Single-component nonpolar continuum and material balances

Adopt a smooth single-component material continuum on a stated macroscopic regime in a Cartesian inertial chart. Let ${\rho }_0\in C^1(\bar{B})$ be positive. On each swept material region, require $\rho ,u,e,T,s,q,\sigma \in C^1$ and $b,r\in C^0$, with $\rho >0$ and $T>0$; the motion has the regularity and positive Jacobian in the preceding definition. A material subregion is ${\chi }_t(V)$ for a nonempty bounded open set $V⋐B$ whose boundary is locally, after a rigid coordinate change, a $C^1$ graph with $V$ on one side; its boundary moves with the material velocity. All volume integrals here use the componentwise Riemann integral by zero extension to a containing closed rectangle. A bounded set is Jordan measurable when its indicator has such a Riemann integral; a set has content zero when, for every positive $\epsilon$, it has a finite cover by axis-parallel boxes whose total volume is less than $\epsilon$, where box volume is the product of side lengths. For each permitted $V$ and its image ${\chi }_t(V)$, their compact closures are Jordan measurable and their boundaries have content zero. For every continuous scalar integrand $f$ on a neighborhood of ${\chi }_t(\bar{V})$, ${\int }_{{\chi }_t(V)}fdx={\int }_{{\chi }_t(\bar{V})}fdx$; the analogous equality is used on $V$ and $\bar{V}$. Vector and tensor integrals are taken componentwise. These are the earlier $C^1$-region integral conventions, so the displayed balances over open material regions have the same meaning as their compact-closure transport formulas. For each current boundary $S=\mathrm{\partial }{\chi }_t(V)$, surface integration is the finite-chart Riemann convention: for any continuous scalar $f$ on $S$, choose regular $C^1$ graph charts $X_j:U_j\subset {\mathbb{R}}^2\to S\cap O_j$ and a finite nonnegative subordinate smooth ambient partition ${\chi }_j$ with compact support in the chart neighborhoods and sum one on $S$, and put ${\int }_{S}fdS={\sum }_j{\int }_{U_j}({\chi }_{j}f)(X_j(y))\sqrt{\det (D{X}_j(y{)}^{T}D{X}_j(y))}dy$. Each parameter integrand is extended by zero outside $U_j$. These integrals are finite and independent of charts and partition; in graph coordinates the density is $\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}$. The unit normal is the continuous outward normal, and vector surface integrals are componentwise. Thus all traction, work, heat and entropy fluxes below use the same boundary-area convention. When a consuming mathematical theorem separately assumes Countable Choice, these continuous-field surface integrals also agree with its finite-chart Lebesgue surface integrals with the same Gram density; that conditional mathematical agreement adds no choice assumption to this physical postulate. The continuum approximation, its empirical regime, and the following balance laws are physical assumptions of this model, not deductions from the mathematical definition of a motion or from molecular mechanics.

Assume there are no mass sources and no diffusion relative to $u$. Adopt a positive reference mass density ${\rho }_0$, a positive current density $\rho$, a body-acceleration field $b$, a contact traction $t(t,x,n)=\sigma (t,x)n$ depending only on the local outward unit normal $n$, a $C^1$ Cauchy stress field $\sigma$, specific internal energy $e$, positive temperature $T$, specific entropy $s$, heat-flux vector $q$, and specific heat supply $r$, with the types and SI units fixed in the preceding definition. Contact traction and volume-force densities are continuous and locally bounded on the smooth material regions under consideration. There are no surface couples or intrinsic spin angular momentum; a micropolar or couple-stress model is outside this postulate.

For every such material subregion, adopt conservation of mass,

${\int }_{{\chi }_t(V)}\rho (t,x)dx={\int }_V{\rho }_0(a)da\mathrm{.}$

In the same inertial chart, adopt the balances of linear and angular momentum,

$\frac{d}{dt}{\int }_{{\chi }_t(V)}\rho udx={\int }_{{\chi }_t(V)}\rho bdx+{\int }_{\mathrm{\partial }{\chi }_t(V)}t(t,x,n)dS,$

$\frac{d}{dt}{\int }_{{\chi }_t(V)}x\times (\rho u)dx={\int }_{{\chi }_t(V)}x\times (\rho b)dx+{\int }_{\mathrm{\partial }{\chi }_t(V)}x\times t(t,x,n)dS\mathrm{.}$

Adopt the total-energy balance and entropy inequality

$\frac{d}{dt}{\int }_{{\chi }_t(V)}\rho (e+\frac{\mathrm{\mid }u{\mathrm{\mid }}^2}{2})dx={\int }_{{\chi }_t(V)}\rho b\cdot udx+{\int }_{\mathrm{\partial }{\chi }_t(V)}t(t,x,n)\cdot udS-{\int }_{\mathrm{\partial }{\chi }_t(V)}q\cdot ndS+{\int }_{{\chi }_t(V)}\rho rdx,$

$\frac{d}{dt}{\int }_{{\chi }_t(V)}\rho sdx\ge -{\int }_{\mathrm{\partial }{\chi }_t(V)}\frac{q}{T}\cdot ndS+{\int }_{{\chi }_t(V)}\frac{\rho r}{T}dx\mathrm{.}$

The entropy flux uses the local material temperature at the heat-transfer point. A reservoir at a different temperature, a singular surface force, a shock, vacuum, or a weak solution requires a separately stated model and is not covered here. The postulate reports no experiment and asserts no measurement or universal claim that nature obeys these assumptions.

Ain A. Sonin, MIT 2.25 lecture notes, §§1.1–1.3 (material volumes and mass, momentum, energy, and entropy balances)

Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 3, §§3.2–3.4 (momentum, traction, and stress)

Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 4, §§4.1–4.4.4 (mass, momentum, energy, and entropy balances)

David Tong, Fluid Mechanics, Part I, §§1.1–1.1.3 (continuum approximation and mass-density/velocity fields)

The cited continuum laws are adopted physical premises. The cited texts motivate their standard formulations; neither a textbook nor the model equations prove the continuum approximation or empirical validity. The temperature in the entropy flux is explicitly the local material temperature. The postulate does not adopt a constitutive law or an equation of state.