How would you determine the flow rate of a liquid that flows through a narrow microneedle channel and then through the opening of a funnel, i.e. pouring liquid through the other end of a funnel?
I've chosen arbitrary diameters but the pressures are from literature. The diameter of the lumen/tip is 50 $\mu$m and the base diameter is 1 mm. The pressure at the tip is 25 mmHg and the other end is open to the air.
I've tried to split the problem into two parts: first determine the flow rate through the straight part of the needle, and then determine the flow rate through the funnel. I calculated the flow rate in the first part using Poiseville's equation but I think that there must be another model for narrow channels or fluids with capillary action coupled with initial pressure. The Reynold's number is clearly in the turbulent range for the first part and I'm not sure if the numbers should be this high for narrow channels.
Is there a fluid model that's better suited for microneedles?