Calculating viscosity force of oil between 2 surfaces

[levi415]

I'm working on a project at work that requires some tribology/fluids calculations that I don't know if are applicable to me or not.

For this scenario, one of the tests will require me to assume hydrodynamic lubrication where oil is completely separating the 2 surfaces.

I have a sliding steel plate being inserted into the opening of a hollow, steel tube. The plate is being pressed against the tube wall simultaneously as it enters the tube. The steel tube is coated with an oil but of unknown film thickness and distribution (I can assume even distribution and am guessing have to assume a thickness). The steel plate is shaped to the curvature of the tube and will be held against the tube wall by an applied force which I can control.

I want to find out the axial force that is required to overcome friction and pull the plate out of the tube.

I have been given the kinetic viscosity of the oil and am looking for the density to calculate dynamic viscosity. Once I have the dynamic viscosity, I know I can calculate viscosity shear force using the Newton equation of: force = dynamic viscosity x area x (dx/dy).

The problem is, I don't know the velocity gradient dx/dy because the oil is initially stationary and I don't know how it will move when the plate comes into contact with the tube wall. I also don't know the thickness of the oil film at any time.

For the purpose of this test, I am assuming that the majority of the oil is not getting displaced enough to cause surface to surface direct contact (i.e. boundary lubrication).

I am not a tribologist and am a pretty poor ME so any help would be appreciated! Thanks very much!

 

[levi415]

I didn't ask explicitly but is this a proper approach? If not, are there some other equations I can use? I am using the classic friction equation (friction force = coeff of friction x normal force) for the boundary lubrication condition only.

Thanks.

 

[boneh3ad]

You don't need any more information. Take a look at what is called Couette flow. Just go on the assumption that the velocity profile is linear between the two surfaces (if they have been sliding long enough it will be) and go from there. That will give you a value for \frac{\partial u}{\partial y} and the rest is trivial.

 

[AlephZero]

and the rest is trivial.

... but not so trivial if the oil film thickness is not constant over the area. In that case, you want the Reynolds lubrication equation (not to be confused with other Reynolds equations, and beware Wikipedia IS apparently confused here).

IIRC there are standard solutions available for "simple" geometries (e.g. a wedge-shaped oil film) where the flow is basically 1-dimensional (possibly throuh a curved gap). For the general case you need to solve the equation numerically over the area of the oil film.

If the film is a curved shape and variable thickness, pressure gradients in the film may also create forces you need to overcome. (With luck, you won't get into scenarios where negative pressures cause cavititation in the oil and the film breaks down...)

 

[levi415]

Thanks for suggestions; they've prompted me to research areas that I wouldn't have known about. Finally, I've decided to go with the Couette's flow suggestion. I think using Reynold's Lubrication Equation would more closely simulate the hardware I'm using but after some reading, I just don't have the brainpower for the calculus. My problem is similar to a generic journal bearing fluid film pressure distribution problem except not using the whole bearing surface.

But I have some unknowns now, or maybe just some misconceptions that I need help with.

Setup: pneumatic cylinder inserts a metal "shoe" into the opening of a metal tube. Inner surface of tube has an unknown oil film. The shoe is curved to match the inner radius of the tube but contact area is only about 1/8th the circumference of the tube. A known pressure is applied normal to the shoe and tube wall that presses the shoe against the wall.
Problem: calculate an axial force to overcome friction and cause slip of the shoe inside the tube.

Step 1. I want to find air flow rate Q. I know air pressure, piston dimensions, length of travel but NOT velocity. Assuming a small deltaP since L is very short (less than 1 inch). Have found this equation but does it apply to air?
Q = [pi*(r^4)*(deltaP)]/(8*N*L)
where r is radius of cylinder/piston
N is viscosity (doesn't specify but I assume dynamic)
L is length of travel

I need help here: can I use this equation? Have looked for other means to find Q such as via Reynold's number equation but I either need to know Q already or know velocity.

Step 2. I can find velocity once I know Q using v = Q/A.
Step 3: Calculate shear stress using tao = mu(v/h).
where tao is sheer stress, mu is dynamic viscosity, h is oil film thickness.

I need help here too: I run into the unknown from my original post... I don't know h!

Let me know if I'm taking a completely wrong approach or if I'm just missing the forest for the trees; this is really irking me.

Thanks in advance!