Vibrations of a diamagnetically levitated water droplet

We measure the frequencies of small-amplitude shape oscillations of a magnetically levitated water droplet. The droplet levitates in a magnetogravitational potential trap. The restoring forces of the trap, acting on the droplet’s surface in addition to the surface tension, increase the frequency of the oscillations. We derive the eigenfrequencies of the normal mode vibrations of a spherical droplet in the trap and compare them with our experimental measurements. We also consider the effect of the shape of the potential trap on the eigenfrequencies.

Figure 1

Bottom: magnetogravitational potential $U\left(x,z\right)$ of a water droplet levitated by the magnetic field $B\left(x,z\right)$ of a vertical-bore superconducting solenoid magnet (contours at $\Delta U/g=0.05\text{mm}$ intervals); $x$ and $z$ are radial and vertical cylindrical coordinates, respectively, with origin at the geometric center of the solenoid. The field at the center of the solenoid is $B_0=B\left(0,0\right)=16.5\text{T}$. The field profile $B\left(x,z\right)$ used to construct the plot was computed by numerical integration of the Biot-Savart integral. The stable levitation point at a local minimum in the potential is labeled $L_0$. An unstable levitation point at a saddle point in the potential is labeled $L_1$. Top: schematic of the experimental setup. The magnetic field lines are shown in gray. Rays are shown as bold lines.

Figure 2

Bottom: photodiode voltage $\mathcal{E}$ measuring the shape oscillations of an $a=5.8\text{mm}$ water droplet. Center: power spectrum of the oscillations of an $a=5.8\text{mm}$ water droplet, as a function of oscillation frequency $f=\omega /2\pi$. Top: measured frequencies $f$ of the oscillation modes of water droplets with radii between $a=4.5$ and 14 mm, up to $l=16$. The broken lines show the Rayleigh frequency spectrum for a water drop, according to Eq. (1). The measured frequencies (crosses) are slightly higher, by $\sim 1\text{Hz}$, due to the effect of the magnetogravitational potential trap.

Figure 3

Left panel: gradient of the magnetogravitational potential trap $c_0^{\prime }$ obtained from the measured frequencies of two modes, $l=2$ and $l>2$, of a drop with radius $a=r$ at rest. Error bars show the standard error. Filled circles: data obtained in nitrogen atmosphere. Open squares: data obtained in air atmosphere. Dotted line: $c_0^{\prime }$ computed from the solenoid geometry. Right panel: dependence of the experimentally obtained $c_0^{\prime }$ on the mode number with $l>2$ (see text).

Figure 4

Values of the $j=0$ (spherically symmetric) and $j=2,3$ (quadrupole and octopole) coefficients $c_j^{\prime }$ in a multipole expansion of the potential well gradient, Eq. (4), at $B_0=16.5\text{T}$ (in nitrogen atmosphere). Note, $c_1^{\prime }=0$ (see text). The values were determined from the magnetogravitational potential [Eq. (2)] shown in Fig. 1. The $c_0^{\prime }$ line is the same as that shown by the dotted line on Fig. 3.

Figure 5

The three spheres show the computed magnitude of the vector $\mathbf{\Gamma }=-\nabla U$ on the surface of three droplets, radius $a=5.0$, 7.5, and 10.0 mm at $B_0=16.5\text{T}$ (in nitrogen atmosphere); the length and direction of the radiating lines show the direction and magnitude of the (inward pointing) $\mathbf{\Gamma }$ at the surface; $\mathbf{\Gamma }$ was determined from the magnetogravitational potential [Eq. (2)] shown in Fig. 1. The variation in $\left|\mathbf{\Gamma }\right|$ over the surface of the droplet is due to the octopole component of the potential trap.

Figure 6

Top: effect on the eigenfrequencies of an $a=7.5\text{mm}$ water droplet of adding a quadrupole harmonic $c_2^{\prime }$ to the shape of the magnetogravitational potential well ($c_0^{\prime }=0.36\text{m}{\text{s}}^{-2}$, all other $c_j^{\prime }=0$). The thick black lines show the eigenfrequencies (squared) of the $m=0$ modes [i.e., the eigenvalues ${\omega }^2$ of Eq. (19)] corresponding to $l=1–5$; the eigenvalues of Eq. (19) were computed using MATLAB. (Note that we have plotted ${\omega }^2-{\sigma }_T^2$ for clarity). The thin lines show the first-order approximation [Eq. (20)] departing from the numerically computed frequencies at $\left|c_2^{\prime }\right|\gtrsim 1$. Long-dash, medium-dash, short-dash, dot-dash, and dotted lines show the frequencies of the $\left|m\right|=1–5$ modes, respectively (the treatment of the $m\ne 0$ modes is summarized in the Appendix). Bottom: effect on the eigenfrequencies of an $a=7.5\text{mm}$ water droplet of adding a harmonic $c_3^{\prime }$ to the shape of the well ($c_0^{\prime }=0.36\text{m}{\text{s}}^{-2}$, all other $c_j^{\prime }=0$).