Calculate the volumetric flow of a fluid

[CFXMSC]

How could I calculate the volumetric flow of a fluid due to the weight of a piston in a cylinder?

At first moment i I thought the Hagen-Poiseuille equation, but I'm not sure. Any suggestions?

 

[CFXMSC]

For Duct

Continuity:

[itex]\frac{\partial v}{\partial y}=0[/itex]

Momentum:

[itex]\rho g_y-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2}\right)=0[/itex]

For Circular Pipe:

Continuity:

[itex]\frac{\partial}{\partial r}(v_z)=0[/itex]

Momentum:

[itex]g-\frac{1}{\rho}\frac{\partial p}{\partial z}+\nu\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial v_z}{\partial r}\right) \right][/itex]=0