Adopted mass balance: the mass of every material subregion is constant in time

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the general Lebesgue mass integrals of The density field of the continuum model and the mass of a material subregion; this qualifies the mathematical mass construction, not the choice-free kinematics.

Adopt the following assumption for every model of the continuum kinematic branch of Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance with density field of The density field of the continuum model and the mass of a material subregion.

(i) Mass balance. Let $\chi$ be a motion of the model on the time window $I$, let $J_0\subseteq I$ be an open interval with $\overline{J_0}\subseteq I$, and let $B^{\mathrm{\prime }}\subseteq B$ be a material subregion (nonempty bounded open with $\overline{B^{\mathrm{\prime }}}\subseteq B$). Then the model mass of $B^{\mathrm{\prime }}$,

$m(t,B^{\mathrm{\prime }})={\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx(t\in J_0),$

is constant in $t$:

$$\begin{array}{cccc}& \frac{d}{dt}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho (t,x)dx=0(t\in J_0)\mathrm{.}& & \text{(CM)}\end{array}$$

The assumption is imposed for every material subregion and every time in the window; the family of subregions is not restricted to a selected finite list.

(ii) Reading and exclusions. The assumption says that a label carries its matter permanently: the material of the body is a closed mechanical system during the motion. It is not adopted, and no statement of this page covers: open systems in which material is added to or removed from the body; diffusion, mixing or relative transport of distinct constituents between labels; chemical reactions, phase change or mass exchange between constituents; mass sources or sinks distributed in the body; and relativistic mass-energy conversion. In such situations the balance above can fail and a different balance with source terms is required.

(iii) Empirical status. The assumption is the classical conservation of mass, adopted here as model structure. Comparisons of predictions with observed masses of prepared closed systems can support it under stated apparatus and calibration assumptions; such comparisons are evidence about the combined model and apparatus, not a deductive proof of the principle, and no measurement is asserted or used by this postulate.

(iv) Status of the consequences. The local continuity equation derived on this page is a conditional theorem: it follows from (CM) together with the declared regularity of the fields and the transport identity of the kinematic branch. If (CM) is replaced by a balance with sources, the local equation acquires the corresponding source terms and none of the present deductions applies unchanged.