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HYDROMECHANICS
M. P. MIKHAILOVA
MOTION OF A PISTON WITH CONSTANT VELOCITY IN AN INHOMOGENEOUS MEDIUM
(Presented by Academician L. I. Sedov, 10 VII 1961)
Consider the motion of a gas behind a piston which expands with constant velocity \(u\) in a medium whose density \(\rho_n\) varies according to the law
\[ \rho_n=\rho_1[1-\varepsilon r^\chi], \tag{1} \]
where \(r\) is the linear coordinate; \(\varepsilon\) is a small parameter; \(\chi, \rho_1\) are constants. The unknown functions are the pressure \(p\), the density \(\rho\), and the velocity \(v\), which depend on the variables \(t, r\) and on the parameters \(p_1, \rho_1, \varepsilon, \chi, \gamma\), where \(\gamma=c_p/c_v\). From these quantities one can form only two dimensionless variables
\[ \lambda=\frac{\gamma p_1 t^2}{\rho_1 r^2},\qquad \mu=\varepsilon r^\chi . \]
Thus the sought dimensional functions can be represented through dimensionless functions depending on the dimensionless variables:
\[ v=\frac{r}{t}\,\overline V(\lambda,\mu),\qquad p=\rho_1\left(\frac{r}{t}\right)^2\overline P(\lambda,\mu),\qquad \rho=\rho_1\overline R(\lambda,\mu). \tag{2} \]
The problem of the motion of a gas behind a piston moving with constant velocity in a homogeneous medium was solved by L. I. Sedov \({}^{(1)}\). Let \(V_0(\lambda)\), \(P_0(\lambda)\), \(R'(\lambda)\) be the solutions of this problem. Then we represent the linearized sought solutions in the form
\[ \overline V=V_0(\lambda)+\mu V(\lambda),\qquad \overline P=P_0(\lambda)+\mu P(\lambda),\qquad \overline R=R_0(\lambda)+\mu R(\lambda). \tag{3} \]
The basic equations of one-dimensional unsteady motion have the form
\[ \frac{\partial v}{\partial t} +v\frac{\partial v}{\partial r} +\frac{1}{\rho}\frac{\partial p}{\partial r}=0, \]
\[ \frac{\partial \rho}{\partial t} +\frac{\partial \rho v}{\partial r} +(\nu-1)\frac{\rho v}{r}=0, \tag{4} \]
\[ \frac{\partial}{\partial t}\left(\frac{p}{\rho^\gamma}\right) +v\frac{\partial}{\partial r}\left(\frac{p}{\rho^\gamma}\right)=0. \]
Here \(\nu=1,2,3\), respectively, for plane, cylindrical, and spherical waves. After passing to dimensionless variables and varying with respect to \(\mu\), we obtain a system of ordinary differential equations for \(V, P, R\). From the independent variable \(\lambda\) we pass to the independent variable
variable \(V_0\), and we shall seek solutions in the form
\[ V=(1-V_0)^S \sum_{n=0}^{\infty} a_{ni}(1-V_0)^n, \]
\[ P=(1-V_0)^S \sum_{n=0}^{\infty} b_{ni}(1-V_0)^n, \tag{5} \]
\[ R=(1-V_0)^S \sum_{n=0}^{\infty} c_{ni}(1-V_0)^n. \]
As is known, in the case of self-similar motion the solutions for \(V_0\), \(P_0\), \(R_0\) are not expressed in analytic form. After passing to the independent variable \(V_0\), approximate solutions can be found by expanding the desired functions in powers of \((1-V_0)\):
\[ \lambda=\lambda_{\mathrm{p}}\left[1-\frac{2}{\nu}(1-V_0)+\ldots\right], \]
\[ P_0=P_{0\mathrm{p}}\left[1-\frac{2}{\nu}(1-V_0)+\ldots\right], \tag{6} \]
\[ R_0=R_{0\mathrm{p}}\left[1+\chi(1-V_0)^2+\ldots\right], \]
where \(\lambda_{\mathrm{p}}, P_{0\mathrm{p}}, R_{0\mathrm{p}}\) are the values of the functions at the piston.
The characteristic equation of the system, if the equalities (6) are taken into account, will be
\[ S^2\left(S-\frac{\chi}{\nu}\right)=0. \tag{7} \]
The roots \(S_1=S_2=0\) correspond to a solution with a logarithmic singularity \((^2)\).
