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I was wondering if someone could give me some help on how to start this problem, I'm really struggling to get my head around it.

A long line of oil is being squashed between two flat plates of length $L$ by a weight $W$ placed on the top of the upper plate. The initial separation distance $\delta$ is very small compared with the initial width $2L$ of the oil strip so one would expect approximations of a lubrication type to describe the flow of oil in the gap. Clearly stating and justifying the assumptions made, find an estimate for the separation distance $\delta h(t)$ as a function of time.

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  • $\begingroup$ In order for the plates to move, oil has to travel from the middle towards the edges (with laminar flow perhaps). I am sure the shear stress formula is where you start after you have decided what the generic velocity profile of the oil is. $\endgroup$
    – John Alexiou
    Commented Aug 31, 2015 at 12:31
  • $\begingroup$ This is not a homework help site, it is a Q&A site about physics concepts. Please rephrase your question to be about the concept that you do not understand. $\endgroup$
    – Kyle Kanos
    Commented Aug 31, 2015 at 13:58
  • $\begingroup$ @jackwo - some effort on your part is required! What have you so far done for this problem? What don't you get your head around? e.g. what is a lubrication type approximation? $\endgroup$
    – nluigi
    Commented Aug 31, 2015 at 22:33

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As a simplification, you can consider that you have a 2D viscous flow between two boundaries that approach each other. Assuming that the flow is symmetrical about the line (with the line along the Y direction), you can simplify this further to "no flow at x=0".

What you are left with is a pressure distribution $p(x,t)$ whose integral in $x$ should equal the total force on the plates, and whose derivative in $x$ describes the force on the liquid. In viscous flow, the flow rate is proportional to the pressure difference. Now the flow velocity will be proportional to the distance $x$ from the center (the further from $x=0$, the more oil needs to move past the point as the plates come closer together). Assuming "normal" viscous flow, this means the pressure has to be proportional with $x$ also.

Finally, you need to consider the shape of the meniscus at the leading edge of the oil. This shape is determined by the contact angle between the oil and the glass, and will result in an additional force on the liquid - either pushing the oil in, or pulling it out, depending on the sign of the contact angle.

Those are the considerations I think should get you going on solving this.

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  • $\begingroup$ Don't know why you were downvoted but this sounds like the approach (i.e. lubrication approximation) OP should take $\endgroup$
    – nluigi
    Commented Aug 31, 2015 at 22:34
  • $\begingroup$ @nluigi - Thanks. I'm glad I am not the only one who could not understand the downvote. Since the question was "how do I start" rather than "solve this for me" I thought I would do just that - give a nudge in (what I believe to be) the right direction. $\endgroup$
    – Floris
    Commented Aug 31, 2015 at 22:35
  • $\begingroup$ OK that kind of makes sense. Would I then use the Navier-Stokes equations to get the relevant equations? What are the forces on the plates over than the weight acting on the top plate? Apologies for my naivety when it comes to these concepts, it's just I've never solved problems like this before! $\endgroup$
    – jackwo
    Commented Sep 3, 2015 at 12:58
  • $\begingroup$ The weight is the force on top of the plate - the integral of the pressure over area will balance this. $\endgroup$
    – Floris
    Commented Sep 3, 2015 at 14:00
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    $\begingroup$ The pictures in this article might be helpful. I will see if I can sketch out something more detailed. $\endgroup$
    – Floris
    Commented Sep 6, 2015 at 16:52

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