Calorically perfect ideal-gas Euler model for numerical states

Use the Newtonian spacetime and Cartesian inertial-chart convention from the earlier definition titled “Classical frames, primitive quantities and SI units”: time is absolute, $M$ is an oriented Euclidean affine three-space with no distinguished origin, and a Cartesian inertial chart is an event-coordinate map with absolute-time coordinate and spatial maps $X_t(q)=b(q)+tw$, where $b:M\to {\mathbb{R}}^3$ is an orientation-preserving affine isometry and $w\in {\mathbb{R}}^3$ is constant in metres per second. Write $X=(X_1,X_2,X_3)$ for the spatial event coordinates of one selected such chart. Choose the positive $X_1$ axis and restrict this branch to fields depending only on absolute time $t$ and the coordinate $x:=X_1$; $u$ is the signed scalar velocity component along that axis. Here a frame means this adapted Cartesian inertial chart, not an observer congruence or a basis field. Let $\mathrm{\Omega }\subseteq \mathbb{R}$ be an open interval of $x$ measured in metres, let $T_f>0$ be a final time measured in seconds, and take the space-time domain to be $\mathrm{\Omega }\times (0,T_f)$. For this one-dimensional weak branch, adopt Countable Choice (every sequence of nonempty sets admits a choice function) and use the all-dimensional Euclidean Lebesgue convention of the conservative weak-form definition. Space-time fields use the restriction of ${\lambda }_2$ to $\mathrm{\Omega }\times [0,T_f)$ and initial data the restriction of ${\lambda }_1$ to $\mathrm{\Omega }$, with the corresponding trace sigma-algebras, measurable-map and almost-everywhere conventions. Integrals use the signed Lebesgue integral and completed-product almost-everywhere section convention from that definition. The measurable fields are mass density $\rho :\mathrm{\Omega }\times (0,T_f)\to (0,\mathrm{\infty })$ in kilograms per cubic metre, velocity $u:\mathrm{\Omega }\times (0,T_f)\to \mathbb{R}$ in metres per second, and specific entropy $s:\mathrm{\Omega }\times (0,T_f)\to \mathbb{R}$ in joules per kilogram kelvin.

