Look just at position 1.

We have work done from left to right p1A1d1, and work done from right to left p2A2d2.

What will happen if pressure p2 is higher than pressure p1, but A1d1 is bigger than A2d2, and lets say they result with a net work = 0

What equations apply to this situation?

I am really not expert at this, but hope this can help you

 

A₁v₁ = A₂v₂

Net work done on the system = p₁A₁d₁ - p₂A₂d₂

the volumes A₁d₁ and A₂d₂ must be equal. Denote their volumes by V i.e.

V = p₁A₁ = p₂A₂

The mass m of the fluid in sections A₁d₁ and A₂d₂ is given by ρV where ρ is the density of the fluid. Thus

p₁A₁ = p₂A₂ = m/ρ

Consequently,