The finite-regularity open-flow model: moving-control-region mass and momentum balances, their material and fixed special cases, and the inlet/outlet decomposition reproducing the open-body and rocket laws

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. Adopt the finite-regularity open-flow model with the following model data in one fixed inertial frame (the adapted affine chart of Inertial frames, stationary observers and the adopted Galilean transformations between frames) and with SI conventions (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): a nonempty open interval $I$; a material region $B\subseteq {\mathbb{R}}^3$ (bounded open with $C^1$ boundary); a motion $\chi$ of $B$ on $I$ of class $C^2$ whose time slices are diffeomorphisms with positive Jacobian onto their images. Let $U\subseteq \mathbb{R}\times {\mathbb{R}}^3$ be an open common field domain containing the swept region over a compact subinterval $J_0$ with $\overline{J_0}\subseteq I$. On $U$ take a scalar mass-density field $\rho :U\to (0,\mathrm{\infty })$ of class $C^1$ with $[\rho ]=\mu {\mathrm{\ell }}^{-3}$ (Material regions, densities and their mass, where mass on a spatial region $C$ is ${\int }_C\rho dx$); its spatial velocity field $u:U\to {\mathbb{R}}^3$ of class $C^1$ with $[u]=\mathrm{\ell }{\tau }^{-1}$ and $u(t,\chi (t,X))={\mathrm{\partial }}_t\chi (t,X)$ (Continuum motions, spatial velocity, material acceleration and swept regions); a Cauchy stress matrix field $\sigma :U\to M_3(\mathbb{R})$ of class $C^1$ with $[\sigma ]=\mu {\mathrm{\ell }}^{-1}{\tau }^{-2}$, representing the traction by $\mathbf{t}(\nu )=\sigma \nu$ on a surface with unit normal $\nu$ (Traction fields and stress matrices representing them); and a continuous specific body-force vector $b:U\to {\mathbb{R}}^3$, meaning force per unit mass and distinct from the material acceleration, with $[b]=\mathrm{\ell }{\tau }^{-2}$. For the material derivative used below, make the time-dependent convention explicit: $D_{t}u(t,x):={\mathrm{\partial }}_{t}u(t,x)+(D_{x}u(t,x))u(t,x),(D_{t}u{)}_i={\mathrm{\partial }}_t{u}_i+{\sum }_{j=1}^3{u}_j{\mathrm{\partial }}_{x_j}{u}_i,$ where ${\mathrm{\partial }}_t$ differentiates at fixed $x$ and $D_{x}u$ is the spatial Jacobian at fixed $t$ (Euclidean spatial calculus: gradient, divergence and material derivative, Continuum motions, spatial velocity, material acceleration and swept regions). Define the row divergence by $g_i:={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{\sigma }_{ij}$, so $g$ is a vector force density with units $\mu {\mathrm{\ell }}^{-2}{\tau }^{-2}$. Adopt as the model's material balances: for every material subregion $B^{\mathrm{\prime }}\subseteq B$ (a nonempty bounded open subset with $\overline{B^{\mathrm{\prime }}}\subseteq B$ whose images ${\chi }_t(B^{\mathrm{\prime }})$ are bounded $C^1$ domains, Bounded C1 domains and their outward normals) and every $t\in J_0$,

$\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho \mathrm{d}x=0,\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho u\mathrm{d}x={\int }_{{\chi }_t(B^{\mathrm{\prime }})}\rho b\mathrm{d}x+{\int }_{\mathrm{\partial }{\chi }_t(B^{\mathrm{\prime }})}\sigma \nu \mathrm{d}S,$

with $\nu$ the outward unit normal and the surface integral that of Surface integration on compact C1 hypersurfaces. These two balance laws are adopted model assumptions of this item, in the sense of the open-body accounting of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (iii); they are not derived from the particle postulates. Then, for every second motion $\psi$ of a material region $B^{\mathrm{\prime }}\subseteq B$ on $I$ (the control motion) with control regions $C_t:={\psi }_t(B^{\mathrm{\prime }})$ that are bounded $C^1$ domains for $t\in J_0$, spatial velocity $w$ with $[w]=\mathrm{\ell }{\tau }^{-1}$, and outward unit normal $\nu$. Require its closed sweep ${\mathrm{\Sigma }}_C:=\{(t,x):t\in \overline{J_0},x\in {\psi }_t(\overline{B^{\mathrm{\prime }}})\}$ to lie in the common open field domain. Require also either that this sweep lie locally inside the current material region ${\chi }_t(B)$, where localization supplies the local mass and momentum equations, or that those same local equations ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0$ and $\rho D_{t}u=\rho b+g$ be explicitly adopted on a neighbourhood of the control sweep. The control motion need not coincide with $\chi$. Then:

