And if so, is this inertia directly proportional to its viscosity?
Can gases have inertia?
All matter has inertia, since mass itself is a measure of inertia. Since gases and liquids have mass, they also have inertia. The inertia of these fluids would not be related to their viscosity.
And so then, what of the http://en.wikipedia.org/wiki/Moment_of_inertia
Moment of inertia is rotational inertia (as it says in the first few sentences of the link). It also has nothing to do with viscocity.
The very top of the wiki page on moment of inertia says moment of inertia "is the inertia of a rigid rotating body with respect to its rotation". The simplifying rigid body assumptions that lead to the moment of inertia do not apply to fluids. The Navier-Stokes equations get very messy (as if they aren't bad enough already) when the fluid is rotating.
That said, one reasonable simplifying assumption is that a fluid in a vessel has a zero moment of inertia tensor about the fluid's center of mass. The parallel axis theorem dictates the moment of inertia about an axis of rotation that does not pass through the fluid's center of mass.
Example: Hold an empty heavy cast iron pot by its handle and spin it around and back, like a washing machine tub. Now put some water in the pot and do the same. You don't feel much difference because the water isn't affected much by the rotation. The water's moment of inertia about its center of mass is essentially zero in this instance. If, on the other hand, you swing the entire pot like a pendulum you will feel a big difference between the empty pot and the full pot.
This zero moment of inertia is a simplifying assumption. The assumption fails if the changes in the vessel's rotational state couple with the fluid's state. Ignoring that this is a simplifying assumption that is only conditionally valid can lead to problems. It did with STS-49 and the capture of Intelsat 6. The rotating fluid inside the satellite acted like a gyroscope and came close to tearing the robotic arm on the Shuttle apart.
I'm going to quibble with your terminology there, because I think the OP question may be confusing "inertia" and "momentum".
The water's moment of inertia is exactly the same as a the moment of inertia of a solid of the same density and shape. Moment of inertia is a property of mass and geometry only.
But in the spinning pan example, there is no way to apply much force to the water to make it rotate, because it has low viscosity, therefore the water doesn't have much angular momentum compared with a solid pan-shaped object. Since force = rate of change of momentum, that's why you don't feel a difference in the force to spin the pan.
The concept of "moment of inertia" isn't very useful in most fluid flow situations because the fluid does not move like a rigid body. The fact that it doesn't always behave like a rigid body is one way to define what you mean by "a fluid".
As an example where water does show the same (linear) inertia effects as a solid object, consider the shape of a jet of water from a hosepipe, compared with throwing a ball at the same velocity as the water jet leaves the hose. Ignoring air resistance etc, they both follow the same path.
You can have situations where fluid does "rotate" in the the same way as a rigid body. For example, imagine a rigid pipe made into a circular coil, about the same size as the rim of a bicycle wheel. If fluid flows through the pipe and you try to move the pipe around, you will feel the same "gyroscopic forces" as you would get on a rotating bike wheel. (I'm assuming the rigid coil of pipe is connected to a flexible hose so you can move it around). That is similar to DH's satellite example, of course.
In geophysics, our Earth will sometimes be treated as a weakly deformable object, which entails, for example, that its moment of inertia is variable.
However, for liquids, the whole concept of "moment of inertia" becomes useless, because there will in general be no common angular velocity with which the whole fluid rotates with around some axis.
What we use instead, is the concept of the vorticity field of the fluid.
The vorticity at a point is a measure of the angular velocity by which an infinitesemal piece of the fluid rotates about that point in the plane whose normal vector is given by the direction of the vorticity vector, whereas the "vorticity", or "vorticity strength" is the magnitude of that vector.
I'm going to quibble with you. The OP started by asking whether fluids have "inertia" and if this is related to viscosity, and then further asked "what about moment of inertia" in post #3. It appears to me the OP is using the term inertia as shorthand for mass moment of inertia, not momentum. Many others use the same shorthand and use inertia tensor in lieu of mass moment of inertia tensor.
I disagree. Mathematicians could care less about something as simple as the mass moment of inertia. They are interested in much more abstract concepts. The inertia tensor is something of keen interest to physicists and engineers. The mathematical definition of moment of inertia only has valid physical meaning with the rigid body assumption. Erroneous physics result when one applies the rigid body rotational equations of motion (which use the mass moment of inertia tensor) to a non-rigid body.
That is why approximating the fluid's mass moment of inertia tensor as zero works. This is a physics forum, not a math forum. Physicists and engineers make simplifying assumptions. Mathematicians don't.
The concept is useful in the simplifying assumption of inconsequential coupling between the rotation of the vessel that holds the fluid and the fluid itself. In this case, the fluid's mass moment of inertia trivially becomes zero. Spacecraft modelers often use this simplifying assumption. The fidelity is quite good so long as the simplifying assumptions are valid. Adding a slosh model to this simple model (but retaining the trivial mass moment of inertia tensor) results in even better fidelity that remains valid even in the presence of some coupling. Any simple model of fluid motion will fail, in which one may have to resort to something like a full-blown CFD model.
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