Abstract. The variation of flow-variables with distance, in the flow-field behind a shock wave propagating in a dusty gas with exponentially varying density, are obtained at different times. The equilibrium flow conditions are assumed to be maintained, and the results are compared with those obtained for a perfect gas. It is found that the presence of small solid particles in the medium has significant effects on the
variation of density and pressure.
The study of high speed flow of a mixture of gas and small solid particles is of great interest in several branches of engineering and science (Pai et al. [1]). The propagation
of strong shock wave produced on account of sudden explosion in a medium where the density varies as some power of the distance from the point of explosion, has been studied by Christer and Helliwell [2], Verma [3] and many others. Hayes [4], Ray and Bhowmick [5], Verma and Vishwakarma [6] have studied the propagation of plane
shock wave in a medium where density increases exponentially.
In our study, we have generalized the solution of Ray and Bhowmick [5] in gas to the case of two phase flow of a mixture of gas and small solid particles in which the density obeys the exponential law. In order to get some essential features of shock propagation, small solid particles are considered as a pseudo-fluid, and it is assumed
that the equilibrium flow condition is maintained in the flow field, and that the viscous stress and heat conduction of the mixture are negligible [1]. Although the density of
the mixture is assumed to be increasing exponentially, the volume occupied by the solid particles may be very small under ordinary conditions owing to the large density of
the particle material. Hence for simplicity the initial volume fraction of solid particles Z is assumed to be a small constant. Our solutions obtained are non-similar ones and
are valid for the time till Z remains small. Variation of the flow variables with distance, behind the shock front, at different times, are shown in Figures.
The fundamental equations for one dimensional and unsteady flow of a mixture of gas and small solid particles can be written as:
We consider that a strong shock wave is propagated into a medium, at rest, with negligibly small counter pressure. Also the initial density of the medium (the mixture of
a gas and small solid particles) is assumed to obey the exponential law:
where suffces "1" and "2" refer to the values just ahead and just behind of the shock, U = dR/dt is the shock velocity,and R the distance of the shock front from the plane, the line or the point of symmetry. Also the quantity "B" is given by:
The initial volume fraction of the solid particles Z is, in general not a constant. But the volume occupied by the solid particles is very small because the density of the solid particles is much larger than that of the gas (Miura and Glass [8]), hence Z may be assumed as a small constant. The expression for Z is :
Since M comes out to be a constant and p can be taken to be of order zero for a very strong shock, we conclude that the shock retains its great strength even for a large time. Hence our solutions obtained in the next section are applicable for any time t > till Z1 remains small, being the duration of initial impulse.
Hence the total energy of the shock wave is nonconstant and varies as R, where i = 0; 1 or 2 for plane, cylindrical or spherical shock.
In terms of dimensionless variables r; p; P and u the shock conditions take the form:
Equations (3.8) to (3.10) along with the boundary conditions (3.12) give the solution of our problem. The solution thus obtained is a non-similar one, since the motion behind the shock can be determined only when a definite value for time is prescribed.
文中“：”后面表示存在公式这里就没有打出来了，本文的主题是关于冲击波在不同密度的含尘空气中的传播。忘各位大侠帮忙翻译下，还有请帮忙的朋友翻译通顺点~~~~~