	equilibrium speed of sound in the mixture

2. Basic equations

Radiation gasdynamics is a field of science in which we study the interactions between the gasdynamics field (rarefied or continuum) and the radiation field. In radiation gasdynamics, when the gas is considered as a continuum, we add the radiation term to the gasdynamic variables. Since it is not possible to solve the fundamental equations of ordinary gasdynamics in general, we cannot solve the fundamental equations of radiation gasdynamics either [6]. The inclusion of dust particles and the relaxing effects will further complicate the system. To bring about essential features of radiation effects, we have to make reasonable approximations so that the fundamental equations may be simplified into a form which can be analysed. The following are some of the approximations that we make: (i) the nuclear explosion is treated as a point source of energy, where energy is released instantaneously and the volume occupied by the explosive and the mass of the charge are negligible; (ii) the flow is one dimensional, adiabatic and can have planar (m = 0), cylindrical (m = 1) or spherical (m = 2) symmetry; (iii) the dusty gas is grey and opaque; (iv) radiation pressure and radiation energy are very small in comparison with material pressure and energy, hence they are ignored; (v) the particles are spherical, of uniform size, incompressible and occupy less than 5% of the total volume; (vi) their specific heat is constant and the temperature is uniform within each particle; (vii) interaction between particles of different sizes is not considered; and (viii) heat transfer and boundary layer effects with the duct walls are not considered, hence radiation is the only dissipative mechanism [1–3,6]. The equations can now be written as

$$\begin{array}{rl}& {\rho }_t+u{\rho }_x+u_x\rho +\frac{m\rho u}{x}=0,\\ & u_t+u{u}_x+\frac{p_x}{\rho }=0,\\ & p_t+u{p}_x+\frac{p\mathit{\Gamma }}{(1-Z)}(u_x+\frac{mu}{x})+\frac{(\mathit{\Gamma }-1)}{(1-Z)}(\rho Q+F_x)=0\\ \text{and}& {\sigma }_t+u{\sigma }_x=Q,\end{array}\}$$ 2.1

where x is the spatial coordinate, being either axial in flows with planar geometry or radial in cylindrical and spherical symmetric flows, t is the time, u is the particle velocity, ρ is the density, p is the pressure, F is the radiative flux and σ is the vibrational energy. Here, Z = V_sp/V_g is the volume fraction and k_p = m_sp/m_g is the mass fraction of the solid particles in the mixture, where m_sp and V_sp are the total mass and volumetric extension of the solid particles, respectively, and V_g and m_g are the total volume and total mass of the mixture, respectively; Γ = γ(1 + λβ)/(1 + λβγ), λ = k_p/(1 − k_p), β = c_sp/c_p, γ = c_p/c_v, where c_sp is the specific heat of solid particles, c_p is the specific heat of the gas at constant pressure and c_v the specific heat of the gas at constant volume. The entities Z and k_p are related via the expression Z = θρ, where θ = k_p/ρ_sp, with ρ_sp as the species density of the solid particles. The variable G = ρ_sp/ρ is the ratio of the density of the solid particles to the species density of the gas. The quantity Q denotes the rate of change of vibrational energy and is the known function of p, ρ and σ; it is given by the expression

$$Q=\frac{\overline{\sigma }(p,\rho )-\sigma }{\tau (p,\rho )},$$ 2.2

where $\overline{\sigma }$ denotes the equilibrium value of σ and is given by

$$\overline{\sigma }={\overline{\sigma }}_0(x)+(\frac{p(1-\theta \rho )}{(1-k_p)\rho }-\frac{p_0(1-\theta {\rho }_0)}{(1-k_p){\rho }_0})\varphi .$$ 2.3

The quantities τ and ϕ are the relaxation time and the ratio of vibrational specific heat to the specific gas constant, respectively. Here and throughout the article, variables with the subscript 0 refer to the initial rest conditions. The equation of state is given by

$$p=\frac{(1-k_p)}{(1-Z)}\rho RT,$$ 2.4

where T is the temperature of the gas and also that of the solid particles as the equilibrium flow conditions are maintained and R is the specific gas constant. Equation (2.1) can be written in conservation form as follows:

$$\begin{array}{rl}& {\rho }_t+{(\rho u)}_x=\frac{-m\rho u}{x},\\ & {(\rho u)}_t+{(p+\rho u^2)}_x=\frac{-m\rho u^2}{x},\\ & {(\frac{\rho u^2}{2}+\frac{(1-\theta \rho )p}{(\mathit{\Gamma }-1)}+\rho \sigma )}_t+{(u\rho \sigma +pu+\frac{\rho u^3}{2}+\frac{(1-\theta \rho )pu}{(\mathit{\Gamma }-1)}+F)}_x\\ & =-\frac{m\rho u^3}{2x}-\frac{(\mathit{\Gamma }-\theta \rho )mpu}{x(\mathit{\Gamma }-1)}-\frac{mpu\sigma }{x}\\ \text{and}& {(\rho \sigma )}_t+{(\rho \sigma u)}_x=\rho Q-\frac{m\rho u\sigma }{x}.\end{array}\}$$ 2.5

We use the following dimensionless forms for the dimensional variables appearing in equations (2.1)–(2.5):

$$\begin{array}{rl}& \check{x}=\frac{x}{x_c},\check{t}=\frac{t}{t_c},\check{u}=\frac{u{t}_c}{x_c},\check{\rho }=\frac{\rho }{{\rho }_c},\check{p}=\frac{p{t}_c^2}{{\rho }_c{x}_c^2},\check{T}=\frac{T}{T_c},\check{R}=\frac{R{T}_c{t}_c^2}{x_c^2},\\ & \check{Q}=\frac{Q{t}_c^3}{x_c^2},\check{\tau }=\frac{\tau }{t_c},\hspace{0.17em}{\check{\rho }}_{sp}=\frac{{\rho }_{sp}}{{\rho }_c},\hspace{0.17em}\check{\sigma }=\frac{\sigma t_c^2}{x_c^2},\hspace{0.17em}\check{F}=\frac{F{t}_c^3}{{\rho }_c{x}_c^3},\end{array}$$

where x_c, t_c, ρ_c and T_c are some reference values of x, t, ρ and T, respectively. It may be noted that, after using the above dimensionless forms in equations (2.1)–(2.5) and then suppressing the check sign, the equations remain invariant. Hereinafter, we will use the dimensionless form of the system of equations (2.1)–(2.5).

3. The radiative flux

After the explosion, following the energy release, thermal waves are propagated in and heat conduction is then determined mainly by radiation. Radiation can play two roles, i.e. (i) transport energy and (ii) exert pressure. The radiation pressure is found to be negligible compared with the material pressure [5], hence it is the transportation of energy by radiation that is of interest in this study. In a nuclear explosion, the surrounding air is heated to about 1 million kelvin or more. At such high temperatures, the effects of radiation become important. In fact, radiation can transport energy from one place in a material to another very effectively.

The radiative diffusion model for an optically thick grey gas is independent of frequency considerations [3]. In the diffusion limit the photon mean free path is determined by Rossland’s mean opacity. The diffusive flux is taken as

$$F=F(p,\rho ),$$ 3.1

where $F=(-ac/3{\mu }_R)\hspace{0.17em}(T^4-T_0^4)$, with c as the speed of light and a = 7.67 × 10⁻¹⁵ ergs cm⁻³ deg⁻⁴ as the Stefan–Boltzmann radiation constant. The parameters T₀ and T are the temperatures before and after the blast. The term T₀ can be ignored as it is negligible in comparison with T. The absorption coefficient, μ_R, is Rossland’s mean opacity and it is assumed to be a function of ρ and absolute temperature T. The dependence of Rossland’s mean opacity on temperature and density has been discussed in [6]. Let μ*_R = 1/μ_R, then

$${\mu }_R^{\ast }={\mu }_0{\rho }^g{T}^d.$$ 3.2

Since from (2.4) $T=\frac{p(1-\theta \rho )}{R\rho (1-k_p)}$, the diffusive flux F takes the form

$$F(p,\rho )=H{\rho }^{g-4-d}{[p(1-\theta \rho )]}^{4+d},$$ 3.3

where H = ( − acμ₀/(3 [R(1 − k_p)]^4+d)) is taken as a radiation parameter.

