Magnetogasdynamic exponential shock wave in a self-gravitating, rotational axisymmetric non-ideal gas under the influence of heat-conduction and radiation heat-flux

Abstract

Similarity solutions are obtained for one-dimensional cylindrical shock wave in a self-gravitating, rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the presence of conductive and radiative heat fluxes. The total energy of the wave is non-constant. It is obtained that the increase in the Cowling number, in the parameters of radiative as well as conductive heat transfer and the parameter of the non-idealness of the gas have a decaying effect on the shock wave however increase in the value of gravitational parameter has reverse effect on the shock strength. It is manifested that the presence of azimuthal magnetic field removes the singularities which arise in some cases of the presence of axial magnetic field. Also, it is observed that the effect of the parameter of non-idealness of the gas is diminished by increasing the value of the gravitational parameter.

Appendices

Appendices

For interested readers, a detailed description of the mathematical model of the problem under consideration is presented in the Appendix 1, as well as the detailed solution procedure, is reported in the Appendix 2.

Mathematical model and problem description

The motion of piston is assumed to obey an exponential law, namely ( Sahu [30], Rao and Ramana [49])

$$\begin{aligned} r_{p} = A exp(\xi \, t),\quad \xi > 0 , \end{aligned}$$

(A.1)

where $r_{p}$ is the radius of the piston, A and $\xi$ are dimensional constants, and t is the time. ‘A’ represents the initial radius of the piston. It may be, physically, the radius of the stellar corona or the condensed explosive or the diaphragm containing a very high-pressure driver gas, at $t = 0$. By sudden expansion of the stellar corona or the detonation products or the driver gas into the undisturbed ambient gas, a shock wave is produced in the ambient gas. The shocked gas is separated from the expanding surface which is a contact discontinuity. This contact surface acts as a ‘piston’ for the shock wave in the ambient medium (Rosenau and Frankenthal [55], Higashino [58], Liberman and Velikovich [59], Sahu [30]). Also, another justification for the exponential growth of the expanding piston refers to the internal energy source in simulating the ignition process. Ignition delay is a notation characterizing in experiments the time interval between mixture is placed under definite conditions and active energy release beginning accompanied by temperature as well as pressure growth. Ignition delay are often measured in shock tube experiments behind reflected shock waves. The dependence of ignition delay on temperature is always monotonous for constant pressure. Ignition delay decreases exponentially on linear increase of temperature (Smirnov and Nikitin [60], Smirnov et al. [61]). Thus, the energy release following chemical kinetics and turbulence increases after ignition, which brings to the formation of strong accelerating shock waves, which can finally result in detonation (Smirnov and Nikitin [60], Smirnov et al. [61]).

The law of piston motion (A.1) implies a boundary condition on the gas speed at the piston, which is required in the determination of the problem. Since we have assumed self-similarity, we may postulate that the shock propagation follows the exponential law

$$\begin{aligned} R = B exp(\xi \, t), \end{aligned}$$

(A.2)

where R is shock radius, and B is a dimensional constant. B depends on A and non-dimensional position of the piston [see equation (B.1)]. As is often the case in problems of this type, it is more convenient to solve for the piston motion in terms of the shock motion, rather than vice-versa. We shall therefore adopt this point of view forthwith, and consider B a known parameter of the problem, rather than A (Rosenau and Frankenthal [55]). The ‘piston’ is used to replicate blast waves as well as other similar phenomena in a model to simulate actual explosions and their effects, usually on a smaller scale. Thus ‘piston’ problem can be applied to quantify an estimate for the outcome from supernova explosions, sudden expansion of the stellar corona or detonation products, central part of starburst galaxies etc.

We have considered the medium to be a non–ideal gas, which is rotating about an axis of symmetry. We have taken r and t as independent space and time coordinates; u,  v,  and w as the radial, azimuthal and axial components of the fluid velocity $\overrightarrow{X}$ in the cylindrical coordinates $(r,\theta ,z)$. Also, the relation between the angular velocity ‘$C^{*}$’ of the medium at radial distance r from the axis of symmetry and the azimuthal component of velocity is given by

$$\begin{aligned} v = C^{*} r, \end{aligned}$$

(A.3)

where ‘$C^{*}$’ is the angular velocity of the medium at radial distance r from the axis of symmetry. In this case the vorticity vector

$$\begin{aligned} \overrightarrow{\zeta } = \dfrac{1}{2} Curl \overrightarrow{X}, \end{aligned}$$

has the following components

$$\begin{aligned} \zeta _{r} = 0, \quad \zeta _{\theta } = -\dfrac{1}{2} \dfrac{\partial w}{\partial r}, \quad \zeta _{z} = \dfrac{1}{2 r} \dfrac{\partial }{\partial r} (rv). \end{aligned}$$

