Let us have the following system where we put a viscous fluid from the upper inlet and we want this fluid NOT to enter the lower chamber thanks to adhesion & cohesive forces.
What is the max. allowed size of the pores of width $w_1$ that would still prevent the viscous fluid from passing through to the lower chamber?
Addendum:
This problem has arisen from an actual experimental setup, and there the fluid is actually a gel.
I'm afraid there is no answer to this and you'll just have to experiment with different sized pores to see what works.
The problem is that you say the liquid is a gel, and gels are typically non-Newtonian fluids. This means their viscosity changes with their flow strain rate and they are frequently also thixotropic i.e. their viscosity changes with time. Many gels will also have a yield stress i.e. below a certain stress they will not flow at all so their viscosity becomes effectively infinite.
This is a problem because the behaviour of the liquid will depend on exactly what strain rates and shear stresses are present at the pores, and in most cases this is impossible to predict as the flow regimes are too complicated to model. You could attempt a finite element calculation, but they typically don't work well for the low strain rate region that probably applies in your experiment.
I used to work for a company (Unilever) that makes shower gel, and they have exactly this problem. You want the gel to squirt out when you squeeze but not to leak when the bottle isn't being squeezed. Although I didn't work in this area, the same problem applies with foods like tomato ketchup and mayonnaise when they are supplied in squeezy bottles. All of these are non-Newtonian fluids with fiendishly complicated low strain rate behaviour and it makes their flow very hard to model. The only way to approach this is to experiment with different sized holes and see what happens.
