Electromagnetic interaction between a permanent magnet and a sphere moving in liquid metal

Abstract

We present a series of model experiments where an electrically non-conductive solid sphere moves inside a vertical column of liquid alloy GaInSn. The experimental setup consists of the liquid metal container, the sphere driving system and the permanent magnet with the attached force sensor. The sphere moves at a controllable constant velocity $U_0$ and follows a straight route, which in turn generates a liquid metal flow around the sphere. This flow interacts with the localized magnetic field of the permanent magnet, and thus a weak reaction force on the magnet is generated. The force sensor attached on the magnet has a resolution of the order $10^{-6}~\text {N}$. Upon elimination of high frequency noise, reproducible time-dependent signals for the forces on the magnet are obtained in the experiments for several Reynolds numbers Re between 160 and 2000. The force component $F_z$ on the magnet parallel to the direction of particle motion exhibits a typical two-peak structure with different peak heights, whereas the transverse force component $F_x$ resembles an antisymmetric pulse. The results demonstrate that the force sensor can detect the presence of a moving particle in a quiescent conducting liquid. They also show that the structure of the $F_x$ signal can be reproduced with less variation and is less sensitive to the Reynolds number than the $F_z$ signal. Moreover, the structure and magnitude of time-dependent Lorentz force signals can be reasonably predicted by a numerical model.

Graphic abstract

Appendix

1.1 Numerical modeling

The grid size needed for a precise MHD simulation is huge, because there are large gradient of velocity field and magnetic fields in the vicinity of the sphere and magnet, which should be well resolved. The mesh used in the numerical modeling is shown in Fig. 14. It is certainly not enough to fully resolve the flow. Nonetheless, we focus on the integral force signals. The influence of discretization was examined by using meshes with different numbers of elements. Based on the velocity field of Re = 200, we check the variations of $F_z, F_x, F_y$. In Fig. 15 we see that the force differences between the different meshes are small.

Fig. 14

Mesh used in the COMSOL simulation. a Overview. b Zoom of the region near the sphere. The mesh was automatically generated within COMSOL

Fig. 15

Comparison of calculated force using different mesh. a $F_z$, b $F_x$, c $F_y$. The finest, fine, coarse meshes have approximately 3.2 million, 960,000 and 340,000 elements respectively. The meshes are certainly not sufficient to fully resolve the velocity field near the sphere and wall. Nevertheless we observe that the force integrals are generally not affected by mesh quality.

1.2 Analytical model for the induction problem

The model provides a rough estimation of the Lorentz force signal based on analytical solutions of the velocity and magnetic fields. The derivation of the magnetic field is identical to Sect. 5. For the computation of the induced currents and Lorentz force, the sphere is replaced by a round cylinder and the magnetic field is also assumed to be two-dimensional, i.e., uniform along the cylinder axis. Therefore the cylinder axis is parallel to the y-direction since the magnetic field does not vary strongly in this direction. The velocity ${\mathbf {u}}$ and magnetic field ${\mathbf {B}}$ have only two components and depend only on z, x, i.e., we have ${\mathbf {u}} = (u_z,~u_x,~0)$ and ${\mathbf {B}} = (B_z,~B_x,~0)$. The electromotive force

$$\begin{aligned} {\mathbf {u}} \times {\mathbf {B}} = {\mathbf {e}}_y \left( u_z B_x - u_x B_z\right) \end{aligned}$$

(18)

has only $y-$component. Since all components of ${\mathbf {j}}$ depend on z, x only, the planar components of ${\mathbf {j}}$ are specified by the z- and x-derivatives of $\phi$. From Eq. (12) it then follows that

$$\begin{aligned} j_y=\sigma \left( u_z B_x - u_x B_z\right) ,~~\frac{\partial ^2 \phi }{\partial z^2} + \frac{\partial ^2 \phi }{\partial x^2} = 0, \end{aligned}$$