For radiation from nuclear detonation, the range of temperatures of the gas considered is very large, from low temperature of atmospheric air to high temperature of several million kelvin. For such a large range of temperatures, chemical reactions and ionization take place and the composition of air changes greatly. The problem of determination of absorption of air becomes very complicated. The processes which determine the absorption coefficient of air change with temperature; the details are given in [5]. At low temperatures, the air mainly consists of molecular nitrogen and molecular oxygen. Hirschfelder et al. [15] have experimentally observed the equilibrium concentrations of NO₂ in air for temperatures below 5000 kelvin. The amount of NO₂ and vibrational excitation formed in the air is limited to the region in which the shock temperature is 2000 kelvin or higher [5].

4. Chapman–Jouguet conditions

The governing equations for the flow over a Boeing 747, through a subsonic wind tunnel or past a windmill, are the same but their flow fields are different. The difference is due to the boundary conditions, which are different for each of the above examples. The boundary conditions, and sometimes the initial conditions, dictate the particular solutions to be obtained from the governing equations [16]. Here, we model the flow field after the explosion has occurred in the atmosphere and find the similarity solutions using appropriate initial and boundary conditions.

The mean free path of radiation is the mean distance over which a photon travels before it is absorbed by a molecule of the gas. The boundary condition of the radiation field depends upon the mean free path of radiation. When the mean free path of radiation is very small, i.e. the optically thick case, all radiation terms can be expressed in terms of temperature. Under this condition we need only the boundary condition of temperature. Hence the consideration of gasdynamic boundary conditions is sufficient for this case of radiation gasdynamics [6].

Consider a shock front moving forward with a velocity V, propagating into an inhomogeneous medium specified by u₀ ≡ 0, p₀ ≡ 0, ρ₀ = ρ₀(x), σ₀ = σ₀(x), F₀ = 0 and negligible rate of change of vibrational energy Q (figure 2). The boundary conditions immediately behind the shock front for the system (2.5) are given by the conservation laws [9],

$$\begin{array}{rl}& {\rho }_0(V-u_0)=\rho (V-u),\\ & p_0+{\rho }_0{(V-u_0)}^2=p+\rho {(V-u)}^2,\\ & \frac{p_0}{{\rho }_0}+e_0+Q_{\ast }+\frac{{(V-u_0)}^2}{2}\\ & =\frac{p}{\rho }+e+\frac{{(V-u)}^2}{2},\text{(without radiation effects)}\\ & \frac{p_0}{{\rho }_0}+e_0+Q_{\ast }+\frac{{(V-u_0)}^2}{2}+\frac{F_0}{(V-u_0)}\\ & =\frac{p}{\rho }+e+\frac{{(V-u)}^2}{2}+\frac{F}{\rho (V-u)},\text{(with radiation effects)}\\ \text{and}& \sigma ={\sigma }_0.\end{array}\}$$ 4.1

Here, the difference V − u₀ is the particle velocity relative to the shock ahead of the wavefront, F/ρ(V − u) is the net radiative flux in the positive direction, $Q_{\ast }$ is the blast energy released at the shock, e is the internal energy per unit mass of the system and is given by e = σ + ((1 − Z)p/(Γ − 1)ρ). In equations (4.1) we take account of the fact that the value of Γ, ahead of the wave and behind the wave, can be different [9]. In the following subsection, the case of ideal, dusty gas without radiation effects will be discussed, whereas the other cases will be discussed at a later stage.

Figure 2.

Chapman–Jouguet conditions relate the flow variables ahead of and behind the shock front. (Online version in colour.)

(a). Ideal and dusty gas without radiation effects, i.e. F = 0

The internal energy of the system in this case will be given by

$$e=\sigma +\frac{p(1-\theta \rho )}{\rho (\mathit{\Gamma }-1)}.$$ 4.2

Using (4.1)_1,2 and (4.2) we get the following forms of flow variables:

$$\frac{{\rho }_0}{\rho }=1-\frac{u}{V}=y,p=p_0+{\rho }_0{V}^2(1-y).$$ 4.3

Substituting (4.2) and (4.3) in (4.1)₃ we get

$$\begin{array}{rl}& y^2-\frac{2}{\mathit{\Gamma }+1}\{(\mathit{\Gamma }+\theta {\rho }_0)+\frac{\mathit{\Gamma }(1-\theta {\rho }_0)}{{\mathit{\Gamma }}_0}\frac{C_0^2}{V^2}\}y\\ & +\frac{1}{(\mathit{\Gamma }+1)}\{(\mathit{\Gamma }-1+2\theta {\rho }_0)+2(\mathit{\Gamma }-1)\frac{C_0^2}{V^2}[\frac{Q_{\ast }}{C_0^2}+\frac{(1-\theta {\rho }_0)}{({\mathit{\Gamma }}_0-1)}]\}=0.\end{array}$$ 4.4

Here, C = [Γp/ρ(1 − θρ)]^1/2 is the equilibrium speed of sound in the medium. Equation (4.4) is a quadratic equation in y and will have two roots, one of which corresponds to the case when the particle shock velocities behind and ahead of the front are subsonic. This would lead to rarefaction shock [9,17]. The other root will correspond to the condition when the compression shock will have the fluid–particle velocity at the shock as supersonic ahead of the front and subsonic or exactly sonic behind the front. The roots of the equation will coincide when the relation

$$\begin{array}{r}{[(\mathit{\Gamma }+\theta {\rho }_0)+\frac{\mathit{\Gamma }(1-\theta {\rho }_0)}{{\mathit{\Gamma }}_0}\frac{C_0^2}{V^2}]}^2=(\mathit{\Gamma }+1)\{(\mathit{\Gamma }-1+2\theta {\rho }_0)+2(\mathit{\Gamma }-1)\frac{C_0^2}{V^2}[\frac{Q_{\ast }}{C_0^2}+\frac{(1-\theta {\rho }_0)}{({\mathit{\Gamma }}_0-1)}]\}\end{array}$$ 4.5

is satisfied [9]. The relations (4.3) yield

$$\frac{C^2}{{(V-u)}^2}+\frac{\mathit{\Gamma }+\theta \rho }{(1-\theta \rho )}=\frac{1}{(1-\theta \rho )}[\frac{\mathit{\Gamma }{C}_0^2(1-\theta {\rho }_0)}{{\mathit{\Gamma }}_0{V}^2}+\mathit{\Gamma }+\theta {\rho }_0]\left(\frac{\rho }{{\rho }_0}\right).$$ 4.6

Substituting (4.5) in (4.6) we get

$$\begin{array}{r}\{(\mathit{\Gamma }-1+2\theta {\rho }_0)+2(\mathit{\Gamma }-1)\frac{C_0^2}{V^2}[\frac{Q_{\ast }}{C_0^2}+\frac{(1-\theta {\rho }_0)}{({\mathit{\Gamma }}_0-1)}]\}=\frac{1}{(\mathit{\Gamma }+1)}{\left(\frac{{\rho }_0}{\rho }\right)}^2[\frac{C^2(1-\theta \rho )}{{(V-u)}^2}+\mathit{\Gamma }+\theta \rho ].\end{array}$$ 4.7

