A Periodically Reversed Flow Driven by a Modulated Traveling Magnetic Field: Part II. Theoretical Model

Abstract

A flow of liquid GaInSn, which has been investigated experimentally in a companion article,^[1] is modeled analytically in this article. The role of this electromagnetically driven flow is to transport the solute rejected by the solidifying interface at significant distances in the melt, and to periodically reverse its circulation such that macrosegregation is minimized. First, we present an analytical derivation of the electromagnetic force, exhibiting its space and time variations, which are essentially legitimate in the central part of the fluid domain. A model for the recirculating fluid flow is also derived, assuming that in this central region the flow is quasiparallel to the main axis. The narrowness of the fluid domain must be taken into account, to achieve a satisfactory agreement with the measurements.

Appendix

To solve the second-order differential equation

$$ \frac{{M^{\prime \prime} {\left( z \right)}}} {{M{\left( z \right)}}} = {\left( {k^{2} - i\mu \sigma \omega } \right)} $$

(A1)

let us denote r ²=k ² − iμσω, while r = α ± iβ is a complex number, the real and imaginary parts of which satisfy

$$ \left\{ \begin{aligned}{} & \alpha ^{2} - \beta ^{2} = k^{2} , \\ & 2\alpha \beta = - \mu \sigma \omega \\ \end{aligned} \right. $$

(A2)

The two roots are such that the general solution of Eq. [A1] is

$$ M{\left( z \right)} = C_{1} \exp {\left[ {{\left( {\alpha - i\beta } \right)}z} \right]} + C_{2} \exp {\left[ { - {\left( {\alpha - i\beta } \right)}z} \right]} $$

(A3)

To get a zero value of M(z → ∞), we must chose C ₁ = 0; we get

$$ M{\left( z \right)} = \exp {\left[ { - \alpha z + i\beta z} \right]} $$

(A4)

The boundary condition at the bottom wall (j _z (z = 0) = 0) is satisfied, because the real part of M(z) cancels as sin (βz).

With the usual definition $\delta = {\sqrt {\frac{2} {{\mu \sigma \omega }}} }$, ω = 2πf, the solution of [A2] can be written as

$$ \alpha = \frac{k} {{{\sqrt 2 }}}{\left( {1 + {\sqrt {1 + \frac{4} {{k^{4} \delta ^{4} }}} }} \right)}^{{\frac{1} {2}}} ,\quad \beta = - \frac{{\mu \sigma \omega }} {{k{\sqrt 2 }}}{\left( {1 + {\sqrt {1 + \frac{4} {{k^{4} \delta ^{4} }}} }} \right)}^{{ - \frac{1} {2}}} $$

(A5)

In the case f = 50 Hz, $\frac{4} {{k^{4} \delta ^{4} }} \ll 1$, so that we can take the approximate values

$$ \alpha \approx k,\quad \beta = - \frac{{\mu \sigma \omega }} {{2k}} $$

(A6)

This means that, under the conditions of this experiment, the skin depth α⁻¹ is not determined by the frequency but by the pole pitch.

The same technique applies to the case of a narrow liquid domain, with the new equation

$$ \frac{{g^{\prime \prime} {\left( z \right)}}} {{g{\left( z \right)}}} = k^{2} + {\left( {\frac{\pi } {{2a}}} \right)}^{2} - i\mu \sigma \omega $$

(A7)

In order to keep this algebra valid, we simply have to introduce $k^{{ * 2}} = k^{2} + {\left( {\frac{\pi } {{2a}}} \right)}^{2}$ and

$$ \alpha ^{*} = \frac{{k^{*} }} {{{\sqrt 2 }}}{\left( {1 + {\sqrt {1 + \frac{4} {{k^{{*4}} \delta ^{4} }}} }} \right)}^{{\frac{1} {2}}} ,\quad \beta ^{*} = - \frac{{\mu \sigma \omega }} {{k^{*} {\sqrt 2 }}}{\left( {1 + {\sqrt {1 + \frac{4} {{k^{{*4}} \delta ^{4} }}} }} \right)}^{{ - \frac{1} {2}}} $$

(A8)

to make this algebra still valid. The approximation $\frac{4} {{k^{{*4}} \delta ^{4} }} \ll 1$ is still justified, so that the expressions in Eq. [A6] can be generalized as

$$ \alpha ^{*} \approx k^{*} ,\quad \beta ^{*} = - \frac{{\mu \sigma \omega }} {{2k^{*} }} $$

(A9)