(19)

because $j_y(z,x)$ does not contribute to the divergence of the current density. As a consequence, the potential $\phi$ must be constant since it satisfies homogeneous Neumann boundary conditions on the electrically insulating cylinder and other walls. The Lorentz force density ${\mathbf {f}} = {\mathbf {j}} \times {\mathbf {B}}$ generates a total Lorentz force (per unit length in axial direction). Its components are

$$\begin{aligned}&\int f_z \, \text {d}z \,\text {d}x= \int \sigma ( u_x B_z B_x - u_z B_x^2 ) \text {d}z \,\text {d}x, \nonumber \\&\int f_x \, \text {d}z \,\text {d}x= \int \sigma (u_z B_z B_x - u_x B_z^2) \text {d}z \,\text {d}x. \end{aligned}$$

(20)

Another assumption we make is that the main contribution to the force originates from the flow in a narrow band centered around the line of motion of the center of the sphere or cylinder. The solution then only requires the knowledge of the distributions of stream-wise magnetic field (see Fig. 16a) and velocity (see Fig. 16b) along the line of motion. We outline these simplifications in what follows.

We use the analytical solutions of the flow generated by a sphere centered at the origin and moving in the $z-$direction. One option is the analytical solution of the potential flow around a moving sphere. Then the velocity is obtained from the velocity potential

$$\begin{aligned} \phi '=-\frac{U_{0} R^{3} \cos \theta }{2 r^{2}} \Rightarrow u_{z,1} = \frac{\partial \phi '}{\partial z'},~~~ u_{x,1} = \frac{\partial \phi '}{\partial x}, \end{aligned}$$

(21)

where $U_0$ is the velocity of the sphere, R is the sphere radius, $r = \sqrt{x^2+y^2+z^2}$ is the radial coordinate, $\theta = \arccos (z/r)$ is the polar angle between radial direction and flow direction, and $z'$ is the coordinate in the frame of center of sphere (i.e., $z' = z - \xi d$). Another option for velocity field is to use the solution of Stokes flow around a moving sphere. In such case the radial and angular components of velocity are

$$\begin{aligned}&u_{r}=U_{0} \cos \theta \left( \frac{3 R}{2 r}-\frac{R^{3}}{2 r^{3}}\right) , \nonumber \\&u_{\theta }=U_{0} \sin \theta \left( -\frac{3 R}{4 r}-\frac{R^{3}}{4 r^{3}}\right) , \end{aligned}$$

(22)

$$\begin{aligned}&\rightarrow u_{z,2} = u_r \cos \theta - u_{\theta } \sin \theta ,\nonumber \\&u_{x,2} = u_r \sin \theta + u_{\theta } \cos \theta . \end{aligned}$$

(23)

We define the velocity field along the z-axis as

$$\begin{aligned} u_z&= u_{z,1}~~~(z' \ge R),&u_z&= u_{z,2}~~~(z' \le -R). \end{aligned}$$

(24)

This patched velocity distribution turns out to be relatively close to the behavior of laminar flow at low Reynolds numbers Re $< 200$.

Finally, we assume that the integrand in the Lorentz force contributes mainly on a band of typical width H around the $z-$axis and that the variation along the $z-$direction is stronger than in the $x-$direction. We thereby reduce the integration over the area to an integration along the $z-$axis. Assuming $|B_x| \gg |B_z|$ based on Eq. (20), we get the results

$$\begin{aligned} F_z = -\int \sigma u_z B_x^2 \text {d}z,~~F_x =- \int \sigma u_z B_z B_x \text {d}z, \end{aligned}$$

(25)

for the force components per unit length in $z-$ direction and per width H. The latter is presumably somewhat larger than the diameter of the sphere.

Fig. 16

Modeling along the line of motion of the sphere ($z-$ axis). a The magnetic field, b the velocity distribution in the frame of sphere center, c the integral force which is illustrated in Fig. 13

Both $F_z$ and $F_x$ depend on the relative position $\xi$ of sphere center and magnet. They are defined by convolutions of $\sigma u_z$ with the functions $B_z B_x$ and $B_x^2$, respectively. The convolution integrals are computed numerically in MATLAB. The results are shown for both $F_x$ and $F_z$ in Figs. 13 and 16c. They are independent of Re since the model assumes direct proportionality between force and velocity. The quantitative agreement for the magnitude of $F_z$ and $F_x$ with measurements in Fig. 13 indicates that the width H is comparable to the diameter of the sphere.