You learn something new ever day! That is definately not a simple linear function like the capacitor charging case.
I see a sqrt being taken of the pressure difference, so its not linear. Volumetric flow is what i would imagine to be the closest analog to electric current but here its behaving much differently. Very interesting.
In a capacitor being charged from a voltage source by a resistor the current is just a simple case of ohms law across the resistor from vcc to the current capacitor voltage.