(A.4)

The total heat-flux Q,  which appear in the energy equation (5) may be decomposed as

$$\begin{aligned} Q = Q_{C} + Q_{R}, \end{aligned}$$

(A.5)

where $Q_{C}$ is conduction heat flux and $Q_{R}$ is radiation heat flux.

According to Fourier’s law of heat conduction

$$\begin{aligned} Q_{C} = - K \, \dfrac{\partial T}{\partial r}, \end{aligned}$$

(A.6)

where ‘K’ is the coefficient of the thermal conductivity of the gas and ‘T’ is the absolute temperature.

Assuming local thermodynamic equilibrium and using the radiative diffusion model for an optically thick grey gas Pomroning [62], the radiative heat flux $Q_{R}$ may be obtain from the differential approximation of the radiation transport equation in the diffusion limit as

$$\begin{aligned} Q_{R} = - \dfrac{4}{3} \left( \dfrac{\sigma }{\alpha _{R}}\right) \dfrac{\partial T^{4}}{\partial r}, \end{aligned}$$

(A.7)

where $\sigma$ is the Stefan–Boltzman constant and $\alpha _{R}$ is the Rosseland mean absorption coefficient.

The radiation energy flux in local equilibrium (when the radiation at each point of a medium with a non-uniform temperature is close to equilibrium, then the medium is spoken of as being in a state of local thermodynamic equilibrium between the radiation and the fluid. The necessary condition for the existence of local equilibrium-small gradients in an extended, optically thick medium - serves simultaneously as a justification for the use of the diffusion approximation when considering radiative transfer) is proportional to the temperature gradient, and the radiative transfer is similar to heat conduction and is termed radiation heat conduction. The coefficient of thermal conductivity is equal to $\dfrac{16 \, \sigma }{ 3 \, \alpha _{R}} T^{3}$ and is a function of temperature. In many cases it is possible to consider the radiation mean free path $\left( = \dfrac{1}{\alpha _{R}} \right)$ proportional to some power of temperature (It is assumed that the density of the medium is constant) and for the power-law variation of the radiation mean free path the coefficient of the thermal conductivity is also proportional to a power of temperature (Zel’dovich and Raizer [31]). Thus, the above mentioned thermal conductivity ‘K’ and the absorption coefficient $\alpha _{R}$ of the medium are assumed to vary with temperature only and these can be written in the form of power laws, namely (Ghoniem et al. [7], Nath and Sahu [9])

$$\begin{aligned} K = K_{0} \left( \dfrac{T}{T_{0}}\right) ^{\beta _{c}} \left( \dfrac{\rho }{\rho _{0}}\right) ^{\delta _{c}} \quad and \quad \alpha _{R} = \alpha _{R_{0}} \left( \dfrac{T}{T_{0}}\right) ^{\beta _{R}} \left( \dfrac{\rho }{\rho _{0}}\right) ^{\delta _{R}}, \end{aligned}$$

(A.8)

where the subscript ‘0’ denotes a reference state. The exponents in the above equations should satisfy the similarity requirements if a self similar solution is sought.

The electrical conductivity of the gas is assumed to be infinite. Therefore the diffusion term from the magnetic field equation is omitted, and the electrical resistivity is ignored. Also, the effect of viscosity on the flow of the gas is assumed to be negligible. The above system of equations (1) - (7) should be supplemented with an equation of state. In most of the cases, the propagation of shock waves arises in extreme conditions under which the assumption that the gas is ideal is not a sufficiently accurate description. The equation of state for a non-ideal gas is obtained by considering an expansion of the pressure p in powers of the density $\rho$ (Anisimov and Spiner [63], Sahu [29, 30])

$$\begin{aligned} p = \Gamma \rho T \left[ 1 + \rho C_{1}(T) + \rho ^{2} C_{2}(T) + ... \right] , \end{aligned}$$

