## 3 Results and Discussion
### Limitations  
To summarize, a complex theoretical model of rotational magnetic particle alignment for external homogeneous magnetic flux density magnitudes \(B_{0}\) from 5 uT to 5 T in water as an ambient viscous fluid at room temperature has been presented. It has been shown that the significance of the arbitrary viscous torque history term increases in the strong \(B_{0}\) limit. In contrast, in the weak \(B_{0}\) field limit, it diminishes, and the stochastic Brownian torque effect starts to manifest. In addition, the characteristic time of magnetic particle alignment for the entire scale of \(B_{0}\) magnitudes has been estimated. Figure 4: Least-square minimization fit of time evolution \(\varphi(t)\) for the simplest ODE model using function (18) and obtained estimates of characteristic time of magnetic particle alignment for each value of \(B_{0}\) in the range from 5 uT to 5 T shown in panels (**a**–**g**), respectively; and the (**h**) least-square minimization fit of the characteristic time dependence on magnetic flux density norm \(\tau_{\text{char}}(B_{0})\) using Equation (19). The presented models bring several simplifications and limitations to the studied system, as discussed below. The weakening of the \(B_{0}\) field causes prolongation of the time needed for rotational magnetic particle alignment. This implies increasing the argument of the exponential function in the kernel function definition with the lowering of \(B_{0}\), which is limited in the numerical realization of the \(f(x)=\exp(x)\) function. The solution of this problem is simple. The effect of the history term in the weak field is minimal and diminishes, so we do not need to solve the whole problem (SIDE model) in the weak \(B_{0}\) limit and can neglect it and focus only on the solution of the SDE model. Due to the non-locality of the history acceleration torque, the stochastic Brownian torque should be not mathematically represented as a Gaussian white random process in [PERSON] theory [53], but instead as a \"colored\" one [45]. However, the correlation in the time of the random process, due to the neglection of the history term in the weak \(B_{0}\) limit, when the stochastic Brownian torque effect manifests, diminishes. Therefore, the stochastic Brownian torque can be considered as a zero-mean uncorrelated random process. In reality, the system memory can affect the \"color\" of the Brownian random process [45; 51]. The external forces can also affect the memory of the system and thermal force as it was discussed, for example in [54; 55; 56; 57; 58]. A solution of the studied problem in this way is still missing. The particle magnetic moment of the considered models comes from the assumption of its constant value, which in reality is not fulfilled, while the used particle is paramagnetic. Therefore, the involvement of a magnetic moment saturation process in future models will be convenient. The following is available online at [[https://www.mdpi.com/article/10.3390/app11209651/s1](https://www.mdpi.com/article/10.3390/app11209651/s1)]([https://www.mdpi.com/article/10.3390/app11209651/s1](https://www.mdpi.com/article/10.3390/app11209651/s1)), Computer Source Codes S1: particle-alignment-models.py and particle-alignment-side-torques.py (developed in Python 3.8.3) zipped in a single file. Conceptualization, A.K. and O.S. ; methodology, A.K. and P.B. ; software, A.K. ; validation, A.K. ; formal analysis, A.K. ; investigation, A.K. and P.B. ; resources, I.F. and O.S. ; data curation, A.K. ; writing--original draft preparation, A.K. ; writing--review & editing, P.B. and O.S. ; visualization, A.K. ; supervision, P.B. ; project administration, I.F. and O.S. ; funding acquisition, I.F. and O.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the Slovak Research and Development Agency (APVV-19-0032), by the Slovak Scientific Grant Agency (VEGA 2/0003/20), and by the Ministry of Health of the Slovak Republic (2018/11-UKMT-7). Not applicable.