Adopted fluid constitutive branches: incompressible Newtonian, barotropic inviscid, and the periodic-cell idealization

Adopt the following constitutive branches for models of the continuum framework (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum body data, the fixed inertial frame and the dimensions of the fields), with the stress conventions of Traction fields and stress matrices of the continuum model, with units and sign conventions.

(a) Incompressible Newtonian fluid. Model data are a constant density $\rho >0$, a dynamic viscosity $\eta >0$ (constant, dimension $\mu {\mathrm{\ell }}^{-1}{\tau }^{-1}$, in SI measured in $\mathrm{P}\mathrm{a}\mathrm{s}$, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework) and a pressure field $p$ of class $C^1$. The branch consists of the incompressibility constraint

$\mathrm{div}u=0$

on the current regions (Euclidean spatial calculus: gradient, divergence and material derivative) together with the Newtonian stress field

$\sigma =-p{I}_3+\eta (Du+D{u}^{\mathsf{T}}),$

the second term being symmetric. By the continuity equation of the model (The continuity equation of the continuum model: local form of mass balance, the material form, and the incompressibility reading) the constraint $\mathrm{div}u=0$ is equivalent, in this branch, to the density being constant along every label worldline, so the constant-density reading is consistent with the adopted mass balance.

(a0) Constant-density incompressible inviscid fluid. Adopt separately the Euler branch: $\rho >0$ is constant, $\mathrm{div}u=0$, pressure is a Lagrange multiplier enforcing this constraint, and $\sigma =-p{I}_3$. This is the same stress form as (a) with $\eta =0$, adopted as model structure; it is not obtained by imposing a barotropic pressure law on a constant density. No positive-viscosity assumption is used in this branch.

(b) Barotropic inviscid fluid. Model data are a $C^1$ pressure law $p:(0,\mathrm{\infty })\to \mathbb{R}$ and a $C^1$ potential $V$ with the conservative body force $b=-\mathrm{\nabla }V$. The branch's stress field is $\sigma =-p(\rho )I_3$, where $\rho$ is the model density field of The density field of the continuum model and the mass of a material subregion. Its enthalpy function is $h:(0,\mathrm{\infty })\to \mathbb{R}$ defined by $h(r)={\int }_{{\rho }_{\mathrm{r}}}^r{p}^{\mathrm{\prime }}(s)s^{-1}ds$, with a fixed model reference density ${\rho }_{\mathrm{r}}>0$ of the same dimension as $r$. It is characterized by $h^{\mathrm{\prime }}(r)=p^{\mathrm{\prime }}(r)\mathrm{/}r$ and has dimension $[h]=[p]\mathrm{/}[\rho ]={\mathrm{\ell }}^2{\tau }^{-2}$, energy per unit mass; the spatial enthalpy field is $x\mapsto h(\rho (t,x))$ (Incompressible Newtonian and barotropic inviscid fluid constitutive models, Continuum body data, the fixed inertial frame and the dimensions of the fields).

(c) Periodic-cell idealization and scale conventions. This clause applies only to the constant-density incompressible branches (a) and (a0), with constant $\rho >0$ and constant $\eta \ge 0$; in (a0), set $\eta =0$. It makes no incompressible reduction or pressure-gauge assertion for the variable-density barotropic branch (b). Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the normalized product measure and Fourier/Sobolev machinery of this clause; the pointwise stress laws of (a), (a0) and (b) do not use this premise. Fix a cell side length $L_{\ast }>0$ and velocity scale $U_{\ast }>0$ as model data with dimensions $\mathrm{\ell }$ and $\mathrm{\ell }{\tau }^{-1}$, and let $T_{\ast }=L_{\ast }\mathrm{/}{U}_{\ast }$ be the time scale. The physical spatial cell is $(\mathbb{R}\mathrm{/}{L}_{\ast }\mathbb{Z}{)}^3$, with physical volume $L_{\ast }^3$; it describes a spatially periodic ideal medium, not a material boundary or a measured wall law. Write $\tau =t\mathrm{/}{T}_{\ast }$ and $y=x\mathrm{/}{L}_{\ast }\in {\mathbb{T}}^3$, using an optional fixed time origin. For physical velocity $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, pressure $P_{\mathrm{m}}$ and density $\rho$, define dimensionless fields and viscosity by $U(\tau ,y)=\frac{u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(T_{\ast }\tau ,L_{\ast }y)}{U_{\ast }},\mathrm{\Pi }(\tau ,y)=\frac{P_{\mathrm{m}}(T_{\ast }\tau ,L_{\ast }y)}{\rho U_{\ast }^2},{\nu }_{\ast }:=\frac{\eta }{\rho U_{\ast }{L}_{\ast }}\mathrm{.}$ The inviscid incompressible branch (a0) has $\eta =0$ and ${\nu }_{\ast }=0$. The normalized unit-torus measure and $H^s$ norms of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection apply to $U$, not to an unscaled dimensionful velocity. In the absence of body force the physical equation scales exactly to ${\mathrm{\partial }}_{\tau }U+(U\cdot {\mathrm{\nabla }}_y)U=-{\mathrm{\nabla }}_y\mathrm{\Pi }+{\nu }_{\ast }{\mathrm{\Delta }}_{y}U,{\mathrm{div}}_{y}U=0.$ A dimensionless lifespan $T$ is a physical lifespan $T_{\ast }T$, and $P_{\mathrm{m}}=\rho U_{\ast }^2\mathrm{\Pi }$. Pressure has zero spatial mean as the chosen gauge. The periodic geometry and the scales are adopted model data, not observed boundary behavior or measured parameters. A change of inertial frame carries the cell lattice to $R(L_{\ast }{\mathbb{Z}}^3)$; the unit-torus identification is made in the correspondingly rotated cell coordinates, rather than silently keeping a fixed lattice under an arbitrary rotation.

(d) Validity and exclusions. The branches are not adopted for: compressible viscous flow with temperature dependence or thermal coupling; non-Newtonian materials (shear-dependent or viscoelastic response); turbulent closures or averaged models (whose additional terms are not the fields of this branch); surface tension, capillarity and multiphase interfaces; and flows in which the continuum description itself fails at the scales of interest. Wall conditions such as the no-slip condition used later are additional ideal boundary data; they are adopted where stated, not derived, and their empirical validity has limits that are not formalized here.

(e) Status. The constitutive laws, the incompressibility constraint, the pressure law and the periodic-cell geometry are adopted model structure, not consequences of the balance principles. The viscosity $\eta$, the pressure law and the potential are model parameters or model functions: whenever numerical values are used they enter as measured empirical premises with their conditions and uncertainty, and no measured value is asserted by this postulate.