## 2 Mathematical Preliminaries
### Effects of Dielectric Fluids
#### 2.1.2 Dynamics of the Conservation of Mass and Momentum
This relationship is given by  
\[\vec{F_{net}}=m\vec{a}=m*\Delta\vec{v}/(\Delta t) \tag{7}\]  
Here, the product of the mass of the device (\(m\)) and its velocity (\(\vec{v}\)) at a specific time (\(t\)) determines the momentum of the fluid, which can be expressed through three scalar values representing the vector relationships along the X, Y, and Z axes. Figure 1c shows a staggered grid model that stores velocity components at its center. Pressure differences between corner nodes drive the rate of mass change in this configuration. Each grid analysis yields a new speed, which helps determine the successive values of the resulting forces. The force in the X direction is equivalent to the rate of mass change, written as:  
\[\vec{F}(\partial x)=\frac{\partial\vec{m}}{\partial t}(v_{2}cos\theta_{2}-v_{ 1}cos\theta_{1}) \tag{8}\]  
where \(v_{1}\) and \(v_{2}\) are speeds in a defined position. Speed change in the X direction and the force in the Y direction are analogous to the mass change rate from the speed change in the Y direction, shown below:  
\[\vec{F}(\partial y)=\frac{\partial\vec{m}}{\partial t}(v_{2}sin\theta_{2}-v_{ 1}sin\theta_{1}) \tag{9}\]  
The individual resultant force for the flow grid model is given by  
\[F_{Resultant}=\sqrt{\vec{F}(\partial x)^{2}+\vec{F}(\partial y)^{2}} \tag{10}\]  
Hence, the total force exerted on the fluid equals the momentum change rate through the DUT volume and can be calculated with Equation (11). \[F=\frac{\partial\vec{m}}{\partial t}(\vec{v}_{out}-\vec{v}_{in}), \tag{11}\]  
where \(\vec{v}_{out}\) and \(\vec{v}_{in}\) are speeds in a defined time range. Analysts reviewed each grid to derive new speeds and consecutive values of resulting forces. The rate of mass change determines the forces acting in the \(x\) and \(y\) directions. Consequently, the momentum rate of change through the DUT volume dictates the total force on the fluid, which comprises three vector components in the velocity direction. The total force \(F_{T}\) equals the sum of these forces:  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]0  
Here, the device's base area exerts the machine force (\(F_{m}\)) on the fluid. The fluid exerts the buoyant force (\(F_{B}\)) on the device's area in contact with it. Similarly, the fluid pressure exerts the drag force (\(F_{D}\)) on the device's contour. Vectors \(F_{B}\) and \(F_{D}\) oppose the velocity and are represented with a negative sign as:  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]1  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]2where \(\rho\) is the fluid density and \(V_{g}\) is the DUT volume, \(\tilde{v}_{max}^{2}\) is the maximum velocity along the \(Z\)-axis, \(\mu\) is the fluid friction coefficient, and \(A\) is the device's area in contact with the fluid. The resulting force on the \(Z\)-axis depends on the weight in the vertical mechanism parallel to the central axis of the screw, the horizontal friction coefficient in the screw, and the forces opposing linear motion [26]. Figure 2 illustrates the variables and parameters involved in the vertical load movement of our load model using the lead-screw drive [27]. A stepper motor provides the required torque to move the load through the lead screw against the thrust load on the ball nut. The necessary torque needed to move the load is given by  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]3  
where \(T_{w1}\) represents the torque against the external force and \(T_{w2}\) is the torque against the friction force. Hence, the required basic torque can be expressed as  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]4  
Isolating (\(F_{m}\)) from (16), we obtain the machine force on the \(Z\)-axis, given as  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]5  
where \(P_{t}\) is the lead screw pitch (m), \(\mu\) is the friction coefficient on the sliding surface, \(m\) is the overall load mass (Kg), and \(g\) is the gravity 9.81 m/s\({}^{2}\).