Generation of interfacial waves by rotating magnetic fields

Interfacial waves arising in a two-phase swirling flow driven by a low-frequency rotating magnetic field (RMF) are studied. At low RMF frequencies, on the order of 1–$10\mathrm{Hz}$, the oscillatory part of the induced Lorenz force becomes comparable to the time-averaged one and cannot be neglected. In particular, when free surfaces or two-liquid stably stratified systems are subject to a low-frequency RMF, induced pressure variations necessarily excite free-surface/interfacial waves, which can improve mass transfer in different metallurgical processes. In this paper we formulate a linear wave model and derive explicit analytical solutions predicting RMF-driven wave patterns that closely resemble hyperbolic paraboloids. These theoretical predictions are validated against experiments based on a nonintrusive acoustic measurement technique, which measures liquid-liquid interface elevations in a two-phase KOH-GaInSn stably stratified system. A good quantitative agreement is found for nonresonant wave responses in the vicinity of the fundamental resonance frequency. The experiments reveal the additional excitation of several higher harmonics superimposing the fundamental wave oscillation, which are visible even in the linear wave regime.

Figure 1

Sketch of the theoretical setup. An upright cylindrical container of radius $R$ is permeated by an external homogeneous magnetic field $\mathit{B}$ rotating horizontally around the $z$ axis with constant angular frequency $\mathit{\Omega }=\mathrm{\Omega }{\mathit{e}}_z$. The container is filled with two immiscible liquids $i=1,2$ of densities ${\rho }_i$, kinematic viscosities ${\nu }_i$, electrical conductivities ${\sigma }_i$, and layer heights $h_i$, which are stably stratified due to gravity $\mathit{g}$. The origin of the cylindrical coordinate system in placed in the center of the interface $z=\eta (r,\theta ,z)$. The blue arrows schematically show force lines of the conservative part of the induced Lorentz force.

Figure 2

Maximum wave elevation at the tank wall ${\eta }_0=\eta (r=R)$ normalized by the container radius $R$ and the Froude number Fr as a function of the excitation frequency $\mathrm{\Omega }$ normalized by the first natural eigenfrequency ${\omega }_{21}$. Additionally, normalized 3D visualizations of excited interface elevations are shown at three different points in the linear regime and close to the first and second resonance from two different perspectives.

Figure 3

Sketch and photographs of the experimental setup.

Figure 4

(a) Measured local interfacial echo distance $\eta$ of sensor 1 and fitted cosine function as a function of time $t$ for the applied RMF frequency $f=\mathrm{\Omega }/\left(2\pi \right)=1.81\mathrm{Hz}$ and magnetic field $B_0=4.5\mathrm{mT}$. (b) Lomb-Scargle spectrum corresponding to the measured echo signal.

Figure 5

3D visualization of a magnetically excited interface at four different times within one period $T$ on the basis of ten simultaneously applied ultrasound probes. The measured interface elevation $\eta /{\eta }_{\max }$ of all probes are coded in color. Surface fits of the wave mode (2,1) are included and color coded.

Figure 6

Measured wave amplitude ${\eta }_0$ at the sensor position $\eta (r=4.2\mathrm{mm})$ as a function of the RMF frequency $f=\mathrm{\Omega }/\left(2\pi \right)$ for an applied magnetic field of $B_0=4.6\mathrm{mT}$ and layer heights of $h_1=4\mathrm{cm}$ and $h_2=3.5\mathrm{cm}$. Snapshots of the metal surface are included to emphasize different observed wave regimes.

Figure 7

Measured peak frequencies $f$ and theoretical estimates as a function of the applied magnetic field $B_0$. The orange line shows corrected frequencies including swirling frequencies with the correction factor $\chi =0.15$, and the natural resonance frequency $f={\omega }_{21}/\left(4\pi \right)$ is shown by the dotted black line.

Figure 8

Wave amplitudes ${\eta }_0$ for different applied RMF intensities $B_0$ as a function of the RMF frequency $f$ measured in $h_1=3.5\mathrm{cm}, h_2=4\mathrm{cm}$ (a) and $h_1=3\mathrm{cm}, h_2=4.5\mathrm{cm}$ (b) stratifications. The dots mark individual measurements, and the solid lines represent the predicted resonance curves [Eq. (27)] with corrected resonance frequencies due to Eq. (34).

Figure 9

Measured peak amplitudes ${\eta }_0$ as a function of the applied magnetic field $B_0$. The orange line shows a square fit for all measurements accounting from the fourth value onwards with an coefficient of determination of $R^2=0.984$.