I Why volume is conserved but not the surface area?

[Herman Trivilino]

Is it that density experiment (means density would not change) or something else?

No, it's what @jbriggs444 describes in Post #28.

Lavoisier did similar experiments in which he established conservation of mass.

Density is a concept derived from the concepts of mass and volume.

 

[etotheipi]

With all this talk of dividing up water, I thought I might add this very satisfying video...

 

[vanhees71]

Thanks, I'll take a look! I hadn't come across the Lagrangian specification of a field before, but it seems fairly intuitive (quite similar conceptually to the $\vec{\xi} = \vec{x}(t_0)$, $\vec{x} = \chi(\vec{\xi}, t)$ of continuum mechanics).

I'd recommend to start with the Lagrangian description of the electromagnetic field, which is simpler than fluid dynamics.

 

[vanhees71]

The conservation of mass is of course valid only in Newtonian physics. In relativistic physics the invariant mass (in the 21st century we only talk about invariant masses in relativistic physics anyway) of composite objects need not be conserved. E.g., in the fusion reactions of nuclei in the Sun the binding energy of the nucleons is not negligible compared to the masses and thus the masses of the nuclei are smaller than the sum of the masses of the protons and neutrons forming the nucleus. What's conserved here is energy. In relativistic physics there's no additional conservation law for mass.

Within Newtonian physics mass is conserved, and in continuum mechanics it is described by the continuity equation for the mass, i.e.,
$$\partial_t \rho + \vec{\nabla} \cdot \vec{j}=0, \quad \vec{j}=\rho \vec{v}.$$
This holds for both compressible as well as incompressible fluids (e.g., a gas which is easily compressed vs. liquids like water which are very hard to compress).

The question whether a fluid can be considered incompressible (which is always an approximation valid for not too large pressure) is quantifiable by the compressibility

https://en.wikipedia.org/wiki/Compressibility

Particularly for water:

https://en.wikipedia.org/wiki/Properties_of_water#Compressibility