The study of shock waves in the mixture of a gas and small solid particles is of great importance due to its applications to nozzle flows, lunar ash flows, bomb blasts, coal-mine blasts, underground, volcanic and cosmic explosions, metallized propellant rockets, supersonic flight in polluted air, collision of coma with a planet, description of star formation, particle acceleration in shocks, shocks in supernova explosions, the formation of dusty crystals and many other engineering problems (see Refs. [1]–[12]). An analytical solution of a planar dusty gas flow with constant velocities of the shock and the piston moving behind it was obtained by Miura and Glass.^[9] Since the volume occupied by the solid particles mixed into the perfect gas is neglected by them, the dust virtually has a mass fraction but no volume fraction. Their results reflect the influence of the additional inertia of the dust upon the shock propagation. Pai et al.^[1] generalized the well known solution of a strong explosion because of an instantaneous release of energy in gas (Sedov,^[13] Korobeinikov^[14]) to the case of two-phase flow of a mixture of small solid particles and perfect gas, and brought out the key effects due to presence of dusty particles on such a strong shock wave. Pai et al.^[1] have taken non-zero volume fraction of solid particles in the mixture, their results reflect the influence of both the decrease of mixture compressibility and the increase of mixture’s inertia on the shock propagation (see, Pai,^[15] Steiner and Hirschler,^[16] Vishwakarma and Nath^[17]).

In recent years considerable attention has been given to study the interaction between gas dynamics and radiation. When the effects of radiation are taken under consideration in gas dynamics the fundamental non-linear equations are very complicated type and thus it is essential to determine approximations which are physically accurate and afford considerable simplifications. The problems of the interaction of radiation with gas dynamics have been studied by many authors by using the self-similar method developed by Sedov,^[13] (see Marshak,^[18] Elliott,^[19] Wang,^[20] Helliwell,^[21] Nicastro,^[22] Ray and Bhowmick^[23] and many others). Many researchers have investigated the motion of a gas under the action of monochromatic radiation (see, Khudyakov,^[24] Zheltukhin,^[25] Nath and Takhar,^[26] Nath,^[27] Vishwakarma and Pandey,^[28] Nath and Sahu^[29]).

The experimental studies and astrophysical observations show that the outer atmosphere of the planets rotates due to rotation of the planets. Macroscopic motion with supersonic speed occurs in an interplanetary atmosphere and shock waves are generated. Thus the rotation of planets or stars significantly affects the process taking place in their outer layers, therefore question connected with the explosions in rotating gas atmospheres are of definite astrophysical interest. In all of the works, mentioned above, the gas is either perfect or non-ideal under the action of monochromatic radiation. The effects of the presence of small solid particles in rotating medium are not taken into consideration by any of the authors under the action of monochromatic radiation. In the present work, we generalize the solution of Nath^[27] in perfect gas to the case of dusty gas (a mixture of perfect gas and small solid particles) by taking into account the axial component of fluid velocity and the component of the vorticity vector. Singh^[30] has considered same problem with the assumption that medium to be non-rotating, whereas we have considered the medium to be rotating.

The purpose of this study is to obtain similarity solutions for the cylindrical shock wave propagating in rotational axisymmetric dusty gas with monochromatic radiation (Nath,^[31–33] Levin and Skopina^[34]). The components of fluid velocity in the ambient medium are assumed to vary and obey the power laws. Also, the angular velocity of rotation of the ambient medium is assumed to be obeying a power law and to be decreasing as the distance from the axis increases. It is expected that such an angular velocity may occur in the atmospheres of rotating planets and stars. In order to get some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid and the mixture at a velocity and temperature equilibrium with a constant ratio of specific heats (Pai^[15]). Also, the heat conduction and viscous stress of the mixture are assumed to be negligible (Refs. [1–2, 16–17]).

Effects of change in the index for the time dependent energy law, the ratio of the density of solid particles to the initial density of the gas, the mass concentration of solid particles in the mixture and the radiation parameter are worked out in detail. It is observed that shock strength decreases in rotating medium.