From equations (4.4), (4.6) and (4.7), we get

$$\frac{\mathit{\Gamma }p}{\rho (1-\theta \rho ){(V-u)}^2}=1.$$ 4.8

Hence, the coincidence of the roots of (4.4) is attained when the particle velocity just behind the shock front is exactly sonic. In fact, the sonic point protects the shock from the non-self-similar region. Condition (4.5), which is equivalent to (4.8) for compression shocks, is called the Chapman–Jouguet condition. Following the Chapman–Jouguet relation, we get the flow variables at the boundary. For nuclear explosions $V^2/C_0^2>>1$, thereby implying $C_0^2/V^2<<1$ and hence can be ignored. Equation (4.4) gives

$$\frac{\rho }{{\rho }_0}=\frac{\mathit{\Gamma }+1}{\mathit{\Gamma }+\theta {\rho }_0},\frac{u}{V}=\frac{(1-\theta {\rho }_0)}{(\mathit{\Gamma }+1)},\frac{p}{{\rho }_0{V}^2}=\frac{1}{(\mathit{\Gamma }+1)}.$$ 4.9

In the absence of dust particles our expressions for the flow variables agree with those of Sedov [9] and Lee [17].

5. Similarity solutions using the Lie group of transformations

Lie symmetry analysis is one of the most effective methods for finding self-similar solutions of differential equations. It is based on the study of their invariance with respect to a one-parameter Lie group of point transformations whose infinitesimal operators are generated by vector fields. Once the Lie group of transformations that leave the differential equations invariant are known, we can construct a solution that is invariant under the transformation. It can transform the non-autonomous first-order quasilinear hyperbolic system to autonomous ones, as done in [18]. Similarity solutions and strong shocks in extended thermodynamics of rarefied gas were studied in [19]. Some classical results on Lie symmetries can be found in [20]. We study the decay of a blast wave using the self-similarity property of the point blast problem. The application of Lie symmetry analysis to obtain a self-similar solution to the point blast problem in radiation gasdynamics including the dust particles and relaxing effects of gases has not been taken up so far, to the best of our knowledge.

To determine a similarity solution and the similarity curve, we first use the subscript notation of variables, x₁ = t, x₂ = x, u₁ = ρ, u₂ = u, u₃ = p, u₄ = σ. The one-parameter (ϵ) group of transformations of the following form is considered:

$$\begin{array}{rl}& x_j^{\ast }=x_j+ϵ{\mathcal{X}}_j(x_1,x_2,u_1,u_2,u_3,u_4)\\ \text{and}& u_i^{\ast }=u_i+ϵ{\mathcal{U}}_i(x_1,x_2,u_1,u_2,u_3,u_4),\end{array}\}$$ 5.1

where j = 1, 2, i = 1, 2, 3, 4 and ${\mathcal{X}}_j$ and ${\mathcal{U}}_i$ are the infinitesimals of the Lie group of transformations. Using $p_j^i=\mathrm{\partial }{u}_i/\mathrm{\partial }{x}_j$ we rewrite equations (2.1) in the form

$$F_k(x_j,u_i,p_j^i)=0,k=1,2,3,4.$$

Equations (2.1) are constantly conformally invariant under the infinitesimal group of transformations

$$\mathcal{L}{F}_k={\alpha }_{kr}{F}_r,k=1,2,3,4,$$ 5.2

where α_rs (r, s = 1, 2, 3, 4) are constants. The Lie derivative $\mathcal{L}$ in the direction of the extended vector field is given by

$$\mathcal{L}={\xi }_x^j\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}+{\xi }_u^i\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_i}+{\xi }_{p_j}^i\frac{\mathrm{\partial }}{\mathrm{\partial }{p}_j^i},$$

with ${\xi }_x^1=\mathit{\Psi }$, ${\xi }_x^2=\chi$, ${\xi }_u^1=S$, ${\xi }_u^2=U$, ${\xi }_u^3=P$, ${\xi }_u^4=E$ and

$${\xi }_{p_j}^i=\frac{\mathrm{\partial }{\xi }_u^i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{\xi }_u^i}{\mathrm{\partial }{u}_k}{p}_j^k-\frac{\mathrm{\partial }{\xi }_x^l}{\mathrm{\partial }{x}_j}{p}_l^i-\frac{\mathrm{\partial }{\xi }_x^l}{\mathrm{\partial }{u}_n}{p}_l^i{p}_j^n;l=1,2;n=1,2,3,4,$$

being the infinitesimals of the derivative transformation. Following the procedure given in [11] and comparing coefficients of $p_j^i$, we get a system of 36 first-order partial differential equations, the consistency of which leads to determining the infinitesimals ${\xi }_x^j$, ${\xi }_u^i$ and ${\xi }_{p_j}^i$ as per the following:

$$\begin{array}{rl}& \mathit{\Psi }=At+A_1,\\ & \chi =\{\begin{array}{ll}({\alpha }_{22}+2A)x& \text{for}\hspace{0.17em}m=1,2,\\ ({\alpha }_{22}+2A)x+U_{1}t+c_1& \text{for}\hspace{0.17em}m=0,\end{array}\\ & S=\{\begin{array}{ll}({\alpha }_{11}+A)\rho & \text{for}\hspace{0.17em}\theta =0,\\ 0& \text{for}\hspace{0.17em}\theta \ne 0,\end{array}U=({\alpha }_{22}+A)u+U_1,\\ & P=\{\begin{array}{ll}(2{\alpha }_{22}+{\alpha }_{11}+3A)p& \text{for}\hspace{0.17em}\theta =0,\\ (2{\alpha }_{22}+2A)p& \text{for}\hspace{0.17em}\theta \ne 0,\end{array}\\ & E=(2{\alpha }_{22}+2A)\sigma +d_1,S{Q}_{\rho }+P{Q}_p+E{Q}_{\sigma }=(2{\alpha }_{22}+A)Q,\end{array}\}$$ 5.3

where A, A₁, c₁, U₁, α₁₁ and α₂₂ are arbitrary constants. For non planar flows U₁ = 0. The arbitrary constants in the expressions for the generators of the local Lie group of transformations give us the different cases of possible solutions, such as power-law, exponential or logarithmic shock paths.

(a). Case 1: when A ≠ 0, α₂₂ + 2A ≠ 0

The basic equations remain invariant with the space and time translation

$$\tilde{x}=\{\begin{array}{ll}x& \text{for}\hspace{0.17em}m=1,2,\\ x+c_1/({\alpha }_{22}+2A)& \text{for}\hspace{0.17em}m=0,\end{array}\tilde{t}=t+A_1/A.$$ 5.4

The similarity variable and the form of similarity solutions for ρ, u, p, σ and Q follow from the invariant surface condition u_i(x*, t*) = u*_i(x, t), which gives us

$$\begin{array}{rl}& \mathit{\Psi }{\rho }_t+\chi {\rho }_x=S,\mathit{\Psi }{u}_t+\chi u_x=U,\\ & \mathit{\Psi }{p}_t+\chi p_x=P,\mathit{\Psi }{\sigma }_t+\chi {\sigma }_x=E.\end{array}\}$$ 5.5

