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HYDROMECHANICS
S. V. IORDANSKII
THE ZEMPLEN THEOREM IN MAGNETOHYDRODYNAMICS
(Presented by Academician M. A. Lavrent’ev on 29 III 1958)
A number of works have been devoted to discontinuities of various kinds in a medium with infinite conductivity and negligibly small dissipation in the presence of a magnetic field (see the review \((^1)\)). Of special interest are discontinuities of the shock-wave type, in which matter passes through the surface of discontinuity and a change in the thermodynamic quantities occurs. In the book \((^2)\) it is shown that, for small magnitudes of such discontinuities, under the same conditions as for ordinary shock waves, only compression waves are possible. We shall prove this assertion for arbitrary magnitudes of the discontinuities.
Discontinuities of the shock-wave type are described by the equations \((^2)\):
\[ j^2(V_1H_1 - V_2H_2)=\frac{H_n^2}{4\pi}(H_1-H_2), \]
\[ j^2=\frac{8\pi(p_2-p_1)+H_2^2-H_1^2}{8\pi(V_1-V_2)}, \tag{1} \]
\[ \varepsilon(p_2,V_2)-\varepsilon(p_1,V_1)+(V_2-V_1)\frac{p_2+p_1}{2} +\frac{1}{16\pi}(H_2-H_1)^2(V_2-V_1)=0. \]
Here \(p, V, H, j\) are, respectively, the pressure, specific volume, tangential magnetic field, and flux of matter through the discontinuity; \(H_n\) is the normal component of the magnetic field; \(\varepsilon(p,V)\) is the internal energy per unit mass of matter. Index 1 refers to quantities before the jump, index 2 to quantities after the jump.
We shall consider only substances for which
\[ \left(\frac{\partial^2 p}{\partial V^2}\right)_S>0, \qquad \left(\frac{\partial p}{\partial S}\right)_V>0. \]
Equations (1) determine all possible types of shock waves in magnetohydrodynamics.
In order to obtain the curves \(p_2(V_2)\) of the type of Hugoniot curves, it is necessary to eliminate \(H_2\) from the last equation with the aid of the other two equations. Eliminating \(j^2\), we obtain
\[ \left(p_2-p_1+\frac{H_2^2-H_1^2}{8\pi}\right)(V_1H_1-V_2H_2) = \frac{H_n^2}{4\pi}(H_1-H_2)(V_1-V_2). \tag{2} \]
In what follows we shall assume that \(H_n\ne 0,\ H_1\ne 0\). There exist three solutions of equation (2), and it can be shown that for weak discontinuities these solutions are real and give three different Hugoniot curves \(p_2(V_2)\) passing through the point \(p_1, V_1\).
It can be shown, however, that for \(H_1\ne 0\) equation (2) has only one real root for sufficiently large \(p_2\). This can be verified by seeking the solution of (2) by means of two different iteration—
processes:
\[ H_{2k+1}=\frac{1}{V_2}\left[V_1H_1-\frac{2H_n^2(H_1-H_{2k})(V_1-V_2)}{8\pi(p_2-p_1)+H_{2k}^2-H_1^2}\right];\qquad H_{20}=\frac{V_1}{V_2}H_1; \tag{I} \]
\[ H_{2k+1}=\pm\sqrt{-8\pi(p_2-p_1)+H_1^2+ \frac{2H_n^2(H_1-H_{2k})(V_1-V_2)}{V_1H_1-V_2H_{2k}}}; \tag{II} \]
\[ H_{20}=\pm\sqrt{-8\pi p_2}. \]
It can be shown that these processes converge for sufficiently large \(p_2\). (We note that \(V_1H_1\) cannot be neglected in comparison with \(V_2H_2\), since this leads to incorrect conclusions near \(V_2=0\).)
According to (2), the entropy increases near \(p_2=p_1\) with increasing \(p_2\) along each of these three curves. We shall show that along any of these Hugoniot curves the entropy \(S_2\) increases monotonically and that always \(p_2>p_1\). The proof is analogous to the well-known proof in gas dynamics \((^3)\).
It is convenient to introduce the auxiliary quantities:
\[ p_2^*=p_2+\frac{H_2^2}{8\pi},\qquad p_1^*=p_1+\frac{H_1^2}{8\pi}. \]
To any Hugoniot curve \(p_2(V_2)\) there corresponds some curve \(p_2^*(V_2)\), according to the chosen branch \(H_2(p_2,V_2)\). The lines \(j^2=\mathrm{const}\) in the plane \(p_2^*, V_2\) will simply be straight lines issuing from the point \(p_1^*, V_1\).
