Fluid element, conceptual questions

[Rasalhague]

In chapter 4 of A First Course in General Relativity, Bernard Schutz introduces the idea of a fluid element, a different one assigned to each point in a fluid.

When he talks, in section 4.5, about the 3-volume of a fluid element, is this 3-volume defined with respect to the MCRF (Momentarily Comoving Reference Frame)?

Is Schutz's MCRF (1) a chart, or (2) a field of orthonormal basis vectors for the tangent spaces at every point in the fluid, or (3) an orthonormal basis for the tangent space at a particular point?

Is it possible to have two fluids in Minkowski space, each of which occupy a cube of the same 3-volume in some common Lorentz chart (spacetime coordinates alanogous to Cartesian coordinates) - so that there's a bijection between points in one fluid and points in the other - yet every fluid element of one fluid has twice the 3-volume of the corresponding fluid element of the other fluid? I'm guessing yes, although I'm not sure what it means physically for an object defined at each point to have a volume, since a point has no extent, whereas volume does have extent.

Is the 3-volume of a fluid element something one could observe experimentally? Or is it a description one imposes arbitrarily on the fluid, then observes how these arbitrarily imposed 3-volmes change with changes in pressure, energy density, number of particles etc.?

 

[Bill_K]

The usual meaning in continuum mechanics is that a fluid element is just an infinitesimal piece of the fluid, the amount contained in an infinitesimal volume element dxdydz. Gives you something to apply Newton's Laws of Motion to. Usually the fluid element is taken to be comoving with the medium, but you could also consider a stationary element in which fluid flows in one side and out the other. As far as experimentally observing it, no, it's just conceptual.

 

[Rasalhague]

Hi Bill, an infinitesimal piece in the sense that its volume is a real number small enough that, in a given context, the first order series expansion of any (continuously?) differentiable function of volume can be considered accurate, say, because the difference between the approximation and the value of the function is less than the error of measuring devices? (So that for a particular real number to be considered infinitesimal, we'd have to state the degree of accuracy?)

From your final comment, I guess the answer to my 3rd question is yes, since we could simply declare every fluid element to have half its original volume, without any change in the volume of the whole fluid.

Schutz writes, "Now if the element contains a total of N particles, and if this number doesn't change (i.e. no creation or destruction of particles), we can write V = N/n, Delta V = -N Delta n / n²." Do these equations apply only to the case where the number of particles changes not because particles are being created or destroyed, but because we've arbitrarily declared the volume to be different (i.e. we're considering new region, of similar location to the original but different size) so that it contains, say, half as many particles?