In Eulerian coordinates, the system of equations of gas dynamics describing the unsteady, adiabatic and cylindrically symmetric one-dimensional rotational axisymmetric flow of mixture of a perfect gas and small solid particles under the action of monochromatic radiation, may be expressed in the form (c.f. Pai et al.,^[1] Vishwakarma and Nath,^[17] Khudyakov,^[24] Nath,^[27] Nath,^[31–33] Levin and Skopina,^[34] Zedan^[35])

The equation of state of the mixture of a perfect gas and small solid particles can be written as (Vishwakarma and Nath,^[17] Nath,^[36–37] Pai,^[15] Singh^[30])

The specific volume of solid particles is assumed to remain unchanged by variations in temperature and pressure. Therefore, the equation of state of solid particles in the mixture is, simply,

The internal energy per unit mass of the mixture can be written as (see Nath,^[37] Pai,^[15] Pai et al.^[1])

Also,

The absorption coefficient K is considered to vary as (Nath,^[27] Nath and Takhar,^[26] Khudyakov^[24])

A diverging cylindrical shock wave is supposed to be propagating outwards from the axis of symmetry into the mixture of perfect gas and small solid particles with constant density, which has zero radial velocity, variable azimuthal, and axial velocities. The flow variables immediately ahead of the shock front are

The momentum equation (2) in undisturbed state of mixture of perfect gas and small solid particles, gives

Ahead of the shock, the components of the vorticity vector, therefore vary as

The initial angular velocity of the medium at radial distance R is given by, from Eq. (11),

From Eqs. (16) and (22), we find that the initial angular velocity vary as

The expression for the initial volume fraction of the solid particles Z₀ is given by

The Rankine–Hugonite conditions i.e. the jump conditions at the shock wave, which are transparent for the radiation flux, are given by the principle of conservation of mass, momentum and energy across the shock (c.f. Nath,^[27] Vishwakarma and Pandey,^[28] Nath,^[31,33] Zel’dovich and Raizer,^[38] Chaturani^[39]) namely,

The total energy “E” of the flow field behind the shock is not constant, but assumed to be time dependent and varying as (Rogers,^[41] Director and Dabora^[40])

Following Levin and Skopina^[34] and Nath,^[31–32] we obtained the jump conditions for the components of vorticity vector across the shock front as

The dimension of the constant coefficient K₀ in Eq. (13) is given by (Singh,^[30] Vishwakarma and Pandey^[28])

For the self-similar solution (Sedov^[13]) the relation between ρ₀, j₀, p₀ is given as

Also, for self-similarity the radiation absorption coefficient K₀ must be dependent on the dimensions of j₀, ρ₀, which is equivalent to .

By the dimensional analysis of Sedov,^[13] the non-dimensional variable η is defined by

From relation (32), it follows that the motion of the shock front is described by the equation

From relation (33) it can be seen that the value corresponds to uniform expansion of a cylindrical shock. Therefore, the solution of physical significance appears to be those for which m lies in the range 0 to 3.

To obtain the similarity solutions, the field variables describing the flow pattern can be written in terms of the dimensionless functions of η such that (Nath,^[27] Nath and Takhar,^[26] Vishwakarma and Pandey,^[28] Nath^[31])

For the existence of similarity solutions “M^*” should be constant, therefore

Using the similarity transformations (34), the system of governing Eqs. (1)–(6) can be transformed and simplified to the following system of ordinary differential equations

Also, was necessary to use to obtain the similarity solution. The quantity ξ is a constant taken as the parameter which characterizes the interaction between the gas and the incident radiation flux (Nath,^[27] Nath and Takhar,^[26] Khudyakov^[24]).

Solving the above set of differential equations (36)–(41) for dV/dη, dD/dη, dP/dξ, dϕ/dη, dW/dη, and dJ/dξ, we obtain

Applying the similarity transformations (34) on Eqs. (12), we obtain the non-dimensional components of the vorticity vector , , in the flow-filed behind the shock as

Using the self-similarity transformations (34), the shock conditions (26) are transformed into

At the inner boundary surface (piston) of the flow-filed behind the shock, the condition is that the velocity of the surface is equal to the normal velocity of the fluid on the surface. This kinematic condition from Eq. (34) can be written as