Consider the Hugoniot function
\[ \mathscr{H}_+(p_2,V_2)=\varepsilon(p_2,V_2)-\varepsilon(p_1,V_1)+\frac{p_2+p_1}{2}(V_2-V_1)+ \]
\[ +\frac{1}{16\pi}(H_{2+}-H_1)^2(V_2-V_1). \]
Here the plus sign indicates the fact that we have chosen a definite branch \(H_+(p_2,V_2)\) as a solution of equation (2). Computing the differential \(\mathscr{H}_+\), we obtain:
\[ d\mathscr{H}_+=T_2\,dS_2+ \left[\frac{p_1^*-p_2^*}{2}\,dV_2+\frac{1}{2}(V_2-V_1)\,dp_{2+}^*\right]+ \]
\[ +\frac{1}{8\pi}(H_{2+}^2-H_1H_{2+})\,dV_2 -\frac{1}{8\pi}H_1(V_2-V_1)\,dH_{2+}. \tag{3} \]
From equations (1) we have the equalities:
\[ j_+^2=\frac{p_{2+}^*-p_1^*}{V_1-V_2};\qquad H_{2+}=H_1\,\frac{H_n^2-4\pi j_+^2 V_1}{H_n^2-4\pi j_+^2 V_2}. \tag{4} \]
Computing \(dj_+^2\) and \(dH_{2+}\) and substituting in (3), we obtain:
\[ d\mathscr{H}_+=T_2dS_2+ \left[1+\frac{H_1^2H_n^2}{(H_n^2-j_+^2V_2)^2}\right] \left(\frac{p_1^*-p_{2+}^*}{2}\,dV_2-\frac{V_2-V_1}{3}\,dp_{2+}^*\right). \tag{5} \]
Hence we conclude that along the Hugoniot curve \(dS_2\) does not change sign, provided only that the corresponding curve \(p_{2+}^*(V_2)\) nowhere touches the ray \(j^2=\mathrm{const}\).
Along the curve \(j_+^2(p_2,V_2)=\mathrm{const}\), \(d\mathscr{H}_+=T_2dS_2\), therefore \(S_2\) and \(\mathscr{H}_+\) have extrema at the same points. In addition, along this same curve
\[ \frac{dp_{2+}^*}{dV_2}=-j_+^2= \left(\frac{\partial p_2}{\partial V_2}\right)_S+ \left(\frac{\partial p_2}{\partial S_2}\right)_V\frac{dS_2}{dV_2} +\frac{1}{8\pi}\left(\frac{\partial H_{2+}^2}{\partial V_2}\right)_{j^2}. \]
At the point where \(dS_2/dV_2=0\), using (4), we obtain:
\[ \frac{d^2 p_2^*}{dV_2^2} = \left(\frac{\partial^2 p_2}{\partial V_2^2}\right)_S + \left(\frac{\partial p_2}{\partial S_2}\right)_V \frac{d^2 S_2}{dV_2^2} + \frac{12\pi H_2^2+j_+^4}{\left(H_2^2-4\pi j_+^2 V_2\right)^2} =0. \]
Using our assumptions about the function \(p(V,S)\), we conclude that the stationary point of \(S_2\) on the ray \(j_+^2=\mathrm{const}\) is a maximum. Since at the intersection points of the ray \(j_+^2=\mathrm{const}\) with the Hugoniot curve \(\mathcal H_+=0\), \(\mathcal H_+\) has an extremum between these points, it follows that \(S_2\) has a maximum at this same point. Therefore there is no more than one intersection point (not counting the point \(p_1,V_1\)), and \(dS_2/dV_2>0\) at any such point. This proves the impossibility of tangency of the Hugoniot curve by a ray.
Since for small discontinuities \(dS_2>0\), \(dj_+^2>0\), it follows from this that always \(dS_2>0\), \(dj_+^2>0\) along any Hugoniot curve. Hence it follows that \(p_2>p_1\) at any discontinuity (since \((\partial p/\partial S)_V>0\)).
Since we have shown that for very large \(p_2\) the two roots \(H_2(p_2,V_2)\) become imaginary, two of the Hugoniot curves must terminate at a finite \(p_2\) (under the assumption \(H_1\ne0\)).
Received
27 III 1958
REFERENCES CITED
- S. I. Syrovatskii, Usp. Fiz. Nauk, 62, no. 3 (1957).
- L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Moscow, 1957.
- R. Courant, K. Friedrichs, Supersonic Flow and Shock Waves, IL, 1950.