Equations (5.5) on integration yield

$$\begin{array}{rl}& \rho =\{\begin{array}{ll}\hat{S}(\xi )& \text{for}\hspace{0.17em}\theta \ne 0,\\ t^{\kappa \delta }\hat{S}(\xi )& \text{for}\hspace{0.17em}\theta =0,\end{array}u=t^{\delta -1}\hat{U}(\xi ),\\ & p=\{\begin{array}{ll}{t}^{2(\delta -1)}\hat{P}(\xi )& \text{for}\hspace{0.17em}\theta \ne 0,\\ t^{(2+\kappa )\delta -2}\hat{P}(\xi )& \text{for}\hspace{0.17em}\theta =0,\end{array}\\ & Q=\{\begin{array}{ll}{p}^{(2\delta -3)/2(\delta -1)}\hspace{0.17em}q(k_1,k_2)& \text{for}\hspace{0.17em}\theta \ne 0,\\ p^{(2\delta -3)A/((2\delta -1)A+{\alpha }_{11})}\hspace{0.17em}q(k_1,k_2)& \text{for}\hspace{0.17em}\theta =0,\end{array}\end{array}\}$$ 5.6

where

$$k_1=\{\begin{array}{ll}\rho & \text{for}\hspace{0.17em}\theta \ne 0,\\ \rho p^{-({\alpha }_{11}+A)/({\alpha }_{11}+(2\delta -1)A)}& \text{for}\hspace{0.17em}\theta =0,\end{array}\delta =\frac{{\alpha }_{22}+2A}{A}$$

and

$$k_2=\{\begin{array}{ll}p{(\sigma +d^{\ast })}^{-1}& \text{for}\hspace{0.17em}\theta \ne 0,\\ p{(\sigma +d^{\ast })}^{-(({\alpha }_{11}+(2\delta -1)A)/2(\delta -1)A)}& \text{for}\hspace{0.17em}\theta =0,\end{array}\kappa =\frac{{\alpha }_{11}+A}{\delta A},$$

with d* = d₁/2(δ − 1)A. The above form of Q is the general form for which the self-similar solutions exist.

The functions $\hat{S}$, $\hat{U}$, $\hat{P}$ and $\hat{E}$ depend upon the similarity variable ξ, which is given by

$$\xi =\frac{x}{t^{\delta }}.$$ 5.7

Since the shock is a similarity curve, it can be normalized to be at ξ = 1. The shock path and shock velocity are given by

$$\chi =t^{\delta },V=\frac{\delta \chi }{t}.$$ 5.8

At the shock, we have the general conditions on the functions $\hat{S}$, $\hat{U}$, $\hat{P}$ and $\hat{E}$

$$\begin{array}{rl}& \rho {|}_{\xi =1}==\{\begin{array}{ll}\hat{S}(1)& \text{for}\hspace{0.17em}\theta \ne 0,\\ t^{\delta \kappa }\hat{S}(1)& \text{for}\hspace{0.17em}\theta =0,\end{array}\\ & p{|}_{\xi =1}==\{\begin{array}{ll}\hat{P}(1)t^{2(\delta -1)}& \text{for}\hspace{0.17em}\theta \ne 0,\\ t^{\delta (2+\kappa )-2}\hat{P}(1)& \text{for}\hspace{0.17em}\theta =0,\end{array}\\ \text{and}& u{|}_{\xi =1}=\hat{U}(1)t^{(\delta -1)},\sigma {|}_{\xi =1}=\hat{E}(1)t^{2(\delta -1)}-d^{\ast }.\end{array}\}$$ 5.9

Equations (5.9), in view of the invariance of the jump conditions (4.1)₅ and (4.9), give us the following forms of ρ₀(x), σ₀(x) [11]:

$${\rho }_0(x)={\rho }_c,{\overline{\sigma }}_0(x)={\sigma }_0(x)={\sigma }_c{x}^{\omega }+{\sigma }_{0c},$$ 5.10

where ρ_c and σ_0c are constants specific to the medium, ω = 2(δ − 1)/δ and d* = −σ_0c, when σ₀ is varying. Considering

$$\begin{array}{rl}& S^{\ast }(\xi )=\frac{\hat{S}(\xi )}{{\rho }_c},U^{\ast }(\xi )=\frac{\hat{U}(\xi )}{\delta },\\ & P^{\ast }(\xi )=\frac{\hat{P}(\xi )}{({\rho }_c{\delta }^2)},E^{\ast }(\xi )=\frac{\hat{E}(\xi )}{{\delta }^2},\end{array}$$

the flow variables for θ ≠ 0 can be expressed as

$$\begin{array}{rl}& \rho ={\rho }_0(\chi (t))S^{\ast }(\xi ),\hspace{0.17em}u=V{U}^{\ast }(\xi ),\hspace{0.17em}p={\rho }_0(\chi (t))V^2{P}^{\ast }(\xi ),\\ & \sigma =V^2{E}^{\ast }(\xi )-d^{\ast },\hspace{0.17em}Q=p^{(2\delta -3)/(2(\delta -1))}q(k_1,k_2),\hspace{0.17em}{k}_1=\rho ,\\ & k_2=p{(\sigma +d^{\ast })}^{-1},\hspace{0.17em}\xi =x{t}^{-\delta },\hspace{0.17em}\chi =t^{\delta },\hspace{0.17em}V=\frac{\delta \chi }{t}.\end{array}\}$$ 5.11

The form of Q given in (2.2) must be consistent with equation (5.11)₅. So, we assume ϕ and τ to be of the form

$$\varphi ={\varphi }_c{p}^{{\beta }_1}{\rho }^{{\beta }_2},\tau ={\tau }_c{p}^{{\beta }_3}{\rho }^{{\beta }_4},$$ 5.12

with β₁, β₂, β₃, β₄, ϕ_c and τ_c being constants specific to the medium. For the existence of similarity solutions we observe that the following compatibility conditions must be satisfied for β₁, β₂, β₃, β₄ in (5.12) and for d, g in (3.3):

$${\beta }_1=0,\hspace{0.17em}{\beta }_2=0,\hspace{0.17em}{\beta }_3=1/2(\delta -1),\hspace{0.17em}{\beta }_4=-1/2(\delta -1),\hspace{0.17em}d=-5/2,\hspace{0.17em}g=1.$$ 5.13

The requirement for a self-similar flow pattern poses the similarity condition (5.13) on the parameters β₁, β₂, β₃, β₄, d and g. The dependence of relaxation time τ on pressure and density and hence temperature is also summarized in [21], where the variation of bulk viscosity with temperature along with the rotational and vibrational contributions for a selection of well-known ideal gases is investigated. In the absence of dust particles the results obtained in (5.13) for β₁, β₂, β₃ and β₄ are the same as those obtained by Arora et al. [22]. The dependence of the absorption coefficient on temperature and density with the exponents (d = −5/2, g = 1) in the vicinity of a temperature of 20 000 kelvin was also observed by Pai [6]. Hence, it may be noted that for similarity solutions the following form of F is permissible:

$$F=\{\begin{array}{ll}H(p{(1-\theta \rho ))}^{3/2}/{\rho }^{1/2}& \text{for}\hspace{0.17em}\theta \ne 0,\\ H{(p/\rho )}^{3/2}& \text{for}\hspace{0.17em}\theta =0.\end{array}$$ 5.14

