I was going through the lectures about Poiseuille flow and I got curious about the governing equation of overall fluid velocity varying in 2 different conditions.

1) Poiseuille flow between plates, from a reservoir (only hydrostatic pressure no inlet velocity). I think this is also solved in our previous pdfs well. 2) Poiseuille flow between plates, from a pump (inlet flow rate/velocity)

I was also curious how in the second condition the hydrostatic pressure, as well as pump flow pressure, can be represented?