Abstract
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Reports of the Academy of Sciences of the USSR
1958, Volume 123, No. 1
HYDROMECHANICS
F. I. FRANKL’
POTENTIAL STEADY RELATIVISTIC GAS FLOWS
(Presented by Academician L. I. Sedov, 14 VI 1958)
One-dimensional steady relativistic gas flows have recently been considered by K. P. Stanyukovich \((^{1})\) *.
In the present note, the general case of a spatial steady potential relativistic flow is considered and, in particular, the especially simple case of an ultrarelativistic gas.
In the case of a barotropic flow without friction and thermal conductivity, flows are possible in which the so-called pseudovelocity possesses a potential \((^{2,3})\). By pseudovelocity we mean the quantity
\[ v^i=\frac{w}{\rho}u^i=Ju^i, \tag{1} \]
where \(u^i\) is the relativistic 4-velocity; \(\rho\) is the rest mass-energy per unit proper volume; \(w\) is the relativistic heat function per unit proper volume. The quantity \(J\), consequently, is the heat function per unit rest energy-mass, so that \(J \geqslant 1\).
In the case of potential flow we have
\[ v_i=\frac{\partial \varphi}{\partial x^i}. \tag{2} \]
The pseudovelocity potential satisfies the differential equation
\[ \left[g^{ik}+(\bar a^{-2}-1)u^i u^k\right]\frac{\partial^2\varphi}{\partial x^i \partial x^k}=0 \ **, \tag{3} \]
where \(\bar a\) is the relativistic speed of sound (the speed of light is here taken as unity). Owing to the constancy of the specific entropy, the quantity \(\bar a\) is a function of the single quantity \(J\), which is determined by the equation
\[ J^2=g_{ik}v^i v^k. \tag{4} \]
In the case of a steady gas flow we have
\[ \frac{\partial v_i}{\partial x^0}=0 \quad \text{or} \quad \frac{\partial v_0}{\partial x^i}=0 \quad (i=0,1,2,3). \]
* The principal results of this work were obtained independently by me and A. A. Aryanov and were reported at the scientific conference of Kabardino-Balkarian State University on 23 II 1958. The case of plane-parallel steady potential flow was considered as early as 1954—55 in work carried out under my supervision by S. V. Danov. Unfortunately, S. V. Danov did not publish these results.
** In cases of general relativity and curvilinear coordinates, instead of \(\partial^2\varphi/\partial x^i\partial x^k\) one must use the corresponding absolute derivative.
In other words,
\[ v_0=\mathrm{const}. \tag{5} \]
This is the relativistic Bernoulli equation \((^4)\). The ordinary velocity, i.e. the 3-vector \(\hat u_i=dx^i/dx^0\), is then determined by the equation
\[ \hat u_i=-\frac{v_i}{v_0}\quad (i=1,2,3). \tag{6} \]
Thus, the ordinary velocity \(\hat u_i\) in the flows under consideration has a potential, which in what follows we shall denote by \(\hat\varphi\).
We shall henceforth restrict ourselves to a Lorentz coordinate system \((g_{00}=-g_{11}=-g_{22}=-g_{33}=1,\ g_{ik}=0\ \text{for } i\ne k)\). Then equation (3) takes the form
\[ \sum_{i=1}^{3}\left[(1-\hat u^2)-(\bar a^{-2}-1)\hat u_i^2\right] \frac{\partial^2\hat\varphi}{\partial x_i^2} -(\bar a^{-2}-1) \sum_{i,k=1(i\ne k)}^{3} \hat u_i\hat u_k \frac{\partial^2\hat\varphi}{\partial x_i\partial x_k} =0, \tag{7} \]
where
\[ \hat u_i=\frac{\partial\hat\varphi}{\partial x^i},\qquad \hat u^2=\hat u_1^2+\hat u_2^2+\hat u_3^2. \tag{7a} \]
In the case of an ultrarelativistic gas, where \(\bar a^2=1/3\), we have
\[ \sum_{i=1}^{3}\left[\frac12(1-\hat u^2)-\hat u_i^2\right] \frac{\partial^2\hat\varphi}{\partial x_i^2} - \sum_{i,k=1(i\ne k)}^{3} \hat u_i\hat u_k \frac{\partial^2\hat\varphi}{\partial x_i\partial x_k} =0. \tag{8} \]
This equation coincides completely with the corresponding equation for a classical ideal gas if the ratio of specific heats is taken to be \(\varkappa=2\). Indeed, the equation mentioned has the form
\[ \sum_{i=1}^{3}\left[\frac{\varkappa+1}{2}(1-\hat u^2)-\hat u_i^2\right] \frac{\partial^2\hat\varphi}{\partial x_i^2} - \sum_{i,k=1(i\ne k)}^{3} \hat u_i\hat u_k \frac{\partial^2\hat\varphi}{\partial x_i\partial x_k} =0, \]
* if the speed of outflow into vacuum is taken as unity.
The present result remains valid in the case of a photon gas, for which \(\rho=0\). In this case it can be obtained directly from the energy–momentum equations
\[ \frac{\partial T^{ik}}{\partial x^k}=0, \]
where \(T^{ik}=\dfrac{e}{3}(4u^iu^k-g^{ik})\) (\(e\) is the internal energy per unit proper volume).
In the case of plane-parallel flow it follows from this that Chaplygin’s theory of gas jets \((^5)\) can be applied to an ultrarelativistic and photon gas, taking \(\varkappa=2\).
Kabardino-Balkarian
State University
Received
14 VI 1958
CITED LITERATURE
- K. P. Stanyukovich, DAN, 119, No. 2, 251 (1958).
- I. M. Khalatnikov, ZhETF, 27, 529 (1954).
- F. I. Frankl, ZhETF, 31, 490 (1956).
- L. D. Landau, E. M. Lifshitz, Mechanics of Continuous Media, Ch. 15, 1954.
- S. A. Chaplygin, On Gas Jets, Collected Works, 2, 1948.