**5. Conclusions**

Solutions to the simplified Navier–Stokes equation in pressure-driven microfluidics, i.e., Poiseuille flows, are di fficult to derive analytically if the channel cross-sections, the boundary conditions or the initial values do not represent trivial cases. Including time-dependency further complicates the derivation of the analytical solutions. These cases are usually addressed using numerical solvers and dedicated software packages. However, as we have shown, a numerical solver suitable for solving time-dependent microfluidic flow cases in arbitrary channel cross-sections can be conveniently implemented using spreadsheet analysis tools such as, e.g., Microsoft Excel. By making use of a controlled iterative calculation, the solver can be stepped in time by manual input thus allowing a precise study of the evolving velocity profiles over time at discrete time points. No additional software is required for obtaining results that almost exactly correspond to the precise analytical solution. We demonstrated this using the case of time-dependent initiating flow in a rectangular channel cross-section. The spreadsheet developed can be used to implement almost any type of boundary condition or initial condition, as well as channel cross-section as required. We believe that providing researchers with intuitive and widely accessible numerical tools will significantly increase the understanding and the correct derivation of the fluid mechanics in microfluidics with implications for applications in liquid delivery, reaction synthesis and analytical applications [9–11]. Spreadsheet software packages such as, e.g., Microsoft Excel are widely available and most scientists are used to working with these tools in routine lab work. Using these tools e ffectively to provide such detailed insight into fluid mechanics will significantly widen the application range and provide more detailed understanding of phenomena which are generally only accessible with specialized software packages.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2079-6374/9/2/67/s1, Microsoft Excel spreadsheet "TimedependentMicrofluidicFlows.xlsx".

**Author Contributions:** Conceptualization, P.R., F.K. and B.R.; methodology, P.R., D.H. and B.E.R.; software, B.R.; writing—P.R. and D.H.; writing—review and editing, B.R.; visualization, F.K. and B.R.; supervision, D.H. and B.R.; project administration, B.R.; funding acquisition, B.R.

**Funding:** This work was funded by the Bundesministerium für Bildung und Forschung (BMBF), funding code 03X5527 "Fluoropor"and by the Baden-Wuerttemberg Stiftung, program "biofunctional materials and surfaces".

**Conflicts of Interest:** The authors declare no conflict of interest.
