How does a finger pushing down on a waterbed affect its surface?

[K.J.Healey]

Where would I start for modeling something like a waterbed? Say a 2D well-shaped object with an incompressible liquid covered in a surface, like a 2D waterbed.
Basically bound at the edges, has some ability to stretch to allow curvature, but will not compress the liquid(the area/volume of liquid in the picture would remain same). I'm looking to do some statics, not dynamics too, so that should make it easier.

Would stuff like the depth of the well affect displacement of the surface? Or is it all in the elasticity of the waterbed and the length?

Like say you have the situation where the left side is 1m high, the right side is 2m high, and the top part is stretched between the two, filled with liquid in a gravitationally free environment to make a linear gradient between the two, then its placed into gravity where the surface has some mass and elasticity, so it would sag some, and probably be exponential/y shaped from point A to B.

Where would I get started with this? Is this fluid dynamics/statics? Is it materials? Topology? etc.
(Writing the simulation code is the easy part, I just need some governing principles)

 

[NoTime]

If I understand your question correctly then your setup is similar to an elastic string tied to two stationary supports with a uniform force acting along its length.
Or is the intention to have a non uniform force, with the high end experiencing a greater pull?

This might help http://en.wikipedia.org/wiki/Catenary

 

[Andy Resnick]

I would start by modeling a water-filled spherical balloon with no gravity- the boundary conditions are marginally easier. The balloon wall is under some uniform tension and the water under some uniform pressure.

This is absolutely a non-trivial problem to model, as you will soon learn, because perturbing the numerical model (to simulate moving fluid) will not always lead to a stable solution.

 

[K.J.Healey]

Its sort of similar to an elastic string, but with the constraint that the area under the curve must ALWAYS be constant.

If I were to do a spherical/circular balloon, it would be the same idea, with the constraint that the volume remain unchanged. But how can I start modeling where it will distort? If I put a force on one point on the surface, what force will every other point feel to make the balloon bigger?

See I don't care at all about the dynamics, and I'll probably only set up problems that would have a stable solution. I care about the final steady stable non-wave solution.

I could see doing a balloon with a surface of points and connections to the other surface points, but I'm not sure how to constrain the volume.

Lets say a circular(cylinder) waterbed. If I push in the middle a circular force should appear between the middle and the outer wall, distorting it up curved.
I think this appears to be a pressure situation, and it seems like it would just be a conservation of pressure, so the magnitude is easy, but what about the radius? And if its not a central force? And so on. I can start simple and work from there I just need some way to accurately govern the constant volume.

 

[Andy Resnick]

I don't understand what the point is, if all you want is the final steady-state solution. A balloon is spherical, a waterbed is a sphere that defoms into the shape of the container with a flat top... because in the steady-state there are no waves or disturbances. What are you really trying to model?

If you are interested in modeling the shape of a waterbed when someone is lying in it, then your boundary condition is a pressure which is applied to one surface (in a spatial manner). The pressure within the water equilibrates and hydrostatic equilibrium balances the fluid pressure and the bed support in addition to a deformation of the top surface balancing the elastic properties of the sheet and the fluid pressure.

A constant volume boundary condition is an integral boundary condition: the integral over all space of the density of fluid is a constant in time.

 

[K.J.Healey]

Yeah, that basically what I want to model. Say you take a finger and push down hard in the center of a round waterbed, say so hard that you deform the surface down to touch the bottom. I'm trying to model how that would deform the surface in upward direction given different locations of the finger pushing down.
So it really doesn't bring forces into it, time into it, etc. Should be static, and there should be a stable equilibrium for each situation.
Like I said, I don't need to model the dynamics, I just wondering how the top will deform given a certain finger push (lets say center).

And to clarify I meant steady state as in the final time-independent solution of the waterbed's surface after deformation, compared to zero deformation.

I think I know what I have to do, ill give it a shot.