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Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere.

How would one prove that? Is it equivalent to showing that the minimal surface area for a simply connected volume in $\mathbb{R}^3$ with a sufficiently smooth boundary is that of a sphere, i.e., the result of the isoperimetric inequality?

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The droplet wants to minimise its surface energy. This energy is proportional to its surface area. So the equilibrium shape is that which minimises the surface area for fixed volume (the bulk density is fixed by the temperature and pressure).

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  • $\begingroup$ Ok, so it is equivalent to showing that a sphere is the surface-area minimizing volume. Thanks. $\endgroup$ – Simon S Dec 17 '14 at 21:35
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A Community Wiki answer to make some other people's comments permanent and tie some loose ends up.

To add to Mark Mitchison's Answer, the reason that the prevailing shape is the one that minimises surface energy as he states is that, in the case of water, the liquid's total energy is an (almost) constant offset (the potential and kinetic energy of the molecules within the body of the liquid) plus the surface energy, so that minimising the latter is almost equivalent to minimising the former.

Given the experimental fact that most liquids are nearly incompressible, the energy change wrought by the internal pressure field that changes with the body's shape is utterly negligible compared with the changes in the energy associated with the surface tension, so that the latter sets the shape.

As QMechanic's excellent link (the "Isoperimetric Inequality" Wikipedia page) points out the (hyper)sphere is the shape that minimises the surface area of a given enclosed body.

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Another way to look at it is the following.

The main force on the molecules will come from other water molecules and be due to cohesion. The system will try to minimize it's energy and bond the molecules together as much as possible. This means minimizing the surface which results in a sphere.

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This is to add up a little more detail to the discussion. Water as it exists in the form of $H_3O^+$ and $OH^-$ ions are bound together by "Van der waals forces" which is "the sum of the attractive or repulsive forces between molecules other than those due to covalent bonds, or the electrostatic interaction of ions with one another, with neutral molecules, or with charged molecules." So the total sum of the energy due to the different forces acting on every molecule and thus the whole droplet should be maintained minimum which gives the best possible shape of a sphere which has the lowest possible surface area(exactly as was said in previous answers).

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  • $\begingroup$ Water, assuming a pH of 7, has very few $H_3O^+$ and $OH^-$ ions compared to the number of $H_2O$ molecules. So I am not sure why you mention it. $\endgroup$ – fibonatic Dec 18 '14 at 12:29
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A liqiid kept in a container feels no force inside . But at the surface a net inward force acts. This causes surface tension. This surface tension tries to contract the liquid. If the liquid is kept in a container surface tension only acts on the top.But in a free liquid , imagine what will happen due to force inwards from all the side. Contact it to an almost sperical shape. It is not completely sperical because of more typical interaction by gravity.

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