Axisymmetric Ferrofluid Oscillations in a Cylindrical Tank in Microgravity

Abstract

The sloshing of liquids in low-gravity entails several technical challenges for spacecraft designers due to its effects on the dynamics and operation of space vehicles. Magnetic settling forces may be employed to position a susceptible liquid and address these issues. Although proposed in the early 1960s, this approach remains largely unexplored. In this paper, the equilibrium meniscus and axisymmetric oscillations of a ferrofluid solution in a cylindrical tank are studied for the first time while subject to a static inhomogeneous magnetic field in microgravity. Coupled fluid-magnetic simulations from a recently developed inviscid magnetic sloshing model are compared with measurements collected at ZARM’s drop tower during the ESA Drop Your Thesis! 2017 campaign. The importance of the fluid-magnetic interaction is explored by means of an alternative uncoupled framework for diluted magnetic solutions. The coupled model shows a better agreement with experimental results in the determination of the magnetic deformation trend of the meniscus, but the uncoupled framework gives a better prediction of the magnetic frequency response which finds no theoretical justification. Although larger datasets are required to perform a robust point-by-point validation, these results hint at the existence of unmodeled physical effects in the system.

Appendices

Appendix 1 Alternative Numerical Solution Procedure

The problem given by Eqs. 6-10 is solved in Romero-Calvo et al. (2020a) by means of Ritz’s method. However, this approach relies on the definition of appropriate primitive functions, and may not be robust for all possible fluid interfaces (Concus and Crane 1967).

In order to discard procedural errors, the numerical approach presented in Herrada and Montanero (2016) to study the nonlinear dynamics and linear stability of capillary fluid systems is here adopted taking the magnetic Bond number and the meniscus profile as inputs. The spatial physical domain is mapped onto a rectangular domain by means of a nonsingular mapping. Figure 12 shows the mesh produced by this procedure for $I=20$ A. The dimensional equivalent of Eqs. 6-10 (Eq. 19 from Romero-Calvo et al. 2020a) are spatially discretized using Chebyshev spectral collocation points. The temporal derivatives for the perturbations h and $\phi$ are computed assuming a oscillatory behavior of the type $e^{i\omega t}$, with $\omega$ being the natural frequency. The generalized eigenvalue problem resulting from the spatial and temporal discretizacion is finally solved using MATLAB's eig function, returning the different modes of oscillation. For the case under analysis, the eigenvalues obtained with both methods differ by at most a 0.5%, which is attributed to numerical precision errors.

Fig. 12

Mesh employed to discretize the magnetic sloshing model in the alternative numerical solution procedure

Appendix 2 Magnetic Comsol Multiphysics Model

In order to compute $Bo{\text {mag}}$ for a given meniscus profile and coils current intensity, the magnetic ($\mathbf {H}$) and magnetization ($\mathbf {M}$) fields are obtained from Comsol Multiphysics by solving the stationary Maxwell equations

$$\begin{aligned} \nabla \times \mathbf {H}=\mathbf {J}, \end{aligned}$$

(17)

$$\begin{aligned} \mathbf {B}=\nabla \times \mathbf {A}, \end{aligned}$$

(18)

$$\begin{aligned} \mathbf {J}=\overline{\sigma }\mathbf {E}, \end{aligned}$$

(19)

where $\mathbf {J}$ is the current field, $\mathbf {A}$ is the magnetic vector potential produced by the current in the coil and the magnetized materials, $\overline{\sigma }$ is the conductivity of the coil, and $\mathbf {E}$ is the electric displacement field. The constitutive relation

$$\begin{aligned} \mathbf {B}=\mu _0\mu _r\mathbf {H}, \end{aligned}$$

(20)

with $\mu _r$ being the relative permeability of the material, is applied to the aluminum plates ($\mu _r^{Al}=1.000022$), surrounding air ($\mu _r^{air}=1$) and copper coils ($\mu _r^{Cu}=1$). Within the ferrofluid volume, the constitutive relation is defined by the magnetization curve $M(H)$ depicted in Fig. 5, that results in

$$\begin{aligned} \mathbf {B}=\mu _0\left( 1 + \frac{M(H)}{H}\right) \mathbf {H}. \end{aligned}$$

(21)

The current field is computed through

$$\begin{aligned} \mathbf {J} = \frac{NI}{A}\mathbf {u}_{\theta }, \end{aligned}$$

(22)

with $N=200$ being the number of turns, I the current intensity flowing through each wire, $A=509$ mm$^2$ the coils cross section and $\mathbf {e}_{\theta }$ the circumferential vector.

Fig. 13

Mesh of the magnetic field FEM model. The simulation domain is a 1000$\times$3000 mm rectangular region enclosing two identical assemblies

The simulation domain is a rectangular 1$\times$3 m region enclosing the assemblies. An axisymmetric boundary condition is applied to the symmetry axis, while the tangential magnetic potential is imposed at the external faces through $\mathbf {n}\times \mathbf {A}=\mathbf {n}\times \mathbf {A}_d$. $\mathbf {A}_d$ is the dipole term of the magnetic vector potential generated by the current in the coils and the magnetization fields of the ferrofluid volumes. Consequently, $\mathbf {A}_d$ is computed as the potential vector generated by four point dipoles applied at the centers of the magnetization distributions and whose moments are those of said distributions. While the dipoles associated to the coils can be calculated beforehand, the ferrofluid dipoles need to be approximated iteratively by integrating $\mathbf {M}$ in the ferrofluid volume. The relative error in the magnetic vector potential due the dipole approximation is estimated to be below 1.0% at the boundary of the domain with respect to the exact value generated by equivalent circular loops.

The mesh is composed of 167755 irregular triangular elements, as shown in Fig. 13 for the $I=20$ A case. Mean and minimum condition numbers of 0.985 and 0.527 are obtained. An increase of the mesh density from 167755 to 302910 elements results in a fundamental frequency increase of 0.025% for $I=22$ A, reflecting the convergence of the solution.