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.

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.

Yes we can. The way this is done is as follows. One immerses a rigid body of well-specified form and characteristics in the fluid of interest. On rotating this body around axes pointing towards independent directions, one measures the various components of the inertia tensor -- these are deduced from the coefficients of the force linearly proportional to the relevant angular frequencies. The inertia tensor thus obtained consists of that of the rigid body in vacuum complemented by the contribution of the forces exerted on it by the fluid with which it has been brought into contact. In the book Fluid Mechanics by Landau and Lifshitz one encounters several worked-out examples concerning the moment of the frictional forces exerted on rigid bodies immersed in fluids under various conditions (the simplest case concerns a sphere immersed in a fluid, sphere being isotropic). One should bear in mind the notion of 'probe' or 'test object', which in principle need not be a constituent part of the system one studies. Thus while a fluid is not a rigid body (at least not over arbitrary time scales), one can perfectly define moment of inertia in the way I described above, where the well-specified rigid body immersed in the fluid is the 'probe'.

hello

All matter has inertia, since mass itself is a measure of inertia. Since gases and liquids have mass, they also have inertia.

If you have an object with some liquid parts (say an uncooked egg), even if the rigid parts are rotating, the liquid parts may not (depending on the geometry) have to rotate along with the rest. However, over time the friction (viscosity) of the liquid will cause it to rotate along with everything else. So you actually get a frequency-dependent effective moment of inertia. If the object is rotated back and forth quickly, the liquid only partly contributes. If it rotates back and forth on a longer time-scale, the liquid contributes almost fully.

Addition to answers to question.

It is necessary to distinguish between low-density fluid - gas-like - and high density - liquid-like.

 Regards