11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 3 Mass and Momentum Conservation

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Chapter 3 Mass and Momentum Conservation

I. Scope and Objectives

1. Provide the conservation gauges for mass and momentum of a slender filament in a 1D approximation, unifying the relations among line density rho_L(ell,t), flux J(ell,t), tension T_fil(ell,t), and axial speed v(ell,t).
2. Establish integral forms and discrete realizations that are directly measurable and storable, supporting Mx-13 conservation audits and the mechanical closure of Chapter 4 (constitutive relations).
3. Pass criteria
   • Conservation residuals for mass eps_mass and momentum eps_mom simultaneously satisfy thresholds gate.mass and gate.mom.
   • All expressions pass check_dim(expr) and boundary manifests are complete.
   • Flux and acceleration estimated via two independent routes have discrepancies delta_flux, delta_acc within thresholds.

II. Terminology and Symbols

1. Path and measure: centerline gamma(ell), measure d ell; reference arc length ell0 and mapping phi(ell0,t) (see Chapter 2).
2. Line quantities and flux
   • rho_L(ell,t) = rho(ell,t) * A(ell,t) (line density, units kg/m).
   • J(ell,t) = rho_L(ell,t) * v(ell,t) (mass flux, units kg/s).
   • m_L(ell,t) = rho_L(ell,t) * v(ell,t) (line momentum density, units N*s/m).
3. Forces and power
   • T_fil(ell,t): axial tension (units N, strictly collision-controlled).
   • f_ax(ell,t): distributed body-force density along the line (units N/m), may include gravity, aerodynamic/hydrodynamic drag projections, friction.
   • P_b(t) = T_fil(b,t) * v(b,t): instantaneous mechanical power at boundary point b.
4. Time bases: tau_mono for internal metrology; published timestamps ts = alpha + beta * tau_mono (see Preface and Chapter 2).

III. Postulates and Minimal Equations

1. P11-1 (slender-geometry approximation): radial effects are higher order; a 1D centerline description is valid.
2. P11-4 (small-deflection axialization): inertia and forces act primarily along the tangent of gamma(ell); shear and bending contribute only second-order effects to 1D conservation.
3. S12-1 (mass conservation, 1D line form)
   • Differential form: ( d/dt ) rho_L + ( d/dell ) J = 0, with J = rho_L * v.
   • Integral form (control segment [a,b]): ( d/dt ) ( ∫_{a}^{b} rho_L d ell ) = J(a,t) - J(b,t).
4. S12-2 (axial momentum conservation, 1D line form)
   • Differential form: ( d/dt ) ( rho_L * v ) + ( d/dell ) ( rho_L * v^2 ) = ( d/dell ) ( T_fil ) + f_ax.
   • Integral form (control segment [a,b]):
     ( d/dt ) ( ∫_{a}^{b} rho_L * v d ell ) = [ rho_L * v^2 + ( - T_fil ) ]_{a}^{b} + ( ∫_{a}^{b} f_ax d ell ).
5. S12-2q (quasi-steady, low-inertia approximation)
   • When ( d/dt ) ( rho_L * v ) and ( d/dell ) ( rho_L * v^2 ) are below the measurement-noise floor: ( d/dell ) ( T_fil ) + f_ax ≈ 0.
   • If f_ax ≈ 0, then T_fil ≈ const (nearly uniform tension in the drawing zone).
6. S12-2e (power density and dissipation ledger)
   • Axial mechanical power density: p_lin = T_fil * s + v * f_ax, with s = ( d/dell ) v = ( d/dt ) ( ln( lambda ) ).
   • Segment-level energy flow: ( d/dt ) K_lin + D_lin = [ T_fil * v ]_{a}^{b} + ( ∫_{a}^{b} v * f_ax d ell ), where D_lin >= 0 is the equivalent dissipation.
7. Compatibility with Chapter 2
   From S12-1 and P11-3, the general relation rho * A * lambda = rho0 * A0 follows; under geometric incompressibility A = ( A0 / lambda ), one has rho ≈ rho0.

IV. Data Gauges and Manifest

1. Control segments and boundaries
   • segment.{a,b} : float (in ell coordinates), boundary.a.type : "Dirichlet"|"Flux"|"Mixed", boundary.b.type : ....
   • Required boundary observables: J(a,t), J(b,t), T_fil(a,t), T_fil(b,t), v(a,t), v(b,t), f_ax_profile (if measurable).
2. Conservation-audit fields (increment to schema.core.drawing/v1)
   • cons.mass.resid : float, cons.mom.resid : float, cons.power.resid : float (after unit harmonization).
   • cons.window : [t0, t1], cons.method : "integral"|"differential", cons.notes : str.
   • Conservation gates: gate.mass, gate.mom, gate.power, tied to TS.* observable states.
3. Units and dimensions
   • rho_L : kg/m, J : kg/s, v : m/s, T_fil : N, f_ax : N/m, p_lin : W/m.
   • All submissions must pass check_dim(expr) before data-lake ingestion.

