Abstract Generated abstract
This paper examines which equations of state permit one-dimensional motions with homogeneous deformation, extending earlier treatments that assumed a perfect gas. Starting from the momentum equation under the deformation condition and considering either adiabatic motion or spatially uniform temperature, it derives functional forms relating pressure, density, entropy, and temperature. The analysis yields general equations of state compatible with such motions and identifies, for arbitrary initial entropy or temperature dependences, a power-law class of the form pressure equals a coefficient depending on entropy or temperature times density to a constant exponent, plus an allowable additive term.
Full Text
Reports of the Academy of Sciences of the USSR
- Volume 120, No. 3
HYDROMECHANICS
A. G. KULIKOVSKII
ON MEDIA ADMITTING ONE-DIMENSIONAL MOTIONS WITH HOMOGENEOUS DEFORMATION
(Presented by Academician L. I. Sedov, 23 I 1958)
One-dimensional motions with homogeneous deformation, i.e., motions satisfying the condition
\[ r=\mu(t)r_0 \tag{1} \]
(where \(r\) is the distance of a particle either to some plane, or to an axis, or to a point, and \(r_0\) is the same distance at \(t=0\)), were considered in works \((^{1-5})\), the medium being regarded as a perfect gas. Below an answer is given to the question: what equation of state must a medium obey in order to admit one-dimensional motions with homogeneous deformation?
Let us consider two cases: the motion satisfies either the adiabaticity condition \(\partial S/\partial t=0\), or the condition \(\partial T/\partial r=0\) (\(T\) is the temperature). In the first case we shall seek the equations of state in the form \(p=p_1(\rho,S)\), and in the second case in the form \(p=p_2(\rho,T)\).
If external forces are absent, and the internal stresses reduce to pressure, as well as in certain other cases (as, for example, in the presence of gravitation \((^2)\)), it follows from the momentum equation under condition (1) that
\[ \frac{1}{\rho_0}\frac{\partial p}{\partial r_0}=k(t)r_0, \tag{2} \]
where \(\rho_0\) is the initial density.
Equality (2) will be the starting point in the subsequent considerations. From equality (2), taking \(p(r_0,t)=p(p_0,t)\) (\(p_0(r_0)\) is the initial pressure), we obtain
\[ \frac{\partial p}{\partial p_0}=\frac{k(t)}{k(0)}=f(t), \]
whence it follows that
\[ p(r_0,t)=f(t)p_0(r_0)+\varphi(t). \tag{3} \]
Denoting \(\rho/\rho_0=\alpha(t)\) and introducing \(\alpha(t)\) and \(\rho_0(r_0)\) as new variables instead of \(t\) and \(r_0\), we rewrite equality (3) as follows:
\[ \begin{aligned} p_1(\rho,S)&=p_1(\alpha\rho_0,\ S(\rho_0))=f_1(\alpha)p_{01}(\rho_0)+\varphi_1(\alpha),\\ p_2(\rho,T)&=p_2(\alpha\rho_0,\ T(\alpha))=f_2(\alpha)p_{02}(\rho_0)+\varphi_2(\alpha). \end{aligned} \tag{4} \]
Introducing the inverse functions \(\rho_0(S)\) and \(\alpha(T)\), we finally obtain the equations of state in the following form:
\[ \begin{aligned} p_1(\rho,S)&=f_1\!\left(\frac{\rho}{\rho_0(S)}\right)p_{01}(\rho_0(S))+\varphi_1\!\left(\frac{\rho}{\rho_0(S)}\right),\\ p_2(\rho,T)&=f_2(T)p_{02}\!\left(\frac{\rho}{\alpha(T)}\right)+\varphi_2(T), \end{aligned} \tag{5} \]
where \(\rho_0(S)\) and \(\alpha(T)\) are prescribed, while \(f\), \(p_0\), and \(\varphi\) are arbitrary functions.
Now let us find, among these equations, those equations of state which admit one-dimensional motions with homogeneous deformation for arbitrary functions \(\rho_0(S)\) and \(\alpha(T)\). To this end, let us first consider the case \(S(\rho_0)=\mathrm{const.}\) and \(T(\alpha)=\mathrm{const.}\) Then \(p\) in equalities (4) does not depend on the second arguments. Applying to both sides of these equalities the operator
\(\dfrac{\partial^2}{\partial\rho_0\partial\alpha}\ln\dfrac{1}{\alpha}\dfrac{\partial}{\partial\rho_0}\), we obtain:
\[ \left(\frac{p''}{p'}\right)'\rho+\frac{p''}{p'}=0, \]
where the prime denotes differentiation with respect to \(\rho\). After integration these equalities take the form:
\[ \begin{aligned} p_1(\rho,S)&=A_1(S)\rho^{\gamma_1}+B_1,\\ p_2(\rho,T)&=A_2(T)\rho^{\gamma_2}+B_2(T), \end{aligned} \tag{6} \]
where \(A_1\), \(\gamma_1\), and \(B_1\), generally speaking, are functions of \(S\), and \(A_2\), \(\gamma_2\), and \(B_2\) are functions of \(T\). But comparing these expressions with equalities (4), we note that \(\gamma_1=\mathrm{const.}\), \(\gamma_2=\mathrm{const.}\), \(B_1=\mathrm{const.}\) If the deformation is homogeneous, then by direct verification it is easy to see that, under these conditions, equalities (6) indeed satisfy condition (3) for arbitrary functions \(S(\rho_0)\) and \(T(\alpha)\).
Moscow State University
named after M. V. Lomonosov
Received
7 I 1958
References
\(^{1}\) L. I. Sedov, DAN, 90, No. 5 (1953).
\(^{2}\) M. L. Lidov, DAN, 97, No. 3 (1954).
\(^{3}\) A. G. Kulikovskii, DAN, 114, No. 5 (1957).
\(^{4}\) I. M. Yavorskaya, DAN, 114, No. 5 (1957).
\(^{5}\) J. B. Keller, Quart. Appl. Math., 14, No. 2 (1956).