Understanding Minimal Surfaces: Laplace Pressure of a Capillary Bridge

[lionelwang]

Hi all,

I am thinking that, A surface is a minimal surface if and only if the mean curvature is zero. then, for a liquid bridge with a catenoid shape, there should be no Laplace pressure due to the zero mean curvature.

But it is a capillary bridge with concave meniscus, how can it have no Laplace pressure?

Any smart guy tell me what is wrong with my understanding on this? Thanks a lot!

 

[mathwonk]

the way you asked your question greatly narrows the range of available answerers.

 

[lionelwang]

the way you asked your question greatly narrows the range of available answerers.

Yes, I did not reveal the problem in a perfect way. Thanks for your valuable advice.

 

[theorem4.5.9]

I know very little of applied math and have no idea what you mean by Laplace pressure, but there may be some confusion in minimal surface. A catenoid is a 2-dim surface, whereas a liquid bridge is a 3-dim solid (if I understand you correctly).

There is a very big difference geometrically in minimizing surface area vs minimizing volume (soap films vs. soap bubbles/clusters).

Just thinking about pressure physically, it doesn't make sense to talk about the pressure inside or outside the catenoid, because a catenoid is an unbounded surface. More explicitly, a catenoid doesn't bound a finite interior, and so pressure doesn't make sense. On the other hand, when I guess at a picture for a water bridge, I think of the catenoid-like solid that my be produced by placing my finger on a water droplet (is this correct?). It is my guess that the relevant physics involved is to minimize surface tension (other forces being negligible), which may or may not minimize surface area (take for example that energy minimization problems in crystals lead to the familiar crystalline shape). I don't see a priori why minimization of surface tension has anything to do with pressure.

 

[lionelwang]

Thanks for your time on this!
I actually prepared the liquid bridge between two substrates, and its shape is much like the catenoid shape, but I can feel that there is Laplace pressure inside the liquid bridge, For a real catenoid liquid ,there should be no Laplace pressure.
So this bother me a lot.

 

[TrickyDicky]

I can feel that there is Laplace pressure inside the liquid bridge

what do you mean? did you measure it?

 

[Vargo]

So I don't know what half those terms mean, but a surface cannot simultaneously be concave and have mean curvature zero. Catenoids, in particular, are not concave. But perhaps I have misunderstood the question because I don't know about liquid bridges or meniscuses.