## 3 Results
### Volumetric Force Estimation  
Thanks to the linear behavior of the simulated crystal translational velocity as a function of the volumetric force [9], a linear regression was able to estimate a good approximation of the required volumetric force which generated the experimentally observed crystal velocities. In order to calculate the straight-line equation \(y=mx+b\), we used two points. The first point is obtained from the static simulation, where the drag force for a static body (zero velocity) is obtained. The second point comes from the time-dependent simulation, where drag force is minimum and velocity is maximum. The dependent variable is the drag force, and the independent variable the velocity of the body. In this way, we can calculate the required drag force from a given velocity (e.g., estimated from video). For a flow rate of 40 \(\mu\)L/min, the first point is (\(v_{1}\), \(F_{D1}\)) = (0 \(\mu\)m/s, \(2.13\times 10^{-9}\) N) and the second point (\(v_{2}\), \(F_{D2}\)) = (1717.7 \(\mu\)m/s, 457.6 \(\times\) 10\({}^{-15}\) N). The straight line equation yields  
\[F_{D}=-1.24\times 10^{-12}\left[\frac{\text{Ns}}{\mu\text{m}}\right]v+2.13 \times 10^{-9}[\text{N}]. \tag{7}\]  
From recorded video (described in Section 3.4), we take one of the nine recorded events, and a translational velocity of 324.8 \(\mu\)m/s is estimated. Substituting this velocity in Equation (7) yields an opposing force of \(1.73\times 10^{-9}\) [N], or equivalently a _volumetric force_ of 102,359.98 [N/m\({}^{3}\)] = \(\frac{1.73\times 10^{-9}[\text{N}]}{15\times 15\times 15\times 10^{-18} \text{m}^{3}}\). When this volumetric force is applied in the time-dependent simulation, it should decrease the free body velocity from 1717.7 \(\mu\)m/s down to 324.8 \(\mu\)m/s. Using this volumetric force estimation resulted in a slightly slower velocity of 287.9 \(\mu\)m/s. A few attempts with smaller values of opposing force yields a value closer to the desired velocity. We found that a volumetric force of 100,500 [N/m\({}^{3}\)] reduced the velocity to 318 \(\mu\)m/s. The results obtained for the different flow rates are summarized in Table 3.  
Drag forces for flow rates of 60 \(\mu\)L/min and 80 \(\mu\)L/min can be estimated by Equations (8) and (9), respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Flow Rate Q** & **Flow Velocity (m/s)** & **Drag Force (N)** & **Torque (Nm)** \\ \hline
40 & \(3.33\times 10^{-3}\) & \(2.13\times 10^{-9}\) & \(6.49\times 10^{-15}\) \\
60 & \(5\times 10^{-3}\) & \(3.2\times 10^{-9}\) & \(9.75\times 10^{-15}\) \\
80 & \(6.67\times 10^{-3}\) & \(4.27\times 10^{-9}\) & \(13.01\times 10^{-15}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of the fixed crystal simulation.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Simulation**} & \multicolumn{3}{c}{**1 Event of Experimental Data**} \\ \hline
**Flow Rate Q** & **Velocity** & **Opposing** & **Est.** & **Force Error [FOOTNOTE:]Footnote : The force that caused the simulated velocity shown in second column of Table 3. \({}^{2}\) The estimated force from Equation (7). \({}^{3}\) Percentage error between (1) and (2). \({}^{4}\) Percentage error between experimental and simulated velocities.[ENDFOOTNOTE]** & **Velocity** & **Angular** \\
**[\(\mu\)L/min]** & **[\(\mu\)m/s]** & **Force ** & **[\(\mu\)m/s]** & **Error ** \\ \hline
40 & 318 & \(1.69\times 10^{-9}\) & \(1.73\times 10^{-9}\) & \(-2.16\) & 324.8 & \(-2.9\) & 900 \\
60 & 411.9 & \(2.62\times 10^{-9}\) & \(2.7\times 10^{-9}\) & \(-2.85\) & 415 & \(-1.15\) & 1351 \\
80 & 589.7 & \(3.37\times 10^{-9}\) & \(3.52\times 10^{-9}\) & \(-4.38\) & 599.4 & \(-1.61\) & 1621 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulated velocities with opposing force vs. experimental data.  
\[F_{D}=-1.2\times 10^{-12}\left[\frac{\text{Ns}}{\mu\text{m}}\right]v+3.2\times 10^{-9} \left[\text{N}\right] \tag{8}\]  
\[F_{D}=-1.24\times 10^{-12}\left[\frac{\text{Ns}}{\mu\text{m}}\right]v+4.27\times 10^{-9 }\left[\text{N}\right] \tag{9}\]