## 2 Materials and Methods
### Parameter Studies of Valve Seat Design Using FEA  
In order to investigate the impact of the geometrical dimensions of the circular trenches on the fluidic resistance of the valve seat and, hence, on the potential improvement with respect to its fluidic performance, we perform a finite element analysis (FEA). To this end, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Microvalve Design** & **Number of** & **Velve Seat Coating** & **Piezoceramic** & **Metal Diaphragm** \\
**Variant** & **Fabricated Valves** & **Valve Seat Coating** & **Thickness** & **Thickness** \\ \hline Basic design & \(n=10\) & No coating & 0.2 mm & 0.1 mm \\ Coated design & \(n=10\) & (15 \(\pm\) 2.5) \(\upmu\)m & 0.2 mm & 0.1 mm \\ High force design & \(n=5\) & No coating & 0.3 mm & 0.15 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview on fabricated NO microvalve variants. we compare the fluid flow through an axisymmetric, annular channel with concentric trenches to the flow between two unstructured flat and parallel discs. The used simulation model is shown in Figure 3 and neglects the inlet and outlet tubes. Instead, we assume a radially symmetric outflow in the direction of the trench cavities leaving along the outer rim of the annular channel, which resembles the valve chamber. As an additional simplification, we assume a constant height along the channel. These assumptions overestimate the total flow rate compared to the real geometry. However, as we are interested in the relative changes of the fluidic resistance due to geometrical changes in the trench geometries and compared to the trenchless disc geometry, this is deemed as an acceptable error. Figure 3a shows the two-dimensional simulation domain, which represents the cross-section of the partially depicted axisymmetric model of the right part of Figure 3b. The inlet is located at x = 0. The boundary at x = 2.7 mm is an open boundary. All other boundaries are defined as walls with a no-slip condition. For the inflow, the velocity field is specified to be fully developed at an average pressure level of 20 kPa, and we solve for a stationary and laminar flow of water at room temperature. To study the different effects of the geometry on the fluidic resistance of this arrangement when the microvalve is open, we vary the height of the valve chamber, as well as the height and width of the concentric trenches, and determine the fluidic resistance, which is calculated as the ratio of the pressure drop \(\Delta p\) and the flow rate \(\dot{q}\). We use the resistance of two parallel discs as a reference in order to evaluate and quantify the relative changes due to the geometric variation of the valve seat trenches. This resistance is calculated analytically to:  
\[\frac{\Delta p}{\dot{q}}=\frac{6\mu}{\pi h^{3}}\ln\frac{r_{1}}{r_{0}}, \tag{1}\]  
where \(r_{0}\) is the inner radius, \(r_{1}\) the outer radius, \(h\) the height of the valve chamber, and \(\mu\) the viscosity of the fluid. A detailed derivation of this formula is given in Appendix A. The formula is valid for small Reynold's numbers and Newtonian fluids and does not account for the trench geometry. The reason for the latter is that the presence of trenches leads to the formation of vortices, whose size and vorticity depend non-linearly on the surrounding conditions such as the fluid velocities above and below these vortices, the wall distances, as well as the viscosity of the fluid. As the incorporation of these dependencies is challenging, we deem a comparison between a trench geometry and two parallel discs to be sufficient to evaluate the effects caused by the trenches. The FEA is carried out with the software package COMSOL Multiphysics. In addition to the above-mentioned approximations, we replace the right angles at the trenches with small circular arcs.