Some isoperimetric results concerning undirectional flows in microchannels

Three isoperimetric results are treated. (i) At a given pressure gradient, for all channels with given (cross-sectional) area that which maximises the steady flow $Q_{\rm steady}$ has a circular cross-section. (ii) Consider flows starting from prescribed initial conditions developing from a prescribed imposed pressure gradient, either periodic or steady. For such flows, amongst all channels with given area, that which generically has the slowest approach to the long-term, periodic or steady, flow is the circular disk cross-section. (iii) Similar results for polygonal, $n$-gon, channels, with the optimising shape being the regular $n$-gon are discussed