1. (Control-volume identities.) For every $t\in J_0$, $$\begin{array}{cccc}& \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{C_t}\rho \mathrm{d}x=-{\int }_{\mathrm{\partial }{C}_t}\rho (u-w)\cdot \nu \mathrm{d}S,& & \text{(CV1)}\end{array}$$ and componentwise $$\begin{array}{cccc}& \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{C_t}\rho u_i\mathrm{d}x={\int }_{C_t}\rho b_i\mathrm{d}x+({\int }_{\mathrm{\partial }{C}_t}\sigma \nu \mathrm{d}S{)}_i-{\int }_{\mathrm{\partial }{C}_t}\rho u_i(u-w)\cdot \nu \mathrm{d}S(i=1,2,3)\mathrm{.}& & \text{(CV2)}\end{array}$$
2. (Material and fixed special cases.) If $\psi =\chi$ (the control region is material, $w=u$ on the boundary), the flux terms vanish and (CV1), (CV2) reduce to the adopted material balances; if $\psi$ is the constant motion ($w=0$, $C_t=C$ fixed), then $m_C^{\mathrm{\prime }}=-{\int }_{\mathrm{\partial }C}\rho u\cdot \nu \mathrm{d}S,P_C^{\mathrm{\prime }}={\int }_C\rho b\mathrm{d}x+{\int }_{\mathrm{\partial }C}\sigma \nu \mathrm{d}S-{\int }_{\mathrm{\partial }C}\rho u(u\cdot \nu )\mathrm{d}S$ with $m_C:={\int }_C\rho$, $P_C:={\int }_C\rho u$; the last term is the momentum flux through the fixed boundary, and the relation is the fixed-region steady-flow balance of the engineering treatment when the fields are steady and the region fixed.
3. (Inlet/outlet decomposition and the open-body law.) Suppose that for every $t\in J_0$ the boundary decomposes into finitely many $C^1$ pieces, among which ${\mathrm{\Sigma }}_{\mathrm{i}\mathrm{n},t}$ (where $(u-w)\cdot \nu <0$), ${\mathrm{\Sigma }}_{\mathrm{o}\mathrm{u}\mathrm{t},t}$ (where $(u-w)\cdot \nu >0$) and the zero-flux rest, and put $r_{\mathrm{i}\mathrm{n}}:=-{\int }_{{\mathrm{\Sigma }}_{\mathrm{i}\mathrm{n},t}}\rho (u-w)\cdot \nu \mathrm{d}S\ge 0,r_{\mathrm{o}\mathrm{u}\mathrm{t}}:={\int }_{{\mathrm{\Sigma }}_{\mathrm{o}\mathrm{u}\mathrm{t},t}}\rho (u-w)\cdot \nu \mathrm{d}S\ge 0,$ $w_{\mathrm{i}\mathrm{n}}:=\frac{1}{r_{\mathrm{i}\mathrm{n}}}{\int }_{{\mathrm{\Sigma }}_{\mathrm{i}\mathrm{n},t}}\rho u(-(u-w)\cdot \nu )\mathrm{d}S,w_{\mathrm{o}\mathrm{u}\mathrm{t}}:=\frac{1}{r_{\mathrm{o}\mathrm{u}\mathrm{t}}}{\int }_{{\mathrm{\Sigma }}_{\mathrm{o}\mathrm{u}\mathrm{t},t}}\rho u((u-w)\cdot \nu )\mathrm{d}S$ (the flux averages are used only where the corresponding rate is positive; define $r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}=0$ when $r_{\mathrm{i}\mathrm{n}}=0$, and likewise for the outlet term), and write $m:={\int }_{C_t}\rho \mathrm{d}x>0,P:={\int }_{C_t}\rho u\mathrm{d}x,F_{\mathrm{e}\mathrm{x}\mathrm{t}}:={\int }_{C_t}\rho b\mathrm{d}x+{\int }_{\mathrm{\partial }{C}_t}\sigma \nu \mathrm{d}S\mathrm{.}$ The units are $[m]=\mu$, $[P]=\mu \mathrm{\ell }{\tau }^{-1}$, $[F_{\mathrm{e}\mathrm{x}\mathrm{t}}]=\mu \mathrm{\ell }{\tau }^{-2}$, $[r_{\mathrm{i}\mathrm{n}}]=[r_{\mathrm{o}\mathrm{u}\mathrm{t}}]=\mu {\tau }^{-1}$ and $[w_{\mathrm{i}\mathrm{n}}]=[w_{\mathrm{o}\mathrm{u}\mathrm{t}}]=\mathrm{\ell }{\tau }^{-1}$. The control-volume theorem supplies the derivatives of $m$ and $P$; since $\rho >0$ and each $C_t$ is a nonempty open region, $m>0$, so $v:=P\mathrm{/}m$ is differentiable on $J_0$. This verifies the differentiability hypotheses of Variable-mass momentum balance: from $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $(mv{)}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ to $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$. Then the control-volume identities read $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}},P^{\mathrm{\prime }}=F_{\mathrm{e}\mathrm{x}\mathrm{t}}+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}},$ and the lumped body with mass $m$ and velocity $v:=P\mathrm{/}m$ satisfies $m{v}^{\mathrm{\prime }}=F_{\mathrm{e}\mathrm{x}\mathrm{t}}+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v),$ the derived law of Variable-mass momentum balance: from $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $(mv{)}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ to $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$. In particular, a body that only loses mass ($r_{\mathrm{i}\mathrm{n}}=0$, $r_{\mathrm{o}\mathrm{u}\mathrm{t}}=-m^{\mathrm{\prime }}$) has thrust $r_{\mathrm{o}\mathrm{u}\mathrm{t}}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}})=-m^{\mathrm{\prime }}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}})$, and a body whose boundary moves with the material ($w=u$; no flux) has $m^{\mathrm{\prime }}=0$ and $P^{\mathrm{\prime }}=F_{\mathrm{e}\mathrm{x}\mathrm{t}}$.
4. (Status.) The balances displayed before clause 1 are adopted component models of the open flow, exactly as clause (iii) of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting adopts the lumped accounting; this item proves only the two stated deductions (localization followed by the control-volume identities, and the inlet/outlet reading). It asserts no constitutive law for $\rho$ or $\sigma$, no boundary condition, no incompressibility, no regularity beyond that stated, and no empirical claim; the fields, the control motion and the region are model data.