Adopt a local-equilibrium constitutive chart on the open state domain $\mathcal{U}=(0,\mathrm{\infty })\times \mathbb{R}$ with coordinates $(v,s)$ and a $C^2$ fundamental relation $e:\mathcal{U}\to \mathbb{R}$ for specific internal energy in joules per kilogram. Let $c_v>0$ be the adopted specific-heat-at-constant-volume parameter and $R>0$ the adopted specific gas-constant parameter, each in joules per kilogram kelvin. Define the derived specific heat at constant pressure and ratio of specific heats by $c_p:=c_v+R$ and $\gamma :=c_p\mathrm{/}{c}_v=1+R\mathrm{/}{c}_v>1$. Choose $v_{\ast }>0$ in cubic metres per kilogram, $T_{\ast }>0$ in kelvins, and $s_{\ast }\in \mathbb{R}$ in joules per kilogram kelvin as reference data. Adopt the calorically perfect ideal-gas relation $e(v,s)=c_v{T}_{\ast }{\left(\frac{v_{\ast }}{v}\right)}^{\gamma -1}\exp \left(\frac{s-s_{\ast }}{c_v}\right),$ with the conventional additive energy reference for which the specific internal energy is zero at zero temperature. Define the thermodynamic temperature and pressure maps by $T(v,s):={\mathrm{\partial }}_{s}e(v,s)$ and $p(v,s):=-{\mathrm{\partial }}_{v}e(v,s)$, holding the other state coordinate fixed. Direct differentiation gives $T(v,s)=\frac{e(v,s)}{c_v}>0,p(v,s)=(\gamma -1)\frac{e(v,s)}{v}>0.$ For density and entropy coordinates define the state-pressure map $p_{\mathrm{t}\mathrm{h}}(\rho ,s):=p(1\mathrm{/}\rho ,s)$ for $\rho >0$ and $s\in \mathbb{R}$. The spacetime state map is $v(t,x):=1\mathrm{/}\rho (t,x)$ and $(t,x)\mapsto (v(t,x),s(t,x))\in \mathcal{U}$. It assigns the specific internal-energy, temperature, and thermodynamic-pressure fields by $e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x):=e(v(t,x),s(t,x)),T(t,x):=({\mathrm{\partial }}_{s}e)(v(t,x),s(t,x)),p_{\mathrm{t}\mathrm{h}}(t,x):=p_{\mathrm{t}\mathrm{h}}(\rho (t,x),s(t,x))=-({\mathrm{\partial }}_{v}e)(v(t,x),s(t,x))\mathrm{.}$ Consequently $T(t,x)=e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x)\mathrm{/}{c}_v$ and $p_{\mathrm{t}\mathrm{h}}(t,x)=(\gamma -1)e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x)\mathrm{/}v(t,x)=(\gamma -1)\rho (t,x)e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x)=R\rho (t,x)T(t,x)$. At fixed specific entropy $s$, the state-pressure map is proportional to ${\rho }^{\gamma }$, so ${{\mathrm{\partial }}_{\rho }{p}_{\mathrm{t}\mathrm{h}}(\rho ,s)\mid }_s=\gamma p_{\mathrm{t}\mathrm{h}}(\rho ,s)\mathrm{/}\rho$; define the positive isentropic sound-speed map by $c_{\mathrm{t}\mathrm{h}}(\rho ,s):=\sqrt{{{\mathrm{\partial }}_{\rho }{p}_{\mathrm{t}\mathrm{h}}(\rho ,s)\mid }_s}=\sqrt{\gamma p_{\mathrm{t}\mathrm{h}}(\rho ,s)\mathrm{/}\rho }$. Thus $\gamma$ is dimensionless, sound speed has units of metres per second, and pressure has units of pascals. This is a constant-specific-heat ideal-gas approximation, restricted to a regime where that closure is adopted.

Define momentum density $m(t,x):=\rho (t,x)u(t,x)$ in kilograms per square metre second and total energy density $E(t,x):=\rho (t,x)(e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x)+u(t,x{)}^2\mathrm{/}2)$ in joules per cubic metre. The corresponding numerical state is $U(t,x):=(\rho (t,x),m(t,x),E(t,x))\in {\mathbb{R}}^3$. For $\gamma >1$, set ${\mathcal{G}}_{\gamma }=\{(\rho ,m,E):\rho >0,E-\frac{m^2}{2\rho }>0\},p(U)=(\gamma -1)(E-\frac{m^2}{2\rho })\mathrm{.}$ Since $\rho (t,x)>0$ and $e_{\mathrm{i}\mathrm{n}\mathrm{t}}(t,x)>0$, $U(t,x)\in {\mathcal{G}}_{\gamma }$. For this state, $E-m^2\mathrm{/}(2\rho )=\rho e_{\mathrm{i}\mathrm{n}\mathrm{t}}$ and $p(U(t,x))=p_{\mathrm{t}\mathrm{h}}(t,x)$, so the numerical and thermodynamic pressure maps agree. With $u=m\mathrm{/}\rho$, its one-dimensional Euler flux and isentropic sound speed are $F(U)=(m,\frac{m^2}{\rho }+p(U),u(E+p(U))),c(U)=\sqrt{\frac{\gamma p(U)}{\rho }}\mathrm{.}$ The components of $F$ are mass flux in kilograms per square metre second, momentum flux in pascals, and energy flux in watts per square metre. The speed $c$ is in metres per second.

