（冲击波）求翻译成汉语:Propagation of shock waves in a dusty gas with exponentially varying density.

Question

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~~~~~

Answer 1

文摘。flow-variables随着距离的变化,在出口一个冲击波传播背后一个尘土飞扬的气体指数型不同密度,并在不同的时间。平衡流条件下,被认为是维护和结果进行了对比与那些得到了一个完美的气体。这是发现小的固体颗粒的存在在这个媒介有显著作用

变异的密度和压力。

研究高速流动的混合气体和小的固体颗粒极大的兴趣在几个分支的工程和科学(Pai et al。[1])。传播

强烈的冲击波产生的帐户突然爆炸介质中的密度随着一些权力距离的爆炸的角度研究了chirster和短时[2],[3]Verma和许多其他人。海斯[4],雷和Bhowmick[5],[6]Vishwakarma Verma和研究了飞机的传播

冲击波在的介质密度成指数增加。

在我们的研究中,我们已经推广了解决方案的光线和Bhowmick[5]在天然气的案例两相流的混合气体和小的固体颗粒的密度服从指数法。为了得到一些基本特征冲击传播,小的固体颗粒被看作是一个pseudo-fluid,认为它是

平衡的水流条件是维护在流场,粘性压力和热传导的混合物可以忽略不计,[1]。虽然密度的

混合物被认为是倍增,体积占领了固体颗粒可能非常小的一般情况下由于大密度的

粒子材料的。因此为了简单起见初始体积分数固体颗粒的Z是假定是一个小的常数。我们的解决方案是non-similar获得的

对于有效的时间到Z仍然很小。变化的流程变量与距离,在激震前沿,在不同的时期,图。

对一维的基本方程和非恒定流的混合气体和小的固体颗粒可以写成:

我们认为,一个强大的冲击波传播到一个媒介,在休息,对抗压力极小。还初始密度的媒介(的混合物

一个天然气和小的固体颗粒)是假设服从指数法:

在suffces“1”、“2”参考价值的前夕,仅次于震惊、U = dR / dt是冲击速度、和R的距离从飞机上激震前沿、线或点对称的。也量“B”是当:

最初的体积分数的固体颗粒的Z是,一般来说不是一个常数。但所占据的体积非常小的固体颗粒因为的密度远远大于固体颗粒的气体(三浦和玻璃[8]),因此可以假定为一个Z小常数。表达式for Z是:问题补充:

自从米出来是一个持续的和磷很可能是用秩序的零制定一个非常强有力的冲击,我们相信这种冲击保持它的伟大力量甚至对于一个很大的时间。因此我们的解在接下来的部分适用于任何时间t >直到Z1仍然很小,最初的鼓动的持续时间。

因此总能量冲击波下界估计,随着R,在那里我= 0,1或2架飞机,圆柱或球形休克。

在术语的无量纲变量r;p;p和u冲击条件下的表现:

方程(3.8)(3.10)以及边界条件(3.12)给我们的问题的解决方案。因此,解决方案是一个non-similar获得一个,因为运动背后的冲击可能是只有当一个明确的价值决定的时间规定。

Answer 2

文摘。flow-variables随着距离的变化,在出口一个冲击波传播背后一个尘土飞扬的气体指数型不同密度,并在不同的时间。平衡流条件下,被认为是维护和结果进行了对比与那些得到了一个完美的气体。这是发现小的固体颗粒的存在在这个媒介有显著作用
变异的密度和压力。
研究高速流动的混合气体和小的固体颗粒极大的兴趣在几个分支的工程和科学(Pai et al。[1])。传播
强烈的冲击波产生的帐户突然爆炸介质中的密度随着一些权力距离的爆炸的角度研究了chirster和短时[2],[3]Verma和许多其他人。海斯[4],雷和Bhowmick[5],[6]Vishwakarma Verma和研究了飞机的传播
冲击波在的介质密度成指数增加。
在我们的研究中,我们已经推广了解决方案的光线和Bhowmick[5]在天然气的案例两相流的混合气体和小的固体颗粒的密度服从指数法。为了得到一些基本特征冲击传播,小的固体颗粒被看作是一个pseudo-fluid,认为它是
平衡的水流条件是维护在流场,粘性压力和热传导的混合物可以忽略不计,[1]。虽然密度的
混合物被认为是倍增,体积占领了固体颗粒可能非常小的一般情况下由于大密度的
粒子材料的。因此为了简单起见初始体积分数固体颗粒的Z是假定是一个小的常数。我们的解决方案是non-similar获得的
对于有效的时间到Z仍然很小。变化的流程变量与距离,在激震前沿

Answer 3

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