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The continuity equation of the continuum model: local form of mass balance, the material form, and the incompressibility reading

Physical scope

Conditional on the adopted mass-balance postulate, the smooth kinematic model and the Axiom of Countable Choice used by the transport machinery: conservation of the model mass of every material subregion is equivalent to the partial differential equation partial_t rho + div(rho u) = 0 on the current regions; equivalently the material derivative satisfies D_t rho = -rho div u, the product rho times the Jacobian is constant along each label worldline, the relative volume rate is div u, and the incompressibility condition div u = 0 is equivalent to the density being constant along each worldline in this branch. The theorem is a conditional model deduction and asserts no empirical value.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(B, O, I)$ be body data (Continuum body data, the fixed inertial frame and the dimensions of the fields), let $χ$ be a motion of the model with spatial velocity $u$ (Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance, Continuum motions, spatial velocity, material acceleration and swept regions), let $J_{0} \subseteq I$ be an open interval with $\overline{J_{0}} \subseteq I$ , let $Σ$ be the swept region over $\overline{J_{0}}$ , and let $ρ : U \to (0, \infty)$ be a density field of class $C^{1}$ on an open set $U$ containing $Σ$ (The density field of the continuum model and the mass of a material subregion, $C^{k}$ Euclidean maps and diffeomorphisms). Assume the mass-balance postulate (CM) of Adopted mass balance: the mass of every material subregion is constant in time for every material subregion and every $t \in J_{0}$ . Then:

(i) Local form. For every $(t, x) \in J_{0} \times χ_{t} (B)$ ,

$\begin{matrix} \partial_{t} ρ (t, x) + div (ρ u) (t, x) = 0. & (C) \end{matrix}$

Conversely, if (C) holds on an open neighbourhood of $Σ$ , then (CM) holds for every material subregion and every $t \in J_{0}$ .

(ii) Material form. On $J_{0} \times χ_{t} (B)$ the material derivative of the density satisfies

$D_{t} ρ = - ρ div u,$

and writing $J (t, X) = \det D_{X} χ_{t} (X)$ along the motion, the product $ρ (t, χ (t, X)) J (t, X)$ is constant in $t$ for every label $X \in B$ .

(iii) Volume rate. For every $X \in B$ and $t \in J_{0}$ one has $\partial_{t} J (t, X) = J (t, X) div u (t, χ (t, X))$ : the relative rate of change of an infinitesimal material volume is $div u$ , and the identity of (ii) is the statement that density times volume is conserved along a label. In particular $div u = 0$ on the swept region if and only if $ρ$ is constant along every label worldline ( $D_{t} ρ = 0$ ).

The statement is a conditional deduction from the adopted mass balance and the kinematic model; it asserts no equation of motion, no constitutive relation and no empirical value.

Facts & Assumptions

Given: ${A C}_{ω}$ ; body data and the motion $χ$ with spatial velocity $u$ ; the interval $J_{0}$ ; the $C^{1}$ positive density field $ρ$ ; the mass-balance assumption (CM).

[F1]

Mass balance (Adopted mass balance: the mass of every material subregion is constant in time): for every material subregion $B^{'} \subseteq B$ and every $t \in J_{0}$ , $\frac{d}{d t} \int_{χ_{t} (B^{'})} ρ (t, x) d x = 0$ .

[F2]

Local mass balance (Local mass balance: conservation on every material subregion implies $\partial_{t} ρ + div (ρ u) = 0$ ): under the stated hypotheses, (CM) for every material subregion and every $t \in J_{0}$ implies $\partial_{t} ρ + div (ρ u) = 0$ on the current interior; conversely, that equation on a neighbourhood of the swept region implies (CM).

[F3]

Transport and material derivative (The transport identity and the material derivative along label worldlines for the adopted continuum model): for every label $X \in B$ , $\frac{d}{d t} f (t, χ (t, X)) = D_{t} f (t, χ (t, X)) = (\partial_{t} f + \nabla f \cdot u) (t, χ (t, X))$ for $C^{1}$ spatial fields $f$ , under the same axiom and kinematic hypotheses.

