The tubes leading to the vessel on the left will fill first because it is at the lowest point, but because the tube leading to the tube to cup 4 is blocked cup 4 will not get any coffee. And the vessel on the left will fill until it reaches the tube to cup 9 but that is also blocked so no coffee in cup 9.
Then the coffee in the main vessel will reach the higher tube to the upper and lower vessels on the right. The tube to cup 7 is lower and will fill first but it is also blocked so no coffee for cup 7.
After the blocked tube to cup 7 is filled and the level rises to the tube to leading to cup 5 is reached, eventually cup 5 will get the coffee.
But here’s the thing. The tube coming off the vessel on the right has a 90-degree angle to a tube of a much smaller diameter, so I would suspect, depending on the volume and rate of coffee being poured in at the top, that it might overflow at the top before it actually fills cup 5.
Am I wrong? Are there any engineers or plumbers here that can weigh in?
It will, in the physical world depend on the diameter of the two tubes at the right angle, and the speed the coffee is poured. Pour slow enough, with all the joints clean and at right angles (so the there is an air path out of the lower lower tube), and the coffee will flow as described.
Pour it quick enough, and it will trap air and stall out. But the trapped air will bubble through the trapping tubes eventually, if the tubes are big enough not to trap the bubbles with a meniscus effect.
Just a retired ODOT employee here. I had a rather cozy job until one of those Asian people invented a shovel that stood up by itself. #5