Using equations (5.11), (5.12), (5.13) and (5.14) in the governing equations (2.1) and suppressing the asterisk sign, we obtain the following set of ODEs for θ ≠ 0:

$$\begin{array}{rl}& (U-\xi )S^{\mathrm{\prime }}+S{U}^{\mathrm{\prime }}+\frac{mUS}{\xi }=0,\\ & \left(\frac{\delta -1}{\delta }\right)U+(U-\xi )U^{\mathrm{\prime }}+\frac{P^{\mathrm{\prime }}}{S}=0,\\ & 2\left(\frac{\delta -1}{\delta }\right)P+(U-\xi )P^{\mathrm{\prime }}+\frac{P\mathit{\Gamma }}{(1-\theta S)}(U^{\mathrm{\prime }}+\frac{mU}{\xi })+\frac{(\mathit{\Gamma }-1)S\tilde{Q}}{(1-\theta S)}\\ & +\frac{(\mathit{\Gamma }-1)H}{2S(1-\theta S)}{\left(\frac{P(1-\theta S)}{S}\right)}^{1/2}[P{S}^{\mathrm{\prime }}(1+2\theta S)-3S{P}^{\mathrm{\prime }}]=0\\ \text{and}& 2\left(\frac{\delta -1}{\delta }\right)E+(U-\xi )E^{\mathrm{\prime }}-\tilde{Q}=0,\end{array}\}$$ 5.15

where

$$\tilde{Q}=\{\begin{array}{ll}{\varphi }_c(1-\theta S)P^{(1-{\beta }_3)}/({\tau }_c(1-k_p){\delta }^{(1+2{\beta }_3)}{S}^{1+{\beta }_4})& \text{for}\hspace{0.17em}\theta \ne 0,\\ {\varphi }_c{P}^{(1-{\beta }_3)}/({\tau }_c{\delta }^{(1+2{\beta }_3)}{S}^{(1+{\beta }_4)})& \text{for}\hspace{0.17em}\theta =0.\end{array}$$ 5.16

The boundary conditions given by equation (4.1)₄ in view of the permitted form of radiative flux (5.14) give the following form of y:

$$M_1{y}^4-M_2{y}^3+M_3{y}^2-M_{4}y-M_5=0,$$ 5.17

where

$$\begin{array}{rl}& M_1={(\mathit{\Gamma }+1)}^2-H^2{(\mathit{\Gamma }-1)}^2,\hspace{0.17em}{M}_2=2(\mathit{\Gamma }+1)(\mathit{\Gamma }-1+2\theta {\rho }_0)-H^2{(\mathit{\Gamma }-1)}^2(1+3\theta {\rho }_0),\\ & M_3={(\mathit{\Gamma }-1+2\theta {\rho }_0)}^2-3\theta {\rho }_0{H}^2{(\mathit{\Gamma }-1)}^2(1+\theta {\rho }_0),\hspace{0.17em}{M}_4={[H\theta {\rho }_0(\mathit{\Gamma }-1)]}^2(3-\theta {\rho }_0),\\ & M_5={[H(\mathit{\Gamma }-1)]}^2{(\theta {\rho }_0)}^3.\end{array}$$

The boundary conditions for the flow variables ρ, p and u in the presence of radiation effects are obtained on solving the above biquadratic equation (5.17). The self-similar solutions which exist are of the second type [23] and the similarity exponent will be found by solving the system of equations (5.15) on the condition that the correct solution passes through the singular point [24].

Using Cramer’s rule for solving the system of equations (5.15) we get

$$\begin{array}{rl}& \mathrm{△}={(U-\xi )}^2[(U-\xi )(U-\xi -P_a)-\frac{P\mathit{\Gamma }}{(1-\theta S)S}+\frac{S_a}{U-\xi }],\\ & {\mathrm{△}}_1={(U-\xi )}^2[U(\xi -U+P_a)\frac{(\delta -1)}{\delta }+\frac{A_{\ast }}{S}-\frac{mU{S}_a}{(U-\xi )\xi }],\\ & {\mathrm{△}}_2=-[\frac{mU}{\xi }\mathrm{△}+{\mathrm{△}}_1]\frac{S}{(U-\xi )},\\ & {\mathrm{△}}_3=-S[{\mathrm{△}}_1-\left(\frac{\delta -1}{\delta }\right)U\frac{\mathrm{△}}{U-\xi }](U-\xi ),{\mathrm{△}}_4=[\tilde{Q}-2\left(\frac{\delta -1}{\delta }\right)E]\frac{\mathrm{△}}{(U-\xi )},\end{array}\}$$ 5.18

where

$$\begin{array}{rl}& S_a=\frac{(\mathit{\Gamma }-1)(1+2\theta S)HP}{2S}\sqrt{\frac{P}{(1-\theta S)S}},P_a=\frac{3H(\mathit{\Gamma }-1)}{2}\sqrt{\frac{P}{(1-\theta S)S}},\\ & A_{\ast }=2\left(\frac{\delta -1}{\delta }\right)P+\frac{mU\mathit{\Gamma }P}{(1-\theta S)\xi }+\frac{(\mathit{\Gamma }-1)S\tilde{Q}}{(1-\theta S)}.\end{array}$$

The derivatives U^′, S^′, P^′ and E^′ can be written in the following form:

$$U^{\mathrm{\prime }}=\frac{{\mathrm{△}}_1}{\mathrm{△}},\hspace{0.17em}{S}^{\mathrm{\prime }}=\frac{{\mathrm{△}}_2}{\mathrm{△}},\hspace{0.17em}{P}^{\mathrm{\prime }}=\frac{{\mathrm{△}}_3}{\mathrm{△}},\hspace{0.17em}{E}^{\mathrm{\prime }}=\frac{{\mathrm{△}}_4}{\mathrm{△}}.$$ 5.19

The value of the similarity exponent δ is determined in such a way that when $\mathrm{△}=0$ then ${\mathrm{△}}_1=0$. Equation (5.19)₁ was solved for ξ = 1 to obtain the only unknown quantity, called the similarity exponent δ, at the blast front. The system (5.15) is solved subject to the boundary conditions given by (4.1)₅ and (4.9) or the boundary conditions obtained from (5.17), depending upon whether it is without or with radiation effects, respectively. The similarity exponent is given by

$$\delta =\frac{\tilde{\lambda }}{(\tilde{\lambda }+m)},$$ 5.20

where

$$\begin{array}{rl}& \tilde{\lambda }=\frac{2(1-\theta S)}{\mathit{\Gamma }U}-\frac{1}{(U-1)}\text{(in the absence of radiation effects)}\\ \text{and}& \tilde{\lambda }=[\frac{U{S}_a}{{(U-1)}^2}+\frac{P}{S}(2-\frac{\mathit{\Gamma }U}{(1-\theta S)(U-1)})]{[\frac{PU\mathit{\Gamma }}{S(1-\theta S)}-\frac{U{S}_a}{(U-1)}]}^{-1}\\ & \text{(with radiation effects)}.\end{array}\}$$ 5.21

The numerical calculations were performed taking β₃ = 1/2(δ − 1), β₄ = −1/2(δ − 1), τ_c = 0.1 and ϕ_c = 0.1. The Chisnell Chester Whitham rule (CCW) [10] was used to verify the similarity exponent in the case where there were no radiation effects, according to which the characteristic equation for the forward moving characteristic can be written as

$$\frac{\text{d}p}{\text{d}x}+\rho C\frac{\text{d}u}{\text{d}x}+\frac{1}{(u+C)}\frac{m\rho C^{2}u}{x}=0.$$ 5.22

Using the boundary conditions (4.9), we solve equation (5.22) and obtain the similarity exponent given by (5.20), where

$$\tilde{\lambda }=(2+\sqrt{(}1-\theta {\rho }_0))[\frac{1}{\sqrt{(}1-\theta {\rho }_0)}+\frac{(1-\theta {\rho }_0)}{(\mathit{\Gamma }+\theta {\rho }_0)}].$$ 5.23

The similarity exponents obtained on solving (5.21)₁ and (5.23) are presented in table 2 and they are compared with the similarity exponent obtained by Jena & Sharma [25]. The difference in the values is due to the different boundary conditions used in the analysis. Equation (5.17) can be solved numerically to obtain the boundary conditions for the flow variable ρ, u and p in the presence of radiation effects. The similarity exponent can then be obtained using equation (5.21)₂. The system of equations (5.15) is solved subject to the boundary conditions (4.1)₅ and (4.9) or the boundary conditions obtained from (5.17), depending upon whether it is without or with radiation effects, respectively, by the fourth-order Runge–Kutta method using Matlab software. The effects of the dust parameters k_p, β, G and the relaxing parameters τ_c and ϕ_c on the velocity, pressure, density, vibrational energy and radiation flux of the flow are obtained. The results are shown in figures 3–7. The numerical values of dust parameters, the radiation parameter and the relaxing parameters used are as considered in [1,3–6,12], respectively.

