Lubrication Theory: Fluid Flow and Integration

1. Oct 25, 2014

E.T.A.

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w the plane and the cylinder, and subscript c is at the point at which the flow leaves the moving plane-cylinder system and into the infinite-length "coating" mechanism. Diagram below:

and the governing equations:

so, here's where I'm at:

• I don't really get the need for the substitution (or is it some dimensionless parameter?). It feels pretty arbitrary right now and it's frustrating.
• I have subbed in the value for h into the eq. for dp/dx, and then attempted to solve that ODE via direct integration (?), which yields something pretty nasty. And, unfortunately, I'm not quite getting at which stage the given substitution's meant to happen (or -again- why), and quite where the sins are coming from.
I'm sorry this is so long and my attempt is so weak but I'm kinda lost and I'd be madly grateful for any direction and/or help.

Thanks!

2. Oct 25, 2014

Staff: Mentor

This is just a straightforward trig substitution to evaluate the integrals of the terms on the right hand side of the equation with respect to x. If they wrote $x=\sqrt{2Rh_0}tan\theta$, would it make more sense to you?

Chet

3. Oct 25, 2014

E.T.A.

Yeah. Thanks, man. The answer fell out pretty easy when I stopped looking for a difficulty that wasn't there.

4. Oct 27, 2014

E.T.A.

Hey, I'm stuck again!

I'm really not sure where to start with this and I'm not ~quite~ sure where I'm meant to be working to :(

Would really appreciate any help.

5. Oct 27, 2014

E.T.A.

6. Oct 27, 2014

E.T.A.

wait: b/c γ=tan^-1 (x/sqrt (2Rh_o)), and the limit of this as x --> negative infinity will = -pi/2, thefore p (-pi/2) = 0, which would allow for the solution for the unknown c in tbe eq. for pressure (?). then, using p(γ_c)=0 the final answer can then be constructed?

this seems wrong and overly simplistic, and I have neglected the given info about the pressure grad.

help!

7. Oct 27, 2014

E.T.A.

I have actually solved this now :)

8. Oct 31, 2014

E.T.A.

hey, quick question: for this system to work, would the the height at x=0 (h_0) have to be smaller than that at h_inifinty? i.e. the pressure generation is biggest at the start of the system?

9. Nov 1, 2014

Staff: Mentor

I don't fully understand this question. You are approximating the shape of the cylinder by a parabola. h_0 is the minimum clearance, so all other h values are larger. Your original equation tells you that the pressure gradient is not largest when h is large.

Chet