Traveling Magnetic Field in a Cylindrical Channel under Laminar Flow of a Conducting Fluid
Abstract Generated abstract
This paper analyzes the distribution of a traveling electromagnetic field in a cylindrical channel carrying a laminar conducting liquid, with no internal ferromagnetic core and a parabolic axial velocity profile. Starting from the vector-potential equation for the moving conducting medium, the author transforms the governing differential equation and expresses its solution in terms of degenerate hypergeometric functions, with constants fixed by magnetic boundary conditions at the channel axis and wall. The resulting formulas give the normal and tangential components of magnetic induction in the channel and are shown, in the short-circuit limiting case, to reduce to previously known expressions involving modified Bessel functions.
Full Text
UDC 538.4
TRAVELING MAGNETIC FIELD IN A CYLINDRICAL CHANNEL UNDER LAMINAR FLOW OF A CONDUCTING LIQUID
Yu. B. Volynskii
(Presented by the scientific seminar of the Faculty of Physics and Technology)
The problem of the distribution of a traveling electromagnetic field in a cylindrical channel under laminar flow of a conducting liquid is considered. It is shown that, for a parabolic velocity profile, a solution may be obtained that contains degenerate hypergeometric functions and that, in the short-circuit regime, can be reduced to the known solution obtained using modified Bessel functions. Bibliography: 6.
A solution has been obtained for the problem of the distribution of electromagnetic fields in a conducting liquid moving in a plane channel with velocity distributions specified by a power law [1]; however, it is also of interest to study the distribution of electromagnetic fields in cylindrical channels without internal ferromagnetic cores, where, because of the small “penetration depth” of the current, the principal energy losses occur in an annular layer of liquid metal adjacent to the inner surface of the pipeline, and the velocity distribution in this layer may differ substantially from the mean flow velocity of the liquid.
Let us consider the problem of the distribution of electromagnetic fields in a cylindrical channel situated in a traveling magnetic field, under the condition that there is no internal ferromagnetic core, that the velocity of the liquid metal has only one component directed along the axis of the channel, and that the velocity distribution is described by the expression
\[ v = v_{\mathrm{cp}} \cdot 2 \left( 1 - \frac{r^{2}}{R^{2}} \right) = 4 \tau f (1 - S_{\mathrm{cp}})\cdot \left( 1 - \frac{r^{2}}{R^{2}} \right) \tag{1} \]
and does not depend on the magnitude of the induction in the channel. Such an assumption may be valid for small values of the Hartmann numbers [2].
In expression (1)
\(R\) — inner radius of the channel,
\(S_{\mathrm{cp}}\) — mean slip,
\(\tau\) — pole pitch,
\(f\) — frequency.
If the thickness of the channel walls is neglected, then the equation for the vector-potential function [3], taking expression (1) into account, will have the following form:
\[ x^{2}\frac{d^{2}\dot{A}_{m}}{dx^{2}} + x\frac{d\dot{A}_{m}}{dx} - \left[ \dot{m}x^{4} + \dot{n}x^{2} + 1 \right]\dot{A}_{m} = 0, \tag{2} \]
where \(A_m\) is the vector-potential function,
\[ x=\frac{r}{R}, \]
\[ \dot m=f^{2}\varepsilon(1-S_{\mathrm{cp}})\frac{\pi^{2}R^{2}}{\tau^{2}}, \]
\[ \dot n=[1-j\varepsilon(1-2S_{\mathrm{cp}})]\frac{\pi^{2}R^{2}}{\tau^{2}}, \]
\[ \varepsilon=\frac{2\pi f\sigma\mu_{0}}{\pi^{2}}\tau^{2}, \]
\[
\sigma \text{ is the electrical conductivity,}
\]
\[
\mu_{0} \text{ is the magnetic permeability.}
\]
The substitutions \(\dot A_m=U^{-1/2}V(U)\), \(U=x^{2}\), lead to the equation
\[ \dot V''-\left(\frac{\dot m}{4}+\frac{\dot n}{4}U^{-1}\right)\dot V=0. \tag{3} \]
One of the solutions of this equation can be obtained in the form [4]
\[ \dot V=\dot C_{1}\sqrt{\dot m U}\,e^{-\frac{\sqrt{\dot m}}{2}U}\, F\left(1+\frac{\dot n}{4\sqrt{\dot m}},\sqrt{\dot m}\,U,2\right), \tag{4} \]
where
\[ F=1+\sum_{k=1}^{\infty} \frac{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \left(2+\frac{\dot n}{4\sqrt{\dot m}}\right)\cdots \left(1+k-1+\frac{\dot n}{4\sqrt{\dot m}}\right) (\sqrt{\dot m}U)^{k} }{ 2\cdot3\cdot4\cdots(2+k-1)\cdot k! }, \tag{5} \]
is a degenerate hypergeometric function existing for all values of \(\sqrt{\dot m}U\).
