Two-dimensional flow equations
 

This question will probably seem elementary to experts such as you, but I am a mere beginner student working with virtually zero reference material. I am looking for an equation that models the two-dimensional flow of a liquid across a circular chamber. The liquid is fed in through a small rectangular channel, and exits through a similar channel.

I already know that the Hele-Shaw equation models the flow in these rectangular input/output channels, but I don't have the equation for the central circular chamber itself. Will this equation, whatever it is, hold if there is more than one input into the chamber, so that two or more separate flows (potentially of different substances) interact with each other?

Many thanks,

Chris Knowles