Moving control-volume mass and momentum

Use a classical Cartesian inertial frame and SI units. Let $I\subseteq \mathbb{R}$ be a nonempty open time interval in seconds, let $C_0\subseteq {\mathbb{R}}^3$ be a nonempty bounded open region with $C^1$ boundary (locally a $C^1$ graph with the region on one side), and let $X:I\times U_C\to {\mathbb{R}}^3$ be a $C^2$ family of orientation-preserving diffeomorphisms (bijections onto open images with $C^2$ inverses and positive Jacobian determinant) on an open neighborhood $U_C\supseteq \overline{C_0}$. Here $C^r$ means all coordinate partial derivatives through order $r$ are continuous. The moving control region is $C(t):=X_t(C_0)$; its boundary velocity is $w(t,X(t,\xi )):={\mathrm{\partial }}_{t}X(t,\xi )$ in metres per second. Let ${\mathrm{\Omega }}_t$ be the open fluid region and $\mathcal{O}:=\{(t,x):t\in I,x\in {\mathrm{\Omega }}_t\}$ be open; assume $\overline{C(t)}\subset {\mathrm{\Omega }}_t$ for every $t$. The fields $\rho :\mathcal{O}\to (0,\mathrm{\infty })$ and $u:\mathcal{O}\to {\mathbb{R}}^3$ are $C^1$, $b:\mathcal{O}\to {\mathbb{R}}^3$ is continuous, and $\sigma :\mathcal{O}\to {\mathbb{R}}^{3\times 3}$ is $C^1$. Their units are respectively kilograms per cubic metre, metres per second, metres per second squared and pascals. Let $n$ be the outward Euclidean unit normal and use Euclidean surface measure, with chart density $\sqrt{\det (D{Y}^{\mathsf{T}}DY)}$. Set $(u\otimes u{)}_{ij}:=u_i{u}_j$ and $(\mathrm{div}\sigma {)}_i:={\sum }_j{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$. Use traction $t=\sigma n$. Assume the adopted single-component continuum mass and momentum balances in the following local material sense. For every $(t_0,x_0)\in \mathcal{O}$, there exist an open interval $K$ containing $0$ with $t_0+K\subset I$, a nonempty bounded open $C^1$ reference region $B$ containing $x_0$, an open neighborhood $U_B$ of $\bar{B}$, and $\chi \in C^2(K\times U_B;{\mathbb{R}}^3)$ with orientation-preserving $C^2$ diffeomorphic slices, positive Jacobian and $\chi (0,a)=a$. Require $(t_0+s,\chi (s,a))\in \mathcal{O}$ and $u(t_0+s,\chi (s,a))={\mathrm{\partial }}_s\chi (s,a)$ for all $s\in K,a\in U_B$. Write $\tilde{f}(s,x):=f(t_0+s,x)$ for each field $f$. For every nonempty bounded open $V⋐B$ with one-sided $C^1$ graph boundary, put $P_s={\chi }_s(V)$ and let $\nu$ be its outward unit normal. Require

${\int }_{P_s}\tilde{\rho }dx={\int }_V\rho (t_0,a)da,\frac{d}{ds}{\int }_{P_s}\tilde{\rho }\tilde{u}dx={\int }_{P_s}\tilde{\rho }\tilde{b}dx+{\int }_{\mathrm{\partial }{P}_s}\tilde{\sigma }\nu dS\mathrm{.}$

Assume also the Axiom of Choice. These material charts may depend on $(t_0,x_0)$; no global material chart or angular or total-energy balance is required.

Then every such $C(t)$ satisfies

$\frac{d}{dt}{\int }_{C(t)}\rho dx=-{\int }_{\mathrm{\partial }C(t)}\rho (u-w)\cdot ndS,$

$\frac{d}{dt}{\int }_{C(t)}\rho udx={\int }_{C(t)}\rho bdx+{\int }_{\mathrm{\partial }C(t)}\sigma ndS-{\int }_{\mathrm{\partial }C(t)}\rho u((u-w)\cdot n)dS\mathrm{.}$

Only the normal component $w\cdot n$ enters these balances. If $C(t)$ is material, so $w\cdot n=u\cdot n$ on its boundary, the relative mass and advective-momentum fluxes vanish. The conclusion is a conditional model deduction; it asserts no existence of a flow satisfying the balances.

Powers, Lecture Notes on Intermediate Fluid Mechanics, §4.6.1–§4.6.4

Mei, MIT OpenCourseWare, Kinematic Transport Theorem, §1.3

Mei, MIT OpenCourseWare, Law of Momentum Conservation, §1.5