(A.9)

where $\Gamma$ is the gas constant and $C_{1}(T)$, $C_{2}(T)$, ... are virial coefficients. The first term in the expansion corresponds to an ideal gas. The second term is obtained by taking into account the interaction between pairs of molecules, and subsequent terms must involve the interactions between the groups of three, four, etc. molecules. In the high-temperature range the. coefficients $C_{1}(T)$ and $C_{2}(T)$ tend to constant values equal to b and $\frac{5}{8} \, b^{2}$, respectively. For gases $b \rho<< 1$, b being the internal volume of the molecules, and therefore it is sufficient to consider the equation of state in the form (Anisimov and Spiner [63], Sahu [29, 30] )

$$\begin{aligned} p&= \Gamma \rho T (1 + \rho b). \end{aligned}$$

(A.10)

In this equation the correction to pressure is missing due to neglect of second and higher powers of $b \rho$, i.e. due to neglect of interactions between groups of three, four, etc. molecules of the gas. The internal energy $U_{m}$ per unit mass of the non-ideal gas is given by (Anisimov and Spiner [63], Sahu [29, 30] )

$$\begin{aligned} U_{m} = \dfrac{p}{(\gamma - 1) \rho \, (1 + \rho b)}, \end{aligned}$$

(A.11)

where $\gamma$ is the adiabatic index.

Real gas effects can be expressed in the fundamental equations according to Chandrasekhar [64], by two thermodynamic variables, namely by the sound velocity factor (the isotropic exponent) $\Gamma ^{*}$ and a factor $K^{*}$, which contains internal energy as follows,

$$\begin{aligned} \Gamma ^{*}&= \left( \dfrac{\partial \, ln \, p }{\partial \, ln \,\rho } \right) _{S} \quad and \quad K^{*} = - \dfrac{p}{\rho } \, \left( \dfrac{\partial \, U_{m} }{\partial \, ln \,\rho } \right) _{P}, \end{aligned}$$

(A.12)

where the subscripts ‘S’ and ‘P’ refers to the process of constant entropy and constant pressure.

Using the first law of thermodynamics and the Equations (A.10) and (A.11), we obtain

$$\begin{aligned} \Gamma ^{*}&= \dfrac{\gamma \, (1 + 2 b \rho )}{1 + b \rho } \quad and \quad K^{*} = \dfrac{1}{\gamma - 1}, \end{aligned}$$

(A.13)

neglecting the second and higher powers of b. This shows that the isentropic exponent $\Gamma ^{*}$ is non-constant in the shocked gas, but the factor $K^{*}$ is constant for the simplified equation of state of the non-ideal gas in the form equation (A.10).

The isentropic velocity of sound $a_{non}$ is given by

$$\begin{aligned} a_{non}^{2}&= \Gamma ^{*} \, \dfrac{p}{\rho }. \end{aligned}$$

(A.14)

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

$$\begin{aligned} \zeta _{r_{a}}&= 0, \end{aligned}$$

(A.15)

$$\begin{aligned} \zeta _{\theta _{a}}&= - \dfrac{w^{*} \alpha }{2 \, \xi R} exp(\alpha \, t), \end{aligned}$$

(A.16)

$$\begin{aligned} \zeta _{z_{a}}&= \dfrac{v^{*} (\delta + \xi )}{2 \, \xi R} exp(\delta \, t). \end{aligned}$$

(A.17)

The initial angular velocity of the medium at radial distance R is given by, from (A.3),

$$\begin{aligned} C^{*}_{a} = \dfrac{v_{a}}{R}. \end{aligned}$$

(A.18)

From equations (10) and (A.18), we find that the initial angular velocity vary as

$$\begin{aligned} C^{*}_{a} = \dfrac{v^{*}}{R} exp(\delta \, t) = \dfrac{v^{*}}{B} exp \left\{ (\delta - \xi )\, t\right\} . \end{aligned}$$

(A.19)

In addition to equilibrium sound speed of non-ideal gas (A.14), the isothermal speed of sound may also play a role, where thermal radiation is taken into account. The isothermal sound speed in the non-ideal gas is

$$\begin{aligned} a_{iso} = \left( \dfrac{\partial p}{\partial \rho } \right) _{T}^{\frac{1}{2}} = \left[ \dfrac{ (1 + 2 b \, \rho ) p}{\rho \, (1 + b \rho )} \right] ^{\frac{1}{2}}, \end{aligned}$$

(A.20)

where the subscript ‘T’ refers to the process of constant temperature.

