Module 6: Navier-Stokes Equation
  Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow
 
 
Newtonian  and non-Newtonian fluid - macroscopic pressure-drop balance

In the previous lecture, we applied the NS equation to obtain an expression for the pressure–drop in a tube for the condition of the steady-state laminar flow of a Newtonian fluid. We can also derive the same expression, the Hagen- poiseuille equation, by making an integral (macroscopic) force balance.

(Fig. 18a)

If the flow is laminar (Re<2100) the velocity profile is parabolic:

For the Newtonian fluid,  

 
  and,

Therefore,  : shear-stress varies linearly in r-direction of the tube.

Now, make a force balance over  

which is the same as that obtained by applying differential momentum balance (NS equation) and then integrating with BCs.