A. I. MOROZOV, L. S. SOLOV'EV

ON SYMMETRIC FLOWS OF A CONDUCTING FLUID ACROSS A MAGNETIC FIELD

(Presented by Academician L. A. Artsimovich on 6 IX 1963)

In the present work, plane and axially symmetric flows of a conducting compressible fluid are considered. A system of equations describing flows along equipotentials of the electric field is obtained. A class of solutions of these equations is considered, as well as solutions for flows in narrow channels.

For simplicity we shall assume that the fluid is inviscid and non-heat-conducting; then the stationary flow is described by the following equations of magnetohydrodynamics \((^1)\):

\[ \rho(\mathbf{v}\nabla)\mathbf{v}=-\nabla p+[\mathbf{j}\mathbf{H}], \tag{1} \]

\[ \operatorname{div}\rho\mathbf{v}=0,\qquad \operatorname{div}\mathbf{H}=0, \tag{2} \]

\[ \operatorname{rot}\vec{\varepsilon}=0,\qquad \rho T\mathbf{v}\nabla S=\nu_m j^2, \tag{3} \]

where

\[ \vec{\varepsilon}\equiv \frac{c}{\sqrt{4\pi}}\mathbf{E} =\nu_m\mathbf{j}-[\mathbf{v}\mathbf{H}]=\nabla\Phi,\qquad \mathbf{j}\equiv \operatorname{rot}\mathbf{H},\qquad \nu_m=\frac{c^2}{4\pi\sigma}, \]

\(\rho\) is the density, \(p\) the pressure, \(\sigma\) the conductivity, \(T\) the temperature, \(S\) the entropy, \(\mathbf{v}\) the velocity, \(\mathbf{B}\equiv \sqrt{4\pi}\mathbf{H}\) the magnetic field, and \(\mathbf{E}\) the electric field. From (1)—(3) there follows the equation

\[ \operatorname{div}\mathbf{q}=0,\qquad \mathbf{q}=\rho\mathbf{v}\left(W+\frac{v^2}{2}\right)+[\vec{\varepsilon}\mathbf{H}], \tag{4} \]

which expresses conservation of the energy flux \(\mathbf{q}\). Here \(W\) is the enthalpy, related to \(p\) and \(S\) by the thermodynamic relation

\[ dW=\frac{dp}{\rho}+T\,dS. \tag{5} \]

Let us consider flow in an axially symmetric channel across an azimuthal magnetic field: \(H=H_\varphi,\ v_\varphi=0\). Introduce, for \(\rho\mathbf{v}\) and \(\mathbf{j}\), the stream functions \(\psi\) and \(I=rH_\varphi\) according to the relations

\[ \binom{\rho v_z}{j_z} = \frac{1}{r}\frac{\partial}{\partial r} \binom{\psi}{I}, \qquad \binom{\rho v_r}{j_r} = -\frac{1}{r}\frac{\partial}{\partial z} \binom{\psi}{I}. \tag{6} \]

We shall further require that the surfaces \(\psi=\text{const.}\), on which the fluid streamlines lie, coincide with the equipotential surfaces \(\Phi=\text{const.}\), i.e. \(\Phi=\Phi(\psi)\). Then it follows from the first equation (3) that \((\nabla I\nabla\psi)=0\), i.e. the fluid streamlines \(\psi=\text{const}\) and the electric-current lines \(I=\text{const}\) form orthogonal families of trajectories.

For the axially symmetric flows under consideration we obtain the following system of equations:

\[ \frac{1}{\rho r}\frac{\partial}{\partial r}\frac{1}{\rho r}\frac{\partial\psi}{\partial r} + \frac{1}{\rho r}\frac{\partial}{\partial z}\frac{1}{\rho r}\frac{\partial\psi}{\partial z} + I\Phi''(\psi) + T\frac{\nabla S\nabla\psi}{\rho^2 r^2 v^2} - U'(\psi)=0, \tag{7a} \]

\[ W+\frac{v^2}{2}+I\Phi'(\psi)=U(\psi), \tag{7b} \]

\[ \frac{I}{\rho r^2} - \nu_m\frac{\mathbf{v}\nabla I}{\rho r^2 v^2} = \Phi'(\psi), \tag{7c} \]

\[ \rho T\mathbf{v}\nabla S=\frac{\nu_m}{r^2}(\nabla I)^2. \tag{7d} \]

The Bernoulli equation (7b) follows from equation (4), (7c) follows from the equality \(\vec{\varepsilon}=-\nabla\Phi\), and (7a) follows from the Euler equation (1). In system (7) the unknowns are \(\psi,\rho\), and \(I\), while \(\Phi(\psi)\) and \(W(\psi)\) are arbitrary functions of \(\psi\). If \(S\) is regarded as a function of \(\psi\) and \(I\), then the term in (7a) proportional to \(T\) may be written in the form \(T\partial S/\partial\psi\). Since equations (7) determine only the derivative \(\partial S/\partial I\) along the streamlines \(\psi=\mathrm{const}\), \(S\) as a function of \(\psi\) may be prescribed arbitrarily and, in particular, one may set \(\partial S/\partial\psi=0\). For flows of an ideally conducting fluid \((\nu_m=0)\), system (7) reduces to that obtained in [2].

