I. INTRODUCTION

The propagation of a blast wave in an inhomogeneous medium, with ambient density given by ${\rho }_{\mathrm{o}}\alpha x^{-\alpha }$, where $x$ is the distance from the center of explosion and $\alpha$ is a positive number, has been analytically studied by Sedov[1] and Rogers[2]. The energy of the explosion does not vary with time, and the problem is self-similar. The uniform expansion of a piston in a homogeneous medium, taking into account the counterpressure across the shock, was numerically solved by Taylor[3]. Bhatnagar and Lal[4] have also considered explosion problem for finite shock strength in a non-homogeneous medium, both when the self-gravitation of the gas is important, and when it is negligible. The motion produced by a piston moving according to the general law $R=C{t}^{n+1}$, where , for strong shocks, has been considered by Kochina and

Melnikov ${}^{[5]}$ and Rogers ${}^{[6]}$, besides others, by solving numerically the ordinary differential equations obtained through similarity transformations. Wang ${}^{[7]}$ has obtained an approximate solution of a plane radiative piston. Sachdev and Ashraf ${}^{[8]}$ have considered the problems of plane, spherical, and, cylindrical pistons moving in an inhomogeneous medium and have obtained solutions in terms of Incomplete Beta functions. The problem of Sedov ${}^{[1]}$ and Taylor ${}^{[3]}$ was extended by Krashaninikova ${}^{[9]}$ to the case when the piston expands non-uniformly with a velocity U is given by

where $U_o$ is a constant.

In this paper, self-similar solutions for the flow between a piston and the strong shock propagating non-uniformly in an ideal gas at rest obeying the power-law density distribution are investigated. Ranga Rao and Purohit ${}^{[10]}$ have studied this problem numerically. The solutions of this problem are similar to that of the solutions of hypersonic flow past the power-law bodies obtained by Lees and Kubota ${}^{[11]}$. They have shown that the condition for the existence of the solution is , where v = 1, 2, 3 for plane, cylindrical and spherical flows respectively. In all these works, the gas ahead of the shock is assumed to be uniform and at rest. Helliwell ${}^{[12]}$ studied the piston problem, in which the piston velocity is assumed to be of the form (1.1) and the density of gas ahead of the shock satisfies the law

where A is a constant.

This problem has been considered as a particular case of a radiative piston problem in which there exists, by dimensional considerations, a relation between n and w, namely

In this paper, we consider the self-similar piston problem in which the piston velocity obeys the law (1.1). The shock is assumed to be strong and propagating in a medium at rest in which the density obeys equation (1.2). The problem with spherical symmetry has got importance in astrophysics, Parker ${}^{[13]}$. We first observe that there are only independent dimensional constants $U_o$ and A involved in the problem, and hence self-similarity exists. Also, there cannot be, in general, any explicit relation between n and w as given by Helliwell ${}^{[12]}$.

On the other hand, it is shown that for all physically meaningful flows the ranges for n and w are

where $\gamma$ is the ratio of specific heats. These conditions on n ensure the finiteness of density and pressure drag on the piston surface. Also, it is shown that when $n=\frac{w(\gamma -1)}{[w(\gamma -1)+2]}$, the flow becomes homentropic. The problem with $\gamma =\frac{7}{5}$, n = -w/w + 5 considered by Helliwell ${}^{[12]}$ corresponds to homentropic flow. Numerical solutions for v = 3, $\gamma =\frac{7}{5}$, and w = 1.5 are given using the Adam Moulton method.

Chernyii ${}^{[14]}$ has given an expansion technique to solve the gas-dynamics problem. The expansion parameter $\epsilon =\frac{\gamma -1}{\gamma +1}$, which occurs in the shock boundary conditions where $\gamma$ is the density ratio across a strong shock. For gases when $\gamma$ generally varies between 1 and 5/3, this parameter is sufficiently small. He has considered the motion of a piston and the big explosion problem in a uniform medium, restricting mostly to zeroth-order solution. In an earlier paper, Chernyii ${}^{[15]}$ has given a particular solution of the piston problem in a homogeneous medium in an integral form to first order in $\epsilon$. Wang ${}^{[7]}$ has also obtained an approximate analytic solution of a plane radiating piston, using Chernyii's ${}^{[15]}$ technique. In this paper, we have employed Chernyii's ${}^{[15]}$ technique, in which flow variables are expanded in power series in $\epsilon$ and have obtained approximate analytic solutions in closed form to the zeroth approximation. This problem belongs to the class of self-similar motion of the first kind in which the similarity exponent occurring in the law of shock propagation can be determined in advance from physical considerations.

II. BASIC EQUATIONS

The partial differential equations of motion, continuity and energy for one-dimensional unsteady flow of a perfect gas are transformed into a set of ordinary differential equations,

by the following transformations

The similarity variable $\lambda$ is took in the form (2.5) by considering $U_0$ and A as the basic dimensional constants involved in the problem and, it takes values $\overline{\lambda }$ and 1 at the piston surface and behind the shock respectively. V, R, and P are non-dimensional reduced particle velocity, density and, pressure, respectively. The boundary conditions for the strong shock are

and the kinematic condition on the piston gives

For the existence of solutions in the self-similar form, certain similarity conditions are to be satisfied. The total energy of the flow between the piston surface and the shock front, using $(2.4)$ and $(2.5)$, can be written as

where

For the flows driven out by the piston, the energy always increases with time. This is possible only if

The second condition of $(2.5)$ is required to make sure that for all physically meaningful solutions, $\delta$ lies between zero and one. These conditions ensure that the pressure drag on the piston is finite. Further, it is necessary that in the domain of interest. Thus, it follows that the density at the piston surface is finite if

For all gases (with ) the ranges for n and w can be obtained from (2.10) and (2.11) as

These are the conditions on n and w for the existence of physically meaningful solutions.

Following Chernyii's ${}^{[15]}$ technique, we expand reduced flow variables as

We substitute expansions (2.13) to (2.15) into equations (2.1) to (2.3) and obtain the following set of differential equations for the zeroth approximation—

The differential equations for the first and second order approximations were too complicated to be amenable to analytic solutions.

The boundary conditions at the shock become

Integrating equation (2.16) to (2.18) with boundary conditions (2.19), we obtain the solution as

where, $A=\frac{2\delta (\gamma +2\delta )}{(\gamma +1{)}^2}$