Control-volume balance: mass and momentum rates for a moving non-material region

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as used by the transport and divergence theorems. Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let $B\subseteq {\mathbb{R}}^3$ be a material region, and let $\chi :I\times O\to {\mathbb{R}}^3$ be a motion of $B$ on $I$ with spatial velocity $u$. Write

${\mathcal{D}}_{\chi }:=\{(t,x):t\in I,x\in {\chi }_t(O)\}$

for the open domain on which $u$ is defined (Continuum motions, spatial velocity, material acceleration and swept regions). Let $J_0\subseteq I$ be an open interval with compact closure in $I$, and let $U\subseteq {\mathcal{D}}_{\chi }$ be open and contain the swept region of $B$ over $\overline{J_0}$.

Let $\rho :U\to (0,\mathrm{\infty })$ and $\sigma :U\to M_3(\mathbb{R})$ be $C^1$ and let $b:U\to {\mathbb{R}}^3$ be continuous. Define $g_i:={\sum }_j{\mathrm{\partial }}_j{\sigma }_{ij}$ and suppose, on $U$:

$$\begin{array}{cccc}& {\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0\text{and}\rho D_{t}u=\rho b+g\mathrm{.}& & \text{(KM)}\end{array}$$

Let $\psi :I\times O^{\mathrm{\prime }}\to {\mathbb{R}}^3$ be a second motion of a material region $B^{\mathrm{\prime }}\subseteq B$ (with the same reference chart domain $O^{\mathrm{\prime }}=O$ used for $B$; write $w$ for its spatial velocity), and suppose that for every $t\in J_0$ the control region $C_t:={\psi }_t(B^{\mathrm{\prime }})$ is a bounded $C^1$ domain (Bounded C1 domains and their outward normals) with outward unit normal $\nu$ and boundary surface integral as in Surface integration on compact C1 hypersurfaces.

Assume also that the control sweep ${\mathrm{\Sigma }}_C:=\{(t,x):t\in \overline{J_0},x\in {\psi }_t(\overline{B^{\mathrm{\prime }}})\}$ is contained in $U$. Thus the fields and the local equations (KM) are defined on a neighbourhood of every closed control region. No condition that the control motion coincide with the material motion is imposed.

Then for every $t\in J_0$:

$$\begin{array}{cccc}& \frac{d}{dt}{\int }_{C_t}\rho dx=-{\int }_{\mathrm{\partial }{C}_t}\rho (u-w)\cdot \nu dS,& & \text{(CV1)}\end{array}$$

and componentwise

$$\begin{array}{cccc}& \frac{d}{dt}{\int }_{C_t}\rho u_{i}dx={\int }_{C_t}\rho b_{i}dx+({\int }_{\mathrm{\partial }{C}_t}\sigma \nu dS{)}_i-{\int }_{\mathrm{\partial }{C}_t}\rho u_i(u-w)\cdot \nu dS(i=1,2,3)\mathrm{.}& & \text{(CV2)}\end{array}$$

• The control region need not be material: $\psi$ is an arbitrary motion of the reference region $B^{\mathrm{\prime }}$, and $w$ is the velocity of its points. The relative velocity $u-w$ in the flux terms records the transport of mass and momentum across the moving boundary. Taking $\psi =\chi$ recovers the material statements, in which the relative velocity vanishes and both rates reduce to the material integrals of (K) and (KM).

• The identities fix the inertial conventions: the momentum flux uses the spatial momentum density $\rho u$ times the normal component of the relative velocity, and the traction term is the surface integral of $\sigma \nu$. No stress symmetry, constitutive relation or boundary condition is used or asserted; the only regularities required are those stated.