*3.2. Iteration*

As discussed, the numerical scheme of Equation (9) does not require iteration within one time point. This is in contrast to the schemes implemented in [6], which require the steady-state version of the Poiseuille Equation (2) to be fulfilled for each position within the domain. Here, the scheme is required to perform one step in time but not to iterate further.

However, due to the nature of the scheme, we require circular references in the spreadsheet, which means we have to allow iteration. In Microsoft Excel, select "File→Options→Formulas→Calculation options" and check "Enable iterative calculation". Set "Maximum Iterations" to 1. This ensures that the scheme will only perform one single iteration.

#### *3.3. Implementation of the Numerical Scheme*

The scheme is implemented in the spreadsheet by taking the initial value from the left panel and copying the values into the center panel at the beginning of the calculation. The scheme then uses these values to calculate *F*(*<sup>t</sup>*+1,*y*,*<sup>z</sup>*) in the right panel from the values *F*(*<sup>t</sup>*,*y*,*<sup>z</sup>*) of the center panel. For the next step in time, the value of the right panel is copied back into the respective cell of the center panel. This is the single iteration which Microsoft Excel will perform—the value will not be updated further. The formulae of the cells in the right panel implement the numerical scheme. They use the value *F*(*<sup>t</sup>*,*y*,*<sup>z</sup>*) as well as *<sup>F</sup>*(*<sup>t</sup>*,*y*+1,*z*), *<sup>F</sup>*(*<sup>t</sup>*,*y*−1,*z*), *F*(*<sup>t</sup>*,*y*,*z*+<sup>1</sup>) and *F*(*<sup>t</sup>*,*y*,*z*−<sup>1</sup>) from the center panel, as well as the values Ω and Γ. For each step in time, the scheme will update the values in the right panel from the values of the center panel and write these values back to the center grip. By pressing F9 or performing any recorded input in Microsoft Excel an additional step in time will be performed. Below the center and the right panel are iteration counters that increment any time an input key or F9 (which triggers a spreadsheet recalculation) is recorded. By keeping F9 pressed, the evolution of the flow profile in time can be observed. Each step correlates to a step in time of *ht*. An additional field is added to calculate the total number of microseconds passed since the beginning of the calculation.

#### *3.4. Resetting the Calculation and Implementing the Boundary Conditions*

Upon close inspection, the cells in the center panel do not simply copy the values from the right panel. They are linked by a conditional expression. If a certain field below the center panel (the field labeled "Reset") is empty, the value from the right panel is copied. If the "Reset" field is not empty, the value from the left panel is copied. This e ffectively resets the calculation and also clears the iteration counters, which are implemented with a similar conditional copy operation. Writing any letter, value or number into the "Reset" field will thus reset the calculation and copy the initial conditions into the center panel corresponding to the velocity profile at *t* = 0.

#### **4. Analytical Solution for Initiating Two-Dimensional Flow in Rectangular Channel Cross-Sections**
