## 3 Results and Discussion
### Comparison of Models  
Simulations were performed for four different combinations of the considered effects and these were denoted as four different models, as shown in Table 2, where the abbreviated notation of each model is explicitly defined. The magnitude of the external homogeneous magnetic flux density field \(B_{0}\) was used for each model with a scale of 5 \(\upmu\)T to 5 T. The obtained results are shown in Figure 1.  
From the solutions shown in Figure 1, a strong dependence for the speed of magnetic particle alignment on the magnitude of the magnetic flux density acting on the particle can be seen. For strong fields, the process of magnetic alignment is rapid and slowed down with decreasing \(B_{0}\). Strong magnetic fields acting on the magnetic particles in water as the ambient fluid at a temperature of 293.15 K cause rapid magnetic particle alignment with a characteristic timescale of the order of microseconds. In weaker fields, this time rises significantly by approximately one order of magnitude with each decreasing order of magnitude of the external magnetic flux density field. It can be seen that the time evolution of the angle and angular velocity for the strongest \(B_{0}\) field and for all modeled combinations of considered effects exhibit characteristic oscillations and nearly exponential decaying behavior. With a decrease in the magnetic flux density \(B_{0}\) field, the oscillations disappear even though the almost exponential decaying behavior persists. For the SDE and SIDE models, weakening the external \(B_{0}\) fields stochastic Brownian torque results in a random pattern in the evolution of the angle and angular velocity with the appearance of stochastic jumps, firstly for \(\omega(t)\) and also for \(\varphi(t)\) for the weakest \(B_{0}\) fields (see Figure 1e-g). \begin{table}
\begin{tabular}{l c c l} \hline \hline
**Parameter** & **Symbol** & **Value** & **Unit** \\ \hline Finite timestep & \(h\) & \(10^{-8}\) & s \\ Boltzmann constant & \(k_{\text{B}}\) & \(1.3807\times 10^{-23}\) & J K\({}^{-1}\) \\ Thermodynamic temperature & \(T\) & 293.15 & K \\ Fluid a dynamic viscosity & \(\eta\) & \(10^{-3}\) & Pa s \\ Fluid mass density & \(\rho\) & 1000 & kg m\({}^{-3}\) \\ Fluid kinematic viscosity & \(v\) & \(\eta/\rho\) & m\({}^{2}\) s\({}^{-1}\) \\ Magnetic flux density norm & \(B_{0}\) & \(5\times 10^{-6}\) & \(\dots\) & T \\ Particle b diameter & \(2R\) & \(10^{-6}\) & m \\ Particle mass density & \(\rho_{\text{P}}\) & \(1\,792\) & kg m\({}^{-3}\) \\ Particle volume & \(V_{\text{P}}\) & \(\frac{4}{3}\pi R^{3}\) & m\({}^{3}\) \\ Particle weight & \(m_{\text{P}}\) & \(V_{\text{P}}\rho_{\text{P}}\) & kg \\ Particle moment of inertia & \(I_{\text{P}}\) & \(\frac{5}{2}m_{\text{P}}R^{2}\) & kg m\({}^{2}\) \\ Particle saturation magnetization & \(M_{\text{sp}}\) & \(43.0\times 10^{3}\) & A m\({}^{-1}\) \\ Particle magnetic moment c  & \(\mu_{\text{P}}\) & \(2.25\times 10^{-14}\) & A m\({}^{2}\) \\ Stokes coefficient & \(\gamma\) & \(8\pi\eta R^{3}\) & kg m\({}^{2}\) s\({}^{-1}\) \\ \hline \hline \end{tabular}  
* Water as an ambient fluid. b Parameters of commercially available magnetic particle MyOne 1 \(\upmu\)m (ThermoFisher Scientific, Waltham, MA USA, Dynabeads(tm), MyOne(tm)). c Magnetic moment calculated as \(I_{\text{P}}=M_{\text{sp}}V_{\text{P}}\).  
\end{table} 
Table 1: Values of parameters used in simulations. \begin{table}
\begin{tabular}{l l l} \hline \hline
**No.** & **Torques Involved** & **Notation** \\ \hline (i) & inertial, magnetic, quasi-steady viscous, stochastic Brownian & SDE a \\ (ii) & inertial, magnetic, arbitrary viscous, stochastic Brownian & SIDE b \\ (iii) & inertial, magnetic, quasi-steady viscous, no stochastic [PERSON] & ODE c \\ (iv) & inertial, magnetic, arbitrary viscous, no stochastic Brownian & IDE d \\ \hline \hline \end{tabular}  
* Stochastic differential equations. b Stochastic integro-differential equations. c Ordinary differential equations.  
\end{table} 
Table 2: Model notation with specified torques involved in simulations.