Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Local momentum equation from the adopted continuum balance

Physical scope

Smooth positive-density constant-density incompressible Newtonian material regions under the adopted integral momentum balance. It derives the local forced momentum equation by transport and localization; it asserts no solution existence or empirical validity.

Statement

Assume Countable Choice. Let $I \subseteq R$ be an open time interval in seconds and let $B \subseteq R^{3}$ be an open reference region. Let $χ \in C^{2} (I \times B; R^{3})$ be a material motion whose maps $χ_{t}$ are orientation-preserving $C^{2}$ diffeomorphisms onto their open images, and define the Eulerian velocity by $u (t, χ (t, a)) = \partial_{t} χ (t, a)$ . Set $Ω_{t} : = χ_{t} (B)$ and $O : = {(t, x) : t \in I, x \in Ω_{t}}$ . Assume that $\partial_{t} u$ and all spatial derivatives of $u$ through order two are continuous on $O$ , that $p$ and its spatial first derivatives are continuous on $O$ , and that $b \in C^{0} (O; R^{3})$ ; here $u$ is in metres per second, $p$ in pascals, and body acceleration $b$ in metres per second squared. In the selected constant-density incompressible Newtonian branch, let the current density be the constant $ρ > 0$ kilograms per cubic metre, let $div u = 0$ , let the coefficient $μ$ of the adopted Newtonian closure be constant and denote its value by $η > 0$ pascal-seconds. Put

$D (u) : = \frac{\nabla u + (\nabla u)^{T}}{2}, σ : = - p I_{3} + 2 η D (u), (div σ)_{i} : = \sum_{j = 1}^{3} \partial_{x_{j}} σ_{i j} .$

Assume the adopted material linear-momentum balance holds for every nonempty bounded open $V ⋐ B$ with $C^{1}$ boundary, meaning that near each boundary point, after a rigid coordinate change, the boundary is the graph of a $C^{1}$ function and $V$ lies on one side:

$\frac{d}{d t} \int_{χ_{t} (V)} ρ u d x = \int_{χ_{t} (V)} ρ b d x + \int_{\partial χ_{t} (V)} σ n d S,$

where $n$ is the outward normal and $d x, d S$ are Euclidean volume and area measures. Then throughout each $Ω_{t}$ the corresponding local equation is

$ρ (\partial_{t} u + (u \cdot \nabla) u) = div σ + ρ b .$

This is a conditional local form of the adopted balance for the stated smooth material motion and model branch; it makes no existence or empirical-applicability claim.

Statement references

David Tong, Fluid Mechanics, Chapter 2, §§3.1.1–3.1.2

Joseph M. Powers, Lecture Notes on Intermediate Fluid Mechanics, §4.4.5.2

David Tong, Fluid Mechanics, Chapter 2, §3.1.2

Facts & Assumptions

Given: Countable Choice, the $C^{2}$ material motion and its Eulerian velocity, the constant-density incompressible Newtonian branch, and the adopted material momentum balance stated above.

[A1]

Single-component nonpolar continuum and material balances: for every admissible material subregion, the adopted linear-momentum balance differentiates the material momentum integral and equates it to body-force resultant plus Cauchy-traction resultant. This is an adopted physical premise.

[F1]

The local motion and induced velocity in this theorem are explicitly given by its own Statement: $χ$ is $C^{2}$ with orientation-preserving $C^{2}$ diffeomorphic slices, $u (t, χ (t, a)) = \partial_{t} χ (t, a)$ , and $D_{t} f = \partial_{t} f + u \cdot \nabla f$ . These local definitions impose no initial-time normalization. Classical frames, primitive quantities and SI units supplies the classical Cartesian inertial and SI field conventions only; its separate initial-chart normalization is not invoked here.

[F3]

Newtonian stress and Fourier conductive closure: the adopted closure has $σ = - p I_{3} + τ$ and $τ = 2 μ D_{0} + ζ θ I_{3}$ ; for the selected constant-viscosity incompressible branch, $θ = 0$ and $D_{0} = D$ , giving the stress in the Statement.

