ReynoldsBEng 10th August 2026
Intended as Appendix to Project 2 PhD Civil Engineering, judder wave equations for continuous solutions. Navier-Stokes bi-stable solution is a solid introduction.
Appendix A
Bistable Continuum Resolution of the Navier–Stokes Equations and Application to Concrete Tank Loading Practice
(Standalone appendix to Project 2)
A.1 Context
Project 1 restored the missing mechanical closure in classical elasticity by completing A. E. H. Love’s A Treatise on the Mathematical Theory of Elasticity (1892/1893, subsequent editions 1906–1927) with the ring-tension and lamina-judder framework developed by B. W. V. Lewe (1906–1925). The restored continuum is an elastic plenum whose surface admits simultaneous smooth and rough states inside every finite moment duration (h).
The present appendix extends the same continuum operator to the Navier–Stokes equations and introduces the resulting mechanical equations, evaluated at Rest Time \(T = 0\), into updated practice for the loading of concrete liquid-containing structures.
A.2 Geometric Operator
At every instant the continuum occupies one of two complementary geometric states:
- State A = +1 — disc continuum. \(\pi_E\) is the smooth limit (all-slip with Instant grip). Inside the circumference the geometry is realised as a circle of straight lines.
- State B = −½ — sphere continuum. \(\pi_R\) is the exterior mean; \(\pi_E\) and \(\pi_N\) are its limiting values (all-grip with Instant slip).
Thickness is mandatory at the boundary between adjacent spherical temperature zones at particle scale. Relative to the judder-wave speed (c), this thickness appears as the colour spectrum compressed to its geometric limit. Recursive dilatancy hysteresis (the volumetric law) is the direct expression of the binary closures inside the continuum dimension.
A.3 Resolution of the Navier–Stokes Equations
The classical singularities of viscous flow, contact-line stress and volumetric hysteresis arise from treating the fluid surface as mono-stable. Once the surface is recognised as bistable inside every moment duration (h), the equations close without additional empirical coefficients.
The contact-patch disc that seeks spherical closure, the countersnap duration of reassembly, the internal deformation required by any finite thickness, and the path-dependent volumetric response are recovered as geometric necessities of the bistable continuum. Work performed against dilatancy appears as heat; the spherical geometric residue is expressed as light. Conservation is reinforced by the continuum’s insistence on retaining spherical geometry on-site.
The volume-constrained peridynamic formulation of the Navier equation (Du et al., 2013) already approaches the continuum from the elastic side; the bistable Instant at Rest Time \(T = 0\) supplies the missing closure.
A.4 Introduction of Mechanical Equations at Rest Time \(T = 0\) into Concrete Tank Loading Practice
Current practice for liquid-containing concrete structures (ACI 350.3-20, Eurocode 8 Part 4 / EN 1998-4, EN 1991-4, API 650 Appendix E, NZSEE guidelines) evaluates hydrodynamic pressures (impulsive and convective components) under the assumption of a mono-stable fluid continuum. The resulting pressure distributions and base shears are therefore incomplete at the contact-line and free-surface boundaries.
The bistable continuum supplies the corrected mechanical equations evaluated at the sole inertial frame Rest Time \(T = 0\):
- Instantaneous equilibrium density is forced by cancellation of residual dilatancy (State-B countersnap) via the Instant grip/slip toggle.
- Hydrodynamic pressure is obtained from the continuous history of binary closures rather than from an assumed mono-stable velocity field.
- Volumetric hysteresis is retained as a geometric source term, eliminating the need for ad-hoc bulk-viscosity coefficients.
- Wall and free-surface boundary conditions become contact-patch conditions relative to the centre (Rest Time \(T = 0\)).
Implementation requires only the substitution of the classical pressure integrals by the corresponding bistable surface integrals evaluated at \(T = 0\). The resulting loads remain compatible with existing limit-state partial factors while removing the classical singularities at the three-phase contact line.
A.5 Conclusion
The continuum operator restored in Project 1 is sufficient to close the Navier–Stokes equations and to update the hydrodynamic loading of concrete tanks. Rest Time \(T = 0\) is the sole inertial frame. The continuum is closed.
References
ACI Committee 350 (2020). Code Requirements for Seismic Analysis and Design of Liquid-Containing Concrete Structures (ACI 350.3-20) and Commentary. American Concrete Institute.
API (2020). Welded Tanks for Oil Storage, API Standard 650, 13th edition. American Petroleum Institute.
Du, Q., Gunzburger, M., Lehoucq, R.B. & Zhou, K. (2013). Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. Journal of Elasticity, 113(2), 193–217.
EN 1991-4 (2006). Eurocode 1: Actions on structures – Part 4: Silos and tanks. CEN.
EN 1998-4 (2006). Eurocode 8: Design of structures for earthquake resistance – Part 4: Silos, tanks and pipelines. CEN.
Lewe, B.W.V. (1906–1925). Collected papers on elastica, ring-tension and lamina dynamics (historical reconstruction in Project 1).
Love, A.E.H. (1892/1893). A Treatise on the Mathematical Theory of Elasticity, Vols I & II. Cambridge University Press. (Subsequent editions 1906, 1920, 1927.)
NZSEE (2009). Seismic Design of Storage Tanks. New Zealand Society for Earthquake Engineering.
Navier, C.L.M.H. (1822) & Stokes, G.G. (1845). Foundational papers on the equations of viscous fluid motion (standard historical citations). https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html
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