The general solution of equation (2) will have the form
\[ \dot A_m=Y(r)\left[\dot C_{1}+\dot C_{2}\int\frac{dr}{Y^{2}(r)}\right], \tag{6} \]
where
\[ Y(r)=r\cdot e^{-\frac{\sqrt{\dot m}}{2}U}\cdot F. \tag{7} \]
The integration constants \(\dot C_{1}\) and \(\dot C_{2}\) are determined with allowance for the boundary conditions
\(B_{mr}\big|_{r=0}=0\), \(B_{mz}\big|_{r=R}=\mu_{0}\dot\delta_{m}\), where \(\dot\delta_{m}\) is the linear current load.
\[ \dot C_{1}= \frac{\mu_{0}\dot\delta_{m}} { e^{-\frac{\sqrt{\dot m}}{2}} \left\{ 2+\sum_{k=1}^{\infty} \frac{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \left(2+\frac{\dot n}{4\sqrt{\dot m}}\right)\cdots \left(k-1+\frac{\dot n}{4\sqrt{\dot m}}\right) }{ [k!]^{2} } + \frac{ \frac{\dot n}{4\sqrt{\dot m}} \left(k+\frac{\dot n}{2\sqrt{\dot m}}\right) \dot m^{\frac{k}{2}} }{ [k!]^{2} } \right\} }, \tag{8} \]
\[ \dot C_{2}=0. \tag{9} \]
Thus, for a parabolic velocity profile of the conducting liquid in a cylindrical channel without an internal ferromagnetic core, the amplitude of the normal component of the induction is equal to
\[ \dot B_{mr} = j\frac{\pi}{\tau} r e^{\frac{\sqrt{\dot m}}{2}\left(1-\frac{r^2}{R^2}\right)} \frac{ \mu_0 \dot\delta_m F\left(1+\frac{\dot n}{4\sqrt{\dot m}},\, \sqrt{\dot m}\frac{r^2}{R^2},\,2\right) }{ 2+\sum_{\kappa=1}^{\infty} \frac{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \left(2+\frac{\dot n}{4\sqrt{\dot m}}\right)\ldots \left(\kappa-1+\frac{\dot n}{4\sqrt{\dot m}}\right) }{ [\kappa!]^2 } \, \frac{1}{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \cdot \left(\kappa+\frac{\dot n}{2\sqrt{\dot m}}\right) \dot m^{\kappa/2} } } , \tag{10} \]
and of the tangential component,
\[ \dot B_{mz} = \mu_0 \dot\delta_m e^{\frac{\sqrt{\dot m}}{2}\left(1-\frac{r^2}{R^2}\right)} \left\{ 2+ \sum_{\kappa=1}^{\infty} \frac{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \ldots \left(\kappa-1+\frac{\dot n}{4\sqrt{\dot m}}\right) \left(\kappa+\frac{\dot n}{2\sqrt{\dot m}}\right) \dot m^{\kappa/2} r^{2\kappa} }{ [\kappa!]^2 R^{2\kappa} } \right. \]
\[ \left. \bigg/ \left[ 2+ \sum_{\kappa=1}^{\infty} \frac{ \left(1+\frac{\dot n}{4\sqrt{\dot m}}\right) \ldots \left(\kappa-1+\frac{\dot n}{4\sqrt{\dot m}}\right) }{ [\kappa!]^2 } \times \frac{ \left(\kappa+\frac{\dot n}{2\sqrt{\dot m}}\right) \dot m^{\kappa/2} r^{2\kappa} }{ [\kappa!]^2 R^{2\kappa} } \right] \right\} \frac{1}{ \left(\kappa+\frac{\dot n}{2\sqrt{\dot m}}\right)\dot m^{\kappa/2} }. \tag{11} \]
It is easy to show [5] that, in the short-circuit regime, for
\(S_{\mathrm{cr}}=1,\ \dot m=0,\ \dot n=\dfrac{\pi^2}{\tau^2}(1+j\varepsilon)\cdot R^2\), expressions (10–11) are transformed into the known expressions [6]:
\[ \dot B_{mr} = j\mu_0 \dot\delta_m \frac{ \dfrac{\pi}{\tau}\, I_1\!\left(\dfrac{\pi}{\tau}\sqrt{1+j\varepsilon}\cdot r\right) }{ \sqrt{ \dfrac{\pi^2}{\tau^2} + j2\pi f\sigma\mu_0\, I_0\!\left(\dfrac{\pi}{\tau}\sqrt{1+j\varepsilon}\cdot R\right) } } \tag{12} \]
and
\[ \dot B_{mz} = \mu_0 \dot\delta_m \frac{ I_0\!\left(\dfrac{\pi}{\tau}\sqrt{1+j\varepsilon}\cdot r\right) }{ I_0\!\left(\dfrac{\pi}{\tau}\sqrt{1+j\varepsilon}\cdot R\right) }. \tag{13} \]
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