Oscillatory surface deformation of paramagnetic rare-earth solutions driven by an inhomogeneous magnetic field

The deformation of the free surface of a paramagnetic liquid subjected to a nonuniform magnetic field is studied. A transient deformation of the surface caused by the interplay of gravity, magnetic field, and surface tension is observed when a permanent magnet is moved vertically downward to the free surface of the liquid. Different concentrations of rare-earth-metal salt (${\mathrm{DyCl}}_3$) are used and different magnet velocities are studied. The deformation of the interface is followed optically by means of a microscope and recorded with a high-speed camera. The experimental results are compared and discussed with complementary numerical simulations. Detailed results are given for the static shape of the deformed surface and the temporal evolution of the surface deformation below the center of the magnet. The frequency of the surface oscillations is found to depend on the concentration of the salt and is compared with analytical findings. Finally, a potential application of the effects observed is presented.

Figure 1

(a) Sketch of the experimental setup. The transparent glass cuvette contains the paramagnetic ${\mathrm{DyCl}}_3$ solution. The origin of the cylindrical coordinate system is marked by a star. The direction of gravity and the magnet's approach are along the $-z$ direction. (b) Magnetic flux density and the magnetic field gradient term $|(\vec{B}·\vec{\nabla })\vec{B}|(T^2/m)$ for the lowest magnet position at $z_0=4.5$ mm (simulation result). The maximum values of both quantities shown were limited to enhance the presentation.

Figure 2

Close-up of the surface of a ${\mathrm{DyCl}}_3$ solution and its deformation under the influence of a magnetic field near the cell center. (a) Flat surface (vertical position $h_0$) when the magnet is far away. (b) Deformed surface when the magnet has reached the closest position ($t=0$). The point in the center (marked by a star) is $h(t=0,r=0)$. (c) Deformed surface at a later point in time $t>0$ s. The marked distances refer to Eq. (15).

Figure 3

Interface structure of the $0.75\mathrm{M}{\mathrm{DyCl}}_3$ solution, shown at an aspect ratio of $z\text{:}r=1\text{:}1$ (a) without magnetic field influence and (b) 4 s after the magnet reached its end position (fast case). The interface is identified using monodisperse polystyrene particles (darker points in the picture), which rest on the liquid's surface. The figure represents the interface in the central plane of the cuvette. The edge of the cuvette is on the right-hand side of the figure ($r=30$ mm), while the center of the magnet is on the left-hand side of the figure ($r=0$ mm), where the largest elevation of fluid is seen.

Figure 4

Shape of the static surface deformation after the decay of oscillations: (a) experimental result for $0.75\mathrm{M}{\mathrm{DyCl}}_3$ observed at 4 s after the fast approach of the magnet and (b) numerical results for different bulk concentrations of ${\mathrm{DyCl}}_3$ in comparison with the experimental result for the $0.75\mathrm{M}$ solution (fitted curve).

Figure 5

Static surface elevation $\mathrm{\Delta }{h}_{\infty }$ below the center of the magnet after decay of oscillation versus bulk concentration: comparison between experimental and numerical results. The error bars represent the minimum and maximum values of experimental results. Additionally, the fitted value of the initial amplitude $A_1$ of the transient oscillations according to Eq. (15) is given for the fast application of the magnet (see Sec. 5b).

Figure 6

Experimental result of the interfacial dynamics in the center of the cell for all investigated ${\mathrm{DyCl}}_3$ concentrations for (a) the slow and (b) the fast case. The magnet velocity is given in the insert. At $t=0$, the magnet has reached the final position near the surface. At $t=5$ s, the magnet is removed upward again.

Figure 7

Simulated interfacial dynamics in the center of the cell for different ${\mathrm{DyCl}}_3$ concentrations.

Figure 8

Frequency of the surface deformation versus bulk concentration: comparison between experiments and simulations. The solid lines represent predicted frequencies of different modes of surface waves as discussed later in Sec. 6.

Figure 9

Dynamics of the surface of water in the cuvette excited by a falling drop. At $t=0$ s the drop reaches the surface.

Figure 10

The $N$ modes $i=0$ to $i=4$ of a Bessel function expansion, which are radial eigenmodes of the oscillating membrane of a drum.

Figure 11

Temporal behavior of the surface height $\mathrm{\Delta }h$ in experiment with a $0.75\mathrm{M}$ Dy(III) solution. An outer part of the cuvette ($13.5\text{mm}<r<16.5$ mm) is shown, within which the surface wave has a node near $r=15$ mm. The black dots shown mark the temporal position of the node at which $\mathrm{\Delta }h-\mathrm{\Delta }{h}_{\infty }=0$ holds.

Figure 12

Variation in the difference of surface deformation with respect to the static surface shape at later points in time near $t=6.3\text{s}$ where $\mathrm{\Delta }h\left(0\right)$ is minimum, maximum, and approximately zero. The simulation result is $c=0.5\mathrm{M}$.

Figure 13

Principle of an electrical-thermal switch with paraffin oil and paramagnetic liquid: (a) no magnet is applied, (b) the magnet is approaching, (c) the magnet is at its end position and the Dy(III) solution is in contact with the cover slip, and (d) demagnetization phase.