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A fluid is said to have divergence (the ability to flow) and curl {the ability to rotate). Do these two characteristics fully define a fluid, or are there other important properties that I am missing? My understanding is that a fluid has a definite volume, but no definite shape.

If something is "incompressible" (divergence= 0) and irrotational (curl=0), do these properties by themselves make it solid, or are there other essential properties? My understanding is that a solid has both a definite volume and a definite shape.

What, if any, are the properties of a gas (no definite volume or shape) expressed in terms of divergence and curl?

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    $\begingroup$ Gasses can also have divergence and curl, elastic solids can have divergence. Fluids can be (essentially) incompressible and irrotational. So no there isn't any real relationship $\endgroup$ – Triatticus Dec 4 '18 at 18:30
  • $\begingroup$ Note that you seem to be confusing liquid with fluid - the latter includes both liquids and gases. $\endgroup$ – J. Murray Dec 4 '18 at 19:43
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Your interpretation of these operators is incorrect as a compressible fluid ($\nabla \cdot \textbf{u} \neq 0$) can clearly flow and a fluid with a non-zero vorticity ($\omega = \nabla \times \textbf{u}$) can exhibit rotational motion. For example consider the potential vortex where the azimuthal velocity is given by \begin{align} u_\theta = \dfrac{\Gamma}{2\pi r} \end{align} The velocity field is shown in the figure below. In the potential vortex, $\nabla \times \textbf{u} = 0$ and the fluid is rotating. If you look in your fluid mechanics texts, you'll see that vorticity is a measure of the local rotation. For incompressible fluids, divergence of the velocity field can be interpreted as a source or sink of mass (see potential mass source) and the curl of velocity is interpreted as vorticity or the local rotation. In the end I don't think you can relate these operators to the state of matter.

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Your idea is unfortunately wrong, let me show you why:

  1. Divergence is not the ability to flow, it is the ability to compress. Curl is not the ability to rotate, there are curl-free flows that clearly rotate. I think you should revise your course of classical field theories, if you had any.
  2. Divergence and Curl are concepts from vector analysis, they operate on vector fields. Vector fields are sometimes good models for steady-state solids, liquids and gases, but not in all cases. A vector field description is only applicable if the mean-free path of the particles involved is much smaller than the mean distance between those particles.
    So low-density gases, including low-density plasmas which make up 98% of the universe by atom numbers, are not even correctly describable by vector fields.
    Also solids experience displacements, just slow ones, as their viscosities are enormous, so e.e. the solid rock in Earth's mantle very well has some shear that one can assign to div and curl.
  3. If you want the time-evolution of a vector field assigned to a blob of matter, it is not enough to know only the properties div or curl. You need to know the Navier Stokes equations (at least) and their equations of state, which is how they react to stresses and changes in internal energy.
  4. There are more states of matter, MANY more than just the everyday solid/liquid/gas. It seems to me that you're trying to establish a table "if it has curl, but no div, then its a fluid", which as already stated before, can't work. Other states of matter that can never fall into this categories are Bose-Einstein-Condensates, Fermi-gases, Para/Ferromagnets, ...

Summarizing: You're trying to force two different concepts together that have nothing to do with each other.

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    $\begingroup$ This answer is really accurate but lacks quite a bit in niceness. $\endgroup$ – my2cts Dec 4 '18 at 20:29
  • $\begingroup$ @my2cts: except for the first sentence, which I edited, I don't know what you're talking about. $\endgroup$ – AtmosphericPrisonEscape Dec 4 '18 at 20:54

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