The set of ordinary differential equations (43)–(48) have been integrated numerically with the boundary conditions (53)–(54) to obtain the distribution of flow variables between piston () and the shock front () by using the Runge–Kutta method of fourth order. Parameter of the inert mixture (glass or alumina Al₂O₃) are within the following range: dust particle size is of the order of 1 μm−10 μm (Higashino and Suzuki,^[2] Fedrov and Kratova,^[42] Nath,^[12]) the mass fraction (concentration) of solid particles in the mixture is varied from to and the material density of solid particles . This case may be realized in an air flow with a suspension of alumina or glass particles. For the purpose of numerical integration, the values of the constant parameters are taken to be (Pai et al.,^[1] Steiner and Hirschler,^[16] Khudyakov,^[24] Nath,^[27] Vishwakarma and Pandey,^[28] Miura and Glass^[43]) ; ; ; ; ; ; ; ; ; ; . The values , may correspond to the mixture of air and glass particles (Miura and Glass^[9]), and correspond to the dust free case (the solution obtained by Nath^[27]). The value of the shock Mach-number is appropriate, because we have treated the flow of a pseudo-fluid (small solid particles) and a perfect gas at a velocity and temperature equilibrium. Our solution corresponds to the solution given by Nath^[27] in dust free case. We have also considered the axial as well as azimuthal components of fluid velocity and components of vorticity vector (see Figs. 1(c), 1(g), 1(h), and 2(c), 2(g), 2(h)). Also, our work corresponds to the solution given by Singh^[30] in non-rotating dust case. Figures 1 and 2 show that the obtained solution is in good agreement with the existing solutions of Nath^[27] and Singh.^[30]

Fig. 1

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Fig. 1 Variation of the reduced flow variables in the region behind the shock front (a) radial component of fluid velocity u/u1, (b) Azimuthal component of fluid velocity v/v1, (c) Axial component of fluid velocity w/w1, (d) Density ρ/ρ1, (e) Pressure p/p1, (f) Radiation flux j/j1, (g) Non-dimensional azimuthal component of vorticity vector lθ, (h) Non-dimensional axial component of vorticity vector lz: 1. ; 2. ; 3. 4. ; 5. ; 6. .

Fig. 2

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Fig. 2 Variation of the reduced flow variables in the region behind the shock front (a) Radial component of fluid velocity u/u1, (b) Azimuthal component of fluid velocity v/v1, (c) Axial component of fluid velocity w/w1, (d) Density ρ/ρ1, (e) pressure p/p1, (f) Radiation flux j/j1, (g) Non-dimensional azimuthal component of vorticity vector lθ, (h) Non-dimensional axial component of vorticity vector lz: 1. ; 2. ; 3. ; 4. .

Table 1 shows the variation of density ratio across the shock front and the position of the inner expanding surface η_p for different values of m, G₀ and K_p with ; ; ; ; ; ; ; in both the rotating and non-rotating cases. Table 2 shows the variation of density ratio ) across the shock front and the position of the inner expanding surface η_p for different values of ξ and K_p with ; ; ; ; ; ; ; in both the rotating and non-rotating cases. Tables 1 and 2 show that the distance of the inner boundary surface from the shock front is less in the case of non-rotating medium in comparison with that in the case of rotating medium. Physically, it means that the gas behind the shock is less compressed in rotating medium i.e. the shock strength is decreased in rotating medium.

Table 1

Table 1

Table 1 The density ratio β across the shock and the position of the inner boundary surface ηp for different values of Kp, G0 and m with ; ; ; ; ; ; ; . . Kp Γ G0 Z0 β m Position of the inner boundary surface ηp Rotating Non-rotating 0.00 1.4 – 0 0.2 1.5 0.798 611 0.806 107           2.0 0.855 189 0.855 823           2.5 0.880 661 0.880 979 0.1 1.36 1 0.1 0.270 218 1.5 0.757 313 0.761 806           2.0 0.815 944 0.816 629           2.5 0.840 948 0.841 281     10 0.010 989 0.194 785 1.5 0.815 628 0.818 813           2.0 0.860 704 0.861 288           2.5 0.884 498 0.884 792     100 0.001 109 88 0.186 413 1.5 0.811 655 0.814 647           2.0 0.865 474 0.866 048           2.5 0.889 169 0.889 461 0.3 1.28 1 0.3 0.418 045 1.5 0.654 960 0.667 344           2.0 0.723 440 0.724 321           2.5 0.748 889 0.749 283     10 0.041 095 9 0.190 936 1.5 0.826 848 0.830 765           2.0 0.868 328 0.868 817           2.5 0.888 608 0.898 856     100 0.004 267 43 0.158 631 1.5 0.848 534 0.850 650           2.0 0.886 385 0.896 847           2.5 0.906 401 0.916 638

Table 1 The density ratio β across the shock and the position of the inner boundary surface ηp for different values of Kp, G0 and m with ; ; ; ; ; ; ; . .

