Graduate Conservation Laws from Continuity Equations in Fluid Flow

[Stuart_M]

TL;DR

There is a version of Reynolds Transport Theorem for differential forms: what is the conserved quantity?

Consider a fluid flow with density $\rho=\rho(t,x)$ and velocity vector $v=v(t,x)$. Assume it satisfies the continuity equation
$$
\partial_t \rho + \nabla \cdot (\rho v) = 0.
$$
We now that, by Reynolds Transport Theorem (RTT), this implies that the total mass is conserved
$$
\frac{d}{dt}\int_{\Omega_t} \rho dx = 0,
$$
where $\Omega_t$ is some control volume (which I understand in the following way: I fix a region $\Omega$ at time $0$; I let $\Omega$ evolve with the flow at time $t$ and I obtain $\Omega_t=\Phi_t(\Omega)$). I have recently been reading Frankel's "The Geometry of Physics" and I've learnt that using differential forms it is possible to generalise (RTT) in the following way (see Chapter 4, Section 3):
$$
\frac{d}{dt} \int_{S(t)} \alpha = \int_{S(t)} (\partial_t \alpha + L_v \alpha)
$$
where now $\alpha$ are (time-indexed family of) $p$-forms, $S$ is a fixed $p$-manifold and $S(t) = \Phi_t(S)$ is the evolution of $S$ under the flow of $v$, and $L_v$ denotes the Lie derivative. From this we easily deduce that, if a $p$-form satisfies the "continuity equation"
$$
\partial_t \alpha + L_v \alpha = 0
$$
then the quantity $\int_{S(t)} \alpha$ is constant in time. If $\alpha$ is a (top-dimensional) volume form, then we recover (RTT) and thus the conservation of *mass*. However, in the general case what does the quantity $\int_{S(t)} \alpha$ represent?

I have considered some easy toy examples: if $\alpha = f$ is a $0$-form (i.e. scalar function), then the "continuity equation" reads $\partial_t f + v \cdot \nabla f = 0$, whence $f(t,\Phi_t(x)) = f(0,x)$ and indeed this can be expressed as "conservation of the integral of $f$ on 0-dimensional manifolds". However, which quantity is conserved?

Another example [taken from Frankel, ibidem]: in Euler, the vorticity *form* $\omega=d \nu$ ($\nu$ being the velocity *co*vector) is invariant under the flow, i.e. solves the continuity equation in the sense of 2-forms. Therefore,
$$
\int_{S(t)} \omega
$$
is constant for any 2-manifold $S$. Which conservation law am I rediscovering? Which conserved quantity is this?

 

[pasmith]

In fluid mechanics, we have transport equations for the following:

Global quantity	Local quantity
Mass	Density
Momentum	Momentum per unit volume
Energy	Energy per unit volume
Amount of substance	Concentration
(No standard name)	Vorticity

Concentration is either amount per unit mass or amount per unit volume.

Note that \int \alpha is not necessarily conserved itself; it is also necessary to account for sources and sinks of the quantity.

 

[Stuart_M]

In fluid mechanics, we have transport equations for the following:

Global quantity	Local quantity
Mass	Density
Momentum	Momentum per unit volume
Energy	Energy per unit volume
Amount of substance	Concentration
(No standard name)	Vorticity

Concentration is either amount per unit mass or amount per unit volume.

Note that \int \alpha is not necessarily conserved itself; it is also necessary to account for sources and sinks of the quantity.

Thank you for your reply and for the table.
I am not sure I am able to see the link with my question with differential forms, though. Is there a way to phrase the momentum per unit volume (or the energy per unit volume) as a differential form? If so, can we say that this form satisfies the "continuity equation" (in the right formulation) and deduce the conservation of the corresponding global quantity? But this would seem strange to me: by the generalized RTT we would we get the conservation of this quantity only when "integrated" over some p-dimensional manifold (I'm in an ideal world, no sinks/sources for the time being). What does this mean? Is this some sort of generalized flux? I'm sorry for my confusion, I hope somebody can shed some light here. Thanks!