Table 2.

Similarity exponent δ and the ratio of the specific heats Γ for non-planar flows of a dusty gas for different values of k_p, β and G; γ = 1.4 (without radiation effects).

				computed δ				Jena & Sharma [25]
kp	β	G	Γ	m = 1 (CCW)	m = 1	m = 2 (CCW)	m = 2	m = 1	m = 2
0.1	0.1	10	1.39387	0.836658	0.837135	0.719185	0.719889
0.1	0.1	100	1.39387	0.837378	0.837426	0.720249	0.720321	0.83525	0.71686
0.1	0.1	1000	1.39387	0.837450	0.837455	0.720357	0.720364	0.83585	0.71792
0.1	0.5	10	1.37113	0.837580	0.838068	0.720549	0.721272
0.1	0.5	100	1.37113	0.838314	0.838363	0.721636	0.721709	0.83748	0.72033
0.1	0.5	1000	1.37113	0.838388	0.838393	0.721745	0.721752	0.83810	0.72133
0.1	1.0	10	1.34615	0.838617	0.839118	0.722085	0.722827
0.1	1.0	100	1.34615	0.839366	0.839416	0.723196	0.723271	0.84008	0.72425
0.1	1.0	1000	1.34615	0.839441	0.839446	0.723308	0.723315	0.84075	0.72535
0.2	0.1	10	1.38647	0.836163	0.837115	0.718454	0.719859
0.2	0.1	100	1.38647	0.837599	0.837696	0.720577	0.720720	0.83530	0.71695
0.2	0.1	1000	1.38647	0.837745	0.837755	0.720793	0.720807	0.83650	0.71891
0.2	0.5	10	1.34043	0.838034	0.839032	0.721221	0.722701
0.2	0.5	100	1.34043	0.839526	0.839628	0.723434	0.723585	0.83996	0.72398
0.2	0.5	1000	1.34043	0.839678	0.839688	0.723659	0.723674	0.84130	0.72619
0.2	1.0	10	1.29629	0.839907	0.840951	0.723999	0.725552
0.2	1.0	100	1.29629	0.841454	0.841560	0.726302	0.726460	0.84498	0.73163
0.2	1.0	1000	1.29629	0.841611	0.841622	0.726536	0.726552	0.84650	0.73418
0.6	0.1	10	1.33058	0.835260	0.838164	0.717122	0.721413
0.6	0.1	100	1.33058	0.839612	0.839918	0.723561	0.724016	0.83795	0.72057
0.6	0.1	1000	1.33058	0.840068	0.840099	0.724239	0.724285	0.84211	0.72736
0.6	0.5	10	1.19512	0.840937	0.844295	0.725532	0.730545
0.6	0.5	100	1.19512	0.845827	0.846178	0.732842	0.733370	0.85461	0.74580
0.6	0.5	1000	1.19512	0.846338	0.846373	0.733609	0.733663	0.86131	0.75721
0.6	1.0	10	1.12903	0.844004	0.847607	0.730110	0.735519
0.6	1.0	100	1.12903	0.849185	0.849560	0.737899	0.738465	0.86593	0.76314
0.6	1.0	1000	1.12903	0.849725	0.849763	0.738715	0.738772

Figure 3.

Variation of velocity, density, pressure, vibrational energy and radiative flux with the similarity variable ξ for various values of k_p for m = 2, G = 100, β = 0.5, H = 0.2, γ = 1.4, τ_c = 0.1. (Online version in colour.)

Figure 7.

Variation of velocity, density, pressure, vibrational energy and radiative flux with the similarity variable ξ for various values of ϕ_c for m = 2, G = 100, k_p = 0.5, β = 0.5, H = 0.2, γ = 1.4, τ_c = 0.1. (Online version in colour.)

(b). Case 2: when A = 0 and α₂₂ ≠ 0

The basic equations remain invariant with the space translation

$$\tilde{x}=x+\frac{c_1}{{\alpha }_{22}},\tilde{t}=t.$$ 5.24

We obtain from (5.3) the similarity solutions for the flow variables in the following form:

$$\begin{array}{rl}& \rho ={\rho }_0(\chi (t))S^{\ast }(\xi ),\hspace{0.17em}u=V{U}^{\ast }(\xi ),p={\rho }_0(\chi (t))V^2{P}^{\ast }(\xi ),\hspace{0.17em}\sigma =V^2{E}^{\ast }(\xi )-d^{\ast },\\ & Q=pq(k_1,k_2),\hspace{0.17em}{k}_1=\{\begin{array}{ll}\rho & \text{for}\hspace{0.17em}\theta \ne 0,\\ \rho p^{(-{\alpha }_{11})/{\alpha }_{22}}& \text{for}\hspace{0.17em}\theta =0,\end{array}\hspace{0.17em}{k}_2=p{(\sigma +d^{\ast })}^{-1}.\end{array}\}$$ 5.25

The shock is normalized at ξ = 1 to obtain the shock location χ, the shock velocity V and ρ₀(x)

$$\begin{array}{rl}& \xi =x{e}^{-\delta t},\hspace{0.17em}\chi =e^{\delta t},V=\delta e^{\delta t},\hspace{0.17em}\delta ={\alpha }_{22}/A_1,\\ & \rho (x)={\rho }_c,\hspace{0.17em}{\sigma }_0(x)={\sigma }_c{x}^2+{\sigma }_{0c},\hspace{0.17em}{d}^{\ast }=-{\sigma }_{0c}.\end{array}\}$$ 5.26

This case gives us a class of similarity solutions with an exponential shock path. Necessary conditions for self-similarity to exist are

$${\beta }_1=0={\beta }_3.$$ 5.27

Substituting (5.25) and (5.26) in equations (2.1) and using equations (2.2) and (5.12) we obtain the following system of ODEs for θ ≠ 0:

$$\begin{array}{rl}& (U-\xi )S^{\mathrm{\prime }}+S{U}^{\mathrm{\prime }}+mUS/\xi =0,\\ & U+(U-\xi )U^{\mathrm{\prime }}+S^{-1}{P}^{\mathrm{\prime }}=0,\\ & 2P+(U-\xi )P^{\mathrm{\prime }}+\frac{P\mathit{\Gamma }}{(1-\theta S)}(U^{\mathrm{\prime }}+mU/\xi )+\frac{S\tilde{Q}(\mathit{\Gamma }-1)}{(1-\theta S)}\\ & +(\mathit{\Gamma }-1)F_{\ast }\{P{S}^{\mathrm{\prime }}(1+2\theta S)-3S{P}^{\mathrm{\prime }}(1-\theta S)\}=0\\ \text{and}& 2E+(U-\xi )E^{\mathrm{\prime }}=\tilde{Q},\end{array}\}$$ 5.28

where F_* and $\tilde{Q}$ are given by

$$\tilde{Q}=\frac{{\varphi }_{c}P{S}^{{\beta }_2}(1-\theta S)}{{\tau }_c\delta (1-k_p)S^{{\beta }_4}},F_{\ast }=\frac{H}{2S}{\left(\frac{P}{(1-\theta S)S}\right)}^{1/2}.$$ 5.29

The above system of equations (5.28) can be solved subject to the boundary conditions (4.1)₅ and (4.9) or the boundary conditions obtained from (5.17), depending upon whether it is without or with radiation effects, respectively, along with the constraints (5.27) and (5.29) to obtain the flow variables.