[F4]

Divergence on a bounded C1 Euclidean domain: under Countable Choice, for a bounded $C^{1}$ domain and a $C^{1}$ vector field up to its boundary, the volume divergence integral equals its outward boundary flux.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every at most countable family of nonempty sets has a choice function; this is the precise choice hypothesis of [F4].

[F5]

Smooth transport, Jacobian and advected reference density: its Statement gives change of variables and material transport for every compact Jordan set and $C^{1}$ scalar field under the stated $C^{2}$ orientation-preserving motion, componentwise for vector fields. C1 material regions have compatible Jordan volume integrals: its Statement gives the open- $C^{1}$ -region/compact-Jordan integral compatibility and the locally $C^{1}$ graph boundary of the diffeomorphic image. Apply these local identities on a relatively compact neighborhood of $\overline{V}$ , translating time so the time under consideration is zero when necessary.

Proof

Given: The assumptions in the Statement, including the material motion, constant density, incompressibility, regularity, integral balance and Countable Choice.

1.1givenF1F5algebra

Fix an admissible reference region $V$ and a time $t_{0} \in I$ , and put $K = \overline{V}$ . Its $C^{1}$ boundary makes $K$ a compact Jordan set. Apply the material-transport identity in [F5], componentwise to $q = ρ u$ , on a relatively compact neighborhood of $K$ and a time-translated interval containing $0$ at $t_{0}$ . Because $ρ$ is constant and $div u = 0$ , $D_{t} (ρ u) + (ρ u) div u = ρ (\partial_{t} u + (u \cdot \nabla) u)$ . The open-region integral compatibility in [F5] identifies the resulting compact-Jordan integrals with those over $χ_{t} (V)$ , so $\frac{d}{d t} \int_{χ_{t} (V)} ρ u d x = \int_{χ_{t} (V)} ρ (\partial_{t} u + (u \cdot \nabla) u) d x$ at $t = t_{0}$ .

2.1givenA1F3F4F5A2step 1.1

By [F5], the image $χ_{t} (V)$ is a bounded $C^{1}$ domain. The adopted Newtonian stress [F3] is $C^{1}$ up to its boundary by the stated field regularity, since $\overline{χ_{t} (V)}$ is compactly contained in $Ω_{t}$ . Apply [F4] to each row of $σ$ to replace its traction resultant by $\int_{χ_{t} (V)} div σ d x$ . Substitution of step 1.1 into [A1] gives $\int_{χ_{t} (V)} R (t, x) d x = 0$ for every such $V$ , where $R : = ρ (\partial_{t} u + (u \cdot \nabla) u) - ρ b - div σ$ is continuous. The hypotheses of [F4] hold by the stated regularity, the $C^{2}$ motion and [A2].

3.1givenF5step 2.1algebra∎

Fix $t$ and $x_{0} \in Ω_{t}$ . For all sufficiently small $r > 0$ , the closed ball $\overline{B_{r} (x_{0})}$ lies in $Ω_{t}$ . Apply [F5]'s $C^{1}$ -image clause to the inverse diffeomorphism $χ_{t}^{- 1}$ on a neighborhood of this closed ball; then $V : = χ_{t}^{- 1} (B_{r} (x_{0}))$ is bounded, has $C^{1}$ boundary, and satisfies $\overline{V} ⋐ B$ . Step 2.1 gives $\int_{B_{r} (x_{0})} R (t, x) d x = 0$ . If any component $R_{i} (t, x_{0})$ were positive, continuity would make it positive on some sufficiently small ball and force its integral to be positive; if negative, the integral would be negative. Both contradict the zero integral, so $R_{i} (t, x_{0}) = 0$ for each component. Thus $R = 0$ throughout $Ω_{t}$ , which is the claimed local momentum equation.

Authoring review notes

The following model dictionary is context only, not a proof premise.

Constant-density incompressible Newtonian model: the current definition specifies constant positive density, $div u = 0$ , pressure-multiplier stress $- p I_{3} + 2 μ D$ , and rowwise stress divergence. Its displayed momentum equation is unforced and is not used in this result; the force enters through [A1].

Local momentum equation from the adopted continuum balance

Statement

Statement references

Facts & Assumptions

Proof

Authoring review notes

Depends on

Dependency tree · 2 levels

Sources