Adopt the inviscid one-dimensional branch with zero body acceleration, zero specific heat supply, zero conductive heat flux, and isotropic three-dimensional Cauchy stress $\sigma =-p_{\mathrm{t}\mathrm{h}}{I}_3$ in the tension-positive convention in which traction on an outward unit normal $n$ is $\sigma n$; here $I_3$ is the $3\times 3$ identity matrix. Its restriction to the selected axis is the scalar normal stress ${\sigma }_{1\mathrm{D}}:={\sigma }_{11}=-p_{\mathrm{t}\mathrm{h}}$ in pascals, so the one-dimensional momentum flux is $\rho u^2-{\sigma }_{1\mathrm{D}}=\rho u^2+p_{\mathrm{t}\mathrm{h}}$. In the following weak identities, write $p:=p_{\mathrm{t}\mathrm{h}}$. Assume the locally integrable conservative variables, fluxes, entropy density, and entropy flux satisfy the identities in the distributional sense. Let ${\rho }_0,m_0,E_0,S_0\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ be the initial mass density in kilograms per cubic metre, momentum density in kilograms per square metre second, total energy density in joules per cubic metre, and entropy density in joules per cubic metre kelvin, respectively. Require all displayed densities, fluxes, and products to be locally integrable up to $t=0$: their absolute values have finite integrals on every compact subset of $\mathrm{\Omega }\times [0,T_f)$ in the relative topology, using the fields for $t>0$. This requirement assigns no point values at $t=0$ and asserts no strong initial trace. Test functions $\varphi ,\psi ,\eta ,\zeta \in C^1(\mathrm{\Omega }\times [0,T_f))$ are compactly supported in space-time and vanish near $T_f$; $\zeta$ is nonnegative, and tests may be nonzero at $t=0$. Then adopt ${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho {\mathrm{\partial }}_t\varphi +\rho u{\mathrm{\partial }}_x\varphi )dxdt+{\int }_{\mathrm{\Omega }}{\rho }_0\varphi (0,x)dx=0,$ ${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho u{\mathrm{\partial }}_t\psi +(\rho u^2+p){\mathrm{\partial }}_x\psi )dxdt+{\int }_{\mathrm{\Omega }}{m}_0\psi (0,x)dx=0,$ ${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(E{\mathrm{\partial }}_t\eta +u(E+p){\mathrm{\partial }}_x\eta )dxdt+{\int }_{\mathrm{\Omega }}{E}_0\eta (0,x)dx=0,$ and ${\int }_0^{T_f}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho s{\mathrm{\partial }}_t\zeta +\rho us{\mathrm{\partial }}_x\zeta )dxdt+{\int }_{\mathrm{\Omega }}{S}_0\zeta (0,x)dx\le 0.$ The last inequality is a separately adopted physical entropy inequality. It is not deduced from the conservative equations or from smooth-flow identities. This postulate specifies a physical model and its idealized local-equilibrium closure; it does not assert existence or uniqueness of a weak solution, prove that actual matter obeys the model, or identify a computed state with an experimental observation.

Locally integrable conservative weak and initial forms

Specific fundamental relation and valid response charts — defines a $C^2$ specific fundamental relation on an open $(v,s)$ state domain and the derivatives $T=e_s$ and $p=-e_v$.

Classical frames, primitive quantities and SI units — fixes the Newtonian Cartesian-frame and SI field conventions.

Local simple-fluid equilibrium and Gibbs closure — its current Postulate supplies the C2 local state chart, T=e_s>0, and p=-e_v used here. Its actual local-equilibrium chart is used only under the separately adopted weak physical assumptions stated here.

Integrable weak continuum balance and entropy adoption — adopts locally integrable weak mass, momentum, and total-energy balances with initial data and a separate weak physical entropy inequality.

Euler state, flux, positive set and numerical roles — fixes the one-dimensional conserved-state, ideal-gas pressure, Euler-flux, and sound-speed formulas.

Powers, Lecture Notes on Gas Dynamics, §§2.3.3.3–2.3.4, §§2.4.6–2.4.7.2, and §§3.2–3.3

Tong, Fluid Mechanics, Part III: Waves, §§4.4.1–4.4.4 and §4.6.1