Module 6: Navier-Stokes Equation
  Lecture 17: Tubular laminar flow and Hagen-Poiseuille equation
 
 
NS (X-direction):  
From the formulation of the problem, 1st, 2nd, and 3rd terms on the LHS are zero.    
 
r-direction:
O-direction:

Note:
Re-look at last two equations.
  •  and  depend on gravity. There is no viscous effects
  •  depends on viscous forces only. There are no gravitational effects.

Therefore, define non-gravitational pressure, so that
 
or  

Check: this definition of  satisfies the above momentum-balance equation.
Therefore,   
Substitute,
 

Say,  (const )
integrate twice to obtain