(c). Case 3: when α₂₂ + 2A = 0 and A ≠ 0

This case is possible only in planar flows and so a similarity solution does not exist in such flow configurations. The similarity curve is normalized to be at ξ = 0 instead of ξ = 1. The basic equations remain invariant with the following transformation of variables:

$$\tilde{x}=x,\tilde{t}=t+\frac{A_1}{A}.$$ 5.30

Hence the following form of flow variables is possible:

$$\begin{array}{rl}& \rho ={\rho }_0(\chi (t))S^{\ast }(\xi ),\hspace{0.17em}u=V{U}^{\ast }(\xi ),\hspace{0.17em}p={\rho }_0(\chi (t))V^2{P}^{\ast }(\xi ),\hspace{0.17em}\sigma =V^2{E}^{\ast }(\xi )+d^{\ast },\\ & \xi =\chi -\delta \log \hspace{0.17em}t,\hspace{0.17em}\chi =\delta \text{log}\hspace{0.17em}t,\hspace{0.17em}V=\delta /t,\hspace{0.17em}{\rho }_0(x)={\rho }_c,\hspace{0.17em}Q=p^{3/2}q(k_1,k_2),\\ & k_1=\{\begin{array}{ll}\rho & \text{for}\hspace{0.17em}\theta \ne 0,\\ \rho p^{({\alpha }_{11}+A)/2A}& \text{for}\hspace{0.17em}\theta =0,\end{array},k_2=p{(\sigma -d^{\ast })}^{-1},\hspace{0.17em}{\sigma }_0(x)={\sigma }_c\exp (-2x/\delta )+{\sigma }_{0c},\hspace{0.17em}{d}^{\ast }={\sigma }_{0c}.\end{array}$$

The similarity considerations imply the following constraints:

$${\beta }_1=0,{\beta }_3=-1/2.$$ 5.31

The system for θ ≠ 0 reduces to the following set of ODEs:

$$\begin{array}{rl}& (U-1)S^{\mathrm{\prime }}+U^{\mathrm{\prime }}S=0,\\ & (U-1)U^{\mathrm{\prime }}-\frac{U}{\delta }+\frac{P^{\mathrm{\prime }}}{S}=0,\\ & (U-1)P^{\mathrm{\prime }}-\frac{2P}{\delta }+\frac{P\mathit{\Gamma }{U}^{\mathrm{\prime }}}{(1-\theta S)}+\frac{S(\mathit{\Gamma }-1)\tilde{Q}}{(1-\theta S)}\\ & +(\mathit{\Gamma }-1)F_{\ast }\{P{S}^{\mathrm{\prime }}(1+2\theta S)-3S{P}^{\mathrm{\prime }}(1-\theta S)\}=0\\ \text{and}& (U-1)E^{\mathrm{\prime }}-\frac{2E}{\delta }-\tilde{Q}=0.\end{array}\}$$ 5.32

F_* and $\tilde{Q}$ are the same as in the previous case. The above equation (5.32) can be solved together with the boundary conditions (4.1)₅ and (4.9) or the boundary conditions obtained from (5.17), depending upon whether it is without or with radiation effects, respectively, and subject to the constraints (5.31).

(d). Case 4: when α₂₂ = A = 0

This is similar to the previous case. Here again this situation will arise in a planar flow and not in non-planar cases and a similarity solution does not exist in this case. The similarity curve is normalized to be at ξ = 0 instead of ξ = 1. We obtain the flow variables for θ ≠ 0 in the following form:

$$\begin{array}{rl}& \rho ={\rho }_0(\chi (t))S^{\ast }(\xi ),\hspace{0.17em}u=V{U}^{\ast }(\xi ),p={\rho }_0(\chi (t))V^2{P}^{\ast }(\xi ),\hspace{0.17em}\sigma =V^2{E}^{\ast }(\xi )+d^{\ast }t,\\ & \xi =\chi -\delta t,\hspace{0.17em}\chi =\delta t,V=\delta ,\hspace{0.17em}{\rho }_0(x)={\rho }_c,\hspace{0.17em}{\sigma }_0(x)={\sigma }_c,\hspace{0.17em}{d}^{\ast }={\sigma }_c\delta /x,\\ & Q=q(k_1,k_2),\hspace{0.17em}{k}_1=\rho /p,k_2=p^{\tilde{d}}{\text{e}}^{-\sigma },\end{array}\}$$ 5.33

with $\tilde{d}$ as a constant. The conditions for a self-similar solution are

$${\beta }_1+{\beta }_2=0,{\beta }_3+{\beta }_4=0.$$ 5.34

The reduced system of equations for θ ≠ 0 are

$$\begin{array}{rl}& (U-1)S^{\mathrm{\prime }}+U^{\mathrm{\prime }}S=0,\\ & (U-1)U^{\mathrm{\prime }}+\frac{P^{\mathrm{\prime }}}{S}=0,\\ & (U-1)P^{\mathrm{\prime }}+\frac{P\mathit{\Gamma }{U}^{\mathrm{\prime }}}{(1-\theta S)}+\frac{S(\mathit{\Gamma }-1)\tilde{Q}}{(1-\theta S)}+(\mathit{\Gamma }-1)F_{\ast }\{P{S}^{\mathrm{\prime }}(1+2\theta S)-3S{P}^{\mathrm{\prime }}(1-\theta S)\}=0\\ \text{and}& (U-1)E^{\mathrm{\prime }}+d_1^{\ast }-\tilde{Q}=0.\end{array}\}$$ 5.35

Here d*₁ = d*/(δ³ A₁), F_* and $\tilde{Q}$ are the same as in the previous cases. The system of equations (5.35) can be solved together with the boundary conditions (4.1)₅ and (4.9) or the boundary conditions obtained from (5.17), depending upon whether it is without or with radiation effects, respectively, and subject to the constraints given in (5.34).

6. Energy released and damage estimation

The criterion for estimating the damage to structures by nuclear explosion is the peak pressure in the blast wave rather than the impulse (integral of pressure over time) which is the criterion used for an ordinary explosion [5]. So we reduce the pressure and hence the temperature by increasing the density of air (by including dust particles). According to Bethe, if the pressure determines the damage, then the $\text{area of damage}\propto {\text{(energy released)}}^{2/3}$ and the $\text{distance of damage}\propto {\text{(energy released)}}^{1/3}$. Since the total energy is always conserved, the energy released from an explosion is converted to various forms, such as thermal energy, kinetic energy, light energy, etc. Only a part of this energy present in the atmosphere is responsible for causing damage.

