## 2 Background
### Measurement of Viscosity  
The viscosity of a fluid can be measured through the falling-sphere method, where a sphere of known size and density is dropped into a fluid [22]. The motion of a sphere falling through a viscous fluid in a tube is governed by several forces, primarily gravity, buoyancy, and viscous drag [23]. The forces acting on a sphere of radius \(r\) are depicted in Figure 1. The dynamics are described using Stokes' law and the equations of motion for the sphere. It must be noted that for the description of the system, laminar flow, isotropic and Newtonian fluid with constant properties is assumed. Given the experimental approach conducted in this work, it is worthwhile mentioning that classical Stokes laws do not account for potential nonlinear effects that may arise at high concentrations of the ferromagnetic particles included in the carrier fluids, or anisotropies in the magnetic field applied. Therefore, the validity of these equations is primarily limited to conditions where the concentration of ferromagnetic particles is low enough to minimize collective interactions and nonlinear effects and when the applied magnetic field remains uniform. The gravitational force acting on the sphere is given by:  
\[F_{g}=m\cdot g=V\cdot\rho_{s}\cdot g \tag{1}\]  
where \(V\) is the volume of the sphere, \(\rho_{s}\) is the density of the sphere, and \(g\) is the acceleration due to gravity. The volume of the sphere is calculated according to the equation:  
\[V=\frac{4}{3}\cdot\pi\cdot r^{3} \tag{2}\]  
The buoyant force is the upward force exerted by a fluid on an object that is either partially or fully submerged in it. This force occurs because of the pressure difference between the top and bottom of the object caused by the fluid's weight. It is given by the equation:  
\[F_{b}=V\cdot\rho_{f}\cdot g \tag{3}\]  
where \(\rho_{f}\) is the density of the fluid. The viscous drag force is the resistive force exerted by the fluid on the sphere moving through it. This force is a result of the viscosity of the fluid, which is a measure of its resistance to flow or deformation. The viscous drag opposes the motion of the object and increases with the object's speed and the fluid's viscosity. For low Reynolds numbers (\(Re<1\)), the viscous drag of a sphere submerged in the fluid is given by [PERSON] law:  
\[F_{d}=6\cdot\pi\cdot\eta\cdot r\cdot v \tag{4}\]  
where \(\eta\) is the dynamic viscosity of the fluid, and \(v\) is the velocity of the sphere. The viscosity of a fluid can be experimentally determined [23]. Figure 1: Gravity, buoyancy, and viscous drag forces acting on a sphere. The equation of motion of the sphere is governed by the net force acting on it, which, in turn, determines its acceleration. \[F_{g}-F_{b}-F_{d}=m\cdot\frac{d\
u}{dt} \tag{5}\]  
Substituting the forces:  
\[V\cdot\rho_{s}\cdot g-V\cdot\rho_{f}\cdot g-6\cdot\pi\cdot\eta\cdot r\cdot v=V \cdot\rho_{s}\cdot\frac{dv}{dt} \tag{6}\]  
The sphere submerged in a viscous fluid reaches a terminal velocity, which is the constant speed it achieves when the net force acting on it is zero and, therefore, the acceleration equals zero. This occurs when the downward gravitational force is exactly balanced by the upward buoyant force and the resistive viscous drag force:  
\[v_{t}=\frac{2r^{2}\cdot\left(\rho_{s}-\rho_{f}\right)\cdot g}{9\cdot\eta} \tag{7}\]  
This equation shows that the terminal velocity increases with the square of the radius of the sphere, with the difference in density between the sphere and the fluid and with the inverse of the viscosity of the fluid. Before reaching terminal velocity, the motion can be described by the following expression:  
\[\rho_{s}\cdot\frac{dv}{dt}=\left(\rho_{s}-\rho_{f}\right)\cdot g-\frac{6\cdot \pi\cdot\eta\cdot\tau}{V}v \tag{8}\]  
Substituting the term of the volume of the sphere, operating and simplifying further:  
\[F_{g}=m\cdot g=V\cdot\rho_{s}\cdot g \tag{1}\]0  
The term that accompanies the velocity \(v\) can be renamed as \(\tau\). This is a first-order linear differential equation.