The adiabatic compressibility of the non–ideal gas may be calculated as (Moelwyn-Hughes [65])

$$\begin{aligned} C_{adi} = \dfrac{1}{\rho } \left( \dfrac{\partial \rho }{\partial p} \right) _{S} = \dfrac{1}{\rho \, a_{non}^{2}}= \left[ \dfrac{ (1 + b \rho ) }{\gamma p (1 + 2 b \rho )} \right] . \end{aligned}$$

(A.21)

Following Levin and Skopina [45], Nath and Sahu [9]; we obtained the jump conditions for the components of vorticity vector across the shock front as

$$\begin{aligned} \zeta _{\theta _{s}} = \dfrac{\zeta _{\theta _{a}}}{\beta }, \quad \zeta _{z_{s}} = \dfrac{\zeta _{z_{a}}}{\beta }. \end{aligned}$$

(A.22)

Detailed solution procedure

Equations (A.1), (A.2) and (21) yields a relation between B and A in the form

$$\begin{aligned} B = \dfrac{A}{y_{p}}. \end{aligned}$$

(B.1)

Using the similarity transformations (21), the system of governing equations (1)–(7) can be transformed to the following system of ordinary differential equations:

$$\begin{aligned}&(U - y) \, \dfrac{dD}{dy} + D \, \dfrac{dU}{dy} + \dfrac{D \, U}{y} = 0, \end{aligned}$$

(B.2)

$$\begin{aligned}&(U - y) \, D \, \dfrac{dU}{dy} + \dfrac{dP}{dy} + H \, \dfrac{dH}{dy} + \dfrac{i \, H^{2}}{y} + U \,D - \dfrac{D \, V^{2} }{y} + \dfrac{ G^{*} \, N \, D}{y} = 0 , \end{aligned}$$

(B.3)

$$\begin{aligned}&(U - y) \, \dfrac{dV}{dy} + \dfrac{(U + y) \, V }{y} = 0 , \end{aligned}$$

(B.4)

$$\begin{aligned}&(U - y) \, \dfrac{dW}{dy} + W = 0 , \end{aligned}$$

(B.5)

$$\begin{aligned}&\dfrac{2\, P }{(\gamma - 1)\, D(1 + {\overline{b}} \,D)} + \dfrac{(U - y)}{(\gamma - 1)\, D^{2} \, (1 + {\overline{b}} \,D)^{2}} \left[ D \, (1 + {\overline{b}} \,D) \dfrac{dP}{dy} - P \, (1 + 2 {\overline{b}} \,D) \dfrac{dD}{dy} \right] \nonumber \\&\quad - \dfrac{P \, (U - y) }{D^{2}} \dfrac{dD}{dy} + \dfrac{F}{D \, y} + \dfrac{1}{ D} \dfrac{dF}{dy} = 0 \end{aligned}$$

(B.6)

$$\begin{aligned}&(U - y) \, \dfrac{dH}{dy} + H \, \dfrac{dU}{dy} + H + \dfrac{H \, U (1 - i) }{y} = 0, \end{aligned}$$

(B.7)

$$\begin{aligned}&\dfrac{dN}{dy} - 2 \, D \, y = 0, \end{aligned}$$

(B.8)

where $G^{*} = \dfrac{G \pi \rho _{a}}{\xi ^{2}}$ is the gravitational parameter.

Using equations (A.6), (A.7) and (A.8) in equation (A.5), we get

$$\begin{aligned} Q = - \dfrac{K_{0}}{ T_{0}^{\beta _{c}} \rho _{0}^{\delta _{c}}} T^{\beta _{c}} \rho ^{\delta _{c}} \dfrac{\partial T}{\partial r} - \dfrac{16\, \sigma T_{0}^{\beta _{R}} \, \rho _{0}^{\delta _{R}}}{3\, \alpha _{R_{0}}} T^{ 3 - \beta _{R}} \rho ^{- \delta _{R}} \dfrac{\partial T}{\partial r}. \end{aligned}$$

(B.9)

