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I've been thinking about the Navier-Stokes equations and trying to build skill in implementing it in various situations.

In a particular situation, if I have a

**fluid flowing down an inclined surface**such that it forms a film of finite height which is smaller than the length of flow,

**there is no applied pressure. So am I allowed to cross off the pressure term in the momentum conservation equation along the direction of flow?**

In a detailed solution to this problem, our instructor non-dimensionalizes the equation to show that the pressure term goes away.

However, I'm wondering, since there is no applied pressure, can't I just cross it off (like Couette flow)?

Also, in another problem (the situation is that oil is flowing up an inclined surface submerged in water) the instructor

**replaces the pressure gradient by the buoyancy force**. So in the inclined flow case, am I not allowed to simply replace the pressure term by the hysdrostatic pressure?

I think my question boils down to

**whether the pressure term in the NS equation is an external force or internal property of the fluid.**