V. Algorithms and Implementation Bindings

1. Discretization and updates (aligned with I10-1 update_draw_state)
   • Grid and interpolation: store rho_L^n(i), v^n(i) at ell_i; store boundary sensor values on half-nodes a^-, b^+.
   • Mass conservation (upwind or flux-limiter schemes):
     rho_L^{n+1}(i) = rho_L^{n}(i) - ( dt / d ell ) * ( J^n(i+1/2) - J^n(i-1/2) ), with J^n = rho_L^n * v^n.
   • Momentum conservation (semi-explicit):
     m_L^{n+1}(i) = m_L^{n}(i) - ( dt / d ell ) * ( F^n(i+1/2) - F^n(i-1/2) ) + dt * ( ( d/dell ) T_fil^n(i) + f_ax^n(i) ), where F = rho_L * v^2.
   • Stable time step: satisfy dt <= cfl * ( d ell / max | v | ), with 0 < cfl < 1.
   • Positivity and filtering: if rho_L^{n+1} < 0 or A^{n+1} < 0, raise E_CONSERVATION_FAIL and roll back.
2. Conservation residuals (aligned with Mx-13 and I10-5 emit_metrics_drawing)
   • Mass: res_mass = ( d/dt ) ( ∫ rho_L d ell ) - ( J(a,t) - J(b,t) ).
   • Momentum: res_mom = ( d/dt ) ( ∫ rho_L * v d ell ) - ( [ rho_L * v^2 - T_fil ]_{a}^{b} + ( ∫ f_ax d ell ) ).
   • Power: res_power = ( d/dt ) K_lin + D_lin - ( [ T_fil * v ]_{a}^{b} + ( ∫ v * f_ax d ell ) ).
   • Normalized publication:
     eps_mass = ( | res_mass | / max( ε_ref , ( ∫ rho_L d ell ) / tau_ref ) ), with analogous forms for momentum and power.

VI. Metrology Workflow and Run Graph

1. Mx-13 conservation-audit flow
   • Time-base alignment: aggregate on tau_mono, map to ts, and form window [t0,t1].
   • Profile reconstruction: from Chapter 2 lambda(ell,t), A(ell,t) and density gauges, obtain rho_L(ell,t) and integrate with consistent d ell.
   • Boundary acquisition: synchronously record J(a,t), J(b,t), T_fil(a,t), T_fil(b,t), v(a,t), v(b,t).
   • Compute eps_mass, eps_mom, eps_power, compare to gate.*; if any exceed thresholds, roll back to Mx-11 or rebuild f_ax.
   • Publication gauges: write gamma(ell), time window, filter ENBW, and uncertainty budget with the manifest.
2. Alerts and rollback
   • Triggers: eps_mass > gate.mass or eps_mom > gate.mom.
   • Remedies: shrink the window, raise sampling bandwidth, update dt, switch to S12-2q, or augment the f_ax model.

VII. Verification and Test Matrix

1. Minimum required cases
   • Constant speed and section: v = const, A = const ⇒ expect constant J and T_fil, with eps_mass, eps_mom near zero.
   • Linear velocity gradient: v(ell,t) = c1 + c2 * ell ⇒ verify s = c2 and the flux-term balance of S12-2.
   • Body force on a limited segment: apply constant f_ax = f0 on [a,b], verify T_fil(b) - T_fil(a) = - ( ∫_{a}^{b} f0 d ell ) (under S12-2q).
   • Step change at inlet flux: step J(a,t) and verify equality of mass accumulation and boundary net flow.
2. Boundary and extreme scenarios
   Slip and recoil, rapid necking causing abrupt A changes, high-frequency v jitter and derivative amplification, temperature-dependent rho.
3. Suggested gates
   gate.mass <= 1e-3 (relative), gate.mom <= 5e-3, gate.power <= 1e-2; calibrate specific values in benchmark cases.

VIII. Cross-References and Dependencies

• See Chapter 2 for the definitions and metrology gauges of lambda(ell,t), s(ell,t), v(ell,t), A(ell,t).
• See Chapter 4 for constitutive relations of T_fil that close S12-2.
• See Core.Density for rho(x,t), measure conventions, and unit audits.
• See Core.Equations for unified control-volume statements of conservation laws.
• See Core.Metrology for uncertainty synthesis, spectral windows, and ENBW.
• See Core.Threads for TS.* metrics and alert chains.

IX. Risks, Limits, and Open Questions

• At high drawing speeds, inertia terms may be non-negligible: estimating ( d/dt ) ( rho_L * v ) and ( d/dell ) ( rho_L * v^2 ) demands higher bandwidth and denoising strategies.
• When rho couples with temperature and draw rate, rho * A * lambda = rho0 * A0 must be extended via a state equation; otherwise systematic bias arises.
• Unobservable body-force zones: when f_ax is hard to measure directly, infer it from boundary power and dissipation gaps, and declare uncertainty in the manifest.
• For non-straight or strongly curved segments, deviations from pure axialization require curvature–tension coupling correction terms.

X. Deliverables and Version Management

1. Artifacts
   • Mx-13 conservation-audit scripts and report template (including computation and visualization of eps_mass, eps_mom, eps_power).
   • Core equation implementations aligned with I10-1 and I10-5, plus reference cfl, filtering, and positivity-repair strategies.
   • Example datasets: annotated manifests and expected residuals for constant-speed and linear-gradient cases.
2. Version policy
   Any modification to S12-1, S12-2, S12-2q, S12-2e is recorded as MOD, with migration guidance and compatibility flags in Appendix C; added body-force terms or energy ledgers are marked as ADD.