Table 2

Table 2

Table 2 The density ratio β across the shock and the position of the inner boundary surface ηp for different values of Kp and ξ with ; ; ; ; ; ; , . . Kp Γ G0 Z0 β ξ Position of the inner boundary surface ηp                         Rotating Non-rotating 0.00 1.4 – 0 0.2 0.01 0.855 842 0.856 463           0.1 0.855 189 0.855 823           0.5 0.848 140 0.852 995           1 0.847 941 0.850 006 0.1 1.36 10 0.010 989 0.194 785 0.01 0.861 267 0.861 841           0.1 0.860 704 0.861 288           0.5 0.857 237 0.858 821           1 0.847 082 0.848 007 0.3 1.28 10 0.041 095 9 0.158 631 0.01 0.868 748 0.869 229           0.1 0.868 328 0.868 817           0.5 0.862 357 0.863 121           1 0.862 259 0.863 020

Table 2 The density ratio β across the shock and the position of the inner boundary surface ηp for different values of Kp and ξ with ; ; ; ; ; ; , . .

Figures 1(a)–1(h) and 2(a)–2(h) show the variation of the reduced radial component of fluid velocity u/u₁, the reduced azimuthal component of fluid velocityv/v₁, the reduced axial component of fluid velocity w/w₁, the reduced density ρ/ρ₁, the reduced pressure p/p₁, the reduced radiation flux j/j₁, the non-dimensional azimuthal component of vorticity vector l_θ and the non-dimensional axial component of vorticity vector l_z against the similarity variable η for different values of parameters m, G₀, K_p; and with the parameters ξ, K_p respectively. These two figures demonstrate that the flow variablesv/v₁, ρ/ρ₁, and j/j₁ decrease but the flow variablesw/w₁, l_z, and l_θ increase as we move from the shock front to the inner expanding surface.

From Tables 1, 2 and Figs. 1(a)–1(h), 2(a)–2(h) it is found that the effects of an increase in the value of the mass concentration of solid particles K_p in the mixture are as follows: (i)

To decrease the value of β i.e. to increase the shock strength; whereas the value of β increases i.e. the shock strength decreases for (see Tables 1, 2);

From Table 1 and Figs. 1(a)–1(h) it is shown that the effects of an increase in the ratio of the density of the solid particles to the initial density of the gas G₀ are as follows: (i)

To decrease the value of β i.e. to increase the shock strength (see Table 1);

The effects of an increase in the value of index for the time dependent energy law parameter m are as follows: (i)

To increase η_p, i.e. distance between the inner boundary surface from and shock front decreases. This means that shock strength increases (see Table 1);

From Table 2 and Figs. 2(a)–2(h) it is found that the effects of an increase in the radiation parameter ξ are as follows: (i)

To decrease the value of η_p i.e. to decrease shock strength (see Table 2);

The present work investigates the self-similar flow behind a cylindrical shock wave propagating in a rotational axisymmetric dusty gas (a mixture of perfect gas and small solid particles) under the action of monochromatic radiation. On the basis of this work, one may draw the following conclusions: (i)

The shock strength decreases as well as distance between shock front and inner boundary surface increases when radiation parameter ξ increases; however reverse behaviour is observed when m and G₀ increase.

The article concerns with the explosion problem in rotating medium, however the methodology and analysis presented here may be used to describe many other physical systems involving non-linear hyperbolic partial differential equations. The examples we have given make clear the nature of shock waves in rotating dusty gas under the action of monochromatic radiation. However, they serve mainly as illustrations of how the shock waves in dusty medium can be described. In reality, many other processes can be important and a more comprehensive analysis of the shock can be important for applications in astrophysics. The shock waves in a rotational axisymmetric dusty gas with monochromatic radiation and increasing energy can be important for description of shocks in supernova explosions and in the study of star burst galaxies, nuclear explosion, rupture of a pressurized vessel and explosion in the ionosphere etc. Other potential applications of this study include analysis of data from exploding wire experiments in dusty medium, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles (c.f. Hutchens,^[44] Nath^[37]). Also, the present study can be important for the description of the following: