Model based investigation of the sedimentation behavior of magnetorheological fluids under the influence of magnetic and acceleration fields

Abstract

This work investigates the sedimentation behavior of magnetorheological fluids (MRFs) under static and dynamic loading. An analytical approach using the dipole-dipole interaction model is applied to describe the sedimentation behavior of the MRF by calculating the magnetic forces resulting from the particle chains’ shape and examining its stability by defining the stability criterion. The modeling is carried out in two steps. Firstly, the static case is considered. Under static loading conditions, the gravitational force is the factor that determines the chains’ shape. Once the shape of the particle chain is known, the magnetic forces acting on each particle and the resulting force can be calculated. Critical conditions for the chain formation and stability are then defined and calculated, which builds the foundation for extending the model to also include dynamic loading, where centrifugal forces become the decisive stability factor. Analogous to the static case, the critical parameters that trigger the onset of sedimentation and the transition to sedimentation are identified, defined, and calculated. Experimental validation is performed for both static and dynamic conditions.

Keywords

sedimentation of magnetorheological fluids stability criteria centrifugal force angular velocity

Introduction

Magnetorheological fluids (MRFs) are a type of smart material that changes its rheological properties very rapidly when exposed to an external magnetic field; Zhang et al. (2018) report reaction times between 15 and $25 ms$ . These fluids are composed of micron-sized highly magnetizable particles between 3 and 5 µm averaging around $4.25$ µm with volume fractions of up to $50 vol . %$ , dispersed in a non-magnetizable carrier liquid as documented by Kumar et al. (2019) and de Vicente et al. (2011). Work by Cvek et al. (2015) demonstrates that incorporating carbon allotropes (fullerene powder, carbon nanotubes, and graphene nanoplatelets), significantly improves MRF performance, stability, and redispersibility. In contrast, Zeng et al. (2016) use chemical surfactants, that is, stearic acid and sodium dodecyl sulfate, to enhance the stability of MRFs. While the carbon allotropes function as physical spacers that prevent particle packing, chemical surfactants stabilize the fluid at a molecular level by creating a coating on the magnetic particles, thereby reducing aggregation.

Due to their rheological properties, which can be manipulated in real-time, MRFs are adopted in a range of engineering applications. Sohn et al. (2018) summarize several robotics applications, including manipulators, deformable grippers, haptic devices, and climbing robots, by considering two distinct methodologies: the first utilizes valve systems that result in an active control force within a closed-loop system, while the second uses the flow, shear, and squeeze operating modes of MRFs for direct actuation. The valve mode is particularly efficient in the design of dampers, as discussed theoretically by Olabi and Grunwald (2007), who showed how this mode results in a more compact design with no moving parts in the control section. Shock absorbers present another major branch of MRF industrial applications. A high-precision application of this is the work conducted by Kang et al. (2021), which considers both major and minor pressure losses within the valve to improve the model accuracy of an aircraft landing gear system. Other applications extend into the medical field as well; to this end, Kikuchi et al. (2024) developed different types of compact MRF clutches for haptics (H-MRCs) and integrated them into a twin-driven MRF actuator (TD-MRA) to provide precise force feedback for tele operative surgical systems.

However, one major limitation for broader application is their tendency to settle under gravitational or other external forces over time due to the density difference between the CIPs and the carrier liquid, as shown by de Vicente et al. (2011) and Ashtiani et al. (2015). This process, known as sedimentation, drastically affects their performance and the long-term stability of MR fluid-based systems.

To date, several experimental methods have been developed to characterize the sedimentation process of MRFs; however, no international standard method that precisely defines the test conditions and the results evaluation has been established. While the simplest measurement method includes a visual inspection of the sedimentation ratio (Choi, 2021), more sophisticated techniques, such as inductance-based monitoring using sensing coils (Xie et al. 2015), magnetic detection via Hall probes (Luigjes et al., 2012), X-ray transmission profiling (van Silfhout and Erné, 2019), thermal conductivity correlation (Cheng et al., 2016), or even capacitive measurement systems (Xie et al., 2023), have also been implemented.

Analytical modeling of sedimentation on the other hand remains a significant challenge due to the transition from standard gravitational settling to a complex regime of hindered settling. In the off-state (no magnetic field), the sedimentation process can be approximated using Stokes’ Law and its various modifications to account for particle-particle interactions in dense suspensions. However, when an external magnetic field is applied, inter-particle magnetic forces and field–particle interactions create chain-like microstructures, that increase the mechanical resistance against settling, hence reducing sedimentation rates. While advanced macroscopic and microscopic continuum models effectively characterize MRF properties such as static shear stress and normal pressure as shown by Potisk et al. (2019), a dedicated analytical model for on-state sedimentation has yet to be established. As a result, researchers and engineers rely on empirical parametric models and analytical fits (curve-fit models) to describe these kinetics.

Furthermore, external dynamics such as rotational forces can accelerate or trigger the sedimentation process due to the internal centrifugal forces acting on individual particles. As demonstrated by Schreiner et al. (2022), these forces can be intentionally used to precisely influence the mass distribution of the MRF. By controlling the magnetic field and thus the transfer of shear stress, targeted control of the sedimentation process is achieved, which is applied in rotor balancing.

The primary objective of this work is to derive an analytical model for predicting the onset of sedimentation under the influence of magnetic and acceleration fields. The study is structured into two main phases.

Static Loading Conditions

Initially, an analytical model is derived to predict CIPs sedimentation under static loading in the presence of a magnetic field. For this purpose, the acting forces are first described at particle level. Based on these, a critical equilibrium state can be determined at which the inertial forces outweigh the magnetically induced forces, and sedimentation of the particles begins. This derivation yields an analytical result for the critical magnetic flux density $B_{0, crit}$ required to prevent sedimentation due to gravity. Furthermore, the spatial arrangement of particles within the chains is determined to define the initial configuration for the subsequent dynamical analysis.

Dynamical Loading and Continuum Transition

Building upon the static threshold and established boundary conditions, rotational forces are superimposed. This stage presents a significant computational challenge: assuming spherical carbonyl iron particle with a radius of $r_{p} = 5$ µm and a particle volume fraction of $c_{v} = 0.5$ , $1 {mm}^{3}$ MRF contains $\approx 9.5 \times 10^{5}$ particles, so to address the very high computational demand of modeling millions of discrete particles, the model is finally extended into a continuum representation. This implies a transition from discrete inter-particle forces to an asymptotic stress-based approach.

Several distinct modeling frameworks are developed, to describe the behavior of MRFs ranging from macroscopic continuum theories to complex numerical simulations. Continuum rheological models, such as the Bingham plastic and Herschel-Bulkley models, are frequently used to characterize the fluid’s non-Newtonian transition and field-dependent yield stress as reported by Utami et al. (2018). Hydrodynamic (Elsaady et al., 2020) and energy-based models (Yeh and Chen, 1997) further refine this by coupling Navier-Stokes equations with magnetic field theory or by minimizing system potential energy to predict stable particle configurations. The methodology implemented in this work is based on the dipole-dipole interaction model, which describes the particle interactions and defines them in terms of magnetic forces when a magnetic field is applied (Boyer, 1988; Durney and Johnson, 1969; Seleznyova et al., 2016). As already mentioned, one of the key parameters in determining the resulting magnetic force is the configuration of the particle chain. Therefore, the foundation of the calculation is deriving of the geometry of the chain, based on the system design and its physical boundary conditions. In both phases, typical continuum mechanics models are used, whereby the equation(s) of motion are formulated and solved to obtain the form of the chain. Given the form, the resulting magnetic forces are calculated by initially determining a suitable coordinate system and the position vector for each particle. Through the equilibrium formulation, the critical parameters that is, specific to the system and its loading conditions, are identified and calculated. Finally, the analytically derived models are validated experimentally.

Theoretical framework for sedimentation stability of particle chains

The system analyzed in this work consists of a chamber filled with MRF as described by Schreiner and Maas (2024). The chamber can rotate around a fixed axis using a rotor and is, therefore, subjected to both gravitational and centrifugal forces acting on the particle chains. Furthermore, by utilizing permanent magnets, a homogeneous magnetic field, parallel to the axis of rotation, is applied throughout the entire chamber. The gravitational, centrifugal, and magnetic forces superimpose in this system, thereby determining the behavior of the particle chains inside the chamber. A simplified schematic of a single particle chain and the system is shown in Figure 1.

Figure 1.

Simplified illustration of the system showing the chamber setup and single particle chain.

Preliminary considerations and assumptions for modeling particle chain dynamics

Based on the classical dipole–dipole interaction model used in the modeling of magnetorheological and ferrofluid particle chains, the following assumptions are adopted:

(i) The particles are uniform, spherical, and composed of the same material (Rosensweig, 1985).

(ii) As per the numerical results, the magnetic field is considered quasi-static and homogeneous.

(iii) Dipolar particles align into single chains and the distance between chains is sufficiently large - hence, the interactions between chains are negligible (Martin and Anderson, 1996; Güth and Maas, 2016).

(v) The particles within each chain are uniformly distributed (Martin and Anderson, 1996).

While real-world MRFs exhibit polydispersity and complex secondary structures, the single-chain dipole-dipole model provides a robust analytical framework to isolate the fundamental interplay between magnetic restoration and displacement forces. Although the dipole approximation is theoretically most accurate for dilute suspensions, where inter-chain interactions are quite small, the framework is extended via a continuum transition to describe actual multi-chain MRF systems. Experimental validation demonstrates that this modeling approach provides reliable quantitative predictions for MRFs with particle volume fractions of up to $c_{v} = 47 %$ , although the accuracy of the predictions decreases with increasing particle concentration. The model is further limited by magnetic flux densities up to $B_{0} \approx 0.4$ T, ensuring operation within the linear magnetization regime of the particles. Furthermore, although the experimental validation utilizes millimeter-scale particles, the underlying interaction remain theoretically scalable, allowing for a quantitative assessment of the model’s reliability for the micron-scale carbonyl iron particles (rp ≈ 3–5 µm) typical of commercial MRFs.

Finally, it is assumed that drag forces do not influence the formulated critical parameters, as the model specifically considers the onset of sedimentation where relative velocity is negligible. Similarly, the effect of buoyancy is implicitly accounted for within the effective gravitational force and thus does not appear as an independent dynamic factor in the stability criterion.

Magnetic dipole model and force formulation

The derivation of the magnetic forces acting on a particle chain is based on previous work employing the classical dipole-dipole interaction model, as derived by Yi et al. (2010), Zhao et al. (2012) and later Güth and Maas (2016).

The magnetic force $F_{j}^{i}$ acting on particle i due to particle j is given by

F_{j}^{i} = μ_{0} \frac{4}{3} π r_{p}^{3} χ_{eff} \nabla (H_{i} \cdot H_{j}),

(1)

where $μ_{0}$ denotes the permeability of vacuum, $r_{p}$ is the particle radius, and $χ_{eff}$ is the effective magnetic susceptibility. For spherical particles Güth and Maas (2016) show that the effective susceptibility is given by

χ_{eff} = \frac{3 χ_{0}}{χ_{0} + 3} .

(2)

Here $χ_{0}$ represents the initial, low-field susceptibility and is set to $χ_{0} = 1000$ .

The magnetic field within the particle is given by H. By taking into account the contributions of all other particles in the chain, the total magnetic field acting on particle i in a chain consisting of n particles can be expressed as

H_{j}^{i} = H_{0} + χ_{eff} \cdot r_{p}^{3} \sum_{j \neq i}^{n} (\frac{(H_{0} \cdot r_{ij}) r_{ij}}{r_{ij}^{5}} - \frac{H_{0}}{3 r_{ij}^{3}}),

(3)

with $H_{0}$ being the external applied magnetic field.

Sedimentation stability under static loading conditions

Sedimentation stability under static loading conditions is analyzed within a continuum framework. By approximating the particle chain as a continuous filament subjected to a uniform gravitational load, the equilibrium configuration is obtained. In this continuum limit, the chain adopts the well-known catenary profile of a hanging chain under gravity as calculated by Irvine (1981). An outline of the derivation for the governing equations is provided in Appendix 1.

Once the equilibrium shape of the chain is determined, the magnetic force acting on each particle, as well as the resulting total magnetic force, can be evaluated using equation (1). For the static case considered here, the particle positions are described using fixed Cartesian coordinates in the x–y plane. The position vector between two particles i and j along the chain is therefore defined as

r_{ij} = (x_{j} - x_{i}) e_{x} + a (\underset{: = C_{j}}{\underset{︸}{\cosh \frac{x_{j}}{a}}} - \cosh \frac{x_{i}}{a}) e_{y},

(4)

where $C_{j} : = \cosh \frac{x_{j}}{a}$ . This parameter denotes a constant value that represents the relative position of any particle j with respect to a varying reference particle i. The value of $C_{j}$ depends solely on the position of the second particle j. The applied magnetic field vector has only one x-component and can be written as:

H_{0} = H_{0} e_{x} .

(5)

Combining equations (1) and (3), the expressions for the magnetic force component in the x-directions can be obtained using the compact notation as follows:

F_{x, j}^{i} = - A \sum_{j \neq i}^{n} [\frac{B_{ij}}{D_{ij}^{5}} - E_{i} (\frac{F_{ij} \cdot G_{ij}}{D_{ij}^{7}})]

(6)

F_{y, j}^{i} = - A \sum_{j \neq i}^{n} E_{i} [\frac{H_{ij}}{D_{ij}^{5}} - \frac{I_{ij} \cdot G_{ij}}{D_{ij}^{7}}]

(7)

where the constant A represents the physical prefactor:

A = μ_{0} \cdot \frac{8}{3} \cdot H_{0}^{2} \cdot π \cdot r_{p}^{6} \cdot χ_{eff}^{2}

(8)

The geometric auxiliary variables are defined as:

\begin{matrix} Δ x_{ij} = x_{j} - x_{i} \\ Δ y_{ij} = a (C_{j} - \cosh (x_{i} / a)) \\ D_{ij} = \sqrt{Δ y_{ij}^{2} + Δ x_{ij}^{2}} \\ B_{ij} = \frac{1}{3} [8 Δ x_{ij} + 2 a \sinh (x_{i} / a) Δ y_{ij}] \\ F_{ij} = \frac{5}{6} [Δ y_{ij}^{2} + 4 Δ x_{ij}^{2}] \\ G_{ij} = 2 Δ x_{ij} + 2 a \sinh (x_{i} / a) Δ y_{ij} \\ H_{ij} = \sinh (x_{i} / a) Δ x_{ij} + Δ y_{ij} \\ I_{ij} = \frac{5}{2} a Δ x_{ij} Δ y_{ij} \\ E_{i} = 1 - χ_{eff} \cdot r_{p}^{3} \sum_{j \neq i}^{n} [\frac{1}{3 D_{ij}^{3}} - \frac{Δ x_{ij}^{2}}{D_{ij}^{5}}] \end{matrix}

(9)

An outline of the calculation is provided in Appendix 2.

There are two theoretically possible modes by which a particle chain can lose its stability in the static case: it can either (1) break at an arbitrary point, most likely at the middle particle, or (2) the particle chain can slide as an unit. At sufficiently high magnetic fields, the particle chain behaves as an effectively rigid body, sustained by the inter-particle magnetic forces. As the external magnetic field decreases, those forces naturally weaken. However, numerical simulations show a significant concentration of magnetic flux density at the particle interfaces (see Figure 11). This localized intensification ensures that the interaction forces remain high enough to prevent individual ruptures. Both COMSOL modeling and the numerical evaluation of the y-component of the magnetic field equation (7), balanced against gravitational forces, confirm this stability. All experiments conducted in the framework of this study align with these findings, showing that the chain does not break at discrete, random particles but rather maintains its shape and slides as a cohesive unit.

Sliding and deformation under diminishing magnetic fields

During the sliding motion, the chain acts as a rigid structure. As a result, both components of the magnetic force remain constant. The sliding of the chain is modeled in this case using Coulomb (1785)’s law of friction, which states that the friction force is proportional to the normal force acting on the structure. Mathematically, this is expressed as:

F_{f} = μ_{f} \cdot F_{w},

(10)

where $μ_{f}$ is the coefficient of friction, and $F_{w}$ represents the normal force resulting from the particle-wall interaction. This unknown force can be determined using the equilibrium of the forces in the x-direction for the first particle based on equation (6) and yields the following result:

F_{w} = F_{m x 1} = \sum_{i = 2}^{n} F_{m xi} .

(11)

Once the wall force is defined, the equilibrium of the forces for the chain can be formulated using the free body diagram (FBD) for the whole chain shown in Figure 2.

F_{f, 1} + F_{f, n} = m_{t} \cdot g .

(12)

Figure 2.

Free body diagram illustrating the interaction of magnetic, centrifugal, and gravitational forces.

Here, $m_{t}$ denotes the total mass of the particle chain and can be calculated as

m_{t} = n \cdot \frac{4 π r_{p}^{3}}{3} \cdot ρ,

(13)

where n is the number of particles per chain, $r_{p}$ is the particle radius, and ρ is the particle density. The friction forces $F_{f, 1}$ and $F_{f, n}$ are equal due to symmetry and will therefore be referred to simply as $F_{f}$ . The chain starts sliding once the resulting friction force, which decreases as the magnetic field diminishes, becomes weaker than the gravitational force. This relationship is given by the following in equation:

2 μ_{f} \cdot F_{w} < m_{t} \cdot g .

(14)

In the static case, the critical parameter that determines the stability of the particle chain is the applied magnetic field $H_{0}$ . This critical value is derived by substituting the geometric auxiliary variables into the force balance equation, resulting in the following compact form and describes the minimum magnetic field required to generate enough frictional resistance at the chamber walls to counteract the weight of the particle chain and prevent cohesive sliding:

\begin{matrix} H_{0, crit} = {(- \frac{2 μ_{0} r_{p}^{3} χ_{eff}^{2}}{n ρ g})}^{- 1 / 2} \cdot {2 μ_{f} \sum_{i = 2}^{n} \sum_{j \neq i}^{n} [\frac{B_{ij}}{D_{ij}^{5}} - \frac{F_{ij} \cdot G_{ij}}{D_{ij}^{7}}]} \cdot \\ (1 - χ_{eff} r_{p}^{3} \sum_{i = 2}^{n} \sum_{j \neq i}^{n} [\frac{1}{3 D_{ij}^{3}} - \frac{Δ x_{ij}^{2}}{D_{ij}^{5}}]) \end{matrix}

(15)

and the geometric terms $B_{ij}, D_{ij}, F_{ij}, G_{ij}$ and $Δ x_{ij}$ follow the definitions provided in equation (9).

At high chain curvatures, the critical magnetic field strength’s output exhibits bifurcation-like behavior that is identified as a numerical artifact rather than a physical phase transition. This phenomenon originates from the discrete-element formulation of the particle chains and the inherent sensitivity of the dipole-dipole summation to infinitesimal variations in the numerical mesh during significant deformation.

Since, for the experimental evaluation, measuring the magnetic flux density B is easier than directly measuring the magnetic field H, assuming a non-permeable carrier fluid, the relation given by

B = μ_{0} μ_{r} (H) \cdot H,

(16)

is used to compute the results for different values of B.

Figure 3 depicts the results of the calculation for four different particle radii and three different coefficients of friction. The particle radii chosen for the evaluation of the critical magnetic flux density do not directly align with those typically used in MRF applications. This choice was made for an efficient numerical evaluation of equation (15), specifically the various factors influencing the results, such as the coefficient of friction $μ_{f}$ , radius $r_{p}$ , and number of particles n. Selecting larger particles ensures that the computational time remains feasible for a comprehensive parametric study. Although real CIPs differ in size, identifying the individual influence of each parameter is prioritized. The graphs are plotted for a constant horizontal chain length $L = 45 mm$ , so the number of particles needs to be calculated and adjusted accordingly. The evaluation starts with a straight chain, with the minimal number of particles $n_{\min} = \frac{L}{2 \cdot r_{p}}$ .

Figure 3.

The critical value of the magnetic flux density as a function of the number of particles n, for different particle radii $r_{p}$ , coefficients of friction $μ_{f}$ , and a constant horizontal chain length of $L = 45 mm$ .

The influence of the friction force is evident. As expected, lower coefficients of friction $μ_{f}$ lead to higher critical magnetic force values. A key finding from this calculation is the core principle behind using micro-particles in MRFs: smaller particle radii result in improved sedimentation stability. As the particle radius decreases, the magnetic field required to prevent slipping also decreases. These results align closely with the experimental studies on sedimentation due to gravity conducted by Shah and Choi (2012) and Xu and Zhou (2024).

In Figure 4, the results of the numerical calculation of the critical magnetic field as a function of the number of particles are presented, where the focus is placed on the minimum magnetic flux density required for the chain to maintain its structure.

Figure 4.

The critical value of the magnetic flux density as a function of the number of particles n, for a particle radius $r_{p} = 1.5 mm$ and the horizontal length of the chain $L = 45 mm$ and $μ_{f} = 0.42$ .

One of the most significant findings is the clear correlation between the total number of particles and the magnetic field threshold required to keep the chain stable. To put this into perspective, the setup shown in Figure 4 is considered, where $r_{p} = 1$ mm and the span $L = 45$ mm. At the lower limit of $n = 22$ (point a), the chain is stretched into a perfectly straight configuration. As the number of particles increases, the magnetic field reaches a local minimum ( $n = 23$ , point b). At this specific count, the chain reaches its most stable arrangement and its lowest energy state.

As the number of particles increases further, the critical magnetic field required to maintain the chain’s form rises, reaching a maximum at $n = 28$ (point c). Beyond this point, the magnetic field required for stability begins to decrease. As previously explained, this bifurcation implies that two different chain configurations share the same critical magnetic field at which sedimentation occurs.

It was experimentally observed that chains consisting of a very large number of particles can be forced into a hyperbolic shape artificially. However, once released, the chain collapses, breaking into several shorter chains, which due to the very strong multi-chain interactions cannot be represented by the model used here. The derivation of these critical parameters, including the minimum magnetic flux density required to prevent sedimentation under gravitational forces and the corresponding optimal chain configuration in terms of particle number, is the foundation for the subsequent analysis of the system’s behavior under dynamic loading conditions.

The experimental and numerical validations of equation (15) utilized millimeter-sized particles ( $0.1$ –1 mm), however due to the assumptions underlying the derived formula are independent of absolute particle dimensions, and therefore the results remain theoretically scalable. As already discussed, the use of larger particles was a necessary simplification to reduce the computational cost. This method establishes the critical threshold for magnetic flux density, boundary conditions, and initial chain configurations required for the dynamic study. By transitioning to a continuum approach to describe the physics of micron-scale CIPs, the model achieves both computational efficiency and physical accuracy in representing the behavior of Magnetorheological Fluids (MRFs).

Sedimentation stability under dynamic loading conditions

The shape of the chain is determined using principles of continuum mechanics, analogous to the approach employed in the static case. Initially, the chain of particles is modeled as a continuum, from which an infinitesimally small element is analyzed. This chain is subjected to various forces, including gravitational forces, magnetic interactions, and centrifugal forces arising from the rotational motion of the chamber.

The system studied in the dynamic analysis is illustrated in Figure 5. The chamber is mounted on a rotor, which rotates with the angular velocity ω. Since a rotating system is considered, it is described in polar coordinates. In this coordinate system, the radial direction aligns with the length of the chamber, while the z-direction corresponds to both the width of the chain and the direction of rotation as well as the external magnetic field. The rotation of the chamber results in a centrifugal force that pulls the particle chain away from the axis of rotation. This force is decisive in maintaining the chain’s stability and in the onset of sedimentation.

Figure 5.

Free body diagram of an infinitesimal element in a rotating system.

Extending the arguments established in the static case, the particle chain resists internal deformation and instead translates as an effectively rigid structure. As a result, damping associated with oscillatory amplitudes is not relevant for the stability analysis. Moreover, the stability criterion is formulated at the onset of sedimentation, that is, when the chain begins to lose equilibrium, rather than during its following dynamic motion.

By applying Newton’s second law of motion in the z-direction, the following relationship can be derived:

\begin{matrix} dm \overset{\cdot\cdot}{r} (r, t) = F_{c, z} (r, t) - F_{c, t} (r + dr, t) \\ - dmg \cos (ω t) + dmr (r, t) ω^{2}, \end{matrix}

(17)

where $F_{c, z}$ is the tangential chain force, $dm \overset{\cdot\cdot}{r} (r, t)$ the inertial force and $dmr (r, t) ω^{2}$ the centrifugal force due to the rotation. The deformation of the chain is then obtained using separation of variables, yielding a spatial mode shape and a time-dependent oscillatory response. The spatial deformation of the chain is shown to follow a cosine-type profile (see Appendix 3), governed by a system-dependent parameter that incorporates the magnetic chain force, particle mass density, and rotational speed. For the subsequent magnetic force calculations, this cosine profile is further approximated by a parabolic shape using a second-order Taylor expansion. Error analysis demonstrates that this rigid-body approximation remains highly accurate for smaller oscillation amplitudes. As discussed, the exceptionally strong magnetic coupling between particles restricts the movement to small amplitudes. Following that, the analysis of sedimentation stability focuses solely on the chain’s sliding behavior toward the outer chamber wall.

Finally, the parameters governing the chain deformation are expressed explicitly in terms of physical system properties, such as particle size, material density, gravitational acceleration, and rotational speed. The resulting expressions form the basis for evaluating the magnetic forces acting on the chain and for determining stability criteria under dynamic loading conditions. For clarity and readability, the detailed mathematical derivation is provided in Appendix 3.

After the calculation for the configuration of the chain is complete, the magnetic force on each particle, along with the resulting magnetic force, can be evaluated using equation (1). This involves defining the position vector between two particles i andj as follows:

r_{ij} = \frac{κ^{2}}{2} (z_{i}^{2} - z_{j}^{2}) e_{r} + (z_{j} - z_{i}) e_{z} .

(18)

Here, κ denotes a system-dependent constant associated with the spatial mode of the chain. It incorporates the combined effects of the effective magnetic stiffness $F_{c}$ , the mass density $μ_{m}$ , and the imposed oscillatory excitation characterized by the angular frequency ω. Due to the implicit coupling of these parameters, κ cannot be expressed in closed form and is therefore determined numerically as shown in Appendix equation (52). The applied magnetic field vector possesses only an axial component and is expressed as:

H_{0} = H_{0} e_{z} .

(19)

The total magnetic force components in both directions, as formulated in equation (1), are determined by utilizing equation (3) and the position vector, equation (18). he magnetic force components in the radial (r) and axial (z) directions can be respectively obtained in an analogous compact form as follows:

F_{r}^{i} = A \sum_{j \neq i}^{n} J_{i} [\frac{K_{ij}}{R_{ij}^{5}} - \frac{L_{ij} \cdot M_{ij}}{R_{ij}^{7}}],

(20)

F_{z}^{i} = A \sum_{j \neq i}^{n} J_{i} [\frac{N_{ij}}{R_{ij}^{5}} - \frac{O_{ij} \cdot M_{ij}}{R_{ij}^{7}}] .

(21)

where the constant A remains the same as in the Cartesian formulation:

A = μ_{0} \cdot \frac{8}{3} \cdot H_{0}^{2} \cdot π \cdot r_{p}^{6} \cdot χ_{eff}^{2}

(22)

The interaction term $J_{i}$ is defined as:

\begin{matrix} J_{i} = 1 + χ_{eff} r_{p}^{3} \sum_{k \neq i}^{n} \\ [\frac{κ^{2}}{2} (z_{i}^{2} - z_{k}^{2}) (z_{k} - z_{i}) + {(z_{k} - z_{i})}^{2} - \frac{1}{3} R_{ik}^{2}] \end{matrix}

(23)

Unlike the nested sum $\sum_{j \neq i}^{n}$ , here k represents a dummy index used to calculate the total magnetic environment surrounding particle i.

The geometric auxiliary variables are:

\begin{matrix} Δ z_{ij = z_{j} - z_{i}} \\ R_{ij} = | {\vec{r}}_{ij} | \\ M_{ij} = κ^{4} z_{i} (z_{i}^{2} - z_{j}^{2}) - 2 Δ z_{ij} \\ K_{ij} = \frac{κ^{2}}{2} [2 z_{i} Δ z_{ij} - (z_{i}^{2} - z_{j}^{2})] \\ L_{ij} = \frac{5}{4} κ^{2} (z_{i}^{2} - z_{j}^{2}) Δ z_{ij} \\ N_{ij} = \frac{1}{2} M_{ij} - 2 Δ z_{ij} \\ O_{ij} = \frac{5}{2} Δ z_{ij}^{2} \end{matrix}

(24)

Appendix 4 provides the outline of the calculation. Under dynamic loading conditions, the primary force influencing stability is the centrifugal force, making the angular velocity ω the defining parameter for the stability criterion. This critical angular velocity can be calculated by formulating the equilibrium of forces for a single particle chain.

From the equilibrium of forces, shown in Figure 6, it follows:

\underset{\binom{radial component}{of the magnetic force}}{\underset{︸}{F_{m, r}}} + \underset{\binom{contact forces}{F_{w}}}{\underset{︸}{2 μ_{f} F_{m z, i}}} + \underset{\binom{radial component}{of gravity}}{\underset{︸}{F_{g} \cos (ω t)}} = \underset{\binom{centrifugal}{force}}{\underset{︸}{m_{t} r ω^{2}}} .

(25)

Figure 6.

Free body diagram of the particle chain under dynamic loading conditions, where e denotes the eccentricity of the chain relative to the axis of rotation.

The notations used are analogous to the previous calculation, for the static case. The calculation of the friction force is also done using Coulomb’s law of friction equation (10), where the normal contact force depends on the particle-wall interaction and therefore on the tangential component of the magnetic force (equilibrium of forces in the z-direction). Due to symmetry, the friction forces are equal.

\sum_{i = 1}^{n} F_{m, ri} + 2 \cdot μ_{f} \sum_{i = 2}^{n} F_{m z, 1} = m_{t} \cdot ω^{2} \sum_{i = 1}^{n} r_{i} .

(26)

The dynamical parabolic chain loses its stability and starts sliding as soon as:

\sum_{i = 1}^{n} F_{m, ri} + 2 \cdot μ_{f} \sum_{i = 2}^{n} F_{m z, 1} < m_{t} \cdot ω^{2} \sum_{i = 1}^{n} r_{i} .

(27)

This formulation represents the dynamic force balance at the onset of instability, where the magnetic restoration and frictional forces are exactly counterbalanced by the centrifugal load in the rotating frame. The derived formula is evaluated for different values of the friction coefficient $μ_{f}$ , as well as for three distinct particle radius-number of particles combinations in order to assess its predictive capability under varying operating conditions.

The graph in Figure 7 illustrates the dependence of the angular velocity ω, on the magnetic flux density for these varying parameters. As expected, an increase in $μ_{f}$ results in a higher frictional force, which counteracts the centrifugal force. As a result, a higher angular velocity is needed to reach the stability limit under the same magnetic flux density.

Figure 7.

ω as a function of $B_{0}$ , different n- $r_{p}$ combinations and different coefficients of friction $μ_{f}$ .

The region below each curve represents the stable regime of the magnetic chain, where the centrifugal force is insufficient to induce particle chain sliding. As long as the critical angular velocity, $ω_{crit}$ , is not reached, the particle chain remains intact. Furthermore, an increase in particle radius leads to a reduction in the critical angular velocity, which is expected since the centrifugal force is proportional to the mass of the particle chain.

Continuum approach using shear stress

While the previous analysis employed a discrete-particle model, in which the equilibrium between centrifugal and magnetic forces determines the stability of particle chains, practical MRFs contain several micrometer carbonyl iron particle chains. To address this, a continuum-based description is employed, in which the suspension is characterized by macroscopic parameters such as the shear stress, τ. The transition from discrete dipole interactions to a continuum shear-stress model is achieved by extrapolating the magnetic restoring force of a single particle chain using the theoretical number of chains per unit area ( $P_{\max}$ ), thereby averaging microscopic forces into a macroscopic yield stress. This continuum approximation remains physically valid provided the flow remains in the laminar regime as shown by Güth and Maas (2016). Here, the collective effect of the magnetic interactions is represented as a force per unit area, and chain deformation or sliding is considered to occur when this shear stress exceeds a critical threshold, $τ_{crit}$ .

Estimating the shear stress requires determining the number of particle chains per unit area. This can be approximated based on the particle volume fraction, $c_{v}$ , and particle dimensions. Following previous derivations by Güth and Maas (2016), the maximum number of chains per unit area, $P_{\max}$ , can be expressed as:

P_{\max} = c_{V} \frac{3 (2 r_{p} + 2 t + δ)}{4 π {(r_{p} + t)}^{3}},

(28)

where t denotes the particle coating thickness, and δ the thickness of the carrier fluid layer between adjacent particles. For uncoated particles, as considered in the present study, the coating thickness is set to $t = 0$ . Based on this estimate of $P_{\max}$ , the calculated magnetic force components (equation (20) and equation (21)), as derived from the discrete dipole-dipole interaction model, the macroscopic shear stress can be evaluated.

Using an analogous approach to that presented in equation (27), all previously derived forces are reformulated on a per-unit-area basis. Consequently, the critical angular velocity, $ω_{crit}$ , is computed as a function of the shear stress for different magnetic flux densities. Following Rosensweig (1985), the equilibrium equation in the rotating frame of reference, including the time-dependent gravity, is formulated as:

\nabla \cdot σ + f_{b} = 0 .

(29)

Where σ is the total stress tensor and $f_{b}$ is the sum of body forces. Breaking this down into the radial direction:

\frac{\partial τ_{rz}}{\partial z} + \underset{centrifugal force}{\underset{︸}{ρ ω^{2} R (z)}} + \underset{oscillating gravitional force}{\underset{︸}{ρ g \cos (ω t)}} = 0,

(30)

where $τ_{rz}$ is the magnetic shear stress and represents the internal restoring stress. In the continuum model, this can be modeled as a traction boundary condition at the rotor surface, adhering to the same principle as described in equation (10) for the contact faces: $z = \pm L / 2$ :

| τ_{rz} | \leq μ_{f} \cdot σ_{zz} .

(31)

The normal stress $σ_{zz}$ is the “magnetic pressure” pushing the chain against the wall. It is derived from the axial force $F_{m z}$ . Additionally, in this rotating reference frame, the gravitational force appears time-dependent as its orientation relative to the rotating axes cycles periodically.

This continuum stability criterion establishes that sedimentation begins when the macroscopically averaged centrifugal shear stress exceeds the local magnetic pressure and frictional at the rotor interface. Since the particle chains are considered effectively rigid, meaning that the inter-particle spacing remains essentially constant as ω increases. Consequently, angular velocity does not directly cause chain stretching, and the magnetic dipole-dipole interactions within each chain are maintained. Under this assumption, changes in the magnetic shear stress arise primarily from the redistribution of particle chains within the rotor chamber, for example, increasing eccentricity rather than structural deformation of the chains themselves.

As ω increases, the centrifugal force acting on each particle chain grows proportionally to $ω^{2}$ , leading to a progressive radial displacement of the chains within the chamber. This increase continues until the resisting magnetic and rheological stresses can no longer counterbalance the centrifugal load, ultimately causing the onset of sedimentation .

As shown in Figure 8, the shear stress for the initial static configuration ( $ω = 0 \frac{rad}{s}$ ) depends primarily on the applied magnetic field, with shear stress values increasing with rising flux density. These findings exhibit characteristics similar to those reported by Güth and Maas (2016). Furthermore, the coefficient of friction plays a critical role; an increase in the coefficient of friction necessitates a higher critical angular velocity, $ω_{crit}$ , for the onset of sedimentation. As ω increases, the shear stress decreases continuously across all magnetic flux densities and coefficient of friction values. This trend continues until a critical threshold is reached, at which point the chain loses stability and undergoes radial acceleration outward. The proposed analytical model assumes a linear magnetic susceptibility, which is physically valid for flux densities up to 0.4 T. For the high-field cases, the results are modified using a nonlinear saturation model as shown in Simon et al. (2001), to account for the onset of magnetic saturation. As the particles approach saturation, their incremental permeability decreases, causing the actual magnetic restoring force to deviate from linear theoretical predictions.

Figure 8.

Shear stress $τ_{crit}$ versus angular velocity ω for rp = 3.125 µm different magnetic flux densities and coefficients of friction.

Experimental validation

First, the results for the static case of loss of stability in the magnetic chain will be presented. The second part is dedicated to the experimental results obtained from the dynamic system. The experimental validation is conducted using millimeter-scale spheres and is intended as a qualitative evaluation of the model’s underlying physics. However, the results remain physically scalable to micron-scale CIPs because of the governing force ratios.

Experimental validation of the chain stability under static loading conditions

Since there are no physical constraints on the size of the particles, the experiment is conducted using particles with radii ranging from 0.5 to 1.5 mm. To generate the required magnetic field, a C-shaped electromagnet is used. The experiments are conducted according to the following schema:

Demagnetization of the C-magnet

Offset Measurements: Even though each experiment starts with the demagnetization of the particle chain, achieving absolute zero magnetization is not possible. Therefore, the offset values must be measured using a Hall probe and documented; follow these steps:

– Measure the magnetic induction in the air gap.

– Measure the magnetic induction of the spheres.

Positioning of the spheres

Gradual Reduction of current

Reaching the critical value

Measuring and documenting the magnetic field

The first result observed from the experiment is the shape of the chain, which aligns quite well with the analytical prediction as seen in Figure 9.

Figure 9.

Analytical model (a) and experimentally obtained shape of the chain by a photo (b).

The evaluation of the critical magnetic flux density is shown in Figure 10. As expected, the obtained values show good agreement with the analytical results up to a critical number of particles, which highlights the limitations of the dipole-dipole interaction model.

Figure 10.

Comparison of analytical, COMSOL-modified, and experimental results for $r_{p} = 1.5 mm$ and air gap length $L = 36 mm$ illustrating the critical magnetic flux $B_{0, crit}$ as a function of particle number.

While the qualitative behavior is consistent, a notable quantitative offset is observed between the analytical (blue) and measured values (red). As shown in the subsequent analysis, the main reason of this discrepancy is the accumulation of magnetic flux at the particle interfaces. This localized intensification of the magnetic flux density within the active region of the chain enhances the inter-particle coupling, leading to a higher threshold for stability.

To address this, COMSOL Multiphysics is employed to perform the numerical calculation of $B_{0, crit}$ . As can be seen in Figure 11, the particle chain drastically influences the distribution of the magnetic flux. The magnetic field tends to “channel” through the particles, becoming highly concentrated within the chain itself, as illustrated in the simulation of the flux distribution shown in Figure 11(b). The numerically computed magnetic flux density is evaluated along two paths: a horizontal line connecting the end particles of the chain denoted as the linear reference, and a curved path following the particle chain itself, as illustrated in Figure 12.

Figure 11.

Numerical results of the magnetic flux density: (a) 2-D numerical simulation of a particle chain within the air gap of a C-shaped electromagnet, color scheme: magnitude of the magnetic flux density |B| in T and (b) zoomed view showing magnetic flux concentration.

Figure 12.

Comparison between the baseline air gap and the enhanced magnetic flux density within the chain structure.

Based on this simulation, the average magnetic flux density along the particle chain is calculated and used to scale the B–H curve of the CIPs providing the modified results shown in Figure 10.

Experimental validation of the particle chain stability under dynamic loading conditions

As already explained, the decisive criterion for the stability of the magnetic chain under the action of a centrifugal force is the angular velocity ω. Therefore, the experiment is designed to measure the critical velocity at which the chain loses its stability. A suitable alternative for assessing the influence of angular velocity on particle chain motion is the method of measuring rotor unbalance.

Measurement methodology by rotor unbalance analysis

Rotor unbalance refers to the unequal mass distribution along the rotor’s axis of rotation, causing centrifugal forces during rotation. The effect of these forces is demonstrated through system vibrations. As described by Schreiner et al. (2022), they are defined by their magnitude ( $m_{t} \cdot e$ in gmm) and their phase angle (angular position relative to a reference mark on the rotor, in radians), thus the measurements involve complex quantities, that is, magnitude and phase, or real and imaginary components, depending on the measuring system used. Mass unbalances are noticeable in vibrations at rotational frequencies. The calculation of rotor unbalance involves a process of measurement, data processing, and visualization. The experimental setup shown in Figure 13 is described in detail in Schreiner and Maas (2025).

Figure 13.

Rotor design: setup used for experimental validation containing the electric drive, the oscillating base plate, the laser triangulation sensor, one chamber filled with MRF and mounted on the rotor, and the electromagnet (Schreiner and Maas, 2024).

A laser triangulation sensor is fixed next to the rotor to capture the movement of the rotor’s frame and measures the amplitude and frequency of the vibration. The analog output from the sensor is digitized using a dSPACE system, which includes analog-to-digital conversion (ADC) to prepare the signal for further processing. Following the signal processing methodology detailed in Schreiner and Maas (2024), the digitized signal is used to isolate the dominant frequency corresponding to the rotor’s angular velocity. Finally, the vibration amplitude and phase angle are used to define the unbalance vector, and a visual representation of the rotor unbalance is plotted on a polar graph.

The unbalance vector calculating process is done as follows:

Preparation: One rotor chamber is filled with MRF, while the other two contain solid material of equivalent mass distribution. The radial masses are adjusted to perform an initial calculation of unbalance vector performed using a dSPACE real-time platform.

Manually fine-tune the system to balance the rotor.

Baseline measurement: Begin rotation of the system at a low angular velocity (e.g. $ω = 30 \frac{rad}{s}$ ), controlled by the dSPACE system and record the reference unbalance vector, including its magnitude and angle. To isolate the effects of particle centrifugal sedimentation, the rotor was pre-balanced to a residual unbalance of $< 1 gmm$ prior to testing. The use of permanent magnets ensured a time-invariant magnetic field, while the high signal-to-noise ratio of the dSpace-based laser triangulation system ensured that the measured unbalance reflected physical mass redistribution rather than sensor drift or electronic noise.

Incremental increase in angular velocity: Gradually increase the rotor’s angular velocity step by step and monitor changes in the unbalance vector (both magnitude and angle) in real time using the dSPACE system.

Observation of sliding behavior: A noticeable change in the measured unbalance vector serves as an indicator of the onset of sedimentation. The unbalance difference $Δ U$ is the change in eccentricity multiplied by the mass:

Δ U = m \cdot (e_{2} - e_{1}),

(32)

where $e_{1}$ and $e_{2}$ denote the initial and final eccentricities, respectively.

Data analysis: The dSPACE system calculates the reference unbalance and tracks changes as the angular velocity increases. Finally, the relationship between the rotor’s speed and the unbalance changes to quantify the sliding behavior is analyzed.

The experiments were conducted to validate both the discrete and continuous models. Three different sphere radii: 0.6, 0.7, and 1 mm were investigated, along with typical MRFs, one containing CIP with radii $r_{p} = 3.125$ µm, volume fraction $c_{v} = 47 %$ , and the other containing CIP with $r_{p} = 90$ µm particles at $c_{v} = 28 %$ volume fraction (both uncoated). To ensure statistical robustness, each experimental scenario was repeated for a minimum of 18 independent trials. The main measurement uncertainty factor is the micro-displacement of mechanical components, such as the screws used for the initial pre-balancing of the rotor, which may undergo shifts under high centrifugal loading. To isolate the MRF mass redistribution process, the results were averaged across all repetitions. The experimental data points are presented with error bars indicating the standard deviation over the independent test runs. The results are summarized in Table 1.

Table 1.

Comparison of analytical and numerical results.

Particle radius	Measured $B_{0}$ in mT	Analytical $ω_{crit}$ in rad/s	Measured $ω_{crit}$ in rad/s
$r_{p} = 0.6$ mm	175	79.10	75.12
$r_{p} = 0.7$ mm	140	66.17	65.79
$r_{p} = 1$ mm	175	71.03	68.93
$r_{p} = 3.125$ µm @ 47%	140	118.78	142.06
$r_{p} = 90$ µm @ 28%	140	133.52	149.11

The comparison between the analytical predictions and the experimental results demonstrates good agreement for a discrete number of particles arranged in a single chain. In this case, the predicted critical angular velocities differ only slightly from the measured values. However, when a continuum-based model is considered, the discrepancies become more pronounced. The main factor for this increased variation is the assumption of negligible intra-chain interactions in the analytical formulation.

As a consequence of this simplification, the analytical model consistently underestimates the critical angular velocity compared to experiments as shown in Figure 14.

Figure 14.

Stability map for the continuum model depicting critical angular velocity $ω_{crit}$ as a function of magnetic flux density $B_{0}$ for two MRFs with different CIP concentrations.

Conclusion and discussion

This study focused on developing a comprehensive framework for predicting the onset of sedimentation in MRFs by integrating analytical modeling, numerical simulations, and experimental validation. The main objective was to apply the dipole-dipole interaction approach to establish a critical threshold governing the onset stability. This threshold describes the interplay among magnetic, gravitational, and centrifugal forces, and defines the conditions under which sedimentation is initiated. Stability criteria were derived by formulating the equilibrium of forces acting on the chain. In the static case, the magnetic field strength emerged as the critical parameter, because the resulting magnetic force is the only force opposing the gravitational force. The dynamic system introduced centrifugal forces that influence the shape and stability of the particle chain. In this case, the stability of the system depends on the interaction between the magnetic, centrifugal, and friction forces. The angular velocity and the coefficient of friction are critical parameters for system stability. However, since the coefficient of friction is an empirical value, depending on contact pairing, the angular velocity becomes the decisive factor in maintaining stability.

Some key findings regarding the sedimentation behavior and stability of particle chains in both static and dynamic scenarios identified in this work include:

In both static and dynamic scenarios, the optimal number of particles that minimizes the total energy of the system reveals a slight sag in the static case and small amplitudes in the dynamic case.

A decrease in particle size improves the onset of sedimentation of the magnetic particle chain, leading to enhanced stability.

Similarly higher magnetic fields provide better results regarding the onset of sedimentation.

In the dynamic case, a higher coefficient of friction contributes to increased chain stability, while an increase in angular velocity accelerates the onset of the sedimentation process.

The fundamental limitation of the implemented model is the assumption that the magnetized particles align into single chains and the distance between chains is sufficiently large; therefore, future work should focus on addressing this assumption. Semi-analytical or combined experimental and numerical studies could investigate scenarios where multiple chains interact closely, leading to collective responses and deviations from idealized alignment. Accurate determination of the friction coefficient is essential to ensure the reliability and predictive capability of the proposed approach. Future investigations may focus on extending the current framework to explicitly address these limitations, thereby improving the quantitative agreement between analytical predictions and experimental observations and enhancing the overall predictive accuracy of the model.

Footnotes

Appendix 1. Derivation of the catenary governing equations

In this appendix, the full algebraic procedure for the continuum limit of the magnetorheological particle chain is detailed.

Appendix 2. Calculation of the total magnetic field - hyperbolic chain

This appendix details the component-wise calculation of the magnetic force. The derivation follows three steps: computing the vector products, determining the field gradients, and calculating the final scalar products.

The position vector $r_{ij}$ is defined as:

(39)

\begin{matrix} r_{ij} = (x_{j} - x_{i}) e_{x} + a (\cosh \frac{x_{j}}{a} - \cosh \frac{x_{i}}{a}) e_{y} \end{matrix}

(40)

\begin{matrix} = (x_{j} - x_{i}) e_{x} + a (C_{j} - C_{i}) e_{y} \end{matrix}

The magnitude is:

(41)

| r_{ij} | = {[{(x_{j} - x_{i})}^{2} + a^{2} {(C_{j} - C_{i})}^{2}]}^{\frac{1}{2}}

The separation vector $r_{ij}$ is:

(42)

r_{ij} = (x_{j} - x_{i}) e_{x} + a (C_{j} - C_{i}) e_{y}

(43)

| r_{ij} | = {[{(x_{j} - x_{i})}^{2} + a^{2} {(C_{j} - C_{i})}^{2}]}^{\frac{1}{2}}

where $C_{j} = \cosh (x_{j} / a)$ and $C_{i} = \cosh (x_{i} / a)$ .