Here, we discuss the energy released in a blast wave with spherical symmetry using the reduced flow variables obtained in (5.11) of §5. The total energy (per unit mass) of the disturbance is given by Taylor [7,8]

$$E=4\pi {\int }_0^{\chi }(\frac{\rho u^2}{2}+\rho \sigma +\frac{p(1-\theta \rho )}{(\mathit{\Gamma }-1)})x^2\hspace{0.17em}\text{d}x.$$ 6.1

On substituting the similarity form of flow variables given in equation (5.11), we get

$$E=4\pi {\rho }_c{\int }_0^1(\frac{S{U}^2}{2}+SE+\frac{P(1-\theta S)}{(\mathit{\Gamma }-1)})V^2{\chi }^3{\xi }^2\hspace{0.17em}\text{d}\xi .$$ 6.2

The total energy E of the flow field behind the shock is not constant but is assumed to be time dependent [13]; it is given by

$$E=4\pi {\rho }_c{V}^2{\chi }^{3}J=E_0{t}^s,$$ 6.3

where E₀ is a constant, s is a non-negative number and $J={\int }_0^1((S{U}^2/2)+(P(1-\theta S)/(\mathit{\Gamma }-1))+SE){\xi }^2\hspace{0.17em}\text{d}\xi$. From (5.8) and (6.3) we get

$$V{\chi }^{3/2}={\left(\frac{E_0}{4\pi {\rho }_{c}J}\right)}^{1/2}{t}^{s/2}.$$ 6.4

Since, V = dχ/dt, the integration of (6.4) leads to

$$\chi ={\left(\frac{5}{s+2}\right)}^{2/5}{\left(\frac{E_0}{4\pi {\rho }_{c}J}\right)}^{1/5}{t}^{(s+2)/5}$$ 6.5

and

$$V=\left(\frac{s+2}{5}\right){\left(\frac{25E_0}{4\pi {\rho }_{c}J{(s+2)}^2}\right)}^{1/5}{t}^{(s-3)/5}.$$ 6.6

The equilibrium speed of sound is given by C = (PΓ/S(1 − θS))^1/2. Following the sonic conditions (4.8), which must exist for the self-similar solutions, we get from equation (6.6) s = 5δ − 2 and E₀ = 4πρ_cδ² J. Hence, the total energy released is given by

$$E/E_0=t^{5\delta -2}.$$ 6.7

The results depicting the effect of dust parameters k_p and G on the energy released and the damage radius with respect to time are shown in §7.

7. Numerical results, discussions and conclusion

In the present paper, the Lie group of invariance is used to study a point blast explosion in an unsteady flow of a relaxing gas with dust particles and radiation effects. The constants appearing in the expression for the generators of the local Lie group of transformations give us the different cases of possible shock paths. One of these cases is discussed in detail. Chapman–Jouguet relations are used to relate the flow ahead of and behind the blast front. The system of ODEs (5.15) obtained using the Lie group of invariance is solved further using Matlab software to derive the flow variables. The similarity exponent was obtained on solving the system of equations (5.15) and was verified using the CCW rule. An expression for the energy released during the blast, which is a function of time, is derived using the flow variables. An estimation of the reduction in energy released and hence the reduction in damage radius was thus obtained.

From table 2, we observe that an increase in any of the dust parameters k_p, β or G causes the Grüneisen coefficient (Γ) to decrease. For G = 10 an increase in k_p brings about a decrease in the similarity exponent δ, whereas for G = 100 and above an increase in G increases the similarity exponent. Also, as k_p and β increase, the similarity exponent increases. Since V = δχ/t, the increase in the similarity exponent means an increase in velocity and hence increased pressure and density in the flow region behind the shock. Figures 3 and 4 show that with the increase in the dust parameters k_p and β the density, velocity and vibrational energy increase monotonically whereas the pressure and the radiative flux first increase and then decrease in the blast waves. As G increases the pressure and radiative flux increase, whereas the velocity, density and the vibrational energy first increase and then decrease (figure 5). Figure 6 indicates that as τ_c, i.e. relaxation time, increases all the flow variables, i.e. the density, velocity, pressure, vibrational energy and the radiative flux, decrease in the blast waves. The increase in the parameter ϕ_c (ratio of vibrational specific heat to the specific gas constant) increases the vibrational energy. The density and velocity first increase and then decrease whereas the pressure and radiative flux first decrease and then increase with the increase in ϕ_c (figure 7). Figure 8 shows the variation in the flow variables with cylindrical (m = 1) and spherical (m = 2) symmetry. The energy released and the damage radius are found to decrease with time with an increase in the dust parameters, namely k_p and G. The results for the dust parameter β were found to be similar. The combined effects of the dust parameters k_p and G on the energy released and damage radius are shown in figures 9 and 10, respectively. It can be seen that the maximum reduction in damage radius occurs is when k_p = 0.6 and G = 100.

Figure 4.

Variation of velocity, density, pressure, vibrational energy and radiative flux with the similarity variable ξ for various values of β for m = 2, G = 100, k_p = 0.6, H = 0.2, γ = 1.4, τ_c = 0.1. (Online version in colour.)

Figure 5.

Variation of velocity, density, pressure, vibrational energy and radiative flux with the similarity variable ξ for various values of G for m = 2, k_p = 0.2, β = 0.5, H = 0.5, γ = 1.4, τ_c = 0.1. (Online version in colour.)

Figure 6.

Variation of velocity, density, pressure, vibrational energy and radiative flux with the similarity variable ξ for various values of τ_c, m = 2, k_p = 0.4, G = 100, H = 0.2, γ = 1.4, β = 0.5. (Online version in colour.)

Figure 8.

Variation of velocity, density, pressure and vibrational energy with the similarity variable ξ for different values of m for G = 100, k_p = 0.6, β = 0.5, H = 0.2, γ = 1.4, τ_c = 0.1. (Online version in colour.)

Figure 9.

Variation of energy (E/E₀) with time for different values of k_p, m = 2, G = 100, β = 0.5. (Online version in colour.)

Figure 10.

Variation of damage radius with time at various values of k_p and G. (Online version in colour.)

An increase in k_p from 0.1 to 0.6 for G = 100 means that the ideal gas in the mixture constituting 90% of the total mass and occupying 99.99% of the total volume now constitutes only 40% of the total mass and occupies 98.5% of the total volume. The volumetric fraction (k_p) of the dust lowers the compressibility of the mixture but the mass of the dust increases the total mass and hence it adds to the inertia of the mixture. The strong compression of a condensed medium causes the Grüneisen coefficient to decrease and generates internal pressure owing to the repulsive forces between the particles. The medium is very strongly heated by shock waves and this results in the appearance of a pressure that is related to the transfer of momentum by particles in the thermal motion. The high temperatures activate the rotational and vibrational state of the gas, which with time will tend towards the equilibrium state. The addition of dust decreases the mixture’s compressibility and increases its inertia, which ultimately influences the propagation of shock waves. Owing to temperature differences in the atmosphere after the blast, the temperature inversions take place in the horizontal and vertical directions. As a result of advection inversion, the dust particles in the atmosphere do not mix and will form a protective wall, thereby absorbing the effect of the shock waves. The results obtained are in agreement with those obtained by Pai et al. [1], Chadha & Jena [2], Plocoste et al. [14], Jena & Sharma [25] and Anand [26]. The experiments conducted by Chojnicki et al. [27], to investigate the dynamics of rapidly decompressed gas–particle mixtures and associated shock waves with application to volcanic eruptions, suggested that the particles hinder gas motion, thereby reducing the shock strength and velocity.

It is worth mentioning that the present radiative gasdynamic model with 5% dust particles, considering the relaxing effects of the gas, provides a more realistic and better insight into the point blast problem than the one studied in an ideal gas, which actually does not exist in nature. The limitation on dust particles to 5% is crucial from a mathematical point of view, as this complex model could be studied by a set of four equations along with the thermodynamic relationship, i.e. equation of state. The results obtained indicate that, although small, mitigation of nuclear explosion is possible by dust particles. However, a detailed laboratory experiment conducted in a controlled environment can help in validating our claims.

Data accessibility

The data used for analysis are available at 10.6084/m9.figshare.12177366.

Authors' contributions

M.C. carried out the study, performed the calculations and drafted the manuscript. J.J. critically examined the manuscript for its intellectual content and suggested necessary changes.

Competing interests

The authors are not aware of any affiliations, memberships, funding or financial holdings that might be perceived as affecting the objectivity of this review.

Funding

Department of Science and Technology, Government of India, Women Scientist Scheme (SR/WOS-A/PM-82/2016).