2.2.3. Numerical Aspects

Simliar to the work from Cai et al. [21], OpenFOAM-<sup>R</sup> was applied to solve the above system of equations numerically with a finite-volume method. Spatial derivatives are approximated by a high-resolution scheme (Gamma scheme) and time integration is performed by a second-order two time level backward scheme (Gear's method). The time step is chosen such that the maximum Courant number is 0.1. For further details see [21]. As suggested by Zolper et al. [23], PAO is considered to behave as a Newtonian fluid. In OpenFOAM-<sup>R</sup> , it is also possible to model fluids with non-Newtonian behavior. However, the multiphase character of the problem in this study makes this more challenging, see for example [24]. Nevertheless, Niethammer et al. [25] show a rigorous treatment of a gas–liquid system with a non-Newtonian liquid phase. In this study, focus is laid on the steady state solution which does not depend on the transient viscosity behavior.

#### *2.3. Experimental Methods*

#### 2.3.1. Laser Surface Texturing

For the DLIP texturing process, an ultrashort pulsed laser (Trumpf Trumicro 5 × 50) was used, with a wavelength of 1030 nm and a pulse length of 6 ps. The base repetition rate was 400 kHz. A DLIP optical head (Fraunhofer IWS, Dresden, Germany) is installed to automatically create the interference pattern. The setup of the optical head has been already published elsewhere [26]. In this study, the interference of two coherent laser beams was used, which leads to a line-like structure. The objective for the laser texturing is to obtain textures with a period of 4.0 μm and 8.0 μm with two depths for each period and the constraint that one depth should be comparable for both periods. The final period can be adjusted by fixing a pyramid position inside the optical head which influences the optical path. Since the optical head is installed in a fixed position the samples are moved on a linear axis system. They are moved with a speed of 25 mm/s in a meandering shape with an offset given by the hatch distance. This distance needs to be a multiple of the period in order to avoid destruction of the textures. In Table 1, the laser parameters applied to create the DLIP textures are outlined. The laser parameters need to be adapted for the different textures to guarantee good quality because they all influence each other. For example, S2 and S3 should have a period of 8.0 μm and 4.0 μm, respectively, but at the same time the depth should be the same. Therefore, if all other parameters stay the same, the pulse energy must be reduced if the hatch distance for the 4.0 μm period is smaller. As a summary, the most uniform line-like textures can be produced for low pulse energies with small hatch distances and a comparably small pulse overlap. The reservoir and the scale on the fluid transport samples were also textured with an ultrashort pulsed laser (Trumpf Trumicro 5070 Femto Edition).


**Table 1.** Laser parameters for the structures evaluated in this study (S1–S4).
