*2.4. Correcting Units*

Before implementing the numerical scheme we must correct for the units. We assume *hxy* to be given in μm, Δ*p* Δ*L* to be given in mbar/mm, ρ to be given in g/cm3, η to be given in mPa·<sup>s</sup> and *ht* to be given in μs. The unit of the dependent variable *vx* is mm/s and the independent variables *y* and *z* are given in μm. In order to correct for the units, Γ has a prefactor of 0.1, whereas Ω has a prefactor of 1.

#### **3. Implementation in Microsoft Excel**

#### *3.1. Layout of the Spreadsheet*

The scheme given by Equation (9) was implemented in Microsoft Excel in a spreadsheet, which can be downloaded from the supporting material (file "TimedependentMicrofluidicFlows.xlsx"). It is shown in Figure 1. The numerical domain was chosen as a 40 × 40 cell grid panel with no-slip boundary conditions. As demonstrated in an earlier contribution, di fferent boundary conditions can be implemented such as, e.g., flip or Neumann-type boundary conditions [7]. The values in the cells represent the velocity of the flow at the given position in the domain. The sheet consists of three panels:


The panels are color-coded to reflect areas of higher velocity in red and areas of lower velocity in green. Next to the right-most panel, the color scale for the velocity profile for the right-most panel, i.e., *F*(*<sup>t</sup>*+1,*y*,*<sup>z</sup>*) is displayed. Below the color scale is the section for the variables. These values can be changed to modify, i.e., the type of fluid or the properties of the numerical scheme. The numerical scheme is corrected for the following units:

• independent variables *y* and *z*: μm


Changing the value of the step width in space effectively increases the lateral dimension of the channel. Changing the value of the step width in time increases the speed of the calculation by assuming larger steps in the forward Euler scheme. However, as discussed, increasing this value may lead to the numerical scheme becoming unstable. This can be observed by the values of the velocity increasing continually until they overflow. Below the adjustable variables are the two variables used as an abbreviation in Equation (9), i.e., Ω and Γ which are updated dynamically.

**Figure 1.** View of the Microsoft Excel spreadsheets with the three panels: initial conditions (left), current time point (center) and next time point (right). (**a**) The evolution of the velocity profile can be observed by pressing the F9 key. The right panel implements Equation (9) and steps forward in time. The values are copied back to the center panel thus performing one iteration. (**b**) By adding any value into the "Reset" field, the scheme is reset, the iteration counter is cleared and the values of the initial condition (left panel) is copied into the center panel thus setting the velocity profile for time point *t* = 0.