Let us now consider the boundary conditions. At the piston \(\vartheta=u\), and since \(V_0=1\), then \(V(1)=0\). Ahead of the piston, at some distance, a shock wave is moving. We shall assume that the piston velocity is large, and take the conditions at the shock wave in the form
\[ \vartheta_2=\frac{2}{\gamma+1}c,\qquad \rho_2=\frac{\gamma+1}{\gamma-1}\rho_H,\qquad p_2=\frac{2}{\gamma+1}\rho_H c^2, \tag{8} \]
where \(c\) is the velocity of the shock wave.
The conditions at the shock wave, after passage to dimensionless variables and variation with respect to \(\mu\), have the form
\[ V_2=\left[\frac{2(1+\chi)(1+\lambda^*)}{\gamma+1}-V_0(\lambda^*)+2\lambda^*\left(\frac{dV_0}{d\lambda}\right)_{\lambda=\lambda^*}\right]a-\frac{2\lambda^*}{\gamma+1}, \]
\[ P_2=\left[2\lambda^*\left(\frac{dP_0}{d\lambda}\right)_{\lambda=\lambda^*}-2P_0(\lambda^*)+\frac{4(1+\chi)}{\gamma+1}\right]a-\frac{2}{\gamma+1}, \tag{9} \]
\[ R_2=\left[2\lambda^*\left(\frac{dR_0}{d\lambda}\right)_{\lambda=\lambda^*}+\frac{4(\gamma+1)(1+\chi)\lambda^*}{\gamma-1}\right]a-\frac{\gamma+1}{\gamma-1}, \]
where \(\lambda^*\) is the value of \(\lambda\) at the shock wave, and \(a\) is a constant.
The radius vector \(r_2\) at the shock wave is represented in the form
\[ r_2=r_{20}[1+a\mu^*]. \tag{10} \]
The solutions for \(V, P, R\) take the following form if \((1-V_0)^2\) is neglected and \(\chi/\nu\) is not an integer:
\[ V=\frac{\chi}{\nu}\frac{b_{01}}{\gamma P_{0\mathrm{p}}}(1-V_0)\ln(1-V_0) +\frac{2\chi c_{03}}{R_{0\mathrm{p}}(\chi+\nu)}(1-V_0)^{\chi/\nu+1}. \]
\[ P=b_{01}\left\{\left[1-\frac{\chi+2}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{4}{\nu}(1-V_0)\right\}, \tag{11} \]
\[ R=\frac{R_{0\mathrm{p}}}{\gamma P_{0\mathrm{p}}}b_{01} \left\{\left[1-\frac{\chi}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{\nu}{\chi}+\frac{4}{\nu-\chi}(1-V_0)\right\} +c_{03}(1-V_0)^{\chi/\nu}. \]
In these formulas the condition at the piston has already been taken into account. The unknown constants \(b_{01}, c_{03}, a\) are determined from the conditions at the shock wave.
If \(\chi/\nu\) is an integer, then the solutions will have the form
\[ V=\frac{\chi}{\nu}\frac{b_{01}}{\gamma P_{0\text{p}}}(1-V_0)\ln(1-V_0), \]
\[ P=b_{01}\left\{\left[1-\frac{\chi+2}{\nu}(1-V_0)\right]\ln(1-V_0)+1+\frac{4}{\nu}(1-V_0)\right\}, \tag{12} \]
\[ R=\left[\frac{R_{0\text{p}}}{\gamma P_{0\text{p}}}b_{01}+c_{11}(1-V_0)\right]\ln(1-V_0) +\left(1+\frac{\nu}{\chi}\right)\frac{R_{0\text{p}}}{\gamma P_{0\text{p}}}b_{01} +c_{11}(1-V_0) \]
Fig. 1
Fig. 2
Fig. 3
The values of \(V, P, R\) computed by us as functions of \(V_0\) for \(\chi=1\) in the cases \(\nu=1,2,3\) are presented in the form of graphs (Figs. 1, 2, 3).
Received
30 VI 1961
CITED LITERATURE
- L. I. Sedov, Methods of Similarity and Dimensionality in Mechanics, Moscow, 1957.
- G. Piaggio, Integration of Differential Equations, 1933.