## 3 Results and Discussion
### Parameter Study. Impact of Microstructured Valve Seat  
The impact of the trenches on the flow field is an essential element of the working principle of the NO microvalve. We perform a detailed FEA parameter study in order to evaluate the fluidic resistance of the valve comprising the valve seat trenches for varying channel height, trench width, and trench depth. Figure 4 provides an overview of the dependencies of the fluidic resistance on the varied geometric parameters. The dependency of the fluidic resistance on the channel height is presented in Figure 4a, which we obtain by varying the channel height from 5 \(\upmu\)m to 200 \(\upmu\)m, while other geometric parameters are set to the corresponding values of the manufactured design. For larger channel heights, the resistance converges against the analytically calculated resistance given in Equation (1). As shown in Figure 5, vortices develop inside the trenches, but their impact on the fluidic resistance becomes less relevant as the channel height increases. The channel height itself is not a static design parameter of the valve but ultimately varies with the actuation of the piezoceramic actuator as well as with the inlet pressure depending on the elastic constants of the actuator diaphragm; hence, the simulations provide insight into the influences of the trenches in the microvalve's open state. The results indicate that this sealing concept is suitable for functional open state flow through the valve, as this trench design does not impede open state flow by an eventually increased fluidic resistance. Figure 4: (**a**) Fluidic resistance as a function of the height of the channel above the trenches. (**b**) Fluidic resistance depending on the depth of the trenches; the analytically calculated resistance at zero depth amounts to 3.9 \(\times\) 10\({}^{9}\) kg/m\({}^{4}\)s. (**c**) Fluidic resistance for varying width of the trenches. Again, the resistance at zero width amounts to 3.9 \(\times\) 10\({}^{9}\) kg/m\({}^{4}\)s.  
Figure 4b shows the dependency of the fluidic resistance on the depth of the trenches at a channel height of 75 \(\upmu\)m. The width of the individual trench is 100 \(\upmu\)m. It reveals that the width of 100 \(\upmu\)m limits the size of the arising vortices contributing to the resistance. Figure 6a shows the typical flow for shallow trenches with a larger width than depth. In this case, small vortices arise only in the trench corners, and the flow is mostly uniform. Increasing the depth, the fluidic resistance decreases until a depth of the trenches of about 150 \(\upmu\)m (c.f. Figure 4b). For this specific trench geometry, the first vortices are now fully developed. Increasing the depth even further leads to more vortices; however, these also affect each other and exert a shear-stress-based resistance towards the neighboring vortices, as can be seen in Figure 6b. The vortices become weaker the deeper they are located in the trench. Due to this, the resistance does not change much for larger trench depths and saturates at an about 20% lower value than the analytically calculated value for a channel between two parallel discs without trenches. The oscillating behavior of the fluidic resistance in Figure 4b can be explained by the discrete nature of the vortices, as fractional vortices cannot exist. In the beginning, the resistance becomes smaller with deeper trenches, as the fluid experiences less friction with the walls and is less confined. Once the depth reaches the same value as the trench width, the first vortex is formed, causing an increase in resistance. Increasing the depth further leads to a smaller resistance at first until the next vortex establishes. However, the more vortices are created inside the trench, the less they contribute to the overall resistance, as they interact more and more with each other than with the main flow field. Finally, the effect of the trench width on the fluidic resistance is depicted in Figure 4c.