Similarly, plane flows independent of \(z\), with \(H=H_z,\ v_z=0\), are described by the system of equations

\[ \frac{1}{\rho}\frac{\partial}{\partial x}\frac{1}{\rho}\frac{\partial\psi}{\partial x} +\frac{1}{\rho}\frac{\partial}{\partial y}\frac{1}{\rho}\frac{\partial\psi}{\partial y} +H\Phi''(\psi)+T\frac{\nabla S\nabla\psi}{\rho^2 v^2}-U'(\psi)=0, \tag{8a} \]

\[ W+\frac{v^2}{2}+H\Phi'(\psi)=U(\psi), \tag{8b} \]

\[ \frac{H}{\rho}-\nu_m\frac{\mathbf{v}\nabla H}{\rho v^2}=\Phi'(\psi), \tag{8c} \]

\[ \rho T\mathbf{v}\nabla S=\nu_m(\nabla H)^2, \tag{8d} \]

where the stream function \(\psi\) is introduced by the relations \(\rho v_x=\partial\psi/\partial y,\ \rho v_y=-\partial\psi/\partial x\), and the components \(\mathbf{j}\) are determined in terms of \(H\) by the equalities \(j_x=\partial H/\partial y,\ j_y=-\partial H/\partial x\). Equation (8d), taking (8b) and (8c) into account, can be transformed to the form

\[ \rho\mathbf{v}\nabla\frac{v^2}{2} +\mathbf{v}\nabla\left(p+\frac{H^2}{2}\right)=0. \tag{8e} \]

We shall seek a solution of (8), prescribing \(\rho\) as a known function of the velocity, \(\rho=F'(v^2)\); then, according to (8e), we have

\[ \frac{F}{2}+p+\frac{H^2}{2}=G(\psi). \tag{9} \]

Eliminating from (8b) and (9) the pressure \(p=\varkappa\rho W\), where \(\varkappa=(\gamma-1)/\gamma\), we find \(H=H(v^2)\):

\[ H=\varkappa\Phi'F'\pm\sqrt{(\varkappa\Phi'F')^2-F+2G+\varkappa(v^2-2U)F'}. \tag{10} \]

From equation (8c) we obtain the quadrature determining the dependence of \(v\) on the arc length \(s\) along a streamline:

\[ s=2\nu_m\int\frac{H'(v^2)\,dv}{F(v^2)-F'(v^2)\Phi'(\psi)}. \tag{11} \]

To find the streamlines in the general case, it is necessary to solve equation (8a). In the simplest case \(\Phi''=0,\ U'=0,\ \partial S/\partial\psi=0\), and \(\rho=\mathrm{const}\), the problem reduces to solving the Laplace equation \(\Delta\psi=0\) with the normal derivative \(\partial\psi/\partial y\) prescribed by relation (11) at \(y=0\). If one restricts oneself to the approximation in which the parameters of the flow vary slowly along the \(x\)-axis, then, neglecting terms \(\sim(\partial\psi/\partial x)^2\) and \(\partial^2\psi/\partial x^2\), in the indicated case \(\Phi''=U'=\partial S/\partial\psi=0\), from (8a) we obtain \(\psi\simeq F'(v^2)vy\), where the velocity \(v(x)\) may be approximately taken equal to \(v(s)\), determined by equation (11).

If the distance \(f\) between the electrodes is small in comparison with the characteristic dimensions, then equations (7) reduce to the system

\[ \rho r f v=\alpha=\mathrm{const}, \tag{12a} \]

\[ \frac{I}{\rho r^2}-\frac{\nu_m}{\rho r^2 v}\frac{dI}{ds}=\beta=\mathrm{const}, \tag{12b} \]

\[ W+\frac{v^{2}}{2}+\beta I=U=\text{const}, \tag{12в} \]

\[ \rho vT\,\frac{dS}{ds}=\frac{\nu_m}{r^{2}}\left(\frac{dI}{ds}\right)^2 . \tag{12г} \]

Here \(ds\) is an element of the arc of the middle line \(r=r(z)\) (see Fig. 1).

In the case under consideration of a narrow gap between ideally conducting electrodes \(r_1(z)\) and \(r_2(z)\), the requirement \(\Phi=\Phi(\psi)\) is not a restriction, and equations (12) may be obtained directly from (1)—(4). Indeed, the first of equations (3), written as an integral around the contour \(MNN'M'\), gives \(\oint \vec{\varepsilon}\,dl=0\), whence we obtain \(\varepsilon_{\perp} f=\text{const}\), i.e., equation (12б). The continuity equation (2) in integral form \(\oint \rho v\,dS=0\), where the integral is taken over the surface of the toroidal region bounded by the contour \(MNN'M'\), leads to equation (12a). Similarly, equation (4) gives \(q_{\parallel}rf=\text{const}\), since the energy flux \(q\) through the side walls is zero; hence, taking (12б) into account, we obtain equation (12в).