The local magnetic field $H_{i}$ is the vector sum:

(44)

\begin{matrix} H_{i} = H_{0} e_{x} + χ_{eff} r_{p}^{3} \sum_{j \neq i}^{n} [ \\ \frac{H_{0}}{{({(x_{j} - x_{i})}^{2} + a^{2} {(C_{j} - C_{i})}^{2})}^{\frac{5}{2}}} \cdot \\ (\frac{3 {(x_{j} - x_{i})}^{2} - ({(x_{j} - x_{i})}^{2} + a^{2} {(C_{j} - C_{i})}^{2})}{3} e_{x}) \\ + \frac{H_{0} (x_{j} - x_{i}) a (C_{j} - C_{i})}{{({(x_{j} - x_{i})}^{2} + a^{2} {(C_{j} - C_{i})}^{2})}^{\frac{5}{2}}} e_{y}] \end{matrix}

For the x-component $\frac{\partial H_{x}}{\partial x}$ :

(45)

\begin{matrix} \frac{1}{H_{0}} \frac{\partial H_{x}}{\partial x} = \frac{- 2 a (C_{j} - C_{i}) \sinh (\frac{x_{i}}{a}) - 4 (x_{j} - x_{i})}{3 {(a^{2} {(C_{j} - C_{i})}^{2} + {(x_{j} - x_{i})}^{2})}^{\frac{5}{2}}} \\ - \frac{5 (a^{2} {(C_{j} - C_{i})}^{2} + 2 {(x_{j} - x_{i})}^{2})}{6 {(a^{2} {(C_{j} - C_{i})}^{2} + {(x_{j} - x_{i})}^{2})}^{\frac{7}{2}}} \cdot \\ (- 2 a (C_{j} - C_{i}) \sinh (\frac{x_{i}}{a}) - 2 (x_{j} - x_{i})) \end{matrix}

For the y-component $\frac{\partial H_{y}}{\partial x}$ :

(46)

\begin{matrix} \frac{1}{H_{0}} \frac{\partial H_{y}}{\partial x} = - \frac{5 a (x_{j} - x_{i}) (C_{j} - C_{i})}{2 {(a^{2} {(C_{j} - C_{i})}^{2} + {(x_{j} - x_{i})}^{2})}^{\frac{7}{2}}} \cdot \\ (- 2 a (C_{j} - C_{i}) \sinh (\frac{x_{i}}{a}) - 2 (x_{j} - x_{i})) \\ - \frac{(x_{j} - x_{i}) \sinh (\frac{x_{i}}{a})}{{(a^{2} {(C_{j} - C_{i})}^{2} + {(x_{j} - x_{i})}^{2})}^{\frac{5}{2}}} \\ - \frac{a (C_{j} - C_{i})}{{(a^{2} {(C_{j} - C_{i})}^{2} + {(x_{j} - x_{i})}^{2})}^{\frac{5}{2}}} \end{matrix}

Appendix 3. Derivation of the chain shape governing equation

In a rotating reference frame, we consider an infinitesimal element dm. Applying Newton’s second law in the tangential (z) direction, the force balance is:

(47)

\begin{matrix} dm \overset{\cdot\cdot}{r} (r, t) = F_{c, v} (r, t) - F_{c, v} (r + dr, t) \\ - dm \cdot g \cos (ω t) + dm \cdot r (r, t) ω^{2} \end{matrix}

Neglecting the gravitational component, the equation simplifies to:

(48)

\begin{matrix} dm \overset{\cdot\cdot}{r} (r, t) = F_{c, v} (r, t) - F_{c, v} (r + dr, t) \\ + dm \cdot r (r, t) ω^{2} \end{matrix}

Following the transformation from discrete coordinates to a mass density parameter $μ_{m} = \frac{dm}{dl}$ , the PDE characterizing the dynamic behavior is obtained:

(49)

\overset{\cdot\cdot}{r} (z, t) = \frac{F_{c}}{μ_{m}} \cdot r^{″} (z, t) + ω^{2} \cdot r (z, t)

Appendix 4. Calculation of the total magnetic field - parabolic chain

This appendix presents the component-wise calculation of the magnetic field and force for a parabolic particle chain. The derivation follows the same three-step procedure as Appendix 2: evaluation of vector products, construction of the magnetic field, and calculation of the spatial gradients.

(62)

H_{0} = H_{0} e_{z} .

The separation vector between particles i and j along a parabolic chain is

(63)

\begin{matrix} r_{ij} = \frac{κ^{2}}{2} (z_{i}^{2} - z_{j}^{2}) e_{r} + (z_{j} - z_{i}) e_{z} . \end{matrix}

Its magnitude is

(64)

| r_{ij} | = {[{(z_{i} - z_{j})}^{2} + \frac{κ^{4}}{4} {(z_{i}^{2} - z_{j}^{2})}^{2}]}^{\frac{1}{2}} .

The scalar product with the external field reads

(65)

H_{0} \cdot r_{ij} = H_{0} (z_{j} - z_{i}) .

Multiplication yields

(66)

\begin{matrix} (H_{0} \cdot r_{ij}) r_{ij} = H_{0} (z_{j} - z_{i}) [\frac{κ^{2}}{2} (z_{i}^{2} - z_{j}^{2}) e_{r} + (z_{j} - z_{i}) e_{z}] . \end{matrix}

(67)

\begin{matrix} H_{i} = H_{0} e_{z} + χ_{eff} r_{p}^{3} \sum_{j \neq i}^{n} [\frac{(H_{0} \cdot r_{ij}) r_{ij}}{{| r_{ij} |}^{5}} - \frac{H_{0}}{3 {| r_{ij} |}^{3}}] \\ = H_{0} e_{z} + H_{0} χ_{eff} r_{p}^{3} \sum_{j \neq i}^{n} [\frac{(z_{j} - z_{i}) \frac{κ^{2}}{2} (z_{i}^{2} - z_{j}^{2})}{{| r_{ij} |}^{5}} e_{r} \\ + (\frac{{(z_{j} - z_{i})}^{2}}{{| r_{ij} |}^{5}} - \frac{1}{3 {| r_{ij} |}^{3}}) e_{z}] . \end{matrix}

(68)

\nabla = e_{z} \frac{\partial}{\partial z_{i}} .

The derivative of the distance is

(69)

\frac{\partial | r_{ij} |}{\partial z_{i}} = \frac{z_{i} - z_{j}}{| r_{ij} |} (1 + \frac{κ^{4}}{4} {(z_{i} + z_{j})}^{2}) .

Axial Component

(70)

\begin{matrix} \frac{1}{H_{0}} \frac{\partial H_{z}}{\partial z_{i}} = \frac{(z_{i} - z_{j}) [5 {(z_{i} - z_{j})}^{2} - {| r_{ij} |}^{2}]}{{| r_{ij} |}^{7}} . \end{matrix}

Radial Component

(71)

\begin{matrix} \frac{1}{H_{0}} \frac{\partial H_{r}}{\partial z_{i}} = \frac{κ^{2}}{2} [\frac{2 z_{i} (z_{j} - z_{i}) - (z_{i}^{2} - z_{j}^{2})}{{| r_{ij} |}^{5}} \\ - \frac{5 (z_{j} - z_{i}) (z_{i}^{2} - z_{j}^{2}) (z_{i} - z_{j}) (1 + \frac{κ^{4}}{4} {(z_{i} + z_{j})}^{2})}{{| r_{ij} |}^{7}}] . \end{matrix}

ORCID iD

Ina Bregu

Ethical considerations

This article does not contain any studies with human or animal participants.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data will be made available on request.

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