Stability of melt flow during magnetic sonication in a floating zone configuration

This paper considers the linear stability of a liquid metal flow driven by superimposed alternating and static axial magnetic fields in the floating zone configuration. A simple model is constructed that assumes a slight axial variation of the flow-driving radial magnetic force and a low-to-moderate skin effect. This force drives two symmetrical flow tori. Formation of the field-parallel layer is observed for a strong static field. In this regime the instability sets in as a standing azimuthal wave around the circumference of the cylindrical melt volume near its midplane. The length of this wave scales with the thickness of the parallel layer. The instability criterion may be formulated in terms of an interaction parameter reaching its critical value at around 525.

Figure 1

(a) Scheme of the model; (b) variation of the alternating magnetic field induction along the cylinder surface.

Figure 2

Dependency of the alternating magnetic field induced force on the dimensionless frequency. Function $f_0\left(S\right)$ represents the value of the fraction expression on the right-hand side of (8) at $r=r_0$. The maximum of 0.18873 is reached at $S=6.325$. Asymptotic solutions $S/16$ and $1/\sqrt{2S}$ are depicted by dotted and dashed lines, respectively.

Figure 3

Maximum (a) axial and (b) radial velocity vs the driving force at different static field strengths. Boxes, circles, and triangles correspond to $\text{Ha}=200,500,$ and 1000, respectively. Solid lines display the functions $0.048F{\text{Ha}}^{-3/2}$ and $0.3F{\text{Ha}}^{-2}$ in (a) and (b), respectively.

Figure 4

Maximum (a) axial and (b) radial velocity vs the interaction parameter $N=F{\text{Ha}}^{-5/2}$ at different $\text{Ha}$. Solid, dashed, dash-dotted, and dotted lines correspond to $\text{Ha}=1000$, 500, 200, and 100, respectively.

Figure 5

Scaled radial profiles of (a) axial velocity at $z=-0.5$; (b) radial velocity at $z=0$ (open symbols) or near the end wall ($z\approx 1$, filled symbols). Boxes, circles, and triangles correspond to ($\text{Ha}=200, F={10}^8$), ($\text{Ha}=500, F={10}^9$), and ($\text{Ha}=1000, F=5\times {10}^9$), respectively.

Figure 6

Streamlines of the basic flow in near-critical state for (a) $\text{Ha}=200, F=3.3\times {10}^8$; (b) $\text{Ha}=500, F=2.66\times {10}^9$; (c) $\text{Ha}=1000, F=1.48\times {10}^{10}$. The streamline spacing is scaled by ${\text{Ha}}^{1/2}$.

Figure 7

Dependency of the critical parameters on the static magnetic field strength: (a) The critical force; (b) oscillation frequency (triangles) and the maximum axial (circles) or radial (boxes) critical flow velocity. The line in (a) displays the function $525{\text{Ha}}^{5/2}$. Functions $18\text{Ha}$ and $120{\text{Ha}}^{1/2}$ are depicted by solid and dashed lines in (b). Numbers in (b) display the azimuthal wave number $m$ of the critical perturbation.

Figure 8

Dependency of the real part of the leading eigenmode on the perturbation azimuthal wave number $m$ near the instability onset for (a) low $\text{Ha}\le 40$ and (b) intermediate to high $\text{Ha}\ge 70$.

Figure 9

Boundary layer mode: Isolines of the velocity amplitude for the critical perturbation at (a) $\text{Ha}=200, F=3.61\times {10}^8, m=20$; (b) $\text{Ha}=500, F=3.0\times {10}^9, m=35$; (c) $\text{Ha}=1000, F=1.67\times {10}^{10},m=52$.

Figure 10

Isolines of amplitude of the critical velocity perturbation components for $\text{Ha}=1000, F=1.67\times {10}^{10}, m=52$. Radial, azimuthal, and axial velocity perturbation displayed in panels (a)–(c), respectively. Isoline spacing in panel (c) is five times larger than in panels (a) and (b).

Figure 11

Bulk mode: Isolines of amplitude the critical velocity perturbation components for $\text{Ha}=40, F=1.2\times {10}^7, m=4$. The radial, azimuthal and axial velocity amplitude is displayed in panels (a)–(c), respectively, with an equal isoline spacing.