Fig. 1

Adding to system (12) the equation of state \(p=p(\rho,T)\), we obtain 5 equations for 7 unknowns: \(\rho,p,T,v,r,f,I\); consequently, two of these quantities may be prescribed arbitrarily. Let us replace equation (12г) by the momentum equation

\[ \rho v\,dv+\frac{1}{2r^{2}}\,dI^{2}+dp=0 \tag{13} \]

and prescribe the functions \(\rho\) and \(r\) in the form \(\rho=F'(v^{2})\), \(r^{-2}=R'(I^{2})\); then from (13) we obtain \(F/2+R/2+p=G=\text{const}\). Eliminating from (12в) the enthalpy \(W=p/\varkappa\rho\) with the aid of the last relation, we obtain

\[ \varkappa F'\left(\frac{v^{2}}{2}+\beta I-U\right)-\left(\frac{F}{2}+\frac{R}{2}-G\right)=0 . \tag{14} \]

Relation (14) determines \(I=I(v^{2})\). The expression for \(v=v(s)\) is found by integrating (12б):

\[ s=2\nu_m\int \frac{I'(v^{2})\,dv}{I-\beta F'/R'} . \]

Differentiating (12) and eliminating \(dp\) and \(ds\), we obtain an equation of the Hugoniot type

\[ \left(v^{2}-c_T^{2}-\frac{\beta^{*}I}{1-\nu_m I''/vI'}\right)\frac{dv}{v} = c_T^{2}\frac{d(rf)}{rf} + \frac{\rho r^{2}\beta\beta^{*}}{1-\nu_m I''/vI'}\, \frac{d(f/r)}{f/r}, \tag{15} \]

where \(c_T^{2}=\rho(\partial W/\partial\rho)_S\) is the square of the speed of sound, \(\beta^{*}=\beta+\gamma\nu_m I'/\rho r^{2}v\). It follows from (15) that, in the acceleration of a cold plasma \(p\ll H^{2}/2\), the determining quantity is \(f/r\), while for the opposite sign of the inequality it is the quantity \(rf\).

Plane flows in a narrow channel are described by the system of equations

\[ \rho vf=\alpha=\text{const}, \tag{16a} \]

\[ \frac{H}{\rho}-\frac{\nu_m}{\rho v}\frac{dH}{ds}=\beta=\text{const}, \tag{16б} \]

\[ W+\frac{v^{2}}{2}+\beta H=U=\text{const}, \tag{16в} \]

\[ \rho vT\,\frac{dS}{ds}=\nu_m\left(\frac{dH}{ds}\right)^2 . \tag{16г} \]

These equations are analogous to (12), but the number of unknowns here is smaller by one. As before, set \(\rho = F'(v^2)\); then the momentum equation \(\rho v\,dv + d(p + H^2/2)=0\) gives \(F + 2p + H^2 = 2G\), and, eliminating \(p\) from this, with the aid of (16b), we find \(H\) as a function of \(v^2\):

\[ H=\chi\beta F' \pm \sqrt{(\chi\beta F')^2 - F + 2G + \chi (v^2 - 2U)F'} . \tag{17} \]

Let the middle plane of the channel be \(y=0\); then \(v\) as a function of \(x\) is determined by the integral
\[ x=2\nu_m \int \frac{H'(v^2)\,dv}{H-\beta F'(v^2)} . \]
For the one-dimensional problem, an analogous solution is given in the work \({}^{(3)}\).

In conclusion, we give the Hugoniot equation for plane flow, which is obtained from (16):

\[ \left( v^2 - c_T^2 - \frac{\beta^* H}{1-\nu_m H''/vH'} \right)\frac{dv}{v} = \left( c_T^2+ \frac{\rho\beta\beta^*}{1-\nu_m H''/vH'} \right)\frac{df}{f}. \tag{18} \]

Here \(\beta^*=\beta+\gamma\nu_m H'/\rho v\). Equations (18) and (15) show that, for finite conductivity \((\nu_m\ne 0)\), the velocity at which reversal of interactions occurs depends on the derivatives of the magnetic field along the velocity, \(H'\) and \(H''\).

Received
23 VII 1963

References Cited

\({}^{1}\) L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Moscow, 1957.
\({}^{2}\) A. I. Morozov, L. S. Solov’ev, DAN, 149, No. 3 (1963).
\({}^{3}\) A. G. Kulikovskii and G. A. Lyubimov, Magnetic Hydrodynamics, Moscow, 1962.