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Formal parameterized fluid systems and observables

Definition

Let $I \subseteq R$ be a nonempty open interval of time and let $Ω \subseteq R^{3}$ be a nonempty open set of Cartesian spatial coordinates $x = (x_{1}, x_{2}, x_{3})$ , with the standard right-handed orientation. A vector field $u : I \times Ω \to R^{3}$ is in the regularity class used here if $u$ , its first time partial derivatives, its first spatial partial derivatives, and its second spatial partial derivatives all exist and are continuous. A pressure field is a map $p : I \times Ω \to R$ with continuous first spatial partial derivatives; a density field is a map $ρ : I \times Ω \to (0, \infty)$ with continuous first time and first spatial partial derivatives. The scalar field $q : I \times Ω \to R$ used in the rotating system below has continuous first spatial partial derivatives. A scalar function of one or several displayed real variables is $C^{1}$ when its first partial derivatives exist and are continuous; piecewise- $C^{1}$ paths have the meaning in the line-integral definition below. For a scalar field $f$ , set $\nabla f = (\partial_{x_{1}} f, \partial_{x_{2}} f, \partial_{x_{3}} f)$ ; for a vector field $u = (u_{1}, u_{2}, u_{3})$ , set $div u = \sum_{j = 1}^{3} \partial_{x_{j}} u_{j}$ , $curl u = (\partial_{x_{2}} u_{3} - \partial_{x_{3}} u_{2}, \partial_{x_{3}} u_{1} - \partial_{x_{1}} u_{3}, \partial_{x_{1}} u_{2} - \partial_{x_{2}} u_{1})$ , and $Δ f = \sum_{j = 1}^{3} \partial_{x_{j}}^{2} f$ , with $Δ u$ applied componentwise. For $v, w \in R^{3}$ , use the right-handed cross product $v \times w = (v_{2} w_{3} - v_{3} w_{2}, v_{3} w_{1} - v_{1} w_{3}, v_{1} w_{2} - v_{2} w_{1})$ . Define the row-component Jacobian $D_{x} u = (\partial_{x_{j}} u_{i})_{i, j}$ and $(u \cdot \nabla) u = \sum_{j = 1}^{3} u_{j} \partial_{x_{j}} u$ .

For constants $ρ_{0} > 0$ and $μ > 0$ , define the kinematic-viscosity parameter $ν = μ / ρ_{0}$ . The formal constant-density incompressible Navier--Stokes system is

$\partial_{t} u + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0, div u = 0.$

Let $I_{3}$ denote the $3 \times 3$ identity matrix. The associated formal Cauchy stress matrix is $σ = - p I_{3} + μ (D_{x} u + (D_{x} u)^{T})$ . The corresponding steady Stokes system is $ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ together with $div u = 0$ ; it omits both the time derivative and convective term. These are definitions of pointwise mathematical systems and do not assert that a physical fluid has a solution with the stated regularity.

For a barotropic branch, meaning a pressure prescribed as a function of density alone, let $P : (0, \infty) \to R$ have two continuous derivatives and let $V : Ω \to R$ have continuous spatial partial derivatives through order two. The formal compressible Euler system for $u$ and positive density $ρ$ is

$\partial_{t} ρ + div (ρ u) = 0, \partial_{t} u + (u \cdot \nabla) u = - ρ^{- 1} \nabla (P (ρ)) - \nabla V .$

At a constant background density $ρ_{0} > 0$ with $P^{'} (ρ_{0}) > 0$ , the positive parameter $c$ defined by $c^{2} = P^{'} (ρ_{0})$ is the sound-speed parameter of this mathematical branch. A separate rotating linear system, for constant $Ω \in R^{3}$ and scalar field $q$ , is $\partial_{t} u + 2 Ω \times u = - \nabla q$ and $div u = 0$ .

The vorticity field is $curl u$ . For every fixed $t \in I$ , define the spatial sections $u^{[t]} : Ω \to R^{3}$ , $u^{[t]} (x) = u (t, x)$ , and $σ^{[t]} : Ω \to R^{3 \times 3}$ , $σ^{[t]} (x) = σ (t, x)$ . If $γ : [a, b] \to Ω$ is a closed piecewise- $C^{1}$ oriented curve, with $γ (a) = γ (b)$ , its circulation at time $t$ is $\oint_{γ} u^{[t]} \cdot d r$ . If $S$ is a compact oriented regular surface patch in $Ω$ with unit normal $n_{S}$ , its volume-flux observable at time $t$ is $\int_{S} u^{[t]} \cdot n_{S} d S$ . The traction at $(t, x)$ for $x \in S$ is $σ^{[t]} (x) n_{S} (x)$ and the formal resultant force at time $t$ is the componentwise surface integral $\int_{S} σ^{[t]} n_{S} d S$ . When time is fixed, $u (x)$ and $σ (x)$ may denote these spatial sections; for time-independent fields this notation is independent of the chosen time. A steady velocity field is a map $u_{0} : Ω \to R^{3}$ for which $u (t, x) = u_{0} (x)$ for all $t \in I$ and $x \in Ω$ ; a streamline of $u_{0}$ is a $C^{1}$ curve $x : J \to Ω$ on a nondegenerate real interval $J \subseteq R$ (open, closed, half-open or unbounded) satisfying $x^{'} (s) = u_{0} (x (s))$ . Here $C^{1}$ means that $x$ is continuous on $J$ , is continuously differentiable on its interior, and at each included finite endpoint has the corresponding one-sided derivative to which the interior derivative extends continuously. The displayed equation is imposed in the interior and with that one-sided derivative at any included endpoint. The variable $s$ is a curve parameter, not time; no curve-existence assertion is made. A closed material loop on a time interval $J \subseteq I$ is a $C^{1}$ map $X : J \times [0, 1] \to Ω$ with $X (t, 0) = X (t, 1)$ and $\partial_{t} X (t, θ) = u (t, X (t, θ))$ ; the definition asserts no existence of such a map. For the following limit, an exterior spatial domain $Ω$ means that there is $R_{0} > 0$ such that ${x \in R^{3} : ∣ x ∣ > R_{0}} \subseteq Ω$ . For a scalar or vector field $f : Ω \to R$ or $f : Ω \to R^{3}$ , respectively, and $f_{\infty}$ in the same target space, $f (x) \to f_{\infty}$ uniformly in direction as $∣ x ∣ \to \infty$ means $\sup_{∣ e ∣ = 1} ∣ f (r e) - f_{\infty} ∣ \to 0$ in the scalar case and $\sup_{∣ e ∣ = 1} ∥ f (r e) - f_{\infty} ∥_{2} \to 0$ in the vector case, with $r \to \infty$ .

For one optional SI interpretation, the dimensions are $[x] = m$ , $[t] = s$ , $[u] = m s^{- 1}$ , $[ρ] = k g m^{- 3}$ , $[p] = P a$ , $[μ] = P a s$ , $[ν] = m^{2} s^{- 1}$ , $[V] = [q] = m^{2} s^{- 2}$ , and $[Ω] = s^{- 1}$ . Thus $[curl u] = s^{- 1}$ , circulation has units $m^{2} s^{- 1}$ , volume flux has units $m^{3} s^{- 1}$ , stress has units Pa, and resultant force has units N. In this interpretation $ρ_{0}, μ, P, V, Ω$ are inputs to the displayed formulas, while $ν = μ / ρ_{0}$ and $c^{2} = P^{'} (ρ_{0})$ are derived quantities relative to those formulas; this classification is not asserted to be universal across physical theories.