**Analytical Solution of the Time-Dependent Microfluidic Poiseuille Flow in Rectangular Channel Cross-Sections and Its Numerical Implementation in Microsoft Excel**

#### **Patrick Risch, Dorothea Helmer, Frederik Kotz and Bastian E. Rapp \***

Department of Microsystems Engineering (IMTEK), Laboratory of Process Technology | NeptunLab, Albert-Ludwigs University of Freiburg, 79085 Freiburg im Breisgau, Germany; Patrick.Risch@imtek.de (P.R.); Dorothea.Helmer@imtek.de (D.H.); Frederik.Kotz@imtek.de (F.K.)

**\*** Correspondence: Bastian.Rapp@imtek.de

Received: 15 April 2019; Accepted: 22 May 2019; Published: 24 May 2019

**Abstract:** We recently demonstrated that the Navier–Stokes equation for pressure-driven laminar (Poiseuille) flow can be solved in any channel cross-section using a finite di fference scheme implemented in a spreadsheet analysis tool such as Microsoft Excel. We also showed that implementing di fferent boundary conditions (slip, no-slip) is straight-forward. The results obtained in such calculations only deviated by a few percent from the (exact) analytical solution. In this paper we demonstrate that these approaches extend to cases where time-dependency is of importance, e.g., during initiation or after removal of the driving pressure. As will be shown, the developed spread-sheet can be used conveniently for almost any cross-section for which analytical solutions are close-to-impossible to obtain. We believe that providing researchers with convenient tools to derive solutions to complex flow problems in a fast and intuitive way will significantly enhance the understanding of the flow conditions as well as mass and heat transfer kinetics in microfluidic systems.

**Keywords:** microfluidics; numerical techniques; Microsoft Excel; finite di fference method; Navier–Stokes; time dependent flow; dynamic flow; initiation of flow; Poiseuille flow
