Falling Film Problem: Velocity Profile Independent of z

[Wisc17]

Homework Statement

This is not exactly a homework problem. I am trying to review material so that I can tutor it this semester. In the falling film problem, as described in this link: http://www.syvum.com/cgi/online/serve.cgi/eng/fluid/fluid204.html, I am confused why the velocity profile does not depend on z. I understand why there is dependence on the x direction, but do not understand why the fluid is not speeding up as it moves downward toward the Earth due to gravity.

Homework Equations

Final Velocity Profile: v_z(x)=ρgδ²cosβ[1-(x/δ)²]/(2μ)

The Attempt at a Solution

My thought that is perhaps since it is a steady state problem, the fluid shouldn't be speeding up over time, because {accumulation} = {in}-{out} but I am really not sure. I can do the calculations, I just don't understand why gravity does not affect the fluid speed.

 

[Chestermiller]

At steady state, the gravitational force is exactly balanced by the drag force. At locations higher up, before steady state is established, the film is accelerating. This is like one of those terminal velocity problems for a body falling with air drag present.

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