Using equations (A.9) and (21) in equation (B.9), we get

$$\begin{aligned} F= & {} \Bigl [ \dfrac{- K_{0} \, \xi }{T_{0}^{\beta _{c}} \, \rho _{0}^{\delta _{c}} \, \Gamma ^{\beta _{c} + 1}} \, \dfrac{{\dot{R}}^{2 \beta _{c} - 2} P^{\beta _{c}} \, \rho _{a}^{\delta _{c} - 1 } \, D^{\delta _{c} - \beta _{c}}}{(1 + {\overline{b}} \, D)^{\beta _{c}} \,} \nonumber \\&\quad - \dfrac{16\, \sigma T_{0}^{\beta _{R}} \, \rho _{0}^{\delta _{R}} \, \xi }{3\, \alpha _{R_{0}} \, \Gamma ^{4 - \beta _{R}}} \, \dfrac{{\dot{R}}^{4 - 2 \beta _{R}} \, P^{3- \beta _{R}} \, \rho _{a}^{- 1 -\delta _{R} } \, D^{- 3 + \beta _{R} - \delta _{R} }}{(1 +{\overline{b}} \, D)^{3- \beta _{R}} \,} \Bigr ] \dfrac{d}{d y} \left( \dfrac{P }{D \, (1 + {\overline{b}} \, D)}\right) .\nonumber \\ \end{aligned}$$

(B.10)

Equation (B.10) shows that the similarity solution of the present problem exists only when

$$\begin{aligned} \beta _{c} = 1 \quad and \quad \beta _{R} = 2 . \end{aligned}$$

(B.11)

Therefore equation (B.10) becomes

$$\begin{aligned} F = - X \left[ \dfrac{1}{D \, (1 + {\overline{b}} \, D)} \dfrac{dP}{d y} - \dfrac{P \, (1 + 2 \, {\overline{b}} \, D)}{D^{2} \, (1 +{\overline{b}} \, D)^{2}} \dfrac{dD}{d y} \right] , \end{aligned}$$

(B.12)

where $X = \left[ \Gamma _{C} \, D^{\delta _{c} - 1} + \Gamma _{R} \, D^{ - 1 - \delta _{R}}\right] \dfrac{ P}{(1 + {\overline{b}} \, D)}$, $\Gamma _{C}$ and $\Gamma _{R}$ are the conductive and radiative non-dimensional heat transfer parameters, respectively. The parameters $\Gamma _{C}$ and $\Gamma _{R}$ depend on the thermal conductivity K and the mean free path of radiation $\dfrac{1}{\alpha _{R}}$ respectively and they are given by

$$\begin{aligned} \Gamma _{C} = \dfrac{ K_{0} \, \xi }{T_{0} \, \rho _{0}^{\delta _{c}} \, \Gamma ^{2}} \, \rho _{a}^{\delta _{c} - 1} \quad and\quad \Gamma _{R} = \dfrac{16\, \sigma T_{0}^{2} \, \rho _{0}^{\delta _{R}} \, \xi }{3\, \alpha _{R_{0}} \, \Gamma ^{2}} \,\rho _{a}^{- 1 - \delta _{R} }. \end{aligned}$$

Solving the set of differential equations (B.2)–(B.8) and equation (B.12), we have equations (23)–(30).

Applying the similarity transformations (21) on equation (A.4), we obtained the non-dimensional components of the vorticity vector $l_{r} = \dfrac{\zeta _{r}}{{\dot{R}}/R},$   $l_{\theta } \dfrac{\zeta _{\theta }}{{\dot{R}}/R}$,   $l_{z} = \dfrac{\zeta _{z}}{{\dot{R}}/R}$ in the flow-filed behind the shock as

$$\begin{aligned} l_{r}&= 0, \end{aligned}$$

(B.13)

$$\begin{aligned} l_{\theta }&= - \dfrac{W}{2 (y- U)}, \end{aligned}$$

(B.14)

$$\begin{aligned} l_{z}&= \dfrac{V}{(y- U)}. \end{aligned}$$

(B.15)

By using equation (21) the expression for reduce isothermal speed of sound (A.20) becomes

$$\begin{aligned} \dfrac{a_{iso}}{{\dot{R}}} = \left[ \dfrac{ (1 + 2\, {\overline{b}} \, D) P}{ D \, (1 + {\overline{b}} \, D)} \right] ^{\frac{1}{2}}. \end{aligned}$$

(B.16)

By using equation (21) in equation (A.21), we obtain the expression for the adiabatic compressibility $C_{adi}$ as

$$\begin{aligned} \left( C_{adi}\right) \, p_{a} = \dfrac{ (1 + {\overline{b}} \,D)}{P \, \gamma ^{2} \, M^{2} \, (1 + 2 {\overline{b}} \, D)}. \end{aligned}$$

(B.17)