Incompressible Newtonian and barotropic inviscid fluid constitutive models

Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let $O\subseteq {\mathbb{R}}^3$ be open, and let $u:I\times O\to {\mathbb{R}}^3$ be a $C^1$ spatial velocity field with the conventions of Euclidean spatial calculus: gradient, divergence and material derivative. Write $Du(t,x)$ for the total derivative in the spatial variable and $\mathrm{div}u=\mathrm{tr}(Du)$ (The trace $\mathrm{tr}(A)$ as the sum of the diagonal entries). For $A\in M_3(\mathbb{R})$ define its transpose by $(A^{\mathsf{T}}{)}_{ij}:=A_{ji}$ for $1\le i,j\le 3$, using the same one-based component labels; call $A$ symmetric when $A_{ij}=A_{ji}$ for every such pair.

(a) Incompressible Newtonian fluid. Model data are a $C^1$ scalar field $p:I\times O\to \mathbb{R}$ (pressure) and a real number $\eta >0$ (dynamic viscosity; a dimensionless mathematical parameter until a unit system is fixed). The model consists of the incompressibility constraint

$\mathrm{div}u(t,x)=0((t,x)\in I\times O)$

and the Newtonian stress field

$\sigma (t,x):=-p(t,x)I_3+\eta (Du(t,x)+Du(t,x{)}^{\mathsf{T}})\in M_3(\mathbb{R}),$

with $(Du(t,x{)}^{\mathsf{T}}{)}_{ij}={\mathrm{\partial }}_i{u}_j(t,x)$ and $I_3$ the identity (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations). The entries of the second term are $\eta ({\mathrm{\partial }}_j{u}_i+{\mathrm{\partial }}_i{u}_j)$, so $\sigma$ is symmetric for every $(t,x)$.

(b) Barotropic inviscid fluid. Model data are a $C^1$ function $p:(0,\mathrm{\infty })\to \mathbb{R}$ (pressure law) and a $C^1$ scalar map $V:I\times O\to \mathbb{R}$. Calling $V$ a body-force potential means that its associated vector field is defined by $b(t,x):=-{\mathrm{\nabla }}_{x}V(t,x)$; this field is continuous, and in a consuming momentum equation the body-force density is $\rho b=-\rho {\mathrm{\nabla }}_{x}V$. This datum is separate from the stress law. The enthalpy of the pressure law is the $C^1$ function

$h(\rho ):={\int }_1^{\rho }\frac{p^{\mathrm{\prime }}(s)}{s}ds(\rho >0),$

which is characterised by $h(1)=0$ and $h^{\mathrm{\prime }}(\rho )=p^{\mathrm{\prime }}(\rho )\mathrm{/}\rho$ for every $\rho >0$ (The integral function $F(x):={\int }_a^{x}f$ of an integrable $f$, The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive). For a $C^1$ positive density field $\rho :I\times O\to (0,\mathrm{\infty })$ the branch's stress field is

$\sigma (t,x):=-p(\rho (t,x))I_3,$

and the branch is inviscid (no velocity-gradient term) and barotropic (the stress depends on position and time only through the density).

• Both branches are adopted constitutive models. The presence or absence of a viscosity term, the value of $\eta$ (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set conventions do not enter), the pressure law and the incompressibility constraint are model structure, not consequences of the momentum balance or of the transport theorem. In particular, incompressibility is an additional scalar equation imposed by the model, and no material is asserted to obey it.

• The sign convention is that of Traction fields and stress matrices representing them: the traction across a surface with unit normal $n$ is $\sigma n$, so the isotropic term $-p{I}_3$ produces the compressive traction $-pn$; positive $p$ is a pressure in this convention. A viscous stress proportional to $Du+D{u}^{\mathsf{T}}$ is the classical Newtonian form, asserted here for the stated $C^1$ field class only.

• In (b) the pressure law is defined on the positive reals only; any statement using this branch must supply a positive density field, and no continuation of $p$ to $\rho \le 0$ or through a vacuum is part of this definition. The enthalpy is an integral of a continuous integrand on the compact interval $[1,\rho ]$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); no path or choice issue arises.

• Units are interpretive: after a unit system is fixed the values of $p$ and the parameter $\eta$ acquire units, and $h$ acquires energy-per-mass units. The mathematical items of this collection use the fields themselves and attach no unit names; a physical reading belongs to a consuming model.