## 2 Mathematical Preliminaries
### Effects of Dielectric Fluids
#### 2.1.2 Dynamics of the Conservation of Mass and Momentum  
The mass, density, and volume dynamics directly influence the behavior of an IC package submerged in a dielectric fluid. Unlike the DUT, the dielectric fluid can change its shape, but the volume of liquid displaced by the DUT remains constant. As the DUT mass remains constant, the product of density and volume also remains unchanged. When the fluid changes direction upon contact with the moving DUT, a force acts from the center to the corners of the DUT where the fluid bends [19]. To calculate the displaced fluid amount, multiply the device's submerged volume (\(V\)) by the fluid's density (\(\rho\)). Laminar turbulence occurs and bends the fluid at contact points where the displaced fluid exceeds the device's contact angles by 180 degrees, altering the device's original position. Determining the force that the DUT can withstand before misalignment occurs is crucial. Various factors determine the resulting thrust force on the IC. These include the device's weight, the interaction between the friction coefficient and fluid pressure in the horizontal direction, and the frictional forces that oppose rectilinear movement. This analysis requires a flow control model in a coordinate axis system, as referenced in [20; 21; 22]. Figure 1a shows a model that analyzes the dielectric flow around the base of a SOIC package, illustrating the force variation from laminar flow as the package moves within the fluid. Similarly, Figure 1b demonstrates the interaction between the laminar flow and the lower IC surface during downward movement. A grid representation of the lower part of the IC enables the identification of consecutive velocity and pressure values, which aids in solving the steady-state incompressible Navier-Stokes equations by treating the flow as compressible [23]. Each analysis yields a new speed, which helps to determine the consecutive values of the resulting forces. The force in the \(x\) direction is equivalent to the rate of mass change, as expressed in Equation (3),  
\[\vec{F}(\partial x)=\frac{\partial\vec{m}}{\partial t}(v_{2}cos\theta_{2}-v_{ 1}cos\theta_{1}), \tag{3}\]  
where \(v_{1}\) and \(v_{2}\) are speeds in a defined position. In a collision between the DUT and the dielectric fluid, the impulse (\(J\)) acting on the DUT is equal to the change in the device's linear momentum in time \(\Delta t=t_{f}-t_{i}\) and can be expressed as the area under the curve of the function \(\vec{F}(t)\)[24]. The impulse (\(J\)) is given by  
\[J=\int_{t_{i}}^{t_{f}}\vec{F}(t)dt \tag{4}\]  
Utilizing [PERSON]'s second law of motion [25], the net force acting on an object is determined as the summation of all vector forces applied to it, expressed by the following equation:  
\[\vec{F_{net}}=\Delta\vec{P}*(\Delta t)^{-1}, \tag{5}\]  
where the net force (\(F_{net}^{*}\)) in the DUT is directly associated with the change in fluid momentum \(\vec{P}\) over time (\(t\)). Evaluating the momentum between the initial moment \(\vec{P}_{i}\) and the final moment \(\vec{P}_{f}\) in Equation (4), we obtain the change in fluid momentum \(\vec{P}\):  
Figure 1: Analysis of linear momentum: (**a**) pressure coefficient and velocity contours, (**b**) IC isometric model with laminar flow fluid behavior, and (**c**) resulting force from stream function by the pressure and speed grid model. \[\Delta\vec{P}=J=\vec{P}_{f}-\vec{P}_{i} \tag{6}\]  
The momentum remains constant if no external forces act over the DUT. Internal forces (\(\vec{F_{net}}\)) can change the momentum \(\Delta\vec{P}*(\Delta t)^{-1}\rightarrow\vec{P}_{f}=\vec{P}_{i}\) on parts of the system, but they cannot change the total momentum of the entire system. The strain rate in a fluid over time depends on the discrete applied shearing stress and its relation to the velocity gradients.