[F4]

Determinant derivative (The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ): for the matrix curve $t \mapsto A (t) : = D_{X} χ_{t} (X)$ with invertible values one has $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ , and $A^{'} = D u (t, χ (t, X)) A$ ; hence $\partial_{t} J = J tr (D u (t, χ (t, X))) = J div u (t, χ (t, X))$ by the definition of divergence in Euclidean spatial calculus: gradient, divergence and material derivative.

[F5]

Algebra of derivatives (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Euclidean spatial calculus: gradient, divergence and material derivative): $div (ρ u) = \nabla ρ \cdot u + ρ div u$ and $D_{t} ρ = \partial_{t} ρ + \nabla ρ \cdot u$ .

Proof

technique · direct; localize the integral mass balance with the cited equivalence, rewrite the divergence with the product rule, and read the along-label rate through the transport item and the volume-rate lemma

1.1givenF1F2

Part (i), forward: by [F1] the integral conservation (CM) holds for every material subregion and every $t \in J_{0}$ . The hypotheses of [F2] are those of the statement (positive $C^{1}$ density, motion, compactly contained interval, equation domain), so [F2] gives $\partial_{t} ρ + div (ρ u) = 0$ at every point of $J_{0} \times χ_{t} (B)$ .

1.2givenF2

Part (i), converse: if (C) holds on an open neighbourhood of $Σ$ , then [F2] applied in its converse direction gives (CM) for every material subregion and every $t \in J_{0}$ .

2.1step 1.1F5algebra

Part (ii), first identity: by [F5], $div (ρ u) = \nabla ρ \cdot u + ρ div u$ , so (C) reads $\partial_{t} ρ + \nabla ρ \cdot u + ρ div u = 0$ , that is $D_{t} ρ + ρ div u = 0$ , which is $D_{t} ρ = - ρ div u$ .

3.1givenstep 2.1F3F4algebra

Part (iii): fix $X \in B$ . By [F4], $\partial_{t} J (t, X) = J (t, X) div u (t, χ (t, X))$ on $J_{0}$ , and by [F3] the along-label derivative of the spatial field $ρ$ is $\frac{d}{d t} ρ (t, χ (t, X)) = D_{t} ρ (t, χ (t, X))$ . Therefore $\frac{d}{d t} [ρ (t, χ (t, X)) J (t, X)] = (D_{t} ρ) J + ρ \partial_{t} J = (- ρ div u) J + ρ J div u = 0$ on $J_{0}$ , so the product is constant in $t$ .

4.1step 2.1step 3.1F3algebra∎

Part (iii), final clause: if $div u = 0$ on the swept region, then $D_{t} ρ = - ρ div u = 0$ there by step 2.1, and by [F3] $\frac{d}{d t} ρ (t, χ (t, X)) = 0$ for every label, so $ρ$ is constant along every worldline. Conversely, if $ρ$ is constant along every worldline then $D_{t} ρ = 0$ on the current regions by [F3], and step 2.1 together with $ρ > 0$ gives $div u = 0$ there. This proves the equivalence and completes the item.

Remarks

Reading of the equation. The partial differential equation (C) is a balance of the model density with the mass flux $ρ u$ ; the material form $D_{t} ρ = - ρ div u$ says that density decreases along a label exactly when the material element dilates. The identity $ρ J = constant$ along a label is the precise sense in which mass is conserved in the model, and it is the form used by the wave and fluid items below.
Conditionality. The equation is a theorem about the adopted model: it holds because the model is assumed to balance mass, and it fails if the mass-balance assumption is replaced by a balance with sources. No claim is made that any material's density satisfies it beyond the modelling regime of Adopted continuum kinematic model: body data, smooth motions as families of label worldlines, the continuum idealization and frame covariance.

The continuity equation of the continuum model: local form of mass balance, the material form, and the incompressibility reading

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources