Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Abstract

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture $K_p$, and the ratio of the density of solid particles to the initial density of the gas $G_a$ are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.

Appendix

From the basic equations of continuity, momentum and energy in Eulerian co-ordinates for the rotational axisymmetric flow of the mixture of a perfect gas and small solid particles with the similarity transformations

$$\begin{aligned}&\eta =\frac{r}{R}, \end{aligned}$$

(64)

$$\begin{aligned} \begin{aligned}&u=\frac{r}{t}V(\eta ),\,v=\frac{r}{t}\phi (\eta ),\,w=\frac{r}{t}W(\eta ), \\&\rho =\rho _a D(\eta ),\,p=\frac{r^{2}}{t^{2}}\rho _a P(\eta ),\,Z=Z_a D(\eta ), \end{aligned} \end{aligned}$$

(65)

(where the variable $\eta$ assumes the value of 1 at the shock front and $\eta _p$ on the piston surface, such that the piston radius $r_{\mathrm{p}} =\eta _{p} R$, with $R (\sim \! t^{n+1})$ being the shock radius), we obtain

$$\begin{aligned}&\left[ {V-(n+1)} \right] \frac{\mathrm{d}D}{\mathrm{d}\eta }+D\frac{\mathrm{d}V}{\mathrm{d}\eta }+\frac{2DV}{\eta }=0, \end{aligned}$$

(66)

$$\begin{aligned}&\left[ {V\!-\!(n\!+\!1)} \right] \frac{\mathrm{d}V}{\mathrm{d}\eta }\!+\!\frac{1}{D} \frac{\mathrm{d}P}{\mathrm{d}\eta }\!+\!\frac{V(V\!-\!1)}{\eta } \!+\!\frac{(2P\!-\!\phi ^{2}D)}{D\eta }\!=\!0, \nonumber \\ \end{aligned}$$

(67)

$$\begin{aligned}&\left[ {V-(n+1)} \right] \frac{\mathrm{d}\phi }{\mathrm{d}\eta }+\frac{(2V-1)\phi }{\eta }=0, \end{aligned}$$

(68)

$$\begin{aligned}&\left[ {V-(n+1)} \right] \frac{\mathrm{d}W}{\mathrm{d}\eta }+\frac{(V-1)W}{\eta }=0, \end{aligned}$$

(69)

$$\begin{aligned}&\frac{\mathrm{d}P}{\mathrm{d}\eta }-\frac{P \Gamma }{D(1-Z_a D)}\frac{\mathrm{d}D}{\mathrm{d}\eta }+\frac{2(V-1)P}{\eta \left[ {V-(n+1)} \right] }=0. \end{aligned}$$

(70)

The boundary conditions for a strong shock in the mixture at $\eta =1$ are given by

$$\begin{aligned} \begin{aligned}&V(1)=\frac{2(1-Z_a )(n+1)}{(\Gamma +1)}, \quad D(1)=\frac{\Gamma +1}{(\Gamma -1+2Z_a)}, \\&P(1)=\frac{2(1-Z_a)(n+1)^{2}}{(\Gamma +1)},\quad \phi (1)=B\left( {B^{*}} \right) ^{-\frac{1}{n+1}}, \\&W(1)=C\left( {B^{*}} \right) ^{-\frac{1}{n+1}}, \end{aligned} \end{aligned}$$

(71)

where it was necessary to use $\alpha =\delta =\frac{n}{n+1}$ for existence of similarity solutions. In addition to the shock conditions (71), the kinematic condition $V(\eta _p)=(n+1)$ at the piston surface must be satisfied.

From equations (66) and (70), one can get the relation

$$\begin{aligned} \frac{D^{\Gamma + (\frac{n}{n+1})}}{(1-Z_a D)^{\Gamma }}=\frac{P \eta ^{(\frac{2}{n+1})}}{C_1} [V-(n+1)]^{-(\frac{n}{n+1})}, \end{aligned}$$

(72)

where $C_1$ is a constant to be determined from (71).