Reports of the Academy of Sciences of the USSR

1. 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).