Could you post some information about the topic that you mention related to the increase of viscosity?
Apart from the solutions proposed by and I can think of no alternative to solve your flow with maximum accuracy.
If you do not have not enough computational resources, you could try to create gaps of different sizes (of course, bigger than 10 microns) and then obtain the desired quantities in each of these cases. Based on the obtained results, you could make correlations and estimate the output in the 10 micron case.