## 3 Results and Discussion
### Patterned Window  
The concept of flat window patterned with radially symmetrical grooves has been proven to be beneficial to control liquid flow dynamics and alter pressure gradients [29; 30; 31; 32; 33; 34; 35; 36; 37; 38], as well as for the separation force reduction [10; 25; 39; 40], both theoretically and experimentally. These studies revealed that the grooves were capable of increasing flow by augmentation of the cross section of the pathway. We recently proposed a model [25] to estimate the separation force for the patterned window. The model employs the concept of an effective gap, \(h_{e}\):  
\[h_{e}=h_{0}+\left(\frac{S_{gr}}{S_{gap}}\right)^{\gamma_{ij}}h_{gr}, \tag{15}\]  
and is based on the analytical solution for a flat window. Here \(h_{0}\), \(h_{gr}\),\(S_{gr}\), \(S_{gap}\) and \(\gamma_{ij}\) stand for initial gap for the flat window, height of the grooves, total area of the bottom surfaces of the grooves, total area of the flat window, and fitting function dependent on the geometry of the grooves, respectively. The concept capitalizes on mass conservation of the fluid domain formed by the patterned window and the equivalent flat window with the larger initial gap attributed to the presence of grooves. An exact form of fitting parameter \(\gamma_{ij}\) depends on the geometry of the window. For instance, if the window is patterned with radially symmetrical grooves, it depends on the number, depth, and shape of the grooves. In particular, for rectangular and triangular grooves, \(\gamma_{ij}\) can be approximated by [25]:  
\[\gamma_{ij}=1+\varepsilon_{i}\delta_{j}+\frac{1}{2!}\big{(}\varepsilon_{i} \delta_{j}\big{)}^{2}+\frac{1}{3!}\big{(}\varepsilon_{i}\delta_{j}\big{)}^{3}, \tag{16}\]  
where \(\varepsilon_{i}=F_{1}/F_{i}\) with \(F_{1}\) and \(F_{i}\) standing for the maximal value of the separation force for a flat window estimated from (6) and the maximum value of the separation force for the patterned window obtained from the simulation/experiment, respectively. An explicit dependence of \(\gamma_{ij}\) on the window geometry is given by parameter \(\delta_{j}\). For instance, in case of radially symmetrical rectangular grooves, the dependence \(\delta_{j}\) on the number of grooves, \(n\), is given by:  
\[\delta_{j}(n)=1-e^{-\psi_{j}\frac{n}{i\alpha_{0}}}, \tag{17}\]  
Figure 5: Effect of deceleration on separation force reduction: (**a**) non-dimensional parameter \(C\) as a function of non-dimensional parameter \(D\) (**b**) for \(h_{0}=1\) mm, \(V_{0}=0.15\) mm/s, \(a=0.01..0.1\) mm/s\({}^{2}\) (**c**) for \(h_{0}=1\) mm, \(V_{0}=0.3\) mm/s, \(a=0.01..0.1\) mm/s\({}^{2}\).  
where \(n_{\infty}\) corresponds of the window completely covered with grooves and \(\psi_{j}\) is the fitting parameter. Once the concept of effective gap was implemented, an expression for the separation force becomes:  
\[F_{s}=\frac{(V_{0}+at)\pi R^{4}}{4\beta\left[h_{0}+\left(\frac{S_{ gr}}{S_{gap}}\right)^{\tau_{ij}}h_{gr}+V_{0}t+at^{2}/2\right]^{3}\left[1+24\sum_{n=1}^{ \infty}\frac{((-1)^{n}-1)^{2}}{\pi^{4}n^{4}}e^{-\left(\frac{\pi n}{h_{0}+\left( \frac{S_{gr}}{S_{gap}}\right)^{\tau_{ij}}h_{gr}+V_{0}t+at^{2}/2}\right)^{2}vt} \right]^{2}vt} \tag{18}\]  
Table 1 summarizes all the parameters introduced in this study. There are two types of parameters: constants and functions. Elevation speed, \(V\), and gap height, \(h\) are both time-dependent.  
Finally, we would like to emphasize the following. As mentioned above, multiple times, this study is a development of our previous one [25]. The latter provided a numerical verification of the proposed method. An expression for the separation force derived in this study has the functional form similar to that in [25]. Thus, we consider the approach developed here sufficiently justified. An interested reader can refer to [13], where an experimental approach for measuring the separation force in a constrained-surface SL process is described in detail.