Tuning particle settling in fluids with magnetic fields

Abstract

A magnetic field is generated to modify the effective gravity acting on settling particles in a laboratory experiment. When applied to a magnetized spherical particle settling in water-glycerol mixtures, the magnetic field produces a vertical force that counteracts the gravitational field, hence allowing for the magnetic tuning of the settling properties of the particle. While doing so, the spin of the particle around the direction perpendicular to the applied magnetic field is blocked, thus allowing spin solely around the direction of the magnetic field. This method of magnetic modification of the effective gravity is tested on the settling of spherical magnets in quiescent fluids over Galileo numbers in the range [100, 300] and a fixed particle density of 8200 kg/m$^3$. The results obtained by varying the Galileo number via the magnetic modification of effective gravity are compared to those obtained with non-magnetic spheres when the Galileo number is modified by varying the fluid’s viscosity. We show that the same taxonomy of settling regimes with nearly identical geometrical properties (in terms of planarity and obliqueness) of the trajectories is recovered. In addition to proving that it is possible to magnetically tame the settling of particles in fluids preserving the features of the non-magnetic case, this also reveals that blocking the spin of the particles does not produce any significant effect on its settling properties in a quiescent fluid. This novel experimental methodology opens new possibilities to experimentally explore many other subtle aspects of the coupling between settling particles and fluids (for instance, to disentangle the effects of rotation, inertia, and/or anisotropy of the particles) in more complex situations including the case of turbulent flows.

Appendices

Appendix A: Gravity compensation theory

1.1 1. Equations of motion

When applying this method to a particle in a fluid the equations of motion need to include hydrodynamical effects. Neglecting added mass and history forces (Gatignol 1983; Maxey and Riley 1983), the fluid adds drag (Brown and Lawler 2003), torque (Lukerchenko et al. 2008), and buoyancy effects yielding the following equations of motion:

$$\begin{aligned} \textbf{F}&= (m_p-\text {V}\rho _f)g~\hat{\textbf{z}} + \textbf{M} \cdot \nabla \textbf{B} - \frac{1}{8}C_D\pi d_p^2\rho _f \textbf{v}|\textbf{v}|, \end{aligned}$$

(E1)

$$\begin{aligned} \textbf{T}&= -1/64C_{\omega }\rho _f \varvec{\omega } |\varvec{\omega }|d_p^5 -\textbf{M} \times \textbf{B}, \end{aligned}$$

(E2)

with fluid density $\rho _f$, particle volume V, kinematic viscosity $\nu$, translational ($C_D$) and rotational ($C_{\omega }$) drag coefficients, particle velocity $\textbf{v}$, and angular velocity $\varvec{\omega }$. Finally, note that if the Reynolds number is low (typically below order one (Cabrera et al. 2022), i.e., the Stokes regime) the fluid drag and torque are simpler:

$$\begin{aligned} \textbf{F}&= (m_p-V\rho _f)g~\hat{\textbf{z}} - \nabla (\textbf{M} \cdot \textbf{B}) - 3\pi d_p \eta \textbf{v}, \\ \textbf{T}&= \pi \eta d_p^3\mathbf {\omega } -\textbf{M} \times \textbf{B}. \end{aligned}$$

1.2 2. Magnetic field derivation

To homogeneously compensate gravity, the magnetic force on gravity’s direction ($\textbf{F}^{M} \cdot \hat{\textbf{z}}$) needs to be a constant independent of z, here denoted $G_z$. Alongside the previous condition, the external magnetic field $\textbf{B}$ has to be a solution of Maxwell’s equations, leading to the following set of equations:

$$\begin{aligned}&\nabla _z~\text {B} =G_z, \end{aligned}$$

(E3)

$$\begin{aligned}&\nabla \cdot \textbf{B} = 0, \end{aligned}$$

(E4)

$$\begin{aligned}&\nabla \times \textbf{B} = 0. \end{aligned}$$

(E5)

The present work focuses on axisymmetric solutions where a linear magnetic induction in $\hat{\textbf{z}}$ can be proposed, resulting in: $\textbf{B}(r,z) = B_r(r)~\hat{\textbf{r}} + (G_z\ z + B_0)~\hat{\textbf{z}}$, in cylindrical coordinates. $B_r$ can be obtained by solving Eq. (E4), leading to the following magnetic field induction:

$$\begin{aligned} \textbf{B}(r,z) = (-G_z/2\ r)~\hat{\textbf{r}}+ ( G_z\ z + B_0)~\hat{\textbf{z}}. \end{aligned}$$

This magnetic induction respects the irrotational condition (Eq. E5), whereas Eq. (E3) is exactly satisfied only at $r=0$. The latter is an unavoidable consequence of the solenoidal nature of magnetic fields. A dependence on the distance to the system axis (r) and the position on the axis (z) are then present in the forces acting on the particle:

$$\begin{aligned} F_z^M(r,z)= & {} M~\frac{\partial B}{\partial z}= M~\frac{(G_z z+B_0)G_z}{\sqrt{(G_z)^2/4r^2 + (G_z z + B_0)^2}}, \nonumber \\ F_r^M(r,z)= & {} M~\frac{\partial B}{\partial r} = M~\frac{r~(G_z)^2 /4}{\sqrt{(G_z)^2/4r^2 + (G_z z + B_0)^2}}. \end{aligned}$$

(E6)

Note that $F_r^M(r\rightarrow 0) = 0$ and $F_z^M(r\rightarrow 0) = MG_z$. Therefore, gravity can be fully compensated at $r=0$ without any radial force present. Note that this is not in conflict with Earnshaw’s theorem (Earnshaw 1842) because the equilibrium is not stable, i.e., the Laplacian of the magnetic energy is not zero.

1.3 3. Magnetic field homogeneity

Fig. 8

Contour plot of the axial (a) and radial (b) component of the theoretical magnetic force, normalized by the axial force at z=0: $F_z^M(r,z)/F_z^M(0,0)$ and 1-$F_r^M(r,z)/F_z^M(0,0)$, respectively

Figure 8 presents contour plots of $F_z^M(r,z)/F_z^M(0,0)$ and 1-$F_r^M(r,z)/F_z^M(0,0)$. Note that the normalization chosen is $F_z^M(0,0)=MG_z$. Values of $G_z=-250~$G/m, $B_0=26~$G, $z\in [-150,50]~$mm and $M=4.96\times 10^{-8}~$G$^{-1}$m$^2$s$^{-2}$ were used to compute the forces from Equations E6, as these are typical magnitudes for the present experimental setup.

The axial component of the force $F_z^M$ has a weak dependence on z and r, as quantified in Fig. 8a: A maximum axial force variation of 20% is achieved at $z=50$ mm and $r=100~$mm. At $z\in [-150,0]~$mm and $r\in [0,20]~$mm, the ranges used in this work, the axial magnetic force has fluctuations below 2%. On the other hand, the radial force $F_r^M$ has a stronger dependence on r and z (see Fig. 8b). When $r=100~$mm and $z=50$, the radial force becomes as high as 30% of the reference axial force at the center $F_z^M(0,0)$. At the ranges $z\in [-150,0]~$mm and $r\in [0,50]~$mm the maximum value of radial force is reduced to 10% of its axial counterpart.

The relative magnitude of the axial and radial forces can be calculated:

$$\begin{aligned} \frac{F_r^M(r,z)}{F_z^M(r,z)} = \frac{1}{4} \frac{\text {G}_\text {z}~r}{\text {G}_\text {z}~z + B_0}. \end{aligned}$$

(E7)

As the aspiration is to solely counteract gravity, a radial force is not desired and the latter ratio needs to be minimized. There are two ways to achieve it: Keep r small compared to ($z + B_0/\text {G}_\text {z}$); and/or have the largest possible value for $B_0$. The latter approach is ideal because it allows a larger volume (r-z) where the axial force is homogeneous and the radial forces are small. Note that it translates to more current on the coils ($B \propto I$) and, therefore, thicker coil winding that might lead to the necessity of external cooling.

Appendix B: Coils’ input parameters

The coils’ positions ($Z_i$) and currents ($I_i$) given by the fit for both cases are presented in Table 4. Note that for Case $g_0$ only four coils are used as this case does not require more coils to achieve better homogeneity.

For the Cases $g^*$, the coils are not powered symmetrically because of the need to create a gradient in the magnetic field. To do so, the coils in the region with the highest magnetic field need to have a larger current. It is also important to produce a linear profile of magnetic field around a nonzero values in order to avoid the reorientation of the particle: This would happen because the magnetic field would vanish and reverse direction, forcing the magnetic moment of the particle to re-align.

Table 4 Coils’ positions and currents given by the fit method for both Cases $g_0$ and $g^{*} = \tilde{g}/g= 0.65$

Appendix C: Particle material discussion

Equation (1) can be rewritten if one specifies the particle magnetic properties: in the ferromagnetic, paramagnetic, or diamagnetic particle cases $\textbf{M}\propto \textbf{B}$, whereas for a permanent magnet (with $B=|\textbf{B}|$ below its coercive field strength) $M=|\textbf{M}|$ is constant and $\textbf{F}^M = \textbf{M}\cdot \nabla \textbf{B}$. This work focuses on the latter particle case because magnetic moment values are at least two orders of magnitude larger. This translates into lower external magnetic induction intensities (i.e., less power or smaller coils) to achieve a certain magnetic force.

In particular, the magnetic moment M of a permanent magnet can be computed, if one assumes that the magnetic dipolar moment is dominant, in the following manner:

$$\begin{aligned} M = \frac{B_{\text {res}} V}{\mu _0}, \end{aligned}$$

(E8)

where V is the volume of the magnet, $\mu _0$ the vacuum magnetic permeability (note that $\mu _0 \approx \mu _\mathrm{{water}}$) and $B_{\textrm{res}}$ is the remnant magnetic flux density (for the particles used here $B_{\textrm{res}}=1.192$ T), in other words the magnet’s magnetic flux density when the external coercive field strength is zero.