• Inertial conventions. The velocities $u$, $w$, $w_{\mathrm{i}\mathrm{n}}$, $w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and the momentum density $\rho u$ are all measured in one inertial frame; the flux uses the relative normal velocity $(u-w)\cdot \nu$, and choosing a different auxiliary velocity on the interior of $C_t$ does not change the identities because its contribution is converted to a boundary term by the divergence theorem (Control-volume balance: mass and momentum rates for a moving non-material region). The lumped velocity of clause 3 is $v=P\mathrm{/}m$, the momentum-weighted mean velocity of the contents, not the velocity of any particular material point.

• Relation to the lumped postulate. Clause 3 exhibits the two identities of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (iii) as a specialization of the spatial balances, with $r_{\mathrm{i}\mathrm{n}},r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ the mass fluxes through the inlet and outlet and $w_{\mathrm{i}\mathrm{n}},w_{\mathrm{o}\mathrm{u}\mathrm{t}}$ the flux-averaged inertial velocities. Conversely the material accounting of the postulate does not by itself produce the distributed identities (CV1), (CV2), which use the spatial fields.

• No empirical content. The finite-regularity open-flow model is a restricted model class; no physical body is asserted to be one, and the balances are adopted assumptions that a comparison with observation would